A Distributed Coordinated Control Strategy for Large-scale Thermostatically Controlled Loads Considering User Comfort Constraints

Xiangyu Chen; Yujun Lin; Qiufan Yang; Yin Chen; Xia Chen; Jinyu Wen

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A Distributed Coordinated Control Strategy for Large-scale Thermostatically Controlled Loads Considering User Comfort Constraints PDF

- ORCID：
Xiangyu Chen ¹
✉
- ORCID：
Yujun Lin ¹
✉
- ORCID：
Qiufan Yang ¹
✉
- ORCID：
Yin Chen ²
✉
- ORCID：
Xia Chen ¹
✉
- ORCID：
Jinyu Wen ¹
✉

1. State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China； 2. Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G11XW, U.K.

Updated：2025-05-21

DOI：10.35833/MPCE.2024.000430

OUTLINE

Abstract

Thermostatically controlled loads (TCLs) on the demand side have been a vital energy resource in smart grids. To efficiently utilize the large-scale TCLs and enhance the flexibility of micro-community systems, this paper proposes a distributed coordinated control strategy based on the distributed model predictive control (MPC). To achieve the adaptive coordinated control among TCLs and consider user comfort constraints, a distributed dual-layer internal control strategy based on MPC is established on a scalable communication network. This strategy achieves the efficient utilization of TCLs in a distributed manner and notably improves the convergence speed through sparse network communication between neighbors. For external resource utilization of TCLs, a multi-timescale scheduling framework is proposed to realize the pre-allocation of electricity. Furthermore, the feasibility of the proposed distributed coordinated control strategy is confirmed through comparative case analysis.

Keywords

Thermostatically controlled load (TCL); distributed model predictive control (MPC); communication network; demand response; user comfort constraint; micro-community system

I. Introduction

FUTURE trends in the power industry indicate a predominant direction of integrating large-scale renewable energy sources into the power grid [

1]. The inherent randomness and volatility of large-scale renewable energy deployment pose significant challenges to the grid flexibility and stability. To address the challenges posed by the uncertain factors of renewable energy sources, the concept of energy storage systems (ESSs) is proposed [1]-[4]. However, the initial investment and operation costs of ESSs are high [5]. Consequently, recent research advocates the judicious utilization of existing resources on the demand side to provide ancillary services to the power grid, thereby reducing the need for additional investments and enhancing the overall energy efficiency [6], [7]. “Measures for Power Demand Side Management (2023 version)” and “Action Plan on Energy Conservation and Carbon Reduction for 2024-2025” proposed by China encourage the load participation in demand response, promote the provision of ancillary services, and emphasize the importance of precise load control [8], [9]. As significant electricity end-uses, thermostatically controlled loads (TCLs) represent promising demand-side energy flexibility resources for the integration of renewable energy sources [10]-[12].

An individual TCL typically has a low power capacity [

13]. To efficiently provide grid regulation services and mitigate power instability problems, the aggregation of a considerable number of TCLs is required. Numerous studies [14]-[16] discuss the economic benefits of TCLs in terms of energy savings, optimization cost, load shedding cost, and optimal scheduling schemes for long timescales. However, ensuring the efficient collaboration of these TCLs is equally critical for optimizing demand-side resources. Much literature focuses on the control methods of large-scale TCLs, which are primarily categorized into centralized [17], [18] and distributed control methods [19]-[27]. A temperature priority control (TPC) method is proposed in [17], which utilizes a centralized controller to coordinate the continuous regulation of building thermal loads for reserve adjustment. Reference [18] models the mathematical problem as a two-stage robust optimization model and solves it by the column constraint generation algorithm. However, the centralized control methods have significant computational and communication burden and are more suitable for operation scheduling than real-time control. Furthermore, the centralized control methods are vulnerable to single-point failures and communication faults, posing challenges in protecting the user data privacy.

Given the limitations of centralized control methods and the complex structure of TCLs, the highly scalable and plug-and-play distributed control methods are needed. The consensus algorithm is applied to distributed control methods. In [

19], a distributed TPC method that determines the switching states of air conditioner clusters based on a priority list generated by the average consensus algorithm is proposed, thereby providing continuous regulation reserves for smart grids. Nevertheless, the switch on-off control has been employed to impose specific detrimental effects on the lifespan of TCL equipment and lead to the power rebound phenomenon [20]. With the rise in popularity of inverter air conditioners (IACs), TCLs shift the control mode from on-off control to seamless continuous control [21]-[24]. This transformation significantly enhances the distributed fine-grained control of large-scale TCLs. Several studies [25]-[27] embrace consensus algorithms for precisely regulating IACs. A new distributed event-triggered consensus control method for demand response in large-scale IAC systems is proposed [25]. Reference [26] designs a weighted control matrix based on a consensus algorithm to smooth power fluctuations and load changes in renewable energy generation for a group of heterogeneous ESSs. These studies emphasize the power fluctuations associated with TCLs in the adjustment process, but often overlook the unique comfort constraints. Reference [27] addresses the comfort in the adjustment process of TCL and regards it as a single-state variable, enabling power and comfort to be shared among all participated TCLs.

However, the initial value updating methods are centralized and still ignore the role of comfort constraints in regulation. Reference [

28] mentions that the comfort constraints can be transformed into an optimization problem. The model predictive control (MPC) can be incorporated into the optimization problem to enhance the efficiency of control communication [29]. MPC can predict future states and feed the selected signals back into the control process to improve the overall control performance. In recent years, some studies apply MPC to the research on TCLs [30]-[32]. In [30], the future status of the system is predicted, and the control coefficient of the TCL control system is updated to avoid the occurrence of TCL control capability exceeding the limit. In [31], an MPC scheme is proposed to control the number of TCLs for regulation services. In [32], MPC manages the AC inverter, reducing energy consumption and enhancing demand response. It can be observed that most of the existing studies utilize MPC to handle uncertain variables in the control process of TCL, such as the randomness of user behavior, power consumption prediction, and the number of participants. Most of these studies focus on centralized control. Few researches are conducted on the application of MPC in distributed control and comfort constraints of TCLs.

From the aforementioned literature, it is evident that: ① the current research works on utilizing consensus algorithms for TCL regulation predominantly focus on equitable power and comfort distribution among multiple devices, neglecting the constraint effect of comfort levels in the control process; ② few studies focus on fully distributed control methods that address photovoltaic output fluctuations within building systems; and ③ there is a lack of discussion on the stability and convergence efficiency of consensus algorithms.

Aiming at these shortcomings, this paper proposes a distributed coordinated control strategy to optimize the utilization of TCLs by considering user comfort constraints. This strategy manages the internal TCLs with the MPC-based leader-follower consensus algorithm, while a multi-timescale scheduling framework handles external resource allocation, meeting the demands of smart grids. Through case analysis, the effectiveness of the proposed distributed coordinated control strategy in utilizing TCLs to mitigate wind power fluctuations in grid ancillary services is demonstrated. The contributions of this paper are as follows.

1) Addressing the comfort constraints inherent in the distributed control of large-scale TCLs, we integrate the distributed MPC strategy with consensus algorithms to ensure fair power and comfort sharing among available TCLs while considering both comfort and power constraints.

2) The proposed distributed coordinated strategy shows significant improvements in the fair distribution speed of TCL power and comfort levels compared with the consensus algorithm. This enhancement effectively mitigates user comfort fluctuations, thus significantly improving the overall user experience.

3) To tackle the dynamic control challenges posed by power scheduling commands, we utilize the steady-state properties of reference TCLs to convert external energy scheduling signals into setpoint temperature adjustment commands. This enables a distributed response to scheduling commands, which are different from the centralized initial value update methods employed in [

27].

The rest of this paper is organized as follows. Section II proposes the state-space model and an extensible communication network of TCLs for residential buildings. In Section III, a detailed description of the proposed distributed coordinated control strategy for TCLs is depicted. Section IV gives the stability analysis of the distributed MPC strategy. Section V presents several simulations to verify the performance of the proposed distributed coordinated control strategy. Section VI concludes this paper.

II. Modeling and Communication Network of TCLs

A. State-space Model of TCLs

For buildings dominated by TCLs, the heat transfer relationship is expressed as:

c_{A} ρ_{A} V \frac{d T_{i n} (t)}{d t} = H_{i n} - H_{o u t}

(1)

where $c_{A}$ is the specific heat capacity of air; $ρ_{A}$ is the air density; $V$ is the room volume; $T_{i n} (t)$ is the indoor temperature at time t; and $H_{i n}$ and $H_{o u t}$ are the heat gained and lost by the load per unit time, respectively.

IACs modulate power by adjusting the compressor frequency. This allows for the continuous and precise temperature control of load. The frequency modulation process of IAC can be implemented with a continuous variable x(t) [

23], as shown by the dashed line of Fig. 1.

Fig. 1 Schematic diagram of equivalent model of TCL.

Consider the room installed with the IAC within the residential building as a TCL, and $N$ rooms in the building form a cluster. As shown in Fig. 1, the dynamic temperature variation of each TCL can be formally described by a set of differential equations according to (1):

C_{t c l} \frac{d T_{i n} (t)}{d t} = \frac{T_{o u t} (t) - T_{i n} (t)}{R_{t c l}} - η P_{r} x (t) + w_{i} (t)

(2)

where $x (t)$ is the power consumption of TCLs, i.e., the power output percentage at time $t$ ; $T_{o u t} (t)$ is the outdoor temperature at time t; $η$ is the energy efficiency ratio, which is positive for cooling TCLs and negative for heating TCLs; $P_{r}$ is the rated power of TCL; $C_{t c l} = c_{A} ρ_{A} V$ and $R_{t c l} = 1 / (ε A_{t c l})$ are the equivalent thermal capacitance and resistance of the room, respectively, $ε$ is the thermal conductivity of the building, and $A_{t c l}$ is the area of room; and $w_{i} (t)$ is the external disturbance. Cooling IACs are exemplified in this paper.

The first term on the right-hand side of (2) represents the impact of ambient heat conduction, as shown in the outdoor area of Fig. 1. The second term represents the thermal power transfer within the IAC, as illustrated in the pink part of Fig. 1. The blue part depicts the equivalent parameters of the room. The air conditioning units with TCLs employ smart home technology, enabling seamless communication and interaction.

Remark 1: we assume the external disturbances follow a zero-mean Gaussian distribution, i.e., $w_{i} (t) ~ N (0, σ^{2})$ , where the variance $σ^{2}$ reflects the uncertainty in these disturbances. Disturbances within the building, such as door opening and closing, solar radiation, infiltration, and the operation of other loads, are considered. Since $w_{i} (t)$ has a long-term average of zero, its integral over time is zero in the state-space modeling process and can be ignored for simplification.

Remark 2: the equivalent thermal capacitance $C_{t c l}$ represents the thermal capacity of a room, indicating the amount of heat that the air inside the room can store or release for each degree of temperature change. It quantifies the energy required or released during temperature fluctuations in the room. The equivalent thermal resistance $R_{t c l}$ represents the resistance to heat conduction between the interior and exterior of the room. It measures the difficulty of heat transfer between the room and the external environment. For simplification, the intrinsic parameters of all the TCLs are considered to be the same. The output power of the cluster is regarded as a superposition of the output power of each TCL, and the heat transfer between neighbors can be neglected if $T_{i n}$ among TCLs is similar.

The negligible effect of temperature adjustment on user perception within an acceptable comfort range allows load adjustment to take advantage of this comfort margin to meet external regulatory needs. To facilitate the evaluation of whether the room temperature is within the comfort range, a comfort variable $y (t)$ is introduced as:

y (t) = \frac{T_{i n} (t) - T_{i n}^{m i n}}{T_{i n}^{m a x} - T_{i n}^{m i n}}

(3)

where $T_{i n}^{m i n} = T_{s e t} - δ$ , $T_{i n}^{m a x} = T_{s e t} + δ$ , T_set is the set temperature of TCL, and $δ$ is the temperature deviation from T_set within the comfort range. As long as y(t) is within $(0,1)$ , the change in T_in is small enough to be imperceptible to the user.

The control utilizes this characteristic to respond to power demands, achieving grid support services without compromising user experience.

Combining (1) with (2), we can obtain:

\dot{y} (t) = \frac{T_{o u t} (t) + δ - T_{s e t} (t)}{2 δ C_{t c l} R_{t c l}} - \frac{y (t)}{C_{t c l} R_{t c l}} - \frac{η P_{r} x (t)}{2 δ C_{t c l}}

(4)

In this paper, we assume that all users voluntarily participate in the demand response. The dynamics of TCLs can be written in a state-space model as (5), which can be written in the simplified form as (6).

[\begin{array}{l} {\dot{x}}_{i} \\ {\dot{y}}_{i} \end{array}] = [\begin{matrix} 0 & 0 \\ - \frac{η P_{r}}{2 δ C_{t c l}} & - \frac{1}{C_{t c l} R_{t c l}} \end{matrix}] [\begin{matrix} x_{i} \\ y_{i} \end{matrix}] + [\begin{matrix} 1 \\ 0 \end{matrix}] u_{i} + [\begin{matrix} 0 \\ \frac{T_{o u t} (t) + δ - T_{s e t} (t)}{2 δ C_{t c l} R_{t c l}} \end{matrix}]

(5)

\dot{p} = Ι_{N} \otimes A p + I_{N} \otimes B u + I_{N} \otimes C

(6)

where $\otimes$ is the symbol for the Kronecker product of matrices; $p = [p_{1}, p_{2}, \dots, p_{N}]^{T}$ , $p_{i} = [x_{i}, y_{i}]^{T}$ is the vector of state variables, $i = 1,2, \dots, N$ , and N is the number of TCLs; $u = [u_{1}, u_{2}, \dots, u_{N}]^{T}$ is the vector of control variables; $A =$ $[\begin{matrix} 0 & 0 \\ - \frac{η P_{r}}{2 δ C_{t c l}} & - \frac{1}{C_{t c l} R_{t c l}} \end{matrix}]$ is the system matrix, which describes the internal state relationships within the system; $B = {[1, 0]}^{T}$ ; $C = [0, (T_{o u t} (t) + δ - T_{s e t} (t)) / (2 δ C_{t c l} R_{t c l} {)]}^{T}$ ; and $I_{N}$ is an $N \times N$ identity matrix.

As the timescale of control signal is notably smaller than that of the changes in $T_{o u t}$ and $T_{s e t}$ during the control process, $A$ , $B$ , and $C$ can be treated as constants.

Based on the temperature regulation capability of TCL, $T_{s e t}$ can be adjusted to reduce energy consumption in scenarios requiring electricity output. Similarly, when there is a need to consume excess electricity, TCLs can adjust $T_{s e t}$ , while maintaining a stable operational mode and ensuring comfort within a specified temperature range. Consequently, buildings equipped with thermal capacitances can effectively function as virtual ESSs, seamlessly integrated into the operational control of distribution networks and microgrids. According to the notion of state of charge (SOC) utilized in ESSs, this paper puts forward the virtual SOC $V_{S O C}$ for virtual ESSs as:

V_{s o c} = V_{s o c}^{i n i} - \frac{\sum_{i = 1}^{N} P_{r} x_{i} (t)}{N P_{r}}

(7)

where $V_{s o c}^{i n i}$ is the initial $V_{S O C}$ .

B. Scalable Communication Network of TCLs

The communication network of a cluster consisting of N TCLs can be depicted as a graph [

33]. For an undirected communication graph

𝒢

with communication connections established by N TCLs, the adjacency matrix

A_{l} = [a_{i j}] \subseteq R^{N \times N}

is associated with the edges, where

a_{i j} = 1

if the node is in the graph

𝒢

; otherwise,

a_{i j} = 0

. The Laplacian matrix

L = [l_{i j}]

can be represented as:

L = D - A_{l}

(8)

l_{i j} = \{\begin{array}{l} - a_{i j} i \neq j \\ \sum_{j = 1_{i}}^{N} a_{i j} i = j \end{array}

(9)

where D is the in-degree matrix of $𝒢$ , which is defined as $D = d i a g {d_{i}} \subseteq R^{N \times N}$ with $d_{i} = \sum_{j = 1}^{N} a_{i j}$ .

The Laplacian matrix L of an undirected graph possesses a unique zero eigenvalue, while other eigenvalues are all positive. The rate of convergence is dictated by the smallest nonzero eigenvalue of the Laplacian matrix $L$ , which is intricately linked to the graph $𝒢$ .

It is assumed that each floor comprises $m = 3$ households, with each household equipped with $m_{i n} = 5$ TCLs, as shown in Fig. 2, where $𝒢_{1}$ refers to the graph of TCLs in each household on a specific floor; and $𝒢_{2}$ refers to the graph of reference TCLs (TCL 1) across multiple households. The reference node is responsible for the regulation of TCLs on a specific floor.

Fig. 2 Communication topology among households on a particular floor.

The Laplacian matrix of a specific floor $L_{h}$ can be represented as:

L_{h} = I_{m} \otimes L_{1} + L_{2} \otimes Γ_{1}

(10)

where $I_{m}$ is an $m \times m$ identity matrix; $L_{1} \subseteq R^{N \times N}$ is the Laplacian matrix of $𝒢_{1}$ , denoting the local interaction of TCLs in each household; $L_{2} \in R^{N \times N}$ is the Laplacian matrix of $𝒢_{2}$ , denoting the global interconnection among households; and $Γ_{1} \in R^{N \times N}$ is the matrix denoting the main TCL in each household, which is actively engaged in load communication on its floor.

The Laplacian matrix of the entire building L_c under normal communication state is expressed as:

L_{c} = I_{n_{f}} \otimes L_{h} + L_{3} \otimes Γ_{2}

(11)

where $Γ_{2}$ is the matrix denoting the TCL in one floor responsible for communicating with other floors; $L_{3} \in R^{N \times N}$ is the Laplacian matrix denoting the communication network between floors; and $I_{n_{f}}$ is an $n_{f} \times n_{f}$ identity matrix, and $n_{f}$ is the number of adjustment-involved floors.

In the event of communication disconnection between floors, the sole modification is the matrix $L_{3}$ in (11). Likewise, if there is a need to disrupt communication between residents, the modification required is limited to the matrix L₂. When the number of floors in residential building is increased or the scale is expanded, only n_f or Laplacian matrix L_c needs to be modified, thus the topology is extensible.

III. Proposed Distributed Coordinated Control Strategy

A. Dual-layer Internal Control Strategy for TCLs

The dual-layer internal control strategy is depicted in Fig. 3, where the lower layer introduces a leader-follower consensus algorithm to achieve decentralized autonomy of TCLs and organize fair power distribution. To meet the comfort constraints and enhance the consensus algorithm, the distributed MPC strategy is integrated into the upper layer.

Fig. 3 Dual-layer internal control strategy.

1)　Consensus Algorithm in Lower Layer

The communication topology of the building is shown in the lower layer of Fig. 3, where the interconnected circular blue regions signify the communication between different building floors.

The control variable of the system modeled in (5) can be represented as:

\begin{array}{l} u_{i} = - \sum_{j = 1}^{N} a_{i j} [κ_{1} (x_{i} (t) - x_{j} (t)) + κ_{2} (y_{i} (t) - y_{j} (t))] - \\ a_{i 0} [κ_{3} (x_{i} (t) - x_{0}) + κ_{4} (y_{i} (t) - y_{0})] \end{array}

(12)

where $κ_{1}$ , $κ_{2}$ , $κ_{3}$ , and $κ_{4}$ are the communication weights; x₀ and $y_{0}$ are the power and comfort variables of reference TCL, respectively; and $a_{i 0}$ is the communication weight between the $i^{t h}$ TCL and reference TCL.

From (12), the implementation of the secondary control input aims to eliminate the temperature and power deviations among TCLs and synchronize them to the reference TCL. The control matrix form of (12) is expressed as:

u = - L_{c} κ_{α} p - B_{r} κ_{β} p + E_{r}

(13)

where $κ_{α} = I_{N} \otimes [κ_{1}, κ_{2}]$ ; $κ_{β} = I_{N} \otimes [κ_{3}, κ_{4}]$ ; and $E_{r} = B_{r} κ_{β} p_{0}$ , $B_{r} = d i a g {a_{i 0}}$ $(i = 1,2, \dots, N)$ is the leader-follower adjacency matrix of the system, $p_{0} = 1_{N} \otimes [x_{0}, y_{0}]^{T}$ , and 1_N is a N-dimension column vector with all the elements being 1.

Discretize (13) and substitute it into the system modeled in (6), and we can obtain:

p (k + 1) = G p (k) + E_{r r}

(14)

where $G = I + h I_{N} \otimes A - h I_{N} \otimes B L_{c} κ_{α} - h I_{N} \otimes B B_{r} κ_{β}$ , and h is the sampling interval; and $E_{r r} = h I_{N} \otimes B E_{r} + I_{N} \otimes C$ .

To ensure the effectiveness of discretization, it is required that: ① the sampling interval h must be small enough to ensure that the discrete-time system closely approximates the continuous-time system dynamics; ② $η$ , $P_{r}$ , $R_{t c l}$ , $C_{t c l}$ , and $δ$ are time-invariant parameters; and ③ the changing timescale of outdoor and setpoint temperatures is much smaller than that of the control signal.

The Laplacian matrix $L_{c}$ possesses a simple zero eigenvalue, indicating that the graph $𝒢$ includes at least one directed spanning tree. Additionally, at least one node in $𝒢$ is connected to the reference node, and then the system modeled in (5) can reach a steady consensus. The proof process is shown in [

34]. The communication topology depicted in Fig. 2 satisfies the above conditions, so the following results can be drawn:

\underset{t \to \infty}{l i m} ‖x_{i} (t) - x_{j} (t)‖ = 0

(15)

\underset{t \to \infty}{l i m} ‖y_{i} (t) - y_{j} (t)‖ = 0

(16)

x_{f} = x_{0}

(17)

y_{f} = - \frac{η P_{r} x_{0} R_{t c l}}{2 δ} + \frac{T_{o u t} + δ - T_{s e t}}{2 δ}

(18)

where $x_{f}$ and $y_{f}$ are the steady-state values of the power and comfort variables, respectively.

2)　Distributed MPC Strategy in Upper Layer

To optimize the control convergence time and meet user comfort constraints, the distributed MPC strategy is incorporated into the upper layer. Design an auxiliary predictive control term $u_{L} = L_{c} κ_{μ} p (k)$ with the modification factor $κ_{μ} = I_{N} \otimes [κ_{μ}, κ_{μ}]$ and add it to (13), we can obtain the new control term $u_{p}$ :

u_{p} = - L_{c} (κ_{α} + κ_{μ}) p - B_{r} κ_{β} p + E_{r}

(19)

After incorporating the distributed MPC strategy, the state-space model of TCLs described by (6) is written as a discrete state-space model with control term $u_{p}$ , as shown in (20).

p (k + 1) = A_{d} p (k) + B_{d} u_{p} (k) + C (k)

(20)

where $A_{d} = e^{I_{N} \otimes A t_{d}}$ and $B_{d} = \int_{0}^{t_{d}} e^{I_{N} \otimes A t_{d}} I_{N} \otimes B d t$ with the discrete-time interval $t_{d}$ .

Since this model involves multiple steps of a rolling horizon, the dynamic of the system can be extended as:

P (k + 1) = S_{A} P (k) + S_{B} U_{p} (k) + S_{C} C (k)

(21)

where the detailed parameters of the matrices $P$ , $U_{p}$ , $C$ , $S_{A}$ , $S_{B}$ , and $S_{C}$ can be found in Supplementary Material A.

Based on (20), the future evolution of the state derivative vector for the prediction horizon of N_p can be defined as:

\begin{array}{l} Δ P (k + 1) = [Δ p (k + 1), (Δ p (k + 2), . . ., Δ p (k + N_{p})] = \\ Ψ P (k + 1) = A_{Ψ} p (k) + B_{Ψ} u_{p} (k) + C_{Ψ} \end{array}

(22)

where $Δ p (k) = L_{c} κ_{α} p (k)$ ; $Ψ = d i a g {L_{c}} \in R^{N_{p} N \times N_{p} N}$ ; $A_{Ψ} = Ψ A_{d}$ ; $B_{Ψ} = Ψ B_{d}$ ; and $C_{Ψ} = Ψ C$ .

To achieve the step-by-step iterative optimization for obtaining the optimal control coefficients $κ_{μ}$ in (19), considering the power and comfort variables of TCLs along with the associated constraints, an optimization problem $J_{m} (k)$ is formulated as:

J_{m} (k) = {‖Δ p (k)‖}_{Q_{L}}^{2} + {‖U_{p} (k)‖}_{R_{L}}^{2} = q_{l} \sum_{k = 0}^{N_{p}} {[\sum_{j = 1}^{N} a_{i j} (x_{i} (k) - x_{j} (k) + y_{i} (k) - y_{j} (k))]}^{2} + r_{l} \sum_{k = 0}^{N_{c}} u_{p}^{} (k) R_{L} u_{p}^{T} (k)

(23)

where $Q_{L} = d i a g {q_{l}} \in R^{N_{p} \times N_{p}}$ , and $q_{l}$ is the positive scalar weight controlling the influence of position error; $R_{L} = d i a g {r_{l}} \in R^{N_{c} \times N_{c}}$ , $r_{l}$ is the positive scalar weight controlling the influence of control input, and $N_{c}$ is the control horizon.

The first component of the objective function in (23) aims to minimize the differences between neighboring nodes by penalizing the state deviations between each pair of neighboring states over the future $N_{p}$ steps, thereby promoting overall consistency within the network. The second component of the objective function in (23) aims to minimize the control input, avoiding excessive energy input.

By balancing the trade-off between minimizing state deviations and controlling the input amount, the efficient and stable operation are ensured, thereby enhancing the stability of the entire system. $J_{m} (k)$ aims to minimize the power consumption variation and comfort disparity among different TCLs while maximizing the power and comfort sharing among TCLs.

Constraints are imposed on the permissible range of state and control variables, as delineated in (24).

\{\begin{array}{l} 0 \leq x (k) \leq 1 \\ 0 \leq y (k) \leq 0.9 \\ U_{m i n} \leq U_{p} (k) \leq U_{m a x} \end{array}

(24)

where $U_{m i n} = u_{m i n} 1_{N_{c}} \in R^{N_{c} \times 1}$ and $U_{m a x} = u_{m a x} 1_{N_{c}} \in R^{N_{c} \times 1}$ , $1_{N_{c}}$ is a N_c-dimension column vector with all the elements being 1, and $u_{m i n}$ and $u_{m a x}$ are the lower and upper bounds of control variables, respectively.

To limit the output power within the rated range and keep V_soc within the range of $(0,1)$ , we make $0 \leq x (t) \leq 1$ . The comfort variable is restricted within the range of $(0,0.9)$ to strictly consider the user comfort so that the indoor temperature fluctuates within a narrow range near the set point.

Substituting (19) into (23), $u_{p}$ is obtained through iterative optimization to solve the objective function. It can be noticed that the objective function of one TCL is related to other TCLs, which means that it is a coupling programming problem. Thus, the global optimal solution can only be obtained by information exchange and iteration among the controllers in different TCLs. The rules governing the iterative process are outlined as follows.

Step 1: set the iteration index $r = 1$ , the initial state for the $i^{t h}$ TCL $x_{i}^{r} (k) = x_{i} (k)$ , $u_{p, r} = 0$ for all TCL.

Step 2: solve the programming problem (23), and obtain the optimal control signal $u_{p, r} (k)$ .

Step 3: calculate the state $x_{i}^{r + 1} (k)$ by substituting $x_{i}^{r} (k)$ and $u_{p, r} (k)$ into (20).

Step 4: exchange the information of each TCL with others by communication networks, then solve the programming problem (23) and obtain the optimal control signal $u_{p, r + 1} (k)$ .

Step 5: if $‖u_{p, r + 1} (k) - u_{p, r} (k)‖ < ε$ , where $ε$ is a small positive value, the condition of ending the iteration is satisfied; the system has reached a balanced state, and $u_{p, r + 1} (k)$ is the optimal solution for TCLs; otherwise, $r = r + 1$ , and return to Step 3.

Equation (24) constrains the power output range and indoor temperature range during the control process of TCLs. It rigorously safeguards the comfort experience throughout the control proccess. The minimization of (23) is a common quadratic programming (QP) problem that can be solved by MATLAB.

B. External Multi-timescale Scheduling Framework for TCLs

In this subsection, an external multi-timescale scheduling framework is depicted in Fig. 4. The energy scheduling signals are transformed into control signals to regulate TCLs within the building.

Fig. 4 External multi-timescale scheduling framework.

The coordinated utilization of multiple energy storage resources in micro-communities can maximize the community flexibility. This paper considers two load-side resources: ESSs and TCLs. The frequency division strategy is employed to coordinate the utilization of two load-side resources.

Given the potential challenge that frequent charge and discharge cycles may affect the rapid responsiveness of TCLs to track power instructions, it is proper to utilize ESSs as a short-term buffer mechanism and TCLs to serve buffer power changes over an extended period. The external power command of TCLs $P_{t c l}^{*}$ originates from:

\{\begin{array}{l} P_{r e 1, k + 1}^{} = \frac{Δ t}{T_{h i g h}} (P_{r e, k}^{} - P_{r e 1, k}^{}) + P_{r e 1, k}^{} \\ P_{r e 2, k + 1}^{} = \frac{Δ t}{T_{l o w}} (P_{r e 1, k}^{} - P_{r e 2, k}^{}) + P_{r e 2, k}^{} \end{array}

(25)

P_{b a}^{*} = P_{r e 2, k + 1}^{} - P_{r e 1, k + 1}^{}

(26)

P_{t c l}^{*} = P_{r e 2, k + 1} - P_{n l o a d}

(27)

where $P_{r e, k}$ is the source-side energy flow composed of renewable energy and tie-line power flow; $P_{r e 1, k}$ is the long-term energy flow; $P_{r e 2, k}$ is the target energy flow; $P_{b a}^{*}$ is the target power of ESSs; $P_{t c l}^{*}$ is the target power of TCLs; $P_{n l o a d}$ is the power of unadjustable loads, which is not considered in this paper; $T_{h i g h}$ is the high-frequency time constant; $T_{l o w}$ is the low-frequency time constant; and $Δ t$ is the sampling time. Adjusting the filtering time constant provides a means to finely adjust the distribution of the target power.

Utilizing TCLs for grid auxiliary services necessitates a sophisticated control strategy for managing external dispatch signals. This policy coordinates resource allocation, prioritizing energy efficiency goals and consumer comfort. The comfort variable expectation of 0.5 means that the room temperature expectation during the control process is closely aligned with the setpoint temperature and guarantees the user experience.

According to (18), we have:

- \frac{η P_{r} x_{0} R_{t c l}}{2 δ} + \frac{T_{o u t} + δ - T_{s e t}}{2 δ} = 0.5

(28)

x_{0} = \frac{T_{o u t} - T_{s e t}}{η P_{r} R_{t c l}}

(29)

Equation (29) establishes the connection between the power variable x₀ and the physical actual variables. Since T_set is the most intuitive and easily adjustable physical quantity for the TCLs, the external energy signal is converted into a setpoint signal as:

T_{s e t} = T_{o u t} - η R_{t c l} P_{t c l}^{*} / N

(30)

The actual power consumption of TCLs $P_{t t}$ is calculated as:

P_{t t} = \sum_{i = 1}^{N} P_{r} x_{i} (t)

(31)

Hence, we adjust T_set based on $P_{t c l}^{*}$ in (27), which further regulates $P_{t t}$ , enabling their participation in auxiliary services, as depicted by the blue box in Fig. 4, with the second-level control timescale $Δ t_{c}$ .

As depicted in Fig. 4, once the energy scheduling instruction is issued at the minute-level timescale $(Δ t)$ , the TCLs swiftly react to the external power command at the second-level timescale $(Δ t_{c})$ . Subsequently, it progresses to the next scheduling cycle upon achieving the control goal.

IV. Stability Analysis of Distributed MPC Strategy

Assuming that the value of $ε$ is chosen appropriately to ensure the stability of (14) with $ρ (G) < 1$ , where $ρ (\cdot)$ denotes the matrix spectrum radius. Let $λ$ and v denote the eigenvalue and the associated eigenvector of an arbitrary symmetric and positive semi-definite matrix $D \in R^{N \times N}$ . We have Lemma 1: ① $(D + I_{N})^{- n} v = [1 / {(λ + 1)}^{n}] v$ , $\forall n \in N$ ; and ② ${‖(D + I_{N})^{- 1}‖}_{2} \leq 1$ .

The proofs are given in [

35]. To minimize the objective function in (23), we calculate the coefficient

μ

κ_{μ}

by setting

\partial J / \partial μ = 0

. Since obtaining an analytical solution to

μ

may be complex and the terms in

J (k)

are similar to those in

U_{p} (k)

, calculating the optimal control law is equivalent to solve

U_{p} (k)

through

\partial J (k) / \partial U_{p} (k) = 0

. The process can be expressed as:

\frac{\partial J (k)}{\partial U_{p} (k)} = 2 B_{d}^{T} Q_{L}^{*} A_{d} p (k) + 2 (B_{d}^{T} Q_{L}^{*} B_{d} + R_{L}) U_{p} (k) = 0

(32)

where $Q_{L}^{*} = Ψ^{T} Q_{L} Ψ$ .

By the matrix manipulation, $U_{p} (k) = - (B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} \cdot B_{d}^{T} Q_{L}^{*} A_{d} p (k)$ . Only the top N entries of $U_{p} (k)$ are selected to be the control input, and we set $G_{M P C} = - [I_{N}, 0_{N}, \dots, 0_{N}] \cdot (B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} B_{d}^{T} Q_{L}^{*} A_{d}$ .

G_MPC is introduced to modify G to obtain a better dynamic performance based on the communication prediction mechanism. The corresponding closed-loop dynamics can be formulated as:

p (k + 1) = (G + G_{M P C}) p (k) + E_{r r}

(33)

According to Lemma 1, we have:

\begin{array}{l} I_{N} + G_{M P C} G^{- 1} = [I_{N}, 0_{N}, \dots, 0_{N}] (B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} \cdot \\ [(B_{d}^{T} Q_{L}^{*} B_{d} + R_{L}) [I_{N}, 0_{N}, \dots, 0_{N}]^{T} - B_{d}^{T} Q_{L}^{*} A_{d}] \end{array}

(34)

Due to $B_{d} [I_{N}, 0_{N}, \dots, 0_{N}]^{T} = A_{d} G^{- 1}$ , we have:

\begin{array}{l} I_{N} + G_{M P C} G^{- 1} = [I_{N}, 0_{N}, \dots, 0_{N}] (B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} \cdot \\ R_{L} [I_{N}, 0_{N}, \dots, 0_{N}]^{T} \end{array}

(35)

\begin{array}{l} ρ (I_{N} + G_{M P C} G^{- 1}) \leq {‖[I_{N}, 0_{N}, \dots, 0_{N}]‖}_{2} \cdot \\ {‖(B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} R_{L}‖}_{2} {‖[I_{N}, 0_{N}, \dots, 0_{N}]‖}_{2} \end{array}

(36)

Considering that ${‖[I_{N}, 0_{N}, \dots, 0_{N}]‖}_{2} = 1$ and $B_{d}^{T} Q_{L}^{*} B_{d}$ is symmetric and positive semi-definite, we have ${‖(B_{d}^{T} Q_{L}^{*} B_{d} + R_{L})^{- 1} R_{L}‖}_{2} \leq 1$ according to Lemma 1. Therefore, we can derive that:

ρ (I_{N} + G_{M P C} G^{- 1}) \leq 1

(37)

Under the assumption $ρ (G) < 1$ , we have:

ρ (G + G_{M P C}) \leq ρ (G) ρ (I_{N} + G_{M P C} G^{- 1}) \leq 1

(38)

The stability of the closed-loop system in the distributed MPC strategy has been verified.

V. Case Study

A micro-community equipped with renewable energy sources is utilized to validate the synergistic application of TCLs and ESSs for the case study. The micro-community contains a building, several small ESSs, and photovoltaic power stations, as shown in Fig. 5. The TCL model parameters for simulation verification are given in Table I [

36]-[38].

Fig. 5 A schematic diagram of a micro-community.

TABLE I TCL Model Parameter for Simulation Verification

Parameter	Value	Parameter	Value
R_tcl (℃/kWh)	2	C_tcl (kWh/℃)	2
$P_{r}$ (kW)	5	$κ_{1}$	0.3
$η$	2.5	$κ_{2}$	0.005
$T_{s e t}$ (℃)	26	$κ_{3}$	0.8
δ (℃)	2	$κ_{4}$	0.05
$m$	3	$T_{h i g h}$	20
$m_{i n}$	5	$T_{l o w}$	5

Subsequently, Cases I and II are introduced for the case study, where Case I considers a constant reference power variable to represent the constant power command response while Case II extends this to the fluctuating power command response in a micro-community.

A. Case I: Constant Power Command Response

In this subsection, we verify the proposed dual-layer internal control strategy shown in Fig. 3 (hereafter refered to as the proposed internal control strategy for simplicity).

The reference power variable $x_{0}$ is set to be 0.4, while the initial comfort variable $y_{0}$ is set to be 0.6. The building with $n_{f l o o r} = 3$ floors is considered, containing $N = m m_{i n} n_{f l o o r} = 45$ TCLs. To introduce the randomness into the simulation, the power and comfort variables of TCLs are uniformly distributed within the range of 0.3 to 0.7. The convergence criterion for comparing the control strategies is set to be 10^-3, ensuring sufficient precision for evaluating TCLs.

1)　Under Normal Communication Conditions

Figure 6 illustrates the power and comfort variables of TCLs with the consensus algorithm in [

27] under normal communication conditions, and Fig. 7 illustrates those with the leader-follower consensus algorithm and the proposed internal control strategy, where different colors of lines represent different TCLs.

Fig. 6 Power and comfort variables of TCLs with consensus algorithms in [

27] under normal communication conditions. (a) Power variable. (b) Comfort variable.

Fig. 7 Power and comfort variables of TCLs with leader-follower consensus algorithm and proposed internal control strategy under normal communication conditions. (a) Power variable with leader-follower consensus algorithm. (b) Comfort variable with leader-follower consensus algorithm. (c) Power variable with proposed internal control strategy. (d) Comfort variable with proposed internal control strategy.

TCLs share the same power and comfort levels by applying the above strategies. However, as depicted in Fig. 6(b), the comfort variables of TCLs under the consensus algorithm in [

27] significantly drop below zero, indicating that the indoor temperature falls outside the designated comfort range. As indicated by the convergence time of the power and comfort variables in Figs. 6 and 7, introducing a reference TCL shortens the time for achieving state sharing, and the distributed MPC strategy further enhances consistency.

Furthermore, a comparison of Fig. 6(b) and Fig. 7(b) with Fig. 7(d) reveals that the proposed internal control strategy can rigorously maintain the comfort variables of TCLs within the range (0, 1).

2)　Under Abnormal Communication Conditions

This part conducts simulations under the operation conditions of communication interruption, communication delay, and input delay while keeping the initial simulation conditions unchanged.

1)　Communication interruption

In the case of communication interruption among residents, where $L_{2} = 0_{3 \times 2}$ , the comfort variables of TCLs with the consensus algorithm in [

27] and the proposed internal control strategy are illustrated in Fig. 8(a) and (b), respectively. It can be found that compared with the consensus algorithm in [27], the optimization objective in the proposed internal control strategy maintains the consistency even in the event of user communication interruption. However, the final convergence time t_f in Fig. 8(b) is longer compared with that in Fig. 7(d). The comparison shows that the proposed internal control strategy can still achieve consensus convergence compared with the consensus algorithm in [27], although the convergence time becomes longer. Moreover, as the communication network becomes sparser, the convergence time further increases.

Fig. 8 Comfort variables of TCLs with communication interruption. (a) Consensus algorithm in [

27]. (b) Proposed internal control strategy.

2)　Communication delay

To evaluate the impact of communication delays, we modify the protocol (12) as:

\begin{array}{l} u_{i}^{'} = - \sum_{j = 1}^{N} {a_{i j} [κ_{1} (x_{i} (t - τ_{i j}) - x_{j} (t - τ_{i j})) + κ_{2} (y_{i} (t - τ_{i j}) - \\ y_{j} (t - τ_{i j}))] - a_{i 0} [κ_{3} (x_{i} (t - τ_{i j}) - x_{0}) + κ_{4} (y_{i} (t - τ_{i j}) - y_{0})]} \end{array}

(39)

Considering a state delay $τ_{i j} = 0.4$ s, the power and comfort variables with the proposed internal control strategy under communication delay are shown in Fig. 9(a) and (b), respectively. The system stability decreases with the increase of communication delay. The primary cause is the injection of uncertain energy into the control system, rendering it an active network. To ensure the system stability and reduce oscillations, the power and comfort variables only need to be transformed as $x_{i j}^{'} (t) = (x_{i j} (t) - x_{i j} (t - τ_{i j})) / (2 c_{i j})$ and $y_{i j}^{'} (t) = (y_{i j} (t) - y_{i j} (t - τ_{i j})) / (2 c_{i j})$ , respectively, where c_ij is the input parameter, which is set to be 1 in this context, $x_{i j}^{} (t) = x_{i} (t) - x_{j} (t)$ , and $y_{i j}^{} (t) = y_{i} (t) - y_{j} (t)$ . The transformation converts the delayed active communication network into a passive one, thereby enhancing system stability. As observed in Fig. 9(c) and (d), the system stability improves with the scattering transformation.

Fig. 9 Power and comfort variables of TCLs with proposed internal control strategy under communication delay. (a) Power variable without scattering transformation. (b) Comfort variable without scattering transformation. (c) Power variable with scattering transformation. (d) Comfort variable with scattering transformation.

3)　Input delay

To explore the influence of the mechanical delay between the IAC response to the setpoint command in TCL on the proposed internal control strategy, we introduce a delay factor into (4) and obtain:

\dot{y}' (t) = \frac{T_{o u t} (t) + δ - T_{s e t} (t)}{2 δ C_{t c l} R_{t c l}} - \frac{y' (t)}{C_{t c l} R_{t c l}} - \frac{η P_{r} x (t - τ_{i n})}{2 δ C_{t c l}}

(40)

where $y' (t)$ is the control variable with input delay; and $τ_{i n}$ is the input delay of the compressor mechanical process, indicating the buffer time required by the IAC compressor to adjust the cooling power output in response to changes in the outdoor temperature. The comfort variables with the proposed internal control strategy when $τ_{i n} = 1$ s and $τ_{i n} = 0.3$ s are presented in Fig. 10.

Fig. 10 Comfort variables of TCLs with input delay under proposed internal control strategy. (a) $τ_{i n} = 1$ s. (b) $τ_{i n} = 0.3$ s.

It can be observed that TCLs can achieve the control goal under the proposed internal control strategy considering $τ_{i n}$ .

However, as $τ_{i n}$ is extended, the comfort variables of the TCLs exceed the range of (0,1), adversely affecting the comfort of the TCL cluster and negatively impacting user experience.

This phenomenon is attributed to the significant delay in adjusting the setpoint temperature, which does not immediately result in power changes. Consequently, the indoor temperature gradually rises under the influence of external temperature without any cooling measures, thus exceeding the comfort range. As illustrated in Fig. 10(b), by reducing the delay time, the range of temperature fluctuations is minimized, thereby enhancing user experience.

B. Case II: Fluctuating Power Command Response

Considering the fluctuating power command response of the photovoltaic station in a real micro-community, as shown in Fig. 5, a multi-timescale scheduling framework is added to the proposed internal control strategy, which is namely the proposed distributed coordinated control strategy.

To meet the capacity requirements, the building with $n_{f l o o r} = 10$ floors is utilized for simulating the TCL cluster, comprising $N = m m_{i n} n_{f l o o r} = 150$ TCLs. The sampling duration is from 08:00-20:00, with a timescale of $Δ t = 1$ min and a control sampling interval of $Δ t_{c} = 1$ s. Assume that the outdoor temperature follows a typical sinusoidal pattern of a summer day. The temperature sampling interval is the same as t_c. The simulation results with the proposed distributed coordinated control strategy in Case II are shown in Fig. 11.

Fig. 11 Simulation results with proposed distributed coordinated control strategy in Case II.

1)　External Power Command Response

To accommodate the renewable energy generation within the community, TCLs within the building serve as the primary accommodating units for low-frequency portion of generation, while ESSs, due to the rapid response characteristics, absorb the high-frequency portion.

The temperature curves with the proposed distributed coordinated control strategy in Case II are illustrated in Fig. 12, from which we can observe that the indoor temperature of TCLs swiftly converges to a consensus, closely tracking the setpoint temperature with minor thermal inertia lag.

Fig. 12 Temperature curves with proposed coordinated strategy in Case II.

Figure 13(a) illustrates the power and comfort variables of TCLs with the proposed distributed coordinated control strategy in Case II. It is evident that under varying outdoor temperatures and external scheduling instructions, both the power and comfort variables of TCLs remain within the range of (0,1). This indicates that the proposed distributed coordinated control strategy effectively maintains user comfort levels. Additionally, the enlarged views in Fig. 13(a) and (b) show that TCLs achieve control consistency in the regulation process, facilitating the power consumption shared among numerous TCLs.

Fig. 13 Power and comfort variables of TCLs with proposed distributed coordinated control strategy in Case II. (a) Power variable. (b) Comfort variable.

V_SOC of TCLs and SOC of ESSs with the proposed distributed coordinated control strategy during the control process are presented in Fig. 14. Both profiles remain bounded within (0,1), confirming that their operation remains within the control capabilities. In scenarios of increasing outdoor temperatures or power demands, potential measures include augmenting the battery capacity of ESSs and enhancing the control capabilities of TCLs. This objective may be achieved by elevating the rated power of IACs or augmenting the TCL population. In practical implementations, the latter is typically preferred, emphasizing that greater TCL population is associated with enhanced control provisioning capability.

Fig. 14 V_SOC of TCLs and SOC of ESSs with proposed distributed coordinated control strategy.

2)　Comparison of Different Control Strategies

The comparison between the proposed distributed coordinated control strategy and the traditional control strategy is illustrated in Fig. 15. It is evident that the proposed distributed coordinated control strategy significantly mitigates fluctuations in the comfort variable, thereby enhancing the overall user comfort experience. As indicated by Fig. 15(b) and (c), the proposed distributed coordinated control strategy markedly shortens the time taken for TCLs to achieve equitable power and comfort sharing, facilitating a close alignment between the indoor and setpoint temperatures of TCLs.

Fig. 15 Comparison results between proposed distributed coordinated control strategy and traditional control strategy. (a) Comfort variable from 08:00-20:00. (b) Power variable at initial stage. (c) Comfort variable at initial stage.

3)　Scalability of Communication Network

To evaluate the scalability of the communication network and the effectiveness of controlling TCLs at different scales, the building scale expands from $n_{f l o o r} = 10$ to $n_{f l o o r} = 20$ .

As shown in Fig. 16, the proposed distributed coordinated control strategy maintains uniform power distribution and comfort equalization throughout the day, thereby demonstrating the scalability of communication network. By comparing Fig. 16(a) with Fig. 13(a), it can be observed that with the increase in building scale, the power consumption response curve remains unchanged under the proposed distributed coordinated control strategy. The larger the building scale, the lower the energy consumption of each TCL, i.e., the lower the power variable.

Fig. 16 Power and comfort variables of TCLs with proposed distributed coordinated strategy in Case II when $n_{f l o o r} = 20$ . (a) Power variable. (b) Comfort variable.

As illustrated in Fig. 17, with more TCLs participating in load demand response, the capacity of TCLs as generalized energy storage is expanded, leading to reduced fluctuations of setpoint temperature. Each TCL handles smaller tasks, thereby the impact on individual users is reduced. Hence, greater involvement of load-side resources in load demand response can improve the system stability and resilience, particularly during periods of high demand or unexpected system stressors.

Fig. 17 Temperature curves with proposed distributed coordinated strategy in Case II when $n_{f l o o r} = 10$ and $n_{f l o o r} = 20$ .

VI. Conclusion

This paper proposes a distributed coordinated control strategy for large-scale distributed TCLs on the load side of micro-communities. This control strategy consists of internal regulation and external energy scheduling. A dual-layer internal control strategy based on MPC is proposed for internal regulation within TCLs, which integrates the user comfort limit to enable the power and comfort sharing in the dynamic control process of TCLs while ensuring the privacy of users. External multi-timescale scheduling framework utilizes a frequency division approach to achieve the coordinated utilization of various resources. Simulation results demonstrate that:

1) The proposed internal control strategy significantly reduces the time for TCLs to reach consensus and ensures the comfort range of TCLs rigorously.

2) As the number of regulated TCLs participating in load demand response increases, the regulation capacity also increases, resulting in smaller fluctuations in indoor temperature for users.

3) The external multi-timescale scheduling framework utilizes various energy sources across different frequency bands, effectively tracking target power commands and enhancing energy efficiency.

Future research will expand the scope of this study to larger communities and explore the coordinated control of multiple energy sources. We will also consider user behavior preferences, economic costs, and electricity prices to more comprehensively investigate the applications of demand response. Introducing learning mechanisms [

39], [40] to address uncertainty in the regulatory process will also be included.

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