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.
FUTURE trends in the power industry indicate a predominant direction of integrating large-scale renewable energy sources into the power grid [
An individual TCL typically has a low power capacity [
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 [
However, the initial value updating methods are centralized and still ignore the role of comfort constraints in regulation. Reference [
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 [
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.
For buildings dominated by TCLs, the heat transfer relationship is expressed as:
(1) |
where is the specific heat capacity of air; is the air density; is the room volume; is the indoor temperature at time t; and and 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) [

Fig. 1 Schematic diagram of equivalent model of TCL.
Consider the room installed with the IAC within the residential building as a TCL, and rooms in the building form a cluster. As shown in
(2) |
where is the power consumption of TCLs, i.e., the power output percentage at time ; is the outdoor temperature at time t; is the energy efficiency ratio, which is positive for cooling TCLs and negative for heating TCLs; is the rated power of TCL; and are the equivalent thermal capacitance and resistance of the room, respectively, is the thermal conductivity of the building, and is the area of room; and 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
Remark 1: we assume the external disturbances follow a zero-mean Gaussian distribution, i.e., , where the variance 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 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 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 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 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 is introduced as:
(3) |
where , , Tset is the set temperature of TCL, and is the temperature deviation from Tset within the comfort range. As long as y(t) is within , the change in Tin 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:
(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).
(5) |
(6) |
where is the symbol for the Kronecker product of matrices; , is the vector of state variables, , and N is the number of TCLs; is the vector of control variables; is the system matrix, which describes the internal state relationships within the system; ; ; and is an identity matrix.
As the timescale of control signal is notably smaller than that of the changes in and during the control process, , , and can be treated as constants.
Based on the temperature regulation capability of TCL, 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 , 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 for virtual ESSs as:
(7) |
where is the initial .
The communication network of a cluster consisting of N TCLs can be depicted as a graph [
(8) |
(9) |
where D is the in-degree matrix of , which is defined as with .
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 , which is intricately linked to the graph .
It is assumed that each floor comprises households, with each household equipped with TCLs, as shown in

Fig. 2 Communication topology among households on a particular floor.
The Laplacian matrix of a specific floor can be represented as:
(10) |
where is an identity matrix; is the Laplacian matrix of , denoting the local interaction of TCLs in each household; is the Laplacian matrix of , denoting the global interconnection among households; and 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 Lc under normal communication state is expressed as:
(11) |
where is the matrix denoting the TCL in one floor responsible for communicating with other floors; is the Laplacian matrix denoting the communication network between floors; and is an identity matrix, and is the number of adjustment-involved floors.
In the event of communication disconnection between floors, the sole modification is the matrix 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 nf or Laplacian matrix Lc needs to be modified, thus the topology is extensible.
The dual-layer internal control strategy is depicted in

Fig. 3 Dual-layer internal control strategy.
The communication topology of the building is shown in the lower layer of
The control variable of the system modeled in (5) can be represented as:
(12) |
where , , , and are the communication weights; x0 and are the power and comfort variables of reference TCL, respectively; and is the communication weight between the 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:
(13) |
where ; ; and , is the leader-follower adjacency matrix of the system, , and 1N 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:
(14) |
where , and h is the sampling interval; and .
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 , , , , and 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 possesses a simple zero eigenvalue, indicating that the graph includes at least one directed spanning tree. Additionally, at least one node in is connected to the reference node, and then the system modeled in (5) can reach a steady consensus. The proof process is shown in [
(15) |
(16) |
(17) |
(18) |
where and are the steady-state values of the power and comfort variables, respectively.
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 with the modification factor and add it to (13), we can obtain the new control term :
(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 , as shown in (20).
(20) |
where and with the discrete-time interval .
Since this model involves multiple steps of a rolling horizon, the dynamic of the system can be extended as:
(21) |
where the detailed parameters of the matrices , , , , , and can be found in Supplementary Material A.
Based on (20), the future evolution of the state derivative vector for the prediction horizon of Np can be defined as:
(22) |
where ; ; ; ; and .
To achieve the step-by-step iterative optimization for obtaining the optimal control coefficients in (19), considering the power and comfort variables of TCLs along with the associated constraints, an optimization problem is formulated as:
(23) |
where , and is the positive scalar weight controlling the influence of position error; , is the positive scalar weight controlling the influence of control input, and 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 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. 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).
(24) |
where and , is a Nc-dimension column vector with all the elements being 1, and and are the lower and upper bounds of control variables, respectively.
To limit the output power within the rated range and keep Vsoc within the range of , we make . The comfort variable is restricted within the range of to strictly consider the user comfort so that the indoor temperature fluctuates within a narrow range near the set point.
Substituting (19) into (23), 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 , the initial state for the TCL , for all TCL.
Step 2: solve the programming problem (23), and obtain the optimal control signal .
Step 3: calculate the state by substituting and 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 .
Step 5: if , where is a small positive value, the condition of ending the iteration is satisfied; the system has reached a balanced state, and is the optimal solution for TCLs; otherwise, , and return to Step 3.
In this subsection, an external multi-timescale scheduling framework is depicted in

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 originates from:
(25) |
(26) |
(27) |
where is the source-side energy flow composed of renewable energy and tie-line power flow; is the long-term energy flow; is the target energy flow; is the target power of ESSs; is the target power of TCLs; is the power of unadjustable loads, which is not considered in this paper; is the high-frequency time constant; is the low-frequency time constant; and 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:
(28) |
(29) |
(30) |
The actual power consumption of TCLs is calculated as:
(31) |
Hence, we adjust Tset based on in (27), which further regulates , enabling their participation in auxiliary services, as depicted by the blue box in
As depicted in
Assuming that the value of is chosen appropriately to ensure the stability of (14) with , where denotes the matrix spectrum radius. Let and v denote the eigenvalue and the associated eigenvector of an arbitrary symmetric and positive semi-definite matrix . We have Lemma 1: ①, ; and ② .
The proofs are given in [
(32) |
where .
By the matrix manipulation, . Only the top N entries of are selected to be the control input, and we set .
GMPC 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:
(33) |
According to Lemma 1, we have:
(34) |
Due to , we have:
(35) |
(36) |
Considering that and is symmetric and positive semi-definite, we have according to Lemma 1. Therefore, we can derive that:
(37) |
Under the assumption , we have:
(38) |
The stability of the closed-loop system in the distributed MPC strategy has been verified.
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 A schematic diagram of a micro-community.
Parameter | Value | Parameter | Value |
---|---|---|---|
Rtcl (℃/kWh) | 2 | Ctcl (kWh/℃) | 2 |
(kW) | 5 | 0.3 | |
2.5 | 0.005 | ||
(℃) | 26 | 0.8 | |
δ (℃) | 2 | 0.05 | |
3 | 20 | ||
5 | 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.
In this subsection, we verify the proposed dual-layer internal control strategy shown in
The reference power variable is set to be 0.4, while the initial comfort variable is set to be 0.6. The building with floors is considered, containing 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 1

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

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
Furthermore, a comparison of
This part conducts simulations under the operation conditions of communication interruption, communication delay, and input delay while keeping the initial simulation conditions unchanged.
In the case of communication interruption among residents, where , the comfort variables of TCLs with the consensus algorithm in [

Fig. 8 Comfort variables of TCLs with communication interruption. (a) Consensus algorithm in [
To evaluate the impact of communication delays, we modify the protocol (12) as:
(39) |
Considering a state delay s, the power and comfort variables with the proposed internal control strategy under communication delay are shown in

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.
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:
(40) |
where is the control variable with input delay; and 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 s and s are presented in

Fig. 10 Comfort variables of TCLs with input delay under proposed internal control strategy. (a) s. (b) s.
It can be observed that TCLs can achieve the control goal under the proposed internal control strategy considering .
However, as 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
Considering the fluctuating power command response of the photovoltaic station in a real micro-community, as shown in
To meet the capacity requirements, the building with floors is utilized for simulating the TCL cluster, comprising TCLs. The sampling duration is from 08:00-20:00, with a timescale of min and a control sampling interval of s. Assume that the outdoor temperature follows a typical sinusoidal pattern of a summer day. The temperature sampling interval is the same as tc. The simulation results with the proposed distributed coordinated control strategy in Case II are shown in

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

Fig. 13 Power and comfort variables of TCLs with proposed distributed coordinated control strategy in Case II. (a) Power variable. (b) Comfort variable.
VSOC of TCLs and SOC of ESSs with the proposed distributed coordinated control strategy during the control process are presented in

Fig. 14 VSOC of TCLs and SOC of ESSs with proposed distributed coordinated control strategy.
The comparison between the proposed distributed coordinated control strategy and the traditional control strategy is illustrated in

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.
To evaluate the scalability of the communication network and the effectiveness of controlling TCLs at different scales, the building scale expands from to .
As shown in

Fig. 16 Power and comfort variables of TCLs with proposed distributed coordinated strategy in Case II when . (a) Power variable. (b) Comfort variable.
As illustrated in

Fig. 17 Temperature curves with proposed distributed coordinated strategy in Case II when and .
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 [
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