Abstract
This paper proposes a hybrid control strategy of air-conditioning loads (ACLs) for participating in peak load reduction. The hybrid control strategy combines the temperature setpoint adjustment (TSA) control and on/off control together to make full use of response potentials of ACLs. The primary free transport model of ACLs has been established in literature at or near a fixed temperature setpoint. In this paper, a wide-range transport (WRT) model suitable for larger value of TSA is proposed. The WRT model can be constructed easily through the parameter of devices and indoor and outdoor temperature. To modulate the aggregate response characteristics of ACLs more friendly to the power grid, the safe protocol (SP) is adopted and integrated into the WRT model, which achieves a good unification of oscillation suppression and efficient modeling. Moreover, the hybrid control strategy is implemented based on the WRT model, and the model predictive control (MPC) controller is designed considering the tracking error and control switch cost. At last, the superiority of the hybrid control strategy is verified and the performance of ACLs for peak load reduction under this controller is simulated. The simulation results show that the hybrid control strategy could exploit the load reduction potential of ACLs fully than the TSA mode and track the reference signal more accurately.
WITH the development of economy in China, the proportion of residential electricity consumption increases year by year [
Since the capacity of a single ACL is too small to have an effective impact on grid operation, the load aggregators take the role of the agency to aggregate large-scale ACLs for participating in ancillary services [
Among the aggregate modeling methods of ACLs, the classic one is the Monte-Carlo method based on the equivalent thermal parameter (ETP) model [
The commonality of the above aggregation models is that they can only describe the situation where ACLs operate at or near a single temperature setpoint. However, in practice, it is a simple and efficient method to adjust the temperature setpoint of the TCLs within the customers’ comfort level to tap the demand response [
Besides, the power rebound and oscillation are inevitable when ACLs are controlled in the centralized control mode [
However, the aggregate power cannot be precisely adjusted if only a single temperature adjustment control method is adopted [
The evolution process of the aggregate ACLs operating at a steady state in the temperature dead band can be described by the original FTM.
When the temperature setpoint of ACLs remains constant, the indoor temperature will reciprocate between the upper and lower boundaries of the temperature dead band. The entire population of ACLs can be considered as a load flux which flows between the temperature dead band on the temperature axis [

Fig. 1 Load flowing process of FTM.
Let Lon/off(t,θ) denote the load flux going through temperature θ at time t. Non/off(t, θ) is the number of loads in on/off state at time t and temperature θ. The relationship between and can be expressed as:
(1) |
(2) |
where is the load transporting rate in the on/off state at time t and temperature θ. In the FTM of TCLs, this parameter is approximately considered as a constant, which is aon ≈()/(CR) and aoff ≈()/(CR), where is the outdoor temperature; P is the cooling or heating power; and C and R are the equivalent thermal capacitance and thermal resistance, respectively. This approximation ignores the change of temperature around the initial temperature setpoint.
By substituting (1) into (2), the governing partial differential equation (PDE) of the FTM can be written as:
(3) |
The transporting processes at the temperature limits are:
(4) |
(5) |
This population of the aggregate power of TCLs Pagg at time t is represented as:
(6) |
where is the coefficient of performance (COP) of ACLs.
In order to establish an aggregation model that can describe the large range adjustment of the temperature setpoint, this subsection improves the FTM and establishes the WRT model.
When the temperature setpoint of ACLs is changed, the flow process of the load flux will no longer follow the FTM, as shown in

Fig. 2 Flow process of load flux.
The load originally in the off state will no longer flow upward through the temperature boundary into the on state, but will flow through the boundary to the right along the red arrow to the higher temperature dead band.
The load in TTD will flow into NTD after temperature adjustment through the red arrow. The load flux will follow the free transport process and gradually recover to a stable state.
It can be observed that the rule of load flow at the boundary has changed when the temperature setpoint is adjusted, so the key to construct a WRT model is to define the boundary conditions of the TTD and the NTD:
(7) |
(8) |
(9) |
(10) |
Equations (
Finally, can be rewritten as:
(11) |
The load flow process described by PDEs is not convenient for practical engineering applications. Therefore, we will use the finite difference method [
First, divide the temperature axis into small temperature segments of equal width, as shown in

Fig. 3 Finite-difference discretization of WRT model.
Once the TSA signal is received, the temperature setpoint will shift to the right to and the TTD will disappear when the response process is over. Therefore, the number of the temperature segments must be converted and the load number in each segment must be initialized.
Let and denote the load numbers at time k in the
Let and denote the load numbers at time k in the temperature segment in the off state and the on state in the NTD, respectively. The initialization method is shown in (14) and (15).
(12) |
(13) |
(14) |
(15) |
where and are the load numbers at the initial state in the
The first-order upwind difference scheme is used to deal with (3) and boundary conditions (7)-(10). The difference expression is shown in Appendix A. Furthermore, the state space model is derived as:
(16) |
where x(k) is a column vector, which denotes the number of ACLs in different segments at time k; is the sampling time; is a constant variable, which denotes the aggregate power of ACLs at time k; I is a -dimension identity matrix; A is a state transition matrix; and C is a output vector.
(17) |
(18) |
where Af is the matrix describing the load flow relationship between the temperature segments in the NTD; is the matrix describing the load flow relationship between the temperature segments in the TTD; and A21 is the matrix describing the load flow relationship at the junction of the TTD and NTD. Thus, it only contains one non-zero element and its position and value.
(19) |
(20) |
where is the grid ratio.
It is worth nothing that the size of is determined by the temperature adjustment sθ, which can be written as:
(21) |
where As0 and As1 can describe the dynamic evolution process of the load in the off and on states in the TTD, respectively; and As2 is the matrix describing the boundary conditions that flow from the lower boundary of off state to on state. When the change of the temperature setpoint is larger than the width of the temperature dead band, there will be a gap between the initial temperature dead band and NTD. Therefore, the method in [
(22) |
(23) |
where is the load transporting rate around in off state and . Similarly, is the load transporting rate around in on state and . The arrangement of the smallest unit is as follows:
(24) |
For different initial temperature setpoints and temperature adjustment values, the corresponding As0 and As1 can be obtained according to (25) and (26).
(25) |
(26) |
matrix contains only one non-zero element:
(27) |
where as1 is the load transporting rate around in the on state and .
Suppose that there are 1000 ACLs in one zone and the initial temperature setpoint of them is 25 ℃, the width of temperature dead band is 1.0 ℃. The initial indoor temperature is subject to the uniform distribution on [24.5 ℃, 25.5 ℃]. The parameters listed in
This simulation verifies the correctness and accuracy of the WRT model by comparing it with the Monte-Carlo method. Assume that the temperature setpoint of ACLs is increased by 0.3 ℃, 1 ℃, and 1.5 ℃ at 0.3 hour, respectively. The aggregate power curve is shown in

Fig. 4 Accuracy verification of WRT model. (a) . (b) . (c) .
It can be observed that WRT model can accurately describe the aggregate power response curve of ACLs in all the above three circumstances, but the accuracy of the model will decrease as time marches.
From the simulation verification in Section II, it can be observed that the aggregate power of ACLs after temperature adjustment will show varying degrees of load rebound and oscillation. We adopt the idea of SP [
SP refers to the embedded intelligent unit in the local thermostat of ACL which aims to control the upper or lower limit of the temperature dead band. References [
Suppose that the TTD is within , the width of temperature dead band is db, and the adjustment value of temperature setpoint is , where s is an integer variable.
Once the ACL receives the TSA command, the intelligent thermostat will make a judgement according to the current working status.
1) If the device is working in the on state, the thermostat will temporarily lock the lower limit of the dead band, and the temperature dead band at this time will change to , . Once the indoor temperature continues to drop to θmin, the lock of the lower limit will be released and the temperature dead band will be updated to .
2) If the device is working in the off state, the temperature dead band will be directly updated to .
The evolution process of the ACLs with SP is shown in

Fig. 5 Evolution process of ACLs with SP.
(28) |
(29) |
The simulation compares the response characteristics of ACLs with and without SP when the temperature adjustment is 1 ℃, as shown in

Fig. 6 Response characteristics of ACLs after modulation.
Although the modulated aggregate response curve of ACLs avoids power rebound and oscillation, the shape of this curve has little plasticity. In order to fully tap the response potential of ACLs, this paper further introduces on/off control on the basis of the above-mentioned TSA mode, and proposes a hybrid control strategy that can effectively participate in peak reduction.
For any temperature segment in the temperature dead band, there are ACLs in either on or off state. On/off control [
Let , the control variable u(k) acts on the system through the input matrix B, and the state space model of the system based on hybrid control is expressed as:
(30) |
where is a column vector, which represents the number of switched loads in the corresponding temperature segments; and B is the matrix describing the state switching relationship between the temperature segments under the on/off control. When the element of is positive, it means that the state of the device is switched from off to on, and vice versa.
Thus,

Fig. 7 Mechanism of hybrid control strategy.
The expression of is:
(31) |
(32) |
where and have the same structure and , . The sum of the elements in each column is 0, which ensures that the number of from on to off or from off to on at the upper and lower positions corresponding to the temperature segments is equal.
MPC is a closed-loop control method that could track a certain reference signal through rolling optimization in a given time domain. It is widely used because it can comprehensively consider multiple objective functions and constraint conditions. In this paper, the state space model based on the hybrid control is used as the predictive model of MPC. On this basis, the objective function and constraint conditions are designed by considering the system tracking error and control cost.
Assuming that a load aggregator needs to follow the market clearing peak reduction power Pref with a duration of within a day, the optimization control problem of the ACL system can be constructed as:
(33) |
(34) |
(35) |
where Jobj is the value of the objective function; Q is the tracking error weight coefficient matrix; R is the weight coefficient matrix of the control variable; and is the predicted output power of ACLs, which can be calculated by (16).
The elements of state variable in constraint (34) are integer and non-negative.
Constraint (35) ensures that the number of controlled loads at each time step cannot exceed the load number of the corresponding temperature segment at the last time step. is used to represent the corresponding temperature segments that u(k) acts on, which is only half of all the segments.
Considering that if each group of the corresponding temperature segments can be switched into the opposite direction, it will cause relatively serious wears and tears. Therefore, we can replace the inequality constraint (35) with the
(36) |
Since the number of ACLs is integer, this optimal control problem is a mixed-integer quadratic programming problem. It can be calculated by MATLAB and Gurobi.
Assume that an ACL aggregator receives a peak load reduction command, which is to reduce the power consumption of ACLs by m% and maintain it for n min [
In order to verify the superiority of the proposed hybrid control strategy under SP, the performance of the hybrid controller is compared with that of the controller in [
First, the extreme ability of the ACLs to follow the peak load reduction signal is observed by setting . Note that n is unlimited. The tracking performance and switching wear-outs are compared with those of the controller in [

Fig. 8 Performance comparison of controllers with and without SP.

Fig. 9 Switching number comparison with and without SP.
Moreover, to verify the effectiveness of the controllers, three different shapes of peak load reduction market clearing curve are adopted with the response time limited to 30 min.

Fig. 10 Control outputs under different commands. (a) Under command A. (b) Under command B. (c) Under command C.

Fig. 11 Tracking error under different commands.
In order to minimize the impact on users’ comfort, the adjustment value of the temperature setpoint is set to be +1 ℃, and the hybrid control strategy is implemented on this basis. In this scenario, this strategy can ensure that the users’ indoor temperature is always within the range of [23.5 ℃, 25.5 ℃] during the response process. To analyze the peak reduction performance of the ACLs with the proposed hybrid control strategy, the following three cases are set for simulation verification:
1) Case 1: ,, with double-direction constraint (35).
2) Case 2: ,, with single-direction constraint (36).
3) Case 3: ,, with double-direction constraint (35).
Case 1 is designed to verify the control performance of the hybrid control strategy. At the beginning of the response, a 30-min peak reduction signal is issued to ACL aggregator.
From case 1 in

Fig. 12 Control outputs of case 1 to case 3.
In order to further analyze the effect of the controller on the dynamic evolution process of ACLs, Figs.

Fig. 13 Number of switched loads in temperature segments under control in case 1. (a) Detailed switched load in TTD. (b) Detailed switched load in NTD.

Fig. 14 Load evolution details in temperature segments. (a) Off-state load number in TTD. (b) On-state load number in TTD. (c) Off-state load number in NTD. (d) On-state load number in NTD.
When the target power is tracked, part of the loads in off state is switched to the on state under the action of the controller. Therefore, s14-s20 in
Subsequently, the loads in NTD still evolve under the influence of FTM and the on/off control and the specific flowing process is no longer repeated here. After the control is released, the ACLs will eventually evolve to a stable state, i.e., all ACLs flow into the NTD, and the proportion of the load in the on state will gradually reach a stable state, as shown in
Case 2 is set to compare and analyze the influence of single- or double-direction constraint on the performance of the proposed controller and the switching number of devices. The peak reduction signal in this case is the same as in case 1. With the limitation of single-direction constraint (29), only the off-state ACLs are controllable, which will cause the ACLs to be unable to quickly track the reference power in the initial stage of the response. As shown by the purple dashed line in
It can be observed from

Fig. 15 Number of switched loads in temperature segments under control in case 2. (a) Detailed switched load in TTD. (b) Detailed switched load in NTD.
Case 3 is set to analyze how the power reduction ratio influences the peak shaving performance of the controller. In case 3, the reduction command of ACLs is 50% of its steady-state power, which is much deeper than that of case 1.
Hence, for a specific ACL cluster, its load reduction performance is not infinite. Based on this fact, we further analyze the maximum response time of the ACL cluster under different power reduction ratios, as shown in

Fig. 16 Relationship between the maximum response time and reduction ratio.
When the ratio of peak reduction is higher, its maintenance time will also be shortened. It can be observed that there is a turning point when the reduction ratio exceeds 20%-30%.
This paper provides a theoretical reference for the ACLs participating in the ancillary service of peak load reduction. The suitable range of the ACL aggregate model is extended, and an MPC controller is designed based on a hybrid control strategy. The following conclusions can be obtained through theoretical analysis and simulation verification.
1) A WRT model suitable for large-scale adjustment of the temperature setpoint is established, which enlarges the application scope of the existing aggregate model. Compared with Monte-Carlo simulation, the WRT model greatly reduces the amount of calculation, which is convenient for practical engineering applications. Besides, the SP is incorporated into the aggregate model, which effectively suppresses the rebound and oscillation phenomenon in the response characteristics of ACLs.
2) A hybrid control strategy combining TSA control and on/off control is proposed, and an optimized controller based on MPC is designed. Simulations verify that ACLs can accurately track the power reduction signal with the proposed hybrid control strategy.
3) The load reduction performance of ACLs with the hybrid control strategy is further analyzed. The response potential of ACLs with the hybrid control strategy can be fully utilized, and the power reduction performance is more controllable. However, when the load reduction ratio is high, the maximum response time of ACL to track the signal will be shortened. The turning point is about 20%-30%. For a certain number of ACL clusters, there is an upper limit for the schedulable capacity.
Appendix
Let , we have:
(A1) |
(A2) |
where xj(k) is the load number in segment j at time k; is the load transport rate in TTD in the off or on state; and is the load transport rate in NTD in the off or on state.
The discretization form of (7)-(10) is:
(A3) |
(A4) |
(A5) |
(A6) |
The aggregate power of ACLs at time k can be written as:
(A7) |
where y(k) is the aggregate power of ACLs at time k.
References
C. Shao, Y. Huang, J. Nie et al., “Power peak load can be controlled through source-load coordination,” China Energy News, vol. 4, pp. 1-3, Jun. 2020. [Baidu Scholar]
Y. Sun, S. Wang, F. Xiao et al., “Peak load shifting control using different cold thermal energy storage facilities in commercial buildings: a review,” Energy Conversion and Management, vol. 71, pp. 101-114, Jul. 2013. [Baidu Scholar]
P. C. Reiss and M. W. White, “Household electricity demand, revisited,” Review of Economic Studies, vol. 72, no. 3, pp. 853-883, Jul. 2005. [Baidu Scholar]
Q. Xu, C. Yang, and Q. Yan, “Strategy of day-ahead power peak load shedding considering thermal equilibrium inertia of large-scale air conditioning loads,” Power System Technology, vol. 10, no. 1, pp. 156-163, Jan. 2016. [Baidu Scholar]
C. Pei, “Study on demand response and optimization interaction of large scale air conditioning load cluster,” M.S. thesis, School of Electric Engineering, Beijing Jiaotong University, Beijing, China, 2018. [Baidu Scholar]
Y. Yang, Q. Yan, S. Xu et al., “Thinking of public building air-conditioning load participating in grid with virtual peak clipping,” Automation of Electric Power Systems, vol. 39, no. 17, pp. 103-107, Sept. 2015. [Baidu Scholar]
B. Wang, X. Hu, W. Gu et al., “Hierarchical control architecture and decentralized cooperative control strategy for large scale air conditioning load participating in peak load regulation,” Proceedings of the CSEE, vol. 39, no. 12, pp. 137-151, Jun. 2019. [Baidu Scholar]
J. Zheng, G. Laparra, G. Zhu et al., “Aggregate power control of heterogeneous TCL populations governed by Fokker-Planck equations,” IEEE Transactions on Control Systems Technology, vol. 28, no. 5, pp. 1915-1927, Sept. 2020. [Baidu Scholar]
M. Franceschelli, A. Pilloni, and A. Gasparri, “Multi-agent coordination of thermostatically controlled loads by smart power sockets for electric demand side management,” IEEE Transactions on Control Systems Technology, vol. 29, no. 2, pp. 731-743, Mar. 2021. [Baidu Scholar]
M. Song, C. Gao, and W. Su, “Modeling and controlling of air-conditioning load for demand response applications,” Automation of Electric Power Systems, vol. 40, no. 14, pp. 158-167, Jul. 2016. [Baidu Scholar]
A. Pahwa and C. W. Brice, “Modeling and system identification of residential air conditioning load,” IEEE Transactions on Power Apparatus and Systems, vol. 104, no. 6, pp. 1418-1425, Jun. 1985. [Baidu Scholar]
J. H. Kampf and D. Robinson, “A simplified thermal model to support analysis of urban resource flows,” Energy and Buildings, vol. 39, no. 4, pp. 445-453, Apr. 2007. [Baidu Scholar]
Y. Ding, W. Cui, S. Zhang et al., “Multi-state operating reserve model of aggregate thermostatically controlled-loads for power system short-term reliability evaluation,” Applied Energy, vol. 241, pp. 46-58, May 2019. [Baidu Scholar]
W. Cui, “Power system reliability evaluation considering operating reserve provided by air conditioners,” M.S. thesis, School of Electric Engineering, Zhejiang University, Hangzhou, China, 2019. [Baidu Scholar]
J. L. Mathieu, S. Koch, and D. S. Callaway, “State estimation and control of electric loads to manage real-time energy imbalance,” IEEE Transactions on Power Systems, vol. 28, no. 1, pp. 430-440, Feb. 2013. [Baidu Scholar]
M. Liu and Y. Shi, “Model predictive control of aggregated heterogeneous second-order thermostatically controlled loads for ancillary services,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 1963-1971, May 2016. [Baidu Scholar]
N. Lu and D. P. Chassin, “A state-queueing model of thermostatically controlled appliances,” IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1666-1673, Aug. 2004. [Baidu Scholar]
L. Zhou, Y. Li, and C. Gao, “Improvement of temperature adjusting method for aggregated air-conditioning loads and its control strategy,” Proceedings of the CSEE, vol. 34, no. 31, pp. 5579-5589, Nov. 2014. [Baidu Scholar]
C. H. Wai, M. Beaudin, H. Zareipour et al., “Cooling devices in demand response: a comparison of control methods,” IEEE Transactions on Smart Grid, vol. 6, no. 1, pp. 249-260, Jan. 2015. [Baidu Scholar]
R. Malhame and C. Y. Chong, “Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system,” IEEE Transactions on Automatic Control, vol. 30, no. 9, pp. 854-860, Sept. 1985. [Baidu Scholar]
D. S. Callaway, “Tapping the energy storage potential in electric loads to deliver load following and regulation, with application to wind energy,” Energy Conversion and Management, vol. 50, no. 5, pp. 1389-1400, May 2009. [Baidu Scholar]
S. Bashash and H. K. Fathy, “Modeling and control of aggregate air conditioning loads for robust renewable power management,” IEEE Transactions on Control Systems Technology, vol. 21, no. 4, pp. 1318-1327, Jul. 2013. [Baidu Scholar]
L. Chen, Y. Yang, Q. Xu et al., “Coordinated control and recovery strategy for aggregated air-conditioner based on time-variant complementary characteristics,” Automation of Electric Power Systems, vol. 44, no. 13, pp. 39-47, May 2020. [Baidu Scholar]
J. Hu, J. Cao, T. Yong et al., “Demand response load following of source and load systems,” IEEE Transactions on Control Systems Technology, vol. 25, no. 5, pp. 1586-1598, Sept. 2017. [Baidu Scholar]
W. Zhang, J. Lian, C. Chang et al., “Aggregated modeling and control of air conditioning loads for demand response,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4655-4664, Nov. 2013. [Baidu Scholar]
M. Liu, X. Chu, W. Zhang et al., “Dispatch and control strategies for air-conditioning load groups based on maintenance of load diversity,” Proceedings of the CSEE, vol. 34, no. 22, pp. 3674-3682, Aug. 2014. [Baidu Scholar]
N. A. Sinitsyn, S. Kundu, and S. Backhaus, “Safe protocols for generating power pulses with heterogeneous populations of thermostatically controlled loads,” Energy Conversion and Management, vol. 67, pp. 297-308, Mar. 2013. [Baidu Scholar]
N. Mehta, N. A. Sinitsyn, S. Backhaus et al., “Safe control of thermostatically controlled loads with installed timers for demand side management,” Energy Conversion and Management, vol. 86, pp. 784-791, Oct. 2014. [Baidu Scholar]
Y. Ma, Y. Yu, Y. Fan et al., “Improvement and modeling of dynamic response characteristics for aggregate thermostatically controlled loads,” in Proceedings of 2020 IEEE 3rd Student Conference on Electrical Machines and Systems (SCEMS), Jinan, China, Dec. 2020, pp. 625-630. [Baidu Scholar]
A. Radaideh, U. Vaidya, and V. Ajjarapu, “Sequential set-point control for heterogeneous thermostatically controlled loads through an extended Markov chain abstraction,” IEEE Transactions on Smart Grid, vol. 10, no. 1, pp. 116-127, Jan. 2019. [Baidu Scholar]
Smart Grid Consumer Automatic Demand Response-Distributed Air Conditioning System Terminal Technical Condition, Standard GB/T 34116-2017, 2017. [Baidu Scholar]
Economy and Information Commission of Jiangsu Province, Jiangsu Price Bureau. (2018, Jun.). Implementation rules for electricity demand response in Jiangsu Province, China. [Online]. Available: http://gxj.yangzhou.gov.cn/yzjingxw/tzgg/201807/ed8d318588104c33b9ba45aef6537238.shtml [Baidu Scholar]
Yunfeng Ma received the B.S. degree in electrical engineering from North China Electric Power University, Baoding, China, in 2017. She is currently pursuing the Ph.D. degree at the School of Electrical and Electronic Engineering, North China Electric Power University, Beijing, China. Her research interests include optimal dispatch and control of thermostatically controlled load. [Baidu Scholar]
Zengqiang Mi received the M.S. degree in electrical engineering from North China Electric Power University, Baoding, China, in 1986. He is currently a Professor at North China Electric Power University, Beijing, China. His research interests include power system operation and control, electrical energy storage technology, and optimal dispatch of flexible load. [Baidu Scholar]
Ruifeng Zhang received the B.S. degree in electrical engineering from China University of Petroleum, Qingdao, China, in 2019. He is currently pursuing the M.S. degree at the School of Electrical and Electronic Engineering, North China Electric Power University, Baoding, China. His research interests include demand response and active distribution network. [Baidu Scholar]
Huowen Peng spent her first two years of university majoring in electrical engineering at North China Electric Power University, Baoding, China, from 2018 to 2020, and is now perusing her B.S. degree at the University of Edinburgh, Edinburgh, UK. Her research interests include electronics and electrical engineering. [Baidu Scholar]
Yulong Jia received the B.S. degree in software engineering from North China Electric Power University, Baoding, China, in 2013, the M.S. degree in electrical engineering from Taiyuan University of Technology, Taiyuan, China, in 2016, the Ph.D. degree in electrical engineering from North China Electric Power University, Beijing, China, in 2020. He is a Researcher with the Planning & Design Institute at State Grid Integrated Energy Services Group Co., Ltd., Beijing, China. His research interests include demand response, flexible load scheduling and smart grid. [Baidu Scholar]