Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Hybrid Control Strategy for Air-conditioning Loads Participating in Peak Load Reduction Through Wide-range Transport Model  PDF

  • Yunfeng Ma
  • Zengqiang Mi
  • Ruifeng Zhang
  • Huowen Peng
  • Yulong Jia
the School of Electrical & Electronic Engineering, North China Electric Power University, Beijing, China; the Department of Electrical Engineering, North China Electric Power University, Baoding, China; the University of Edinburgh, Old College, South Bridge, Edinburgh, EH8 9YL, UK; the Planning & Design Institute at State Grid Integrated Energy Services Group Co., Ltd., Beijing 100032, China

Updated:2022-11-19

DOI:10.35833/MPCE.2021.000108

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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.

I. Introduction

WITH the development of economy in China, the proportion of residential electricity consumption increases year by year [

1], [2]. By increasing the investment at the power generation side, the costs will be very high. Considering that the cooling devices such as air-conditioning loads (ACLs) account for a large proportion of peak loads in summer [3], [4], it can be economical and effective to reduce the peak load by controlling the large-scale ACLs.

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 [

5]-[9]. Hence, the premise of ACL participation in demand response is to establish an aggregation model that can describe the aggregate response characteristics accurately and efficiently [10].

Among the aggregate modeling methods of ACLs, the classic one is the Monte-Carlo method based on the equivalent thermal parameter (ETP) model [

11], [12]. Although this method can analyze the aggregation characteristics of ACLs accurately, it will be very time-consuming due to the million-level load scale in practice [13], [14]. In view of this, many studies have adopted different methods to establish a concise and efficient aggregation model. References [15] and [16] regard the state transition process in the temperature dead band of the thermostatically controlled loads (TCLs) as a Markov process, and establish a state bin transition model. Reference [17] establishes a state queueing (SQ) model to describe the operating state of ACLs, and [18] and [19] apply this model to different scenarios. References [8], [20] and [21] describe the dynamic process of TCLs based on the Fokker-Planck equations. Reference [22] further derives the free transport model (FTM) of TCLs and gives a bilinear state space 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 [

19], [23], [24]. The above models have the limitation that none of them could describe the dynamic characteristics of ACLs during the process of wide-range adjustment of the temperature setpoint. Therefore, this paper intends to expand on the classic FTM [22] and establish a wide-range transport (WRT) model that can describe the dynamic aggregation characteristics of ACLs after adjusting the temperature setpoint.

Besides, the power rebound and oscillation are inevitable when ACLs are controlled in the centralized control mode [

25], [26]. The safe protocol (SP) [27], [28] is a method of modulating the aggregate response characteristics of ACLs in the temperature setpoint adjustment (TSA) mode. In order to make the response curve of ACLs smoother after adjustment, SP is adopted to modulate its aggregation characteristics, and an aggregation model suitable for SP is given in this paper.

However, the aggregate power cannot be precisely adjusted if only a single temperature adjustment control method is adopted [

27], which causes that the peak shaving potential of ACLs cannot be fully utilized. Therefore, this paper proposes a hybrid control strategy by introducing on/off control on the basis of the TSA control method. Furthermore, a controller is designed based on model predictive control (MPC), so that it can track the response signal of a given response time and the power reduction more accurately, and fully tap the peak reduction potential of ACLs.

II. Aggregate Response Characteristic Modeling of ACLs

The evolution process of the aggregate ACLs operating at a steady state in the temperature dead band can be described by the original FTM.

A. FTM of ACLs

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 [

22], [29], as shown in Fig. 1 [29], where θmax and θmin are the upper and the lower limits of temperature dead band, respectively; and θset denotes the temperature setpoint of ACLs.

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 Lon/off(t,θ) and Non/off(t,θ) can be expressed as:

Lon/off(t,θ)=aon/off(t,θ)Non/off(t,θ) (1)
Non/off(t,θ)t=-Lon/off(t,θ)θ (2)

where aon/off(t,θ) 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 ≈(θa-θset)/(CR) and aoff ≈(θa-θset-RP)/(CR), where θa 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:

Nofft+aoffNoffθ=0Nont+aonNonθ=0 (3)

The transporting processes at the temperature limits are:

Lonθθ=θmin+Loffθθ=θmin=0 (4)
Lonθθ=θmax+Loffθθ=θmax=0 (5)

This population of the aggregate power of TCLs Pagg at time t is represented as:

Paggt=PηθminθmaxNon(t,θ) dθ (6)

where η is the coefficient of performance (COP) of ACLs.

B. WRT Model Suitable for TSA Mode

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.

1) Mechanism Analysis of Load Flow During TSA Control

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. 1. Taking the upward adjustment of the temperature setpoint as an example, the load in the on state will no longer flow to the left along the blue arrow, but will instantly drop from the blue area shown in Fig. 2 along the red dashed arrow to the green area, where the initial temperature dead band is called as transition temperature dead band (TTD). In Fig. 2, θmin,1 and θmax,1 are the lower and upper temperature limits of TTD, respectively; and θmin,2 and θmax,2 are the lower and upper temperature limits of the new temperature dead band (NTD), respectively.

Fig. 2  Flow process of load flux.

The load originally in the off state will no longer flow upward through the temperature boundary θmax,1 into the on state, but will flow through the boundary θmax,1 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:

Lonθθ=θmin,1+Loffθθ=θmin,1=0 (7)
Lonθθ=θmax,1+Loffθθ=θmax,2=0 (8)
Lonθθ=θmin,2+Loffθθ=θmin,2+Loffθθ=θmax,1=0 (9)
Lonθθ=θmax,2+Loffθθ=θmax,2=0 (10)

Equations (7) and (10) show that the interchange of loads at the boundaries is equivalent to equating the entering and the exiting fluxes at the boundaries.

Equation (8) expresses that the off-state loads flow out the TTD through the boundary θmax,1 and all flow into NTD through the boundary θmin,2.

Equation (9) indicates that the load increment at the boundary θmin,2 of NTD is equal to the sum of the off-state load decrement at the boundary θmax,1 and the on-state load decrement at the boundary θmin,2.

Finally, Pagg(t) can be rewritten as:

Pagg(t)=Pηθmin,1θmax,2Non(t,θ)dθ (11)

2) Construction of WRT Model

The load flow process described by PDEs is not convenient for practical engineering applications. Therefore, we will use the finite difference method [

22] to discretize the PDE to obtain a WRT model in a discrete form.

First, divide the temperature axis into small temperature segments of equal width, as shown in Fig. 3, where s and M are the numbers of segments at TTD and NTD, respectively; Δθ is the temperature step size; nj is the number of loads in the jth temperature segment in the initial state; θs,1 is the initial temperature setpoint; and θs,2 is the new temperature setpoint. According to the distribution of ACLs on the temperature axis at the initial state, each segment contains a certain number of ACLs.

Fig. 3  Finite-difference discretization of WRT model.

Once the TSA signal is received, the temperature setpoint will shift to the right to θs,2 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 xjTDD-off(k) and xjTTD-on(k) denote the load numbers at time k in the jth temperature segment in the off state and the on state in the TTD, respectively. The initialization method is shown in (12) and (13).

Let xjNTD-off(k) and xjNTD-on(k) denote the load numbers at time k in the jth temperature segment in the off state and the on state in the NTD, respectively. The initialization method is shown in (14) and (15).

xjTDD-off(0)=njoff+n2M-j+1on    jM, j[1,2,,s]0        j>M, j[1,2,,s] (12)
xjTDD-on(0)=0    j[1,2,,s] (13)
xjNTD-off(0)=nj+soff    jM-s, j[1,2,,M]0        j>M-s, j[1,2,,M] (14)
xjNTD-on(0)=0           js, j[1,2,,M]nM+j-son    j>s, j[1,2,,M] (15)

where njoff and njon are the load numbers at the initial state in the jth segments in the off and on states, respectively.

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:

x(k+1)=(I+AΔt)x(k)y(k)=Cx(k) (16)

where x(k) is a (2s+2M)×1 column vector, which denotes the number of ACLs in different segments at time k; Δt is the sampling time; y(k) is a constant variable, which denotes the aggregate power of ACLs at time k; I is a (2s+2M)-dimension identity matrix; A is a (2s+2M)×(2s+2M) state transition matrix; and C is a 1×(2s+2M) output vector.

A=As|2s×2s0|2s×2MA21|2M×2sAf |2M×2M (17)
C=Pη0,,0|s 1,,1|s 0,,0|M 1,,1|M (18)

where Af is the matrix describing the load flow relationship between the temperature segments in the NTD; As 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.

Af=λ-aoff-aonaoff-aoffaoff-aoffaoffaon-aonaon-aonaon (19)
A211,s=as0λ (20)

where λ=Δt/Δθ is the grid ratio.

It is worth nothing that the size of As is determined by the temperature adjustment sΔθ, which can be written as:

As=As0s×sAs2s×s0s×sAs1s×s (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 [

30] is referred to set a matrix Ws that can traverse all temperature segments. Ws is composed of several square matrix as the smallest unit arranges along the main diagonal. The smallest unit with the temperature setting value around θs,1 is shown in (22) and (23).

Wsoffθs=θs,1=λ-as0,1as0,1-as0,1as0,1-as0,1as0,1-as0,1M×M (22)
Wsonθs=θs,1=λas1,1-as1,1as1,1-as1,1as1,1-as1,1as1,1M×M (23)

where as0,1 is the load transporting rate around θs,1 in off state and as0,1=(θa-θs,1)/(CR). Similarly, as1,1 is the load transporting rate around θs,1 in on state and as1,1=(θa-θs,1-RP)/(CR). The arrangement of the smallest unit is as follows:

Ws=Wsoffθs=θs,1Wsoffθs=θs,n-1Wsonθs=θs,n-1Wsonθs=θs,1 (24)

For different initial temperature setpoints and temperature adjustment values, the corresponding As0 and As1 can be obtained according to (25) and (26).

As0=Ws(θs,1-θs,min)M+1, (θs,2-θs,min)M (25)
As1=Ws(2θs,max-θs,min-θs,2)M+1, (2θs,max-θs,min-θs,1)M (26)

As2 matrix contains only one non-zero element:

As21,s=-as1λ (27)

where as1 is the load transporting rate around θs,1 in the on state and as1=(θa-θs,1-RP)/(CR).

3) Accuracy Verification of WRT Model

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 Table I are all subject to normal distribution [

18], [19]. And it can be obtained that there are 50.7% of the ACLs over the on state under this initial condition.

TABLE I  Simulation Parameters of ACLs
ParameterMeanStandard deviation
C (kWh/℃) 0.21 0.02
R (℃/kW) 5.92 0.06
P (kW) 3.00 0.30
η 3.00 0.30
θa (℃) 34.00 0.34

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.

Fig. 4  Accuracy verification of WRT model. (a) s<M. (b) s=M. (c) s>M.

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.

III. Modulation of Aggregate Response

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 [

27], [28] to modulate the dynamic response characteristics of ACLs.

A. 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 [

13], [27], and [28] have proven that this method could avoid the rebound and large oscillation of aggregate power after response.

Suppose that the TTD is within [θmin, θmax], the width of temperature dead band is db, and the adjustment value of temperature setpoint is sΔθ, 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 [θmin, θmax+sΔθ] . 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 [θmin+sΔθ, θmax+sΔθ].

2) If the device is working in the off state, the temperature dead band will be directly updated to [θmin+sΔθ, θmax+sΔθ].

The evolution process of the ACLs with SP is shown in Fig. 5. The SP is adopted to modulate the response characteristics of ACLs and is incorporated into the proposed WRT model.

Fig. 5  Evolution process of ACLs with SP.

B. WRT Model Imbedded with SP

Figure 5 shows that the introduction of SP mainly affects the load number in temperature segments of TTD at the beginning of the evolution process, which means that all loads in the on state will be locked without dropping, as shown by the red dotted arrow in Fig. 2, into the off state instantaneously. Hence, (12) and (13) should be replaced with the following two equations:

xjs0=njjM, j1,2,,s0  j>M, j1,2,,s (28)
xjs1=njjM, j1,2,,s0  j>M, j1,2,,s (29)

The simulation compares the response characteristics of ACLs with and without SP when the temperature adjustment is 1 ℃, as shown in Fig. 6. It can be observed that the response characteristics after the modulation of SP have been significantly improved, and the power bounce and oscillation phenomena have been well suppressed. Besides, the accuracy of the proposed WRT model is verified by comparison with Monte-Carlo simulation.

Fig. 6  Response characteristics of ACLs after modulation.

IV. Hybrid Control Strategy of ACLs for Peak Load Reduction

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.

A. State Space Model Based on Hybrid Control

For any temperature segment in the temperature dead band, there are ACLs in either on or off state. On/off control [

15] refers to that the state of ACLs in certain segments can be switched from on to off or from off to on by the direct load control method. In order to accurately track the peak reduction signal of a specific response time and power, this paper introduces on/off control on the basis of the TSA control described in (16).

Let G=I+AΔt, 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:

x(k+1)=Gx(k)+Bu(k)y(k)=Cx(k) (30)

where u(k) is a (s+M)×1 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 u(k) is positive, it means that the state of the device is switched from off to on, and vice versa.

Thus, Fig. 7 shows the entire mechanism of hybrid control strategy, where the TSA control process is represented by the red arrow and the on/off control process is represented by the purple dotted arrows between the corresponding on and off states.

Fig. 7  Mechanism of hybrid control strategy.

The expression of B is:

B=B11|2s×s0|2s×M0|2M×sB22|2M×M (31)
B11=B22= -1-1-1111 (32)

where B11 and B22 have the same structure and rank(B11)=s, rank(B22)=M. 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.

B. Objective Function and Constraints

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 Г within a day, the optimization control problem of the ACL system can be constructed as:

minJobj=(P(k)-Pref)TQ(P(k)-Pref)+u(k)TRu(k) (33)
x(k)0    k=1,2,,Γ-1 (34)
u(k)xu(k-1)    k=1,2,,Γ-1 (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 P(k) is the predicted output power of ACLs, which can be calculated by (16).

The elements of state variable x(k) 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. xu(k) 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 formula (36), so that the ACLs can only be switched in one direction while the temperature setpoint is adjusted. The operation frequency of ACLs is thus reduced.

0u(k)xu(k-1)    k=1,2,,Γ-1 (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.

V. Simulation and Discussion

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 [

31]. According to the demand-side energy management system, the aggregator knows that a total of 10000 ACLs can be controlled at this time, and the equipment operating parameters are shown in Table I. The outdoor temperature is assumed to be 34 ℃. All the initial temperature setpoint of the selected ACLs is supposed to be 24 ℃. It can be calculated that about 56.32% of the ACLs are in the on state in this circumstance, and their steady aggregate power is about 5632 kW. The superiority of the hybrid control strategy is first verified through the comparison with the existing methods. Moreover, the performance of the ACL cluster is analyzed.

A. Superiority Verification of Hybrid Control Strategy

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 [

30] without SP. The load reduction is usually between 0% and 5% of the peak load consumption of the aggregator [32]. And the respond signal lasts up to 30 min at a time. Therefore, the performance of the controller is tested by the following simulation experiments.

First, the extreme ability of the ACLs to follow the peak load reduction signal is observed by setting m=10. Note that n is unlimited. The tracking performance and switching wear-outs are compared with those of the controller in [

29], what is called as “without SP”. Figure 8 shows that when the reduction command is 10% of the steady operation power, the controlled ACL cluster could track the command up to 0.714 hour, which could satisfy the response qualification of Jiangsu Province in China. Due to the integer attribute of the equipment, there is an unavoidable tracking error. Besides, the proposed hybrid control strategy is faster than the controller without SP. This is because the latter controller suppresses the power oscillations all by the MPC process. Therfore, the aggregate power has to pass through the zero point at the very beginning. Moreover, the large amounts of actions switch on/off state of ACLs in the first few control steps, as shown in Fig. 9. Once the aggregate power is close to the command, the switching numbers tend to be stable and small.

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. Figure 10(a)-(c) shows the control output under different commands. It can be observed that the controller is effective for different command signals. Figure 11 presents the tracking error under different commands. For simplicity, Fig. 10 only presents the error after t=0.004 hour. It can be directly observed that the performance of the controller varies depending on the tracking signals. Relatively, command A is the strictest one for the ACLs because the response characteristics without on/off control tend to be a hook shape in Fig. 6. This also explains why the tracking error is a “raised” one. Comparatively, command B is the friendliest one to ACL cluster and the tracking effect is the best.

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.

B. Performance of ACL Cluster with Hybrid Control

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: m=20,n=30, with double-direction constraint (35).

2) Case 2: m=20,n=30, with single-direction constraint (36).

3) Case 3: m=50,n=30, 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, it can be observed that the MPC controller can track and keep up with the reduction signal within 0.004 hour (about 0.24 min) for case 1. The tracking power can remain stable within the whole 30 min. After the control is released at t=0.5 hour, the aggregate power will go through slight fluctuations and approach to the blue line (which is the response curve in the single TSA mode), and finally return to a stable operating state. It is worth noting that there is a certain gap between the final stable operating state (case 1) and the steady aggregate power without control (line SA). This is because the TSA part of the hybrid control strategy reduces the power consumption of the load.

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. 13 and 14 show the number of switched loads in temperature segments under control in cases 1 and 2. It can be observed from Fig. 13(a) that in order to make the aggregate power drop rapidly after receiving response signal, the switched ACLs number u(10) can reach -101 at t=0.002 hour. Correspondingly, it means that in TTD, there are 101 devices at temperature segment s11 switched into s10 through on/off control, as shown in Fig. 14(a)-(b). It can be found that the devices in the on state are forced to switch to the off state at the beginning of the response, thereby increasing the power tracking speed of the load reduction signal. Subsequently, to maintain the stability of the aggregate power, the number of switches during the response period is roughly stabilized within [-3, 3]. Since the number of loads in the NTD at the initial stage of response is almost zero, the number of loads in the NTD is relatively stable during the entire control process.

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 Fig. 14(b) all have a slight load rise. As s11-s13 are closer to the upper boundary of the TTD, they are more affected by the FTM and continue to flow through s21 into the NTD, resulting in a continuous reduction in the load number and a rapid increase of s21.

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 Fig. 14(c)-(d).

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 Fig. 10, it takes about 0.04 hour for ACLs to reach the tracking power. After reaching the preset response time, the aggregate power returns to a new stable operating state after a small power oscillation.

It can be observed from Fig. 15 that the number of switched loads in each sampling step is no more than 3 in the response process. And the statistics of the simulation results show that the total switching times in case 2 are 13899, and the total switching times under case 1 are 15848. 12% of the control cost is saved at the expense of a certain tracking rate. In general, the average number of switching times of a single air-conditioning in the above two cases is 1.4 to 1.6 in a complete 30-min response. Hence, the ACL aggregator can choose the control scheme by considering the actual cost and response demand comprehensively.

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. Figure 12 shows that the aggregate power reaches the response limit after being maintained for 15.6 min with the controller. After that, the controller looses its effect, and the aggregate power quickly returns to the original dynamic curve of TSA mode. Finally, the aggregate power returns to steady state after a period of oscillation.

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.

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%.

VI. Conclusion

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

Appendix A

Let Δx(k)=x(k+1)-x(k), we have:

Δxjs0/s1(k)=as0/s1λxj-1(k)-xj(k)j=2,3,,s (A1)
Δxjon/off(k)=aon/offλxj-1(k)-xj(k)j=2,3,,M (A2)

where xj(k) is the load number in segment j at time k; as0/s1 is the load transport rate in TTD in the off or on state; and aon/off is the load transport rate in NTD in the off or on state.

The discretization form of (7)-(10) is:

Δx1s0(k)=-as0x1s0(k)λ-as1xss1(k)λ (A3)
Δx1s1(k)=as1x1s1(k)λ (A4)
Δx1off(k)=(-aoffx1off(k)-aonxMon(k)+as0xMs0(k))λ (A5)
Δx1on(k)=aonx1on(t)λ+aoffxMoff(t)λ (A6)

The aggregate power of ACLs at time k can be written as:

y(k)=j=1sPηxjs1(k)+j=1MPηxjon(k) (A7)

where y(k) is the aggregate power of ACLs at time k.

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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]