Modeling and Control of Air Conditioning Loads for Consuming Distributed Energy Sources

Abstract

This paper aims to tap the potential of air conditioning loads (ACLs) for consuming photovoltaic power (PV) and wind power (WP). By fully considering different thermal comfort of different users, an ACL twice-clustered model based on different ACL parameters and users tolerance values (UTV) is built. Then, a two-stage ACL control method based on both temperature control (TC) and switch control (SC) is proposed, which achieves rapid control of ACLs as well as diminishing users’ discomfort. Widely existent communication time delay in ACL control network causes obvious control error, which leads to ACL consumption deviation from the target. Therefore, on the basis of analyzing errors and impacts of ACLs caused by communication time delay, this paper proposes a time delay compensation method based on a network predictive control system. Applying the ACLs clustered model and the control method into consuming PV and WP, a dual-stage consumption model considering communication time delay is established. Simulations of the PV and WP consumption effects based on ACLs clusters are conducted, and the influence of SC cycle and outdoor temperature on the simulation results are analyzed. The simulation results demonstrate the validity of the model and the methods proposed in this paper, showing a strong adaptability in different circumstances.

1. Introduction

Increasing penetration of photovoltaic power (PV) and wind power (WP) presents a huge challenge for power consumption. Due to the intermittency and instability of distributed power, PV and WP curtailment have been seriously taken to meet the requirements of safety, economic operation and power supply quality of power systems [1]. In order to reduce the PV and WP curtailment, it is feasible to store extra electric power in energy storage devices or to control flexible loads to consume the extra power. Previous studies usually adopted energy storage devices and mostly focused on optimization of the configuration and planning of microgrid energy storage. Studies on microgrid optimization configuration with different control strategies and energy storage characteristics are carried out in [2,3]; the microgrid economic dispatching model based on optimization configuration of energy storage is built in [4]. Energy storage devices have rapid responses and large adjustable ranges of storage abilities. However, construction costs and short lifetime are restricting their development. As a flexible load, air conditioning load (ACL) has rapid response, low construction costs and long lifetime. Conventionally, active power and frequency controls are carried out by generation tracing the loads. However, with the development of smart power systems and the improvement of communication capacity, it has become possible to adjust the operating status of ACLs to make them trace the commands for system power and frequency regulation.

Direct load control (DLC), a kind of demand response, is able to achieve rapid responses of ACLs towards fluctuation of PV and WP. The earliest DLC research built a single objective function based on system operating costs or corporation profits. However, the users’ satisfaction was not taken into account, which reduced users’ enthusiasm for participation [5,6,7]. Based on previous research, improved multi-objective control models and corresponding control plans which take all system operation, corporation profits and user’s satisfaction into account have been discussed in [8,9,10]. At present, the ACL control modes mainly focused on the centralized control mode [11,12,13], distributed control mode [14,15] and load aggregator mode [16]. Due to the high stability and strong predictability, the centralized control mode is adopted in this paper. The ACL control methods basically include switch control (SC) [17,18], temperature control (TC) [19,20] and duty cycle control (DCC) [21]. DCC is generally utilized in peak-shaving, whereas SC and TC can achieve ideal effects in PV and WP consumption. SC is easy to conduct and to change ACL’s state quickly. A depth iterative genetic algorithm is used [17] to solve the optimal switching states of ACLs to achieve peak-shaving. However, frequent ON-OFF switchings may cause ACLs to wear out, reducing users’ enthusiasm. Applying TC to ACLs has great potential for consuming PV and WP by minor adjustments at temperature set-points, which have little influence on users’ thermal comfort. Virtual energy storage ability of ACL clusters was proved in [19], in which a Lyapunov sliding mode controller was also designed to restrain the fluctuation of WP generation; in [22], an operating status model of aggregated loads was achieved based on a first-order physical model of single ACL and Fokker-Planck theorem, and the model was also demonstrated in the experiment to restrain the fluctuation of WP generation. However, it is hard to extend the application of TC on minute-scheduling due to the difficulty of the control algorithm and complexity of the controller [23]. Thus, it is urgent to put forward a kind of control method that integrates the advantages of TC and SC, and has little influence on users’ thermal comfort. In addition, most literature concerning control of ACLs did not refer to the communication time delay due to the acknowledgement of the wide existence of communication time delay in various fields, and researchers tend to ignore it in digital simulations. However, in practice, the communication time delay causes conspicuous error when controlling a large number of ACLs under the centralized control mode. Especially when ACL is at the switching point of ON-OFF, communication time delay will cause obvious impacts on consumption results, which will be further discussed in this paper. The paper [24] designed central controller to forecast and correct the room temperatures every 15 min. This method reduced the communication frequency between ACLs and controller so that the impacts of communication delay were reduced. The impact of communication delay on the ACLs’ control effect was evaluated in [25], whereas a delay compensation method has not been proposed.

Based on the studies above, this paper aims to tap the potential of ACL for consumption of PV and WP. The remainder of this paper is organized as follows. Section 2 presents the ACL models, including the physical model and an ACL twice-clustered model based on different parameters and user tolerance values (UTV). Next, Section 3 proposes a two-stage ACL control method based on both TC and SC and the collaboration mode of TC and SC is discussed. Afterwards, Section 4 analyzes the errors and impacts caused by communication time delay, and proposes a time delay compensation method based on network predictive control system. Applying the ACLs clustered model and control method to consuming PV and WP, Section 5 establishes a dual-stage consumption model considering communication time delay. Numerical simulations are provided to validate the proposed models and methods in Section 6. Finally, Section 7 concludes the whole paper.

2. Modeling of ACLs

2.1. Physical Model of a Single ACL

This paper carries out a single ACL model based on the physical model in [26], which is expressed as (1). This model has been proved able to capture the physical characteristics of an ACL accurately:

$$\frac{d{\theta }_{(i,t)}}{dt}=-\frac{1}{C_i{R}_i}[{\theta }_{(i,t)}-{\theta }_{(a,t)}+s_{(i,t)}{R}_i{P}_i]$$

(1)

where $C_i$ and $R_i$ are thermal capacity and thermal resistance of ACL i. ${\theta }_{(i,t)}$ is the room temperature of ACL i at time t. ${\theta }_{(a,i)}$ is the outdoor temperature which is assumed as the same value for every ACL. $P_i$ is the power of ACL i. $s_{(i,t)}$ is the binary status of ACL i at time t, which equals to 1 when the ACL is ON and equals to 0 when the ACL is OFF.

The difference equation of an ACL could be obtained by using the sample time h, shown as:

$${\theta }_{(i,t_{n+1})}=a_i{\theta }_{(i,t_n)}+(1-a_i)({\theta }_{(a,i)}-s{}_{(i,t_n)}R{}_i{P}_i)$$

(2)

where $a_i=e^{-\frac{h}{C_i{R}_i}}$, which is a constant representing the thermal characteristics of ACL i. ${\theta }_{(d,i)}$ is room temperature set-point of ACL i, ${\xi }_{(i,t_n)}$ is the adjustments on temperature set-point. The room temperature set-point of ACL i after adjustments is:

$${\theta }_{(d,i)}^{*}={\theta }_{(d,i)}+{\xi }_{（i,t_n)}$$

(3)

2.2. Twice-Clustered ACL Control Model and Constraints

2.2.1. Construction of Twice-Clustered ACL Control Model

Some research built primary clustered ACL model based on the same parameters to analyze the dynamic responses or power fluctuation [27,28]. The users’ thermal comfort and control priorities are taken into consideration in this paper. The twice-clustered ACL model is necessary for both consuming PV and WP precisely and guaranteeing users’ satisfaction.

Firstly, to obtain the primary clustered ACL model, this paper extracts ACL’s physical parameter vector which has the most conspicuous impact on consuming PV and WP: $\chi =\left(\begin{array}{ccc}P& R& C\end{array}\right)$. A hierarchical clustering algorithm [29] is used to cluster ACLs according to ACLs’ parameter vectors. In each clustering step, the two clusters which have the minimum Euclidean distance will be clustered into one cluster. The clustering process can be stopped when the distance between any two of the clusters increases sharply, thus the total clusters and ACLs in each clusters can be determined. The finally obtained parameter clusters are $\left(\begin{array}{ccc}{c}_{p1},& \begin{array}{cc}{c}_{p2},& ...\end{array},& c_{pm}\end{array}\right)$, in which the ACLs in the same parameter cluster have identical or similar dynamic responses and the physical parameters of the center point in the cluster are regarded as the physical parameters of all ACLs in that cluster. After conducting the primary clustering based on different ACLs’ parameters, the difficulty of optimization algorithm can be reduced by optimizing among the ACLs’ clusters.

This paper makes a reasonable assumption that on average there is only a single ACL within a single user. This paper further obtains the second clustered model according to UTV. UTV function proposed in this paper is expressed as:

$${\gamma }_i=\ln (\eta \cdot {\beta }_i\cdot T_{i\max }\cdot \left|{\xi }_{i\max }\right|\cdot {\Delta }_i)$$

(4)

where $T_{i\max }$ is the longest control time of ACL i, ${\xi }_{i\max }$ is the largest adjustment of temperature set-point, and ${\beta }_i$ is the grade value of ${\theta }_{(d,i)}$ evaluated by dispatching center. ${\Delta }_i$ is the room temperature fluctuation range of ACL i. $\eta$ is the coefficient of UTV. The differences among ${\beta }_i$s are not huge because ${\theta }_{(d,i)}$s are set by users originally at which users are in the most comfortable thermal conditions. ${\Delta }_i$s are obtained by investigating the ACLs. The dispatching center set various $T_{i\max }$s and ${\xi }_{i\max }$s for users to choose. Then UTVs are calculated after users submit selected $T_{i\max }$s and ${\xi }_{i\max }$s. Since ${\xi }_{i\max }$ is voluntarily chosen by a user, it is deemed that adjustments within ${\xi }_{i\max }$ will not cause discomfort to the user. The ACLs with the same UTV within $c_{pi}$ are clustered into one group, and different groups are of different UTVs. The finally obtained groups within $c_{pi}$ are expressed as $\left(\begin{array}{ccc}{\gamma }_{g1},& \begin{array}{cc}{\gamma }_{g2},& ...\end{array},& {\gamma }_{gz}\end{array}\right)$. The twice-clustering process can be illustrated clearly in Figure 1.

The total power of twice-clustered ACLs at $t_n$ can be expressed as:

$$P_{t_n}={\displaystyle \sum _{c_{pi}=c_{p1}}^{c_{pm}}{\displaystyle \sum _{{\gamma }_{gi}={\gamma }_{g1}}^{{\gamma }_{gz}}{\displaystyle \sum _{i=1}^{X}s{}_{(i,t_n)}*P_{c_{pi},{\gamma }_{gi},i}}}}$$

(5)

where $P_{c_{pi},{\gamma }_{gi},i}$ is the power of ACL i which belongs to ${\gamma }_{gi}$ and $c_{pi}$. By building the twice-clustered ACL model to replace the simple primary clustered model, differential control of ACLs based on different UTVs can be achieved, which could meet various thermal requirements of different users. Therefore, PV and WP consumption and users’ satisfaction can be all guaranteed.

2.2.2. Model Constraints

Constraint conditions of the twice-clustered ACL model include integrated constraints, parameter clusters constraints, group constraints, and ACL constraints, which are shown as below:

• Integrated constraints: the total control time is less than a certain value $t_{\max }$, i.e., $t_N\le t_{\max }$.
• Parameter clusters constraints: firstly, the continuous control time of $c_{pi}$ is less than the maximum continuous control time $t_{pm}$ to avoid excessive control of $c_{pi}$, where $t_{pm}$ is also less than $t_{\max }$.

  $$con(t_{c_{pi}})={\displaystyle \sum _{t_{po}}^{t_n}{\upsilon }_t^{c_{pi}}\cdot h\le t_{pm}}$$

  (6)

  where $t_n$ is current time point, $t_{po}$ is the closest $c_{pi}$ control time point to $t_n$, and ${\upsilon }_t^{c_{pi}}$ is a decision variable where ${\upsilon }_t^{c_{pi}}=1$ indicates $c_{pi}$ is under control at t.
  Secondly, the total number of $c_p$s under control at $t_n$ is less than the maximum number of the allowable clusters:

  $$0\le N_{c_{pi},t_n}\le N_{c_{pi}}$$

  (7)
• Group constraints: firstly, the continuous control time of ${\gamma }_{gi}$ in $c_{pi}$ is less than the maximal continuous control time: $t_{gm}$, and $t_{gm}\le t_{pm}$:

  $$con(t_{c_{gi}})={\displaystyle \sum _{t_{go}}^{t_n}{\upsilon }_t^{{\gamma }_{gi}}\cdot h}\le t_{gm}$$

  (8)

  where $t_{go}$ is the nearest ${\gamma }_{gi}$ control time point to $t_n$, and ${\upsilon }_t^{{\gamma }_{gi}}$ is a decision variable where ${\upsilon }_t^{{\gamma }_{pi}}=1$ indicates ${\gamma }_{gi}$ is under control at t.
  Secondly, the total number of ${\gamma }_g$s under control at $t_n$ is less than the maximum number of the allowable groups:

  $$0\le N_{{\gamma }_{gi},t_n}\le N_{{\gamma }_{gi}}$$

  (9)
• ACL constraints: ACL constraints include the total ACL control time constraint, ACL continuous control time constraint, and temperature boundary, which are expressed as:

  $$t_{i\min }<{\displaystyle \sum _{t_n=t_0}^{t_N}{\mu }_t^i\cdot h\le t_{i\max }}$$

  (10)

  $$con(t_{Ai})={\displaystyle \sum _{t_{ao}}^{t_n}{\upsilon }_t^i\cdot h}\le t_{Am}$$

  (11)

  $${\theta }_{(-,i)}^{*}\le {\theta }_{(i,t_n)}^{}\le {\theta }_{(+,i)}^{*}$$

  (12)

  where $t_{i\min },t_{i\max }$ are the minimum and maximal control time of ACL respectively, $t_{Am}$ is the maximal continuous control time of ACL, and ${\theta }_{(-,i)}^{*},{\theta }_{(+,i)}^{*}$ are the upper temperature margin and the lower temperature margin respectively.

3. Two-Stage ACLs Control Method Based on TC and SC

3.1. TC and SC Collaborative Mode Analysis

Combining advantages of TC and SC, this paper proposes a collaborative two-stage control method. As shown in Figure 2, the controlled ACL consumption target is the sum of TC target and SC target, and TC cycle time is longer than SC cycle time.

This paper adopts the TC method developed from the SQ equivalent model in [23]. The method is applied to ACLs with the same or similar parameters. In this paper, TC objects are groups with integrated ACLs. This paper assumes that ACLs’ states are ideally distributed according to the SQ equivalent model [30]. For those ACLs at the ON state before conducting TC, set them operate within $[{\theta }_{(-,new)},{\theta }_{(+,old)}]$ when starting to conduct TC and then adjust the temperature range to $[{\theta }_{(-,new)},{\theta }_{(+,new)}]$ when ACLs turn OFF sequentially; for those ACLs at the OFF state before conducting TC, set them operate within $[{\theta }_{(-,old)},{\theta }_{(+,old)}]$ firstly, then adjust the temperature range to $[{\theta }_{(-,new)},{\theta }_{(+,old)}]$ when ACLs turn ON sequentially and finally adjust the temperature range to $[{\theta }_{(-,new)},{\theta }_{(+,new)}]$ when ACLs turn OFF sequentially. This method is easy to conduct and can reduce the power fluctuation well. The total power change and duration time of ACL groups can be calculated via several Monte Carlo simulations [9] under a certain ${\xi }_{(i,t_n)}$. The dispatching center should set reasonable TC target according to the group response characteristics and total consumption target.

SC is easy to conduct and to change ACL’s state quickly. However, frequent ON and OFF switchings of ACL may reduce the thermal comfort of users. Therefore some users may be unwilling to participate even though financial incentive is provided. By implementing TC, the total power of ACLs is lifted-up, and the power vacancy of SC is greatly diminished and can be met by fewer users. Thus, conducting the TC and SC collaborative mode can not only cause less impact on users’ thermal comfort, but also can achieve following the target precisely.

3.2. Setting of Control Priorities

In this section, parameter clusters priority, group priority and ACL priority are developed as the control priorities. The hierarchical relations among these priorities are shown in Figure 3. The objects participating in TC are integrated groups where total ACLs in this group are contained. The objects participating in SC are single ACL. Thus, every hierarchical control priority is taken into consideration.

Parameter clusters priorities: the parameter clusters combination with less ACLs has higher control priority when total power $P_{\Omega }$s of different candidate combinations are within the certain range $\epsilon$ between the target $P_{t_n}$, as shown in (13), which improves dispatching efficiency:

$$\min {\displaystyle \sum _{{\Omega }_i=1}^{{\Omega }_y}{\displaystyle \sum _{\begin{array}{c}\end{array}{c}_{pi}=c_{p1}}^{c_{pm}}nu{m}_{ACL}({\delta }_{c_{pi}}\cdot c_{pi})}|{}_{t_{n,n=1\to N}}}$$

(13)

where ${\Omega }_i$ is the scheme number of different parameter clusters; $nu{m}_{ACL}(c_{pi})$ denotes the quantities of required ACLs of $c_{pi}$. ${\delta }_{c_{pi}}$ is a 0/1 decision variable and ${\delta }_{c_{pi}}=1$ means the parameter cluster ${\delta }_{c_{pi}}$ is chosen.

Group priorities: when $c_{pi}$ is chosen and sig = sig (TC), group priorities can be neglected because the related ${\gamma }_g$s and $c_p$s have been determined in parameter clusters priorities. When sig = sig (SC), groups are controlled according to $\left\{{\gamma }_1,{\gamma }_2,...{\gamma }_z\right\}$ where the UTVs are ranked in descending order.

ACL priorities: Temperature flexibility value is defined as ${\omega }_i=\frac{\theta (i,t_n)-\theta (-,i)}{\Delta }$ [31]. The ACL with smaller $\omega$ has higher priority when sig (SC) = ON and the ACLs are sequentially turned on according to the ascending sequence of $\omega$:$\left\{{\omega }_1^{on},{\omega }_2^{on},...,{\omega }_x^{on}\right\}$. Conversely, the ACLs are turned off according to the descending sequence of $\omega$:$\left\{{\omega }_1^{off},{\omega }_2^{off},...,{\omega }_x^{off}\right\}$.

3.3. Constraint Conditions of Two-Stage ACLs Control Method

Every ACL is allowed to participate in only one control; otherwise it will cause unexpected power fluctuation and disorder of ACL states. TC is designed to be with higher priority than SC because integrated ACLs in one group are required under TC and its influence on users is less than SC. If the ACLs in one group are controlled under TC at $t_{n1}$, they are not allowed to be controlled under SC until the power changes are finished and ACLs turn into steady states at $t_{n2}$. These constraints are expressed as:

$$0\le {\Re }_{(i,t_n)}\le 1$$

(14)

$$Prior(TC)>Prior(SC)$$

(15)

$${\Re }_{(i,T{C}_{t_{n1}-t_{n2}})}^{S{C}_{t_{n1}-t_{n2}}}=0$$

(16)

where ${\Re }_{(i,t_n)}$ is the number of control modes that ACL i is participating in at $t_n$.

It should be noted that more constraints will be constructed when consuming PV and WP based on the proposed model and control method.

4. Control Error Analysis and Compensation Method Considering Communication Time Delay

4.1. Control Error Analysis of a Single ACL Considering Communication Time Delay

Communication time delay is practically inevitable, since it causes control response to be later than the control signal, leading to noticeable delay and deviation between the actual total power of aggregated ACLs and the desired total power.

Taking a single ACL as an example, the communication time delay has three kinds of influence, which are Situation 1: response delay only, Situation 2: response delay as well as more power consumption, and Situation 3: response delay as well as less power consumption. Because of the limitation of length, this paper only discusses the Situation 2. The time delay in Figure 4 is amplified by 5 times for convenience of analysis.

Situation 2 is likely to happen when ACL is OFF and the room temperature is about to reach ${\theta }_{(+,i)}$. Assume that $\theta$ reaches ${\theta }_{(+,i)}$ at $t_{\max }$, $t_{\max }^{\prime }$, $t_{\max }^{"}$, …, etc. The control signals are given at $t_0$, $t_1$ and $t_2$, and the ACL responds at $t_0^{\prime }$, $t_1^{\prime }$ and $t_2^{\prime }$ respectively. Taking $(si{g}_{t_0},si{g}_{t_1},si{g}_{t_2})=(OFF,OFF,OFF)$ for example, there are 8 possible situations shown in Figure 4. Except the situation of Figure 4a where $t_0^{\prime }<t_{\max },t_1^{\prime }<t_{\max },t_2^{\prime }<t_{\max }$ is Situation 1, other situations all belong to Situation 2. Taking Figure 4b as an example, $t_2^{\prime }$ is later than $t_{\max }$, which means that ACL responds to the OFF signal after it turns from OFF to ON spontaneously. As a result, the ACL’s response delays and cause extra power consumption.

From the discussion above, it can be concluded that the switch of ACL i no longer obeys spontaneous evolution relationship of $\theta$, ${\theta }_{(+,i)}^{*}$ and ${\theta }_{(-,i)}^{*}$ when ACL i is controlled. Instead, the switch status of ACL i can be expressed as (17):

$${\tilde{s}}_{(i,t_n)}=\{\begin{array}{cc}{s}_{(i,t_n)}\hfill & \hfill \theta \ge {\theta }_{+}^{*}-\epsilon or \theta \le {\theta }_{-}^{*}+\epsilon \\ {\dot{s}}_{(i,t_n)}\hfill & \hfill {\theta }_{-}^{*}+\epsilon <\theta <{\theta }_{+}^{*}-\epsilon \end{array}$$

(17)

where ${\dot{s}}_{(i,t_n)}$ is a binary control signal and $\epsilon$ is margin threshold. This paper assumes that controller sends a control signal at a certain sampling point, ACL receives this signal and responds to it immediately at another sampling point. The time interval between two control signals sent from controller equals to x*h, where x is an integer. $kk=0$ means that the ACL does not receive the control signal at the sampling point, and $kk=1$ means that the ACL receives the control signal at the sampling point. Considering the communication time delay, (17) can be rewritten as:

$${\tilde{s}}_{(i,t_n)}=\{\begin{array}{ccc}{s}_{(i,t_n)}\hfill & & \hfill kk=0\\ s_{(i,t_n)}\hfill & \hfill \theta \ge {\theta }_{+}^{*}-\epsilon or \theta \le {\theta }_{-}^{*}-\epsilon & \hfill kk=1\\ \beta *{\dot{s}}_{(i,t_{n-l_n})}\hfill & \hfill {\theta }_{-}^{*}+\epsilon <\theta <{\theta }_{+}^{*}-\epsilon & \hfill kk=1\end{array}$$

(18)

where $l_n$ is the counting number of sampling time steps between a control signal sent and received, then the communication delay equals $l_n*h$. $l_n$ also varies within a certain range at different sampling points. $\beta$ is a binary confirmation number and when the control signal responded at $t_n$ is confirmed sent at $t_{n-l_n}$, $\beta =1$, otherwise, $\beta =0$.

Assuming a control signal sent at $t_n$ is received at $t_{n+m}$, (19) can be formulated by applying recursion to (2):

$${\theta }_{(i,t_{n+m})}=a_i^m{\theta }_{(i,t_n)}+{\displaystyle \sum _{k=0}^{m-1}{a}_i^{m-1-k}}(1-a_i)({\theta }_{(a,i)}-\tilde{s}{}_{(i,t_{n+k})}R{}_i{P}_i)$$

(19)

The temperature flexibility value is:

$${\omega }_{(i,t_{n+m})}=a_i^m{\omega }_{(i,t_n)}+{\displaystyle \sum _{k=0}^{m-1}{a}_i^{m-1-k}}(1-a_i)({\omega }_{(a,i)}-\tilde{s}{}_{(i,t_{n+k})}R{}_i^{\nabla }{P}_i)-{\omega }_{(-,i)}$$

(20)

The advantage of using temperature flexibility value is that all ACLs’ temperature states can be expressed in a unified manner.

4.2. The Forecasting Status Model of Clustered ACLs

In this paper, communication time delay of every ACL at a certain time is considered equal, and the time delay varies within a certain range at different times. In practice, ACLs in one parameter cluster always have the same $\Delta$. Thus, the temperature flexibility values of all ACLs in one parameter cluster can be uniformly expressed. ${\omega }_{{\gamma }_{gi},i,t_n}^{c_{pi}}$ is the temperature flexibility value at $t_n$ of ACL i , which belongs to group ${\gamma }_{gi}$ and parameter cluster $c_{pi}$. Assuming there are y ACLs in $c_{pi}$, the temperature flexibility value of $c_{pi}$ at $t_{n+m}$ can be expressed as:

$${\omega }_{t_{n+m}}^{c_{pi}}=a_{c_{pi}}^m\cdot {\omega }_{t_n}^{c_{pi}}-\tilde{S}\cdot a_f+(1-a_{{\gamma }_q}){\omega }_a^{}B-{\omega }_{-}^{c_{pi}}b$$

(21)

where ${\omega }_{t_{n+m}}^{c_{pi}}=\left[\begin{array}{c}{\omega }_{1,t_{n+m}}^{c_{pi}}\\ {\omega }_{2,t_{n+m}}^{c_{pi}}\\ ⋮\\ {\omega }_{y,t_{n+m}}^{c_{pi}}\end{array}\right]$, $a_f^{}=\left[\begin{array}{c}{a}_{c_{pi}}^{m-1}(1-a_{c_{pi}})R_{c_{pi}}^{\nabla }{P}_{c_{pi}}\\ a_{c_{pi}}^{m-2}(1-a_{c_{pi}})R_{c_{pi}}^{\nabla }{P}_{c_{pi}}\\ ⋮\\ (1-a_{c_{pi}})R_{c_{pi}}^{\nabla }{P}_{c_{pi}}\end{array}\right]$, $\tilde{S}=\left[\begin{array}{ccccc}{\tilde{s}}_{1,t_n}^{c_{pi}}& {\tilde{s}}_{1,t_{n+1}}^{c_{pi}}& {\tilde{s}}_{1,t_{n+2}}^{c_{pi}}& \cdots & {\tilde{s}}_{1,t_{n+m-1}}^{c_{pi}}\\ {\tilde{s}}_{2,t_n}^{c_{pi}}& {\tilde{s}}_{2,t_{n+1}}^{c_{pi}}& {\tilde{s}}_{2,t_{n+2}}^{c_{pi}}& \cdots & {\tilde{s}}_{2,t_{n+m-1}}^{c_{pi}}\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {\tilde{s}}_{y,t_n}^{c_{pi}}& {\tilde{s}}_{y,t_{n+1}}^{c_{pi}}& {\tilde{s}}_{y,t_{n+2}}^{c_{pi}}& \cdots & {\tilde{s}}_{y,t_{n+m-1}}^{c_{pi}}\end{array}\right]$, $a_m=\left[\begin{array}{c}{a}_{c_{pi}}^{m-1}\\ a_{c_{pi}}^{m-2}\\ ⋮\\ 1\end{array}\right]$, $B=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& 1& 1& \cdots & 1\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& 1& 1& \cdots & 1\end{array}\right]$, $b=\left[\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right]$.

The temperature flexibility values of all ACLs at $t_{n+m}$ is:

$${\omega }_{t_{n+m}}={\left[\begin{array}{cccc}{\omega }_{t_{n+m}}^{c_{p1}},& {\omega }_{t_{n+m}}^{c_{p2}},& \cdots ,& {\omega }_{t_{n+m}}^{c_{pv}}\end{array}\right]}^T$$

(22)

4.3. Communication Time Delay Compensation Based on Network Predictive Control System

A kind of network predictive control system [32], shown in Figure 5, is utilized in this paper to compensate the communication time delay.

In order to better focus on addressing the communication time delay problem, this paper ignores the effect caused by physical configuration of different locations, and other inherent control delay like lock-in effect and thermal effect delay which can also be solved by the compensation method proposed in this paper. At first, the dispatching center determines the range of communication time delay. Therefore the forecasting time range of ACLs can be determined. Secondly, at a certain sampling point close to the next switch control point, buffer transmits the newly collected ACLs’ data packet to dispatching center through communication network, including input value: ON-OFF state and output values: power and room temperature. It is assumed that the data packet arrives exactly at the switch control point—if not, counts the time deviation into the forecasting model. Then, the identifier recognizes and obtains the forecasting status model according to the latest data packet by (21) and (22). Besides, the control prediction generator determines the forecasting control sequence according to the forecasting status model and consumption target at every future sampling point within the delay range. The control sequence is sent to delay compensator on the user side. Then, the appropriate control signal is selected according to the real delay time. Therefore, power consumption delay and deviation are eliminated.

5. PV and WP Consumption Based on Clustered ACLs

The twice-clustered ACL model and the two-stage control method are applied to consume PV and WP in this section. It is known that in the grid-connected situation, excessive adverse power delivery will affect power system stability [33]. Therefore, the PV and WP are prior to be used locally, and the extra power of PV and WP are prior to be consumed by ACLs. When PV and WP are lower than total load power, ACLs will be uncontrolled and operate spontaneously.

5.1. The Objective Functions of PV and WP Consumption

The objective functions include the first-stage consumption function and the second-stage consumption function.

The first-stage consumption target equals to the sum of TC target and the base load power, and the first-stage consumption function is aimed to diminish the gap between TC response and the target, which can be expressed as:

$$F_1=\min \mathrm{var}({\displaystyle \sum _{t_n=t_0}^{t_{N1}}{P}_{t_n}^{*}}-(P_{{}_{(agg,t_n)}}^T+P_{(base,t_n)}^{bl}))$$

(23)

$$P_{(agg,t_n)}^T={\displaystyle \sum _{{\Omega }_i={\Omega }_1}^{{\Omega }_y}{\displaystyle \sum _{{\gamma }_{gi}=1}^{{\gamma }_{gz}}{\delta }_{{\Omega }_i}\cdot {\delta }_{{\gamma }_{gi}}\cdot P_{{}_{{\gamma }_{g_i}}}^T}}|{}_{t_n}$$

(24)

where $P_{t_n}^{*}$ is the first-stage consumption target, $P_{(agg,t_n)}^T$ is the total power of ACLs under TC, and $P_{(base,t_n)}^{bl}$ is the base-line power of the base load. ${\delta }_{{\Omega }_i}$ and ${\delta }_{{\gamma }_{gi}}$ indicate whether the plan ${\Omega }_i$ is implemented and the group ${\gamma }_{gi}$ is controlled. $P_{{}_{{\gamma }_{gi}}}^T$ is the power change of the group ${\gamma }_{gi}$ conducting the plan ${\Omega }_i$ at $t_n$.

The second-stage consumption target is the total PV and WP consumption target. Considering the communication time delay, it can be expressed as:

$$F_2=\min \mathrm{var}({\displaystyle \sum _{t_n=t_0}^{t_{N2}}{P}_{t_{n+m}}^{**}}-(P_{(agg,t_{n+m})}^S+P_{(agg,t_{n+m})}^T+P_{(agg,t_{n+m})}^{non}+P_{(base,t_{n+m})}^{bl})$$

(25)

$$P_{(agg,t_{n+m})}^S={\displaystyle \sum _{{\Omega }_i={\Omega }_1}^{{\Omega }_y}{\displaystyle \sum _{c_{pi}=c_{p1}}^{c_{pm}}{\displaystyle \sum _{{\gamma }_{gi}=1}^{{\gamma }_{gz}}{\displaystyle \sum _{i=1}^X{\delta }_{{\Omega }_i}\cdot {\delta }_{c_{pi}}\cdot {\delta }_{{\gamma }_{gi}}\cdot {\delta }_i\cdot P_{{}_{{\Omega }_i,c_{pi},g_i,i}}^{}}}}}|{}_{t_{n+m}}$$

(26)

where $P_{t_{n+m}}^{**}$ is the second-stage consumption target at $t_{n+m}$; $P_{(agg,t_{n+m})}^S$ is the total power of ACLs under SC and $P_{(agg,t_{n+m})}^{non}$ is the total power of uncontrolled ACLs.

5.2. PV and WP Consumption Constraints

Power balance constraints: the power balance equation without considering power loss is expressed as:

$$P_{(TL,t_n)}+(P_{(W,t_n)}+P_{(V,t_n)})=P_{(\mathit{base},t_n)}+P_{(\mathit{agg},t_n)}+P_{(\mathit{abd},t_n)}$$

(27)

$$P_{(\mathit{agg},t_n)}=P_{(agg,t_n)}^T+P_{(agg,t_n)}^S+P_{(agg,t_n)}^{non}$$

(28)

where $P_{(TL,t_n)}$ is the tie-line power flow; $P_{(W,t_n)}$ and $P_{(V,t_n)}$ are wind power and photovoltaic power; $P_{(agg,t_n)}$ is the total power of aggregated ACLs, and $P_{(abd,t_n)}$ is the power of PV and WP curtailment.

Reversed power flow constraints: to eliminate impacts on the grid, the tie-line reversed power flow is limited to be less than a certain number, which is shown as:

$$P_{out}=P_{t_n}^{**}-P_{(agg,t_n)}-P_{(base,t_n)}\le P_{(out,\max )}$$

(29)

PV and WP curtailment constraints:

$$P_{(abd,t_n)}=P_{t_n}^{**}-P_{(agg,t_n)}-P_{(base,t_n)}-P_{(out,\max )}\le P_{(abd,\max )}$$

(30)

6. Simulation Analysis

6.1. Simulation Preprocessing

To validate the twice-clustered ACLs model, the two-stage ACL control method and effects of PV and WP consumption with communication delay compensation, 5000 ACLs and a microgrid where PV and WP largely exceed total load power during a certain period are selected as the object in this paper. The predicted outdoor temperature of the consumption day is shown in Figure 6a. 5000 ACLs are clustered into 6 different clusters, shown in Figure 6b. Assuming the total load power, base load power, PV and WP forecasted by dispatching center of the microgrid are accurate, accordingly, the corresponding TC, SC, the first-stage consumption and the second-stage consumption targets are obtained, which are shown in Figure 7. The red solid line in Figure 7b represents the periods when the total consumption target exceeds the total load power and the ACLs are involved into consumption. The red dotted line in Figure 7b shows PV and WP are less than total load power, and thus the ACLs operate naturally. 5000 ACLs are randomly distributed to different types of which the optimum comfort temperature set-points and related adjustment boundaries upon set-points are generated based on thermal comfort grades [9], as shown in

Table 1. Optimum comfort temperature set-points and related adjustments boundaries.

	Type	1	2	3	4	5
Parameters
Temperature set-points		26	26	25	25	24
Setting boundaries		−1	−2	−1	−2	−1
Quantities		413	550	636	659	672
	Type	6	7	8	9	10
Parameters
Temperature set-points		24	23	23	22	22
Setting boundaries		−2	−1	−2	−1	−2
Quantities		621	489	381	309	270

. According to Table 1, dispatching center gives hierarchical $T_{i\max }$s and ${\xi }_{i\max }$s shown in

Table 2. Hierarchical $T_{i\max }$s and ${\xi }_{i\max }$s.

${\mathit{T}}_{\mathit{i}\mathbf{max}}$	${\mathit{\xi }}_{\mathit{i}\mathbf{max}}$	${\mathsf{\Delta }}_{\mathit{i}}$
1 h	−0.5	0.5
5 h	−1	1
9 h	−2	1.5

. Different ${\Delta }_i$s are generated and distributed to different parameter clusters. Therefore, 161 groups within 6 parameter clusters are obtained. By conducting Monte Carlo simulations, the dispatching center obtains the power changes per −0.5 °C on ${\theta }_d$s within adjustments boundaries.

6.2. Simulation results Analysis

6.2.1. Basic Case for Consuming PV and WP

The sampling period is set to 10 s. The TC cycle is 5 min, and TC targets are set per 30 min. The SC cycle is 100 s. The communication time delay follows a normal distribution: $N(40，{10}^2)$. As shown in Figure 8, 95% of time delay is in the range of 20–60 s. Thus, the dispatching center only needs to forecast ACL status model and control sequence from the second to the sixth sampling point after the SC point. As shown in Figure 9a, the total power after control of ACLs can well capture the target curve. It is noted that the ACL response lags during the ramping stage and the tracking curve has conspicuous burrs. This performance has been discussed in Section 4.2. Root-mean-square error (RMSE) is used to evaluate the consuming performance by calculating deviation between the total consuming target and actual response, which is shown as:

$$RMSE=\sqrt{\frac{{\displaystyle \sum _{t_n=t_0}^{t_N}(P_{t_n}^{**}-{(P_{(agg,t_n)}+P_{(base,t_n)})}^2}}{N}}$$

(31)

where N is the number of sampling cycles per 10 s. RMSE = 0.176 in Figure 9a. The PV and WP consumption result after compensating communication delay is shown in Figure 9c and there is no conspicuous lag or burrs. RMSE = 0.053 in Figure 9c.

6.2.2. Impacts of SC Period on Simulation Result

In this section, the SC cycle is set to 70 s and 50 s respectively, other parameters and settings remain unchanged. The respective simulation results of ACLs consuming PV and WP during 8:00–14:30 are shown in Figure 10a,b.

RMSE in Figure 10a decreases to 0.043, while RMSE in Figure 10b slightly increases to 0.048. Figure 11 is used to illustrate this phenomenon. As shown in Figure 11, the longer segment represents SC cycle, the shorter segment represents sampling period. At the end of every SC cycle—marked with blue frames, ACL data packets are transmitted to the dispatching center. The SC cycle is 70 s in Figure 11a and 50 s in Figure 11b. Assuming the control signals sent at $t_1$, $t_2$ and $t_3$ in Figure 11a can be received at $h_1$, $h_2$ and $h_3$, the consumption result can be slightly better than that in Figure 11b. However, if the control signal sent at $t_1$ is responded to at $h_1^{\prime }$, the forecasting status in the second SC cycle will be different from the real status, causing power consumption deviation due to ACLs responding to the wrong control signals. Therefore, the dispatching center should cautiously determine the range of the SC cycle, so as to insure the ACL response is within the same SC cycle. For example, setting the SC cycle as 70 s and 100 s may all obtain desirable results.

6.2.3. Impacts of Outdoor Temperature on Simulation Result

In this section, parameters remain unchanged except setting the outdoor temperature constantly to 28 °C and 33 °C, respectively. The second-stage consumption performances from 0:00–17:00 are shown in Figure 12a1,a2. The respective TC results are shown in Figure 12b1,b2. In addition, respective ACL quantities involved in SC are shown in Figure 12c1,c2. It can be seen that the ACLs well follow the TC target at 28 °C, but the second-stage consumption target is not well captured due to the existing gap between the target and actual response. This is because power changes and fluctuation durations of ACL groups decrease due to low outdoor temperature. Therefore, more groups are prior called to meet the TC target, and a gap between SC target and actual response appears for lacking enough ACLs to participate, as shown in Figure 12a1 from 11:40 to 14:02. There will be abundant reversed power flow, as shown in Figure 12d. The situation at 33 °C is just the opposite. Precision of following TC target decreases and the available ACLs for SC are more than that at 28 °C. As mentioned before, large quantities of ACLs participating in SC will decrease users’ thermal comfort. Thus, the dispatching center should make a proper first-stage consumption target to balance the consumption effects and users’ thermal comfort, and the priority order of TC and SC can be reversed if necessary to meet the total consumption target.

7. Conclusions

This paper firstly presents the ACL twice-clustered model. Afterwards, a two-stage ACL control method is proposed. On the basis of analyzing errors and impacts of ACLs caused by communication time delay, this paper proposes a time delay compensation method based on network predictive control system. Applying the ACL clustered model and control method into consuming PV and WP, a dual-stage consumption model considering communication time delay is established. Simulations are conducted based on ACL clusters, and the influences of SC cycle and outdoor temperature on the simulation results are analyzed. The results suggest that the potential of ACLs for consuming PV and WP are sufficiently tapped, and the TC target as well as SC cycle should be properly set to ensure the consumption result and users’ comfort.

In practice, there are cases of ACLs payback after control and user’s independent dropping out. Studying the strategy of reducing the ACLs rebound and solving the user’s exit will be key tasks in the future.