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Hybrid control of aggregated thermostatically controlled loads: step rule, parameter optimisation, parallel and cascade structures

2016; Institution of Engineering and Technology; Volume: 10; Issue: 16 Linguagem: Inglês

10.1049/iet-gtd.2016.0619

1751-8695

Kai Ma, Chenliang Yuan, Zhixin Liu, Jie Yang, Xinping Guan,

Building Energy and Comfort Optimization

IET Generation, Transmission & DistributionVolume 10, Issue 16 p. 4149-4157 Research ArticleFree Access Hybrid control of aggregated thermostatically controlled loads: step rule, parameter optimisation, parallel and cascade structures Kai Ma, Kai Ma School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorChenliang Yuan, Chenliang Yuan School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorZhixin Liu, Zhixin Liu School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorJie Yang, Corresponding Author Jie Yang jyangysu@ysu.edu.cn School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorXinping Guan, Xinping Guan Department of Automation, Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai Jiao Tong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this author Kai Ma, Kai Ma School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorChenliang Yuan, Chenliang Yuan School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorZhixin Liu, Zhixin Liu School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorJie Yang, Corresponding Author Jie Yang jyangysu@ysu.edu.cn School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004 People's Republic of ChinaSearch for more papers by this authorXinping Guan, Xinping Guan Department of Automation, Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai Jiao Tong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this author First published: 01 December 2016 https://doi.org/10.1049/iet-gtd.2016.0619Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study proposes hybrid control strategies for aggregated thermostatically controlled loads (TCLs) in order to provide ancillary service. The on/off control strategy and the setpoint-regulation strategy are two typical control strategies. In order to reduce the tracking errors of the two control strategies, two improved methods were developed. The first method dispatches the aggregated loads to the automatic generation control signal following a step rule based on the state of charge. The second method optimises the controller parameters of the setpoint-regulation strategy by the Powell optimisation algorithm. Based on these two improved methods, the hybrid control strategies with parallel and cascade control structures were established by dividing the TCLs into two clusters, and the optimal allocations of the loads, the reference signal and the tracking error between the two clusters were obtained by the particle swarm optimisation algorithm. The simulation results demonstrate that the hybrid control strategies can reduce the errors in load following. 1 Introduction Thermostatically controlled loads (TCLs) are devices that consist of a thermostat to modulate the power consumption for heating or cooling. The simple TCLs, such as frequency-fixed air conditioners, usually have two operation states, i.e. 'on' or 'off', each of which corresponds to one output power level, Prated or 0. Due to the characteristic that the TCLs generally operate within a deadband around a temperature setpoint, they can be viewed as thermal cells that store electricity as thermal energy [1]. As the TCLs are widely used in residential and commercial buildings and contribute a large part of the total electricity consumption in power grid, they are becoming promising candidates in enhancing the property of power systems, such as the frequency stability and the voltage stability [2, 3]. The studies on the dynamic modelling and application of TCLs can be traced back to [4, 5], and the loads were applied in cold load pickup. The aggregated power of a large population of TCLs can be controlled to follow a desired trajectory, such as the automatic generation control (AGC) signal or the power fluctuations of renewable energy resources [6]. In [7], the potentials and economics of populations of TCLs for regulation reserve were studied. The typical control strategies of the aggregated TCLs can be classified into two categories: the on/off control strategy and the setpoint-regulation strategy. Recently, several models and control strategies were established to turn the TCLs on or off directly. The authors in [8] focused on developing a finite-space stochastic dynamical model for aggregated TCLs. A temperature-priority-list method, a priority-stack-based control algorithm, and a stochastic ranking method were developed to control the on/off states of the aggregated TCLs in [9–11], respectively. A centralised control framework of the TCLs was presented in [12–14] to provide continuous regulation services, and the operational characteristics were analysed under different system states and communication models. In [15], the aggregated TCLs were viewed as an equivalent battery that is 'charged' or 'discharged' ceaselessly during the regulation period. The idea that regulating the setpoints of the aggregated TCLs to provide ancillary services was first presented in [16]. In this control mode, the TCLs are regulated by changing the setpoints instead of turning on or off directly. For example, the setpoints of the cooling loads are turned up to reduce the energy consumption when the generation is scarce and are turned down to increase the energy consumption when the generation is in surplus. The authors in [17–19] proposed several types of controllers, such as the internal model controller and the linear quadratic controller, to achieve peak shaving or load shifting by regulating the temperature setpoint. In [20], a continuous-time bilinear model of aggregated TCLs was established based on the partial differential equation and the temperature setpoints were regulated to support the power grid. In [21], the imbalance between power supply and demand was mitigated through a universal sliding mode controller for regulating temperature setpoints. The authors in [22] analysed and compared the on/off control strategy and the setpoint-regulation strategy. It was suggested that the on/off control strategy is constrained by the energy capacity, and the setpoint-regulation strategy is unable to achieve a precise tracking performance. To reduce the errors in tracking the AGC signals, a hybrid control strategy was developed by combining the two types of control strategies based on the parallel and cascade structures. Specifically, the TCLs are divided into two clusters, where the on/off control strategy is used in one cluster, and the setpoint-regulation strategy is adopted in the other cluster. To implement the hybrid control strategy, two problems have to be addressed: How to improve the tracking performance for each type of control strategy? and How to allocate the loads and control information to each cluster? This work focuses on these problems and achieves the following contributions: To relieve over-charging and deep-discharging of TCLs, a step rule was established for the on/off control strategy. The continuous-time bilinear model was transformed into the discrete-time form, and the optimal control parameters were obtained by the Powell optimisation algorithm to adjust the setpoint of TCLs. Two cluster-based control structures (parallel control and cascade control) were proposed, and the optimal allocations of the loads, the reference signal, and the tracking error between each cluster were obtained by the particle swarm optimisation (PSO). The rest of the paper is organised as follows. Section 2 introduces the system model of the TCLs. Section 3 proposes a step rule to improve the performance of the on/off control strategy, optimises the controller parameters of the setpoint-regulation strategy by the Powell optimisation algorithm, and develops the parallel and cascade control structures by combining the two improved control strategies. The simulation results are shown in Section 4, and the conclusions are summarised in Section 5. 2 System modelling To model the dynamics of an individual TCL, two state variables are essential: the internal temperature θi and the discrete state of the thermostat (on or off) si. In this study, a discrete-time difference equation was used to model the temperature evolution of a TCL [23]. For aggregated N TCLs, the dynamic temperature evolution of the i th TCL (i ∈ N) in the cooling mode can be formulated as (1) where (2) (3) (4) θa denotes the ambient temperature, θi is the internal temperature, and is the temperature setpoint of the i th TCL. Ri is the thermal resistance, Ci is the thermal capacitance, Pi is the energy transfer rate, and si is a binary variable, which represents whether the TCL is on (si = 1) or off (si = 0). and are the upper and lower temperature bounds, respectively. The load is turned on when the temperature reaches the upper bound and turned off when the temperature reaches the lower bound. Δ is the width of the temperature deadband, which means the difference between the upper and lower temperature bounds. k denotes the iterations, and h is defined as the time interval in each step. The aggregated power consumption of aggregated TCLs can be calculated by (5) where ηi denotes the i th TCL's efficiency coefficient, which is greater than 1. 3 Hybrid control strategies 3.1 Step rule of the on/off control As a typical on/off control method for aggregated TCLs, a temperature-priority control strategy was developed in [24]. As illustrated in Fig. 1, the aggregated loads are ordered by their internal temperatures. The 'on' loads with lower internal temperatures have higher priority to turn off, and the 'off' loads with higher indoor temperatures have higher priority to turn on. Then according to the priority stack, the loads will be turned on or off in sequence until the desired objective is achieved. Fig. 1Open in figure viewerPowerPoint Temperature-priority control strategy In the on/off control mode, the tracking errors are small when the temperatures of TCLs are uniformly distributed in the deadband. It means that the equivalent battery (i.e. the aggregated TCLs) have enough charge and discharge capacity. However, once the loads are centring at the upper or lower temperature boundaries, the synchronisation occurs and the loads cannot track the regulation signal. This is because the equivalent battery is over charged or deeply discharged and cannot provide any ancillary service. In this case, it is adverse for supporting frequency regulation service, and the wear of the TCLs is serious. To reduce the probability of the loads' centring at the boundaries, a step rule is presented to improve the tracking performance. To establish the step rule, an index named SOC is used to describe the state of charge of the aggregated TCLs. The SOC of the aggregated TCLs under the cooling mode can be defined as The SOC is a variable between 0 and 1, which represents the percentage of the remaining energy stored in the battery. For aggregated TCLs, if all of them reach their lower temperature boundaries, the SOC is 1 which means that the stored energy is full; if all of them reach their upper temperature boundaries, the SOC is 0 which means that the stored energy is used up. It is noted that the loads are centred at the boundaries and turned on or off frequently when the SOC is 1 or 0. A step rule is introduced to determine a ratio K according to the SOC and use (K · PAGC + PBL) as the actual reference signal, where PAGC is the AGC signal and PBL denotes the power baseline of the aggregated TCLs When PAGC > 0, the aggregated TCLs are to be charged. Then if the SOC is between 0 and κ, which means the battery has enough charging capacity, the TCLs can be controlled to track the reference signal and K = 1. If the SOC is greater than κ, the tracking ratio K should be reduced properly to deal with the situation that the charging ability is decreasing. When PAGC ≤ 0, the aggregated TCLs are to be discharged. Then if the SOC is between κ to 1.0, which means the battery has enough discharging capacity, the TCLs can be controlled to track the reference signal and K = 1. If the SOC is smaller than κ, the tracking ratio K could be reduced properly to deal with the situation that the discharging ability is decreasing. The detailed rule is described in Figs. 2 and 3, where λ is the step width. Fig. 2Open in figure viewerPowerPoint Tracking ratio K when PAGC > 0 Fig. 3Open in figure viewerPowerPoint Tracking ratio K when PAGC ≤ 0 3.2 Parameter optimisation of the setpoint-regulation control For a system consisting of N TCLs that satisfy the dynamics described in Section 2, the thermostats operate according to their internal temperature and temperature boundaries independently. Considering the 'flow' of TCLs along the temperature, the aggregated model can be derived by reasonable approximations [21]. The temperature deadband is uniformly divided into n bins, and each bin contains the 'on' loads and 'off' loads. As time evolves, the loads enter and leave each temperature bin. In that case, 2n states that denote the numbers of 'on' loads and 'off' loads in each bin can be obtained. The aggregated model can be described as the discrete-time system (6) (7) where x is a 2n × 1 vector that represents the numbers of loads in each state, I is the identity matrix, H = [0, …, 0|n, P/η, …P/η] is a 1 × 2n vector, and y is the total power consumption. Under the assumption that the temperature setpoint can be regulated collectively, the change of the temperature setpoint in unit time u (k) (° C/h) is added to the model, which is a scalar variable. A is the state matrix (8) where the parameters in A are detailed as (9) (10) where R, C, P, and θset are the average values of Ri, Ci, Pi, and , respectively. Here, denotes the initial temperature setpoint of the i th TCL. The input matrix B is given by (11) To regulate the temperature setpoints of the aggregated TCLs, a sliding-mode controller u in the discrete-time form is derived as (12) where (13) , Pr is the reference signal which denotes the summation of the power baseline and the AGC signal. ϕ is the boundary layer of the sliding-mode controller, and ρ is the control gain. Theorem 1.The discrete-time controller (12) is asymptotically stable if (14) where , , and . Proof.The proof is shown in Appendix 1. □ In the robustness condition (14), φ and ψ are time-varying according to the variation of the ambient temperature and the temperature setpoint, and ψ can be proved to be negative. Once happens, the Lyapunov stability is not satisfied and the tracking error is large. Remark 1.The final control law for the temperature setpoint is obtained as: . When e (k) > 0, the loads need to increase the power consumption, thus the temperature setpoint should be decreased (u (k) < 0). When e (k) < 0, the loads need to reduce the power consumption, thus the temperature setpoint should be increased (u (k) > 0). The sliding mode controller for the setpoint regulation is robust but not precise, which is demonstrated in the simulation. It is necessary to search for the optimal controller parameters to reduce tracking errors and improve the tracking performance. According to (14), the value of the control gain ρ should be adjusted in real time to guarantee the stability of the system. However, the AGC signal is stochastic and cannot be predicted, and thus ρ is usually chosen large enough to satisfy the robust condition, which results in serious sliding-mode chattering effect and large tracking errors. The boundary layer ϕ and the saturation function of the sliding mode controller in (12) can be used to alleviate the chattering effect, and the tracking error can be bounded by ϕ. If ϕ is too small, the chattering effect cannot be mitigated efficiently. If ϕ is too large, the stability of the system cannot be guaranteed. Therefore it is better to choose proper ρ and ϕ to reduce the computation complexity, guarantee the stability and improve the tracking performance. In this study, the root-mean-square error (RMSE) is formulated to evaluate the tracking performance (15) where Ns is the number of control cycles, and are the maximum and minimum values of the reference signal, respectively. In fact, the tracking performance is determined by both the control gain ρ and the boundary layer ϕ. To find the optimal controller parameters , the relationship among RMSE and need to be analysed at first. The actual RMSE values at different are calculated and the results are shown in Fig. 4. From the figure, it can be observed that the relationship among them is convex and there exists a minimum RMSE value. However, the exact analytic expression cannot be established, and the analytical solution at the minimum RMSE cannot be obtained. Next, the Powell optimisation algorithm is utilised to obtain the numerical solution. Fig. 4Open in figure viewerPowerPoint Relationship among RMSE and The Powell optimisation algorithm is an efficient direct search method and first proposed in [25]. The basic idea is utilising one-dimensional search methods to determine the conjugate directions without calculating derivatives. The algorithm is easy to be implemented and has a fast convergence rate. Hence it is suitable to search for the optimal controller parameters (16) and the seeking process of the Powell optimisation algorithm is shown in Fig. 5 and Algorithm 1 (see Fig. 6). Fig. 5Open in figure viewerPowerPoint Diagram of the Powell optimisation algorithm Fig. 6Open in figure viewerPowerPoint Powell optimisation 3.3 Parallel and cascade control structures In this subsection, the N TCLs are divided into two clusters. The loads in one cluster are controlled by the on/off control strategy with the step rule, and the loads in the other cluster are controlled by the setpoint-regulation strategy with the optimised parameters. The objective is to improve the tracking performance by determining the control structures of the two clusters and allocating control information to them. The basic control structures of the clusters are parallel and cascade, which are shown in Figs. 7 and 8, respectively. In the parallel control, the reference signal is divided into two parts proportionally, and the two parts are allocated to each cluster, respectively. The loads in each cluster are controlled to track the respective reference signal. Then, the total tracking error can be calculated according to the power consumption of the loads in the two clusters. The total error is also divided into two parts proportionally as the feedback information of each cluster. The final tracking error is Pr − y1 − y2. Hence, it is important to obtain the optimal allocations of the loads, the reference signal, and the total error to achieve best tracking performance. The problem can be formulated as (17) where α, β, and γ denote the proportions of the loads, the reference signal, and the total error for the cluster using on/off control strategy, respectively. 1 − α, 1 − β, and 1 − γ denote the corresponding proportions for the cluster using the setpoint regulation strategy. Fig. 7Open in figure viewerPowerPoint Parallel control structure Fig. 8Open in figure viewerPowerPoint Cascade control structure In the cascade control, it also involves the allocation of the reference signal. The loads under the setpoint regulation mode are controlled first to track their reference signal. Then, the tracking error and the other part of reference signal are added as the signal to be tracked by the loads under the on/off control mode with the step rule. The final tracking error is also Pr − y1 − y2. In this control structure, only the optimal allocations of the loads and reference signal are required. The parameters optimisation can be expressed as (18) where α and β denote the proportions of the loads and the reference signal for the cluster using on/off control strategy, respectively. 1 − α and 1 − β denote the corresponding proportions for the cluster using the setpoint regulation strategy. However, no matter in which control structure, it is difficult to establish the analytic model of RMSE. Thus, the PSO algorithm is adopted to solve the problem [27]. The details of the PSO algorithm are shown in Appendix 2. In the parallel control structure, three parameters need to be optimised. Thus, it is a three-dimensional optimisation problem and Li in Fig. 13 can be expressed as . In the cascade control structure, the problem becomes searching the optimum over a two-dimensional parameter space, and Li is . To solve the problems (17) and (18), the location Li and velocity Vi of each particle should be initialised randomly. According to the basic PSO algorithm shown in Fig. 13, Li, Vi and fitness = RMSE are updated, then the optimal parameters can be obtained within limited iterations. 3.4 Control framework The control framework of the frequency regulation is shown in Fig. 9. The aggregator decides the control structure (parallel or cascade control) of the TCLs in advance and calculate the optimal control parameters based on the historical regulation data before the regulation. When the regulation starts, the independent system operator communicates the AGC signal to the aggregator. The aggregator collects the state information of the loads, such as the temperature θi and the on/off state si, and publishes control commands to the loads. The loads change their power consumption in response to the control commands. Fig. 9Open in figure viewerPowerPoint Control framework of the frequency regulation Remark 2.The loads under the on/off control and the setpoint regulation provide state information to the aggregator, and the aggregator publishes the control commands to the loads. The two-way communications can be implemented based on the advanced metering infrastructure. The information to be communicated is the same as the existing works about the centralised control [8–21]. In the proposed control approach, the information is communicated in the offline optimisation. Thus, the optimisation result, which is utilised in real-time control, will not increase the complexity. 4 Simulation results The loads are simulated based on the software MATLAB and the related parameters of loads are from [15]. The simulation results are obtained under the assumption that the loads can be observed perfectly. In the simulations, 104 TCLs are chosen and the simulation parameters are shown in Table 1, where R, C, and P are assumed to follow Gaussian distributions and the standard deviations are 0.1. The initial temperatures of the loads are assumed to be distributed uniformly in the deadband. Table 1. Parameters setting Parameters Meaning Value R average thermal resistance 2°C/kW C average thermal capacitance 2 kWh/°C P average energy transfer rate 14 kW η efficiency coefficient 2.5 θset initial temperature setpoint 20°C θa ambient temperature 32°C Δ thermostat deadband 0.5°C The AGC signal is taken from the Pennsylvania-New Jersey-Maryland Interconnection electricity markets [26]. The power deviation is defined as the difference between the actual aggregated power and the baseline power. The actual aggregated power means the total power consumption Ptotal of all the loads, i.e. y1 + y2 in the cascade and parallel control structures. 4.1 Evaluation of the step rule In this section, the parameters of the step rule are set as κ = 0.5 and λ = 0.1. Because κ = 0.5 represents the aggregated TCLs have both enough charge and discharge capacity. The tracking ratios K under different system state are summarised in Table 2, and the tracking results without and with the step rule are shown in Fig. 10. It is observed that there are two periods during which the TCLs are centring at temperature edges when the step rule is not used. In these two periods, the tracking error is extremely large and the TCLs cannot provide regulation service. After the step rule is utilised, the interval of the first period is reduced and the second period is eliminated. As a result, the RMSE is decreased and the tracking performance is improved. Fig. 10Open in figure viewerPowerPoint AGC signal tracking under the on/off control strategy without and with the step rule Table 2. Parameters setting of the step rule PAGC>0 SOC 0–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1.0 K 1 0.9 0.8 0.7 0.6 0.5 PAGC<0 SOC 0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–1.0 K 0.5 0.6 0.7 0.8 0.9 1 4.2 Optimisation of the control parameters The Powell optimisation algorithm is used to obtain the optimal controller parameters based on the historical AGC data. It is observed that the optimal parameters are close in different days by utilising the daily signals, and hence it is reasonable to take the average of them as the optimal parameters. In this case, the optimal parameters are . 4.3 Computation of the allocation proportions In the hybrid control strategy, the loads in one cluster use the on/off control strategy with the step rule, the loads in the other cluster employ the setpoint regulation strategy with the optimal control parameters. The control cycle is 4 s in the parallel control structure, while it is 8 s in the cascade control structure. Because the controller of each cluster makes control commands simultaneously in the parallel structure, whereas the controller in the on/off control mode makes decisions after the setpoint regulation is finished under the cascade control structure, which takes more time than the parallel control. The optimal allocation proportions in the parallel and cascade control structure are obtained by the PSO algorithm. The convergence of the PSO algorithm under the two control structures are shown in Figs. 11 and 12. It is observed that the algorithm has fast convergence rate, and the optimal parameters can be obtained within a few iterations. Different daily AGC signals are used to obtain the average values of the parameters. As a result, the optimal parameters under the parallel structure are (0.25, 0.72, 0.11), which denotes that 25% loads are utilised to follow the 72% reference signal with the on/off control strategy, which uses the 11% tracking errors as the feedback. Meanwhile, 75% loads are utilised to follow the 28% reference signal with the setpoint regulation strategy, which uses the 89% tracking errors as the feedback. The optimal parameters under the cascade structure are (0.36, 0), which means that 64% loads with the setpoint regulation strategy are used to follow the whole reference signal in the first stage, and then 36% loads with the on/off control strategy are used to absorb the tracking errors in the second stage. Fig. 11Open in figure viewerPowerPoint Convergence of RMSE under the parallel control structure Fig. 12Open in figure viewerPowerPoint Convergence of RMSE under the cascade control structure 4.4 Comparison of the control strategies The confidence interval is introduced to estimate the distribution of the tracking errors. The confidence interval is expressed as , where is the mean of the tracking errors, σe is the standard deviation, and z* is the critical value which is dependent on the confidence level. For example, when the confidence level is 95%, the critical value z* is 1.96, then the confidence interval can be obtained by the statistical tracking errors. It means that the probability is 95% that the tracking error is in the interval. In this work, the greater confidence interval means that the more system reserve is needed for the system stability, and the more cost is to be invested. Based on the optimal parameters obtained by the PSO algorithm, the RMSE, the setpoint range, and the confidence interval of the actual tracking errors with different control strategies are compared in Table 3, and the following conclusions can be obtained: The smaller RMSEs can be achieved by the hybrid control strategies under both the parallel and cascade structures than using the on/off control or the setpoint regulation strategy individually. It demonstrates that the hybrid control strategies can improve the tracking performance of the aggregated TCLs. The best tracking performance is obtained under the cascade control structure. However, the setpoint variation of the loads is greater than that under the parallel control structure, which causes greater sacrifice of the consumers' comfort. Greater control cycle means less communication overhead. Thus, th