diff --git "a/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt" "b/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EdE1T4oBgHgl3EQfqQXv/content/tmp_files/load_file.txt" @@ -0,0 +1,976 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf,len=975 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 1 Real-time Feedback Based Online Aggregate EV Power Flexibility Characterization Dongxiang Yan, Shihan Huang, and Yue Chen, Member, IEEE Abstract—As an essential measure to combat global warming, electric vehicles (EVs) have witnessed rapid growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Meanwhile, thanks to the flexibility of EV charging, vehicle-to-grid (V2G) interaction has captured great attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the direct con- trol of individual EVs is challenging due to their small capacity, large number, and private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, it is the aggregator that interacts with the grid on behalf of EVs by characterizing their aggregate flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In this paper, we focus on the aggregate EV power flexibility characterization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' First, an offline model is built to obtain the lower and upper bounds of the aggregate power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It ensures that any trajectory within the region is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, considering that parameters such as real-time electricity prices and EV arrival/departure times are not known in advance, an online algorithm is developed based on Lyapunov optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that the charging time delays for EVs always meet the requirement even if they are not considered explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, real-time feedback is designed and integrated into the proposed online algorithm to better unlock EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Comprehensive performance comparisons are carried out to demonstrate the advantages of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Index Terms—Aggregate flexibility, charging station, electric vehicle, Lyapunov optimization, online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' INTRODUCTION T HANKS to the low carbon emissions, electric vehicles (EVs) have been considered a promising solution to climate change and proliferate in recent years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the uncontrolled charging of a large number of EVs can cause voltage deviation, line overload, and huge transmission loss [2], threatening the reliability of the power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Unlike inelastic loads, the charging power and charging period of EVs are more flexible [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, unlocking the power flexibility hidden in EVs is a promising way to lessen the adverse impact of EVs on the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' There are extensive literature aiming to design coordinated charging strategies to optimally schedule EV charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For example, to promote local renewable generation consumption, a dynamic charging strategy was proposed to allow the EV charging power to dynamically track the PV generation [4] and wind generation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To save the electricity cost, a deterministic optimal charging strategy was proposed for a home energy management system based on the time-of-use tariffs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A model predictive control (MPC) algorithm was proposed to minimize the operational cost of EV charging stations [7] relying on short-term forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To address the uncertainties related to EV charging, reference [8] proposed a D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Chen are with the Department of Me- chanical and Automation Engineering, the Chinese University of Hong Kong, Hong Kong SAR, China (e-mail: dongxiangyan@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk, shhuang@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk, yuechen@mae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' stochastic charging strategy based on the probabilistic model related to EV daily travels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A combined robust and stochastic MPC method was developed in [9] to handle the uncertain EV charging behaviors and renewable generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A multi-stage energy management strategy including day-ahead and real- time stages was developed for a charging station integrated with PV generation and energy storage [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, a pricing mechanism was suggested in [11] to guide EVs for economical charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A double-layer optimization model was built to reduce the voltage violations caused by EV charging [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Despite the efforts mentioned above that intend to determine the EV charging power, it is challenging to directly control a large number of individual EVs due to the high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To get around this problem, some other literature en- deavored to characterize the EV charging power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Reference [13] proposed to model the aggregate EV charging flexibility region by the lower and upper bounds of power and cumulative energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This aggregate EV model was adopted by [14] to evaluate the achievable vehicle-to-grid capacity of an EV fleet and by [15] to quantify the value of EV flexibility in terms of maintaining distribution system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Reference [16] further considered the spatio-temporal distribution of the probability that an EV is available for charging during the aggregation and clustering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' An EV dispatchable region was proposed to allow charging stations to participate in market bidding [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, the aggregate flexibility issue was also studied in the fields of thermostatically controllable loads (TCLs) [18], distributed energy resources [19], and virtual power plant [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For example, a geometric approach was utilized to model the aggregate flexibility of TCLs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' An inner box approximation method was proposed to charac- terize the power flexibility region of various distributed energy resources [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The above works provide sound techniques for evaluating EV flexibility in an offline manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It means that the aggregator is assumed to have complete information of future uncertainty realizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', EV arrival/departure time, and electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In practice, those data are usually unavailable or inaccurate, making the obtained region fail to reflect the actual EV aggregate flexibility in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Thus, an online algorithm is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A straightforward approach is the greedy algorithm that decomposes the offline problem into subproblems in each time slot by neglecting the time-coupling constraints [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Obviously, the result could be far from optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, we resort to another approach, Lyapunov optimization, that can run in an online manner but with an outcome near to the offline optimum [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Lyapunov optimization has been used in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='03342v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='OC] 9 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 2 microgrid control [25], energy storage sharing [26], and data center energy management [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For EV charging, a charging strategy based on Lyapunov optimization was proposed to minimize the total electricity cost [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, it cannot guarantee that EVs will depart with desired amount of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To meet the EV charging requirement, virtual delay queues were introduced to minimize the charging cost under uncertain renewable generations and electricity prices [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though the optimal online EV charging strategy has been widely studied as above, the online aggregate EV power flexibility characterization problem has not been well explored yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The latter problem is more complicated than the former one, requiring a new model and algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This paper proposes a real-time feedback based online aggregate EV power flexibility characterization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Our main contribu- tions are two-fold: 1) Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We first propose an offline optimization model to characterize the aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It decomposes the time-coupled flexibility region into each time slot and gives their lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that any trajectory within the region is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, by categorizing the EVs according to their allowable time delays, we develop a counterpart of the offline model that enables the further utilization of the Lyapunov optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed model has not been reported in previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2) Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A real-time feedback based online algorithm is developed to derive the aggregate EV power flexibility region sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' First, to fit into the Lyapunov optimization framework, charging task queues and delay-aware virtual queues are introduced to reformulate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, a drift-plus-penalty term is constructed and by minimizing its upper bound, an online algorithm is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that the charging time delays for EVs will not exceed their maximum allowable values even if they are not explicitly considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The bound of optimality gap between the offline and online outcomes is provided theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, real-time dispatch strategy based feedback is designed and integrated into the online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed real-time feedback based online algorithm is prediction free and can adapt to uncertainties such as random electricity prices and EV charging behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, it can make use of the most recent information, allowing it to even outperform the offline model with full knowledge of future uncertainty realizations but without the updated dispatch information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Section II formulates the offline model for deriving aggregate EV charging power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Section III and IV introduce the Lyapunov optimization method and real-time feedback design, respectively, to generate flexibility region in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Simulation results are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Finally, Section VI concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we first introduce the concept of aggregate EV power flexibility and then formulate an offline optimization problem to approximate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Aggregate EV Charging Power Flexibility As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1, when an EV v ∈ V arrives at the charging station, it submits its charging task to the aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The task is described by (ta v, td v, ea v, ed v), where ta v is its arrival time, td v is its departure time, ea v is the initial battery energy level at ta v, and ed v is the desired energy level when it leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For the EV v, the maximum allowable charging time delay is Rv = td v−ta v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EV charging task needs to be finished within this declared time duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' With the submitted information, the aggregator can flexibly schedule the EV charging to meet the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Two possible trajectories to meet the EV charging need are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let {pc v,t, ∀t} be the charging power of EV v over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The range that the charging power can vary within is called the power flexibility of EV v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' If we sum the power flexibility of all EVs in a charging station up, we can get the aggregate EV power flexibility of the charging station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (𝑡1 𝑎, 𝑡1 𝑑, 𝑒1 𝑎, 𝑒1 𝑑) (𝑡2 𝑎, 𝑡2 𝑑, 𝑒2 𝑎, 𝑒2 𝑑) (𝑡𝑣𝑎, 𝑡𝑣𝑑, 𝑒𝑣𝑎, 𝑒𝑣𝑑) Aggregator Distribution System Operator Aggregate dispatch power Aggregate power flexibility region Ƽ𝑝𝑑,𝑡, Ƹ𝑝𝑑,𝑡 𝑝𝑑,𝑡 𝑑𝑖𝑠𝑝 𝑝1,𝑡 𝑑𝑖𝑠𝑝 𝑝2,𝑡 𝑑𝑖𝑠𝑝 𝑝𝑣,𝑡 𝑑𝑖𝑠𝑝 EV 1 EV 2 EV v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Generate EV aggregate power flexibility region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Disaggregation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' System diagram and illustration of EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, it is difficult to characterize the EV power flexibil- ity for each time slot due to the temporal-coupled EV charging constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EV power flexibility in the current time slot is affected by those in the past time slots and further affects those in the future time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is different from the traditional controllable generators whose flexibility can be described by the minimum and maximum power outputs in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In the following, we aim to derive an aggregate EV power flexibility region that: 1) is time-decoupled so that it can be used in real-time power system operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' and 2) any trajectory within it can meet the EV charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Offline Problem Formulation Suppose there are T time slots, indexed by t ∈ T = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The desired time-decoupled aggregate EV power flexibility region can be represented by a series of intervals [ˇpd,t, ˆpd,t], ∀t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The intervals can be specified by a lower power trajectory {ˇpd,t, ∀t} and an upper power trajectory {ˆpd,t, ∀t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To obtain the lower and upper power trajectories, we formulate the following offline optimization problem: P1 : max ˆpd,t,ˇpd,t,∀t lim T →∞ 1 T T � t=1 E � Ft � , (1a) Power Aggregate power flexibility region flexibilitJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 3 where Ft = πt(ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t − ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1b) subject to ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈V ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1c) 0 ≤ ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1d) ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t+1 = ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t + ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t ̸= T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1e) ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tav = eini v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tdv ≥ ed v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1f) emin v ≤ ˆev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ emax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1g) ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � n∈V ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1h) 0 ≤ ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1i) ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t+1 = ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t + ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t ̸= T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1j) ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tav = eini v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='tdv ≥ ed v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1k) emin v ≤ ˇev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ emax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1l) ˇpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t ≤ ˆpd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1m) ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t/ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � ˆpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t/ˇpc v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' if t ∈ [ta v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' td v] 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' if t < ta v ∪ t > td v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1n) In the objective function (1a)-(1b), πt, ∀t are the real-time electricity prices, showing the unit value of power flexibility in different time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the objective function aims to maximize the value of total aggregate EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (1c) defines the upper bound of aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The charging power of an EV v is limited by (1d), where pmax v is the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (1f) defines the EV’s initial energy level and the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1e) and (1g) describe the EV’s energy dynamics and battery capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, (1h)-(1l) are the constraints related to the lower bound of the aggregate EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1m) is the joint constraint to ensure that {ˆpd,t, ∀t} and {ˇpd,t, ∀t} provide the upper and lower bounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (1n) limits that charging only happens during the EV’s declared parking time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 1: Any aggregate EV charging power trajectory within [ˇpd,1, ˆpd,1] × · · · × [ˇpd,T , ˆpd,T ] is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of Proposition 1 can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Despite this nice property, the offline optimization problem above cannot be solved directly since it requires complete knowledge of the future EV charging tasks and future elec- tricity prices, which are usually not available in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, an online algorithm is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To this end, in the next section, we will first propose a closely related but more flexible form of the problem studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we adopt the Lyapunov optimization framework to reformulate the offline problem into an online one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We construct charging task queues and delay-aware virtual queues to ensure the satisfaction of charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, considering the impact of real-time dispatch decisions on the future aggregate EV power flexibility, a real-time feedback based online flexibility characterization method is developed in Section IV to avoid the potential underestimate of EV power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ONLINE ALGORITHM In this section, we adopt the Lyapunov optimization frame- work to solve the offline problem P1 in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proposed algorithm can output an aggregate EV power flexibility value with an economic value close to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Problem Modification As mentioned above, the charging station serves dozens of EVs every day, and each EV arrives along with a charging task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', (ta v, td v, ea v, ed v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Those EV charging tasks can be first stored in a queue and be served later according to a first-in-first- out basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since different EVs may have different allowable charging time delays, we use multiple queues to classify and collect the EV charging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Suppose there are G types of charging time delays Rgs, each of which is indexed by g ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Correspondingly, we construct G queues to collect the respective charging tasks, and each queue is denoted by Qg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For queue Qg, Qg,t refers to its charging task backlog in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The queue backlog growth is described by Qg,t+1 = max[Qg,t − xg,t, 0] + ag,t, (2) where xg,t is the charging power for EVs in group g at time t, and ag,t is the arrival rate of EV charging tasks of group g at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In particular, ag,t sums up the energy demand of all EVs that arrive at the beginning of time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ag,t = � v∈Vg ag,v,t, (3) where ag,v,t is the charging demand of EV v of group g in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Vg is the set of EVs in group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Recalling that our target in P1 is to derive an upper bound and a lower bound for the aggregate EV power flexibility region, we correspondingly define the upper bound queue ˆQg,t and the lower bound queue ˇQg,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similar to (2), we have ˆQg,t+1 = max[ ˆQg,t − ˆxg,t, 0] + ˆag,t, (4) ˇQg,t+1 = max[ ˇQg,t − ˇxg,t, 0] + ˇag,t, (5) where ˆxg,t and ˇxg,t are the charging power for upper and lower bound queues, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆxg,t = � v∈Vg ˆpc v,t and ˇxg,t = � v∈Vg ˇpc v,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper and lower bounds of arriving charging demand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t and ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' are determined by ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈Vg ˆag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � v∈Vg ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (6) Particularly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' the lower bound of arriving charging demand ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t can be determined in the following charging as soon as possible way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t = � � � pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ta v ≤ t < ⌊ˇtmin v ⌋ + ta v ˇecha v /ηc − ⌊ˇtmin v ⌋pmax v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' t = ⌊ˇtmin v ⌋ + ta v 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' otherwise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (7) where ˇecha v = ed v −ea v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ˇtmin v is the minimum required charging time determined by ˇtmin v = ˇecha v pmax v ηc ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' and ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='⌋ means rounding down to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 4 Different from ˇag,v,t, the upper bound of arrival charging demand ˆag,v,t is determined using the maximum charging demand emax v instead of ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Denote ˆecha v = emax v − ea v, ˆtmin v = ˆecha v pmax v ηc , and then ˆag,v,t can be determined by ˆag,v,t = � � � pmax v , ta v ≤ t < ⌊ˆtmin v ⌋ + ta v ˆecha v /ηc − ⌊ˆtmin v ⌋pmax v , t = ⌊ˆtmin v ⌋ + ta v 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (8) We then formulate the aggregate EV power flexibility char- acterization problem as follows: P2 : min ˆxg,t,ˇxg,t lim T →∞ 1 T T � t=1 E � − Ft � , (9a) subject to lim T →∞ 1 T T � t=1 E[ˆag,t − ˆxg,t] ≤ 0, ∀g (9b) lim T →∞ 1 T T � t=1 E[ˇag,t − ˇxg,t] ≤ 0, ∀g (9c) 0 ≤ ˆxg,t ≤ min{xg,max, ˆQg,t}, ∀g (9d) 0 ≤ ˇxg,t ≤ min{xg,max, ˇQg,t}, ∀g (9e) ˆxg,t ≥ ˇxg,t, ∀g (9f) where xg,max = � v∈Vg pmax v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (9b) ensures that if using the upper bound trajectory {ˆxg,t, ∀t}, the total charg- ing requirement can be satisfied in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraint (9c) poses a similar requirement for the lower bound trajec- tory {ˇxg,t, ∀t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Based on (9b), (9c), and the definitions of ˆQg,t+1, ˇQg,t+1 in (4)-(5), we can prove that the queues ˆQg,t and ˇQg,t are mean rate stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To be specific, ˆQg,t+1 − ˆag,t ≥ ˆQg,t − ˆxg,t, ∀t (10) Summing (10) up over all t and divide both sides by T yields 0 ≤ E[ ˆQg,T ] T ≤ �T t=1 E[ˆag,t − ˆxg,t] T (11) Hence, lim T →∞ E[ ˆ Qg,T ] T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, lim T →∞ E[ ˇ Qg,T ] T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Constraints (9d) and (9e) give the upper and lower bounds of the aggregate EV charging power for group g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper bound is no less than the lower bound, as shown in (9f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The P2 provides a counterpart problem for P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similar to the proof of Proposition 1, we can prove that any trajectory between [ˇxg,t, ˆxg,t] is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, the allowable charging delay is not considered in P2, which may result in unfulfilled EV charging tasks upon departure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Construct Virtual Queues To overcome the aforementioned charging delay issue, we introduce delay-aware virtual queues, ˆZg,t+1 = max{ ˆZg,t + ηg Rg I ˆ Qg,t>0 − ˆxg,t, 0}, ∀g, ∀t (12) ˇZg,t+1 = max{ ˇZg,t + ηg Rg I ˇ Qg,t>0 − ˇxg,t, 0}, ∀g, ∀t (13) where I ˆ Qg,t>0 and I ˇ Qg,t>0 are indicator functions of ˆQg,t and ˇQg,t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' They are equal to 1 if there exists unserved charging tasks in the queues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQg,t > 0 and ˇQg,t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Using ηg Rg to times it, this whole term constitutes a penalty to the virtual queue backlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ηg is a user-defined parameter that can adjust the growth rate of the virtual queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For instance, increasing the value of ηg leads to a fast queue growth and a larger backlog value, calling for more attention to accelerate the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that, when Qg,t and Zg,t have finite upper bounds, with a proper ηg, the charging time delay for EVs in group g is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 2: Suppose ˆQg,t, ˇQg,t, ˆZg,t, and ˇZg,t have finite upper bounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQg,t ≤ ˆQg,max, ˇQg,t ≤ ˇQg,max ˆZg,t ≤ ˆZg,max, and ˇZg,t ≤ ˇZg,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The charging time delay of all EVs in group g is upper bounded by ˆδg,max and ˇδg,max time slots, where ˆδg,max := ( ˆQg,max + ˆZg,max)Rg ηg , (14) ˇδg,max := ( ˇQg,max + ˇZg,max)Rg ηg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (15) The proof of Proposition 2 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It ensures that the charging tasks can always be fulfilled within the available charging periods by properly setting the parameters ηg, ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Lyapunov Optimization Based on the charging task queues and delay-aware virtual queues, the Lyapunov optimization framework is applied as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1) Lyapunov Function: First, we define Θt = ( ˆ Qt, ˆ Zt, ˇ Qt, ˇ Zt) as the concatenated vector of queues, where ˆ Qt = ( ˆQ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆQG,t), (16a) ˆ Zt = ( ˆZ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˆZG,t), (16b) ˇ Qt = ( ˇQ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˇQG,t), (16c) ˇ Zt = ( ˇZ1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ˇZG,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (16d) The Lyapunov function is then defined as L(Θt) = 1 2 � g∈G ˆQ2 g,t + 1 2 � g∈G ˆZ2 g,t + 1 2 � g∈G ˇQ2 g,t + 1 2 � g∈G ˇZ2 g,t, (17) where L(Θt) can be considered as a measure of the queue size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A smaller L(Θt) is preferred to push (virtual) queues ˆQg,t, ˆZg,t, ˇQg,t, and ˇZg,t to be less congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2) Lyapunov Drift: The conditional one-time slot Lyapunov drift is defined as follows: ∆(Θt) = E[L(Θt+1) − L(Θt)|Θt], (18) where the expectation is taken with respect to the random Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The Lyapunov drift is a measure of the expectation of the queue size growth given the current state Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Intuitively, by minimizing the Lyapunov drift, virtual queues are expected to be stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, only minimizing the Lyapunov drift may lead to a low aggregate EV power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 5 Therefore, we include the expected aggregate flexibility value (1b) for the time slot t to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The drift-plus-penalty term is obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', ∆(Θt) + V E[−Ft|Θt], (19) where V is a weight parameter that controls the trade-off between (virtual) queues stability and aggregate EV power flexibility maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3) Minimizing the Upper Bound: (19) is still time-coupled due to the definition of ∆(Θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To adapt to online imple- mentation, instead of directly minimizing the drift-plus-penalty term, we minimize the upper bound to obtain the upper and lower bounds of aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We first calculate the one-time slot Lyapunov drift: L(Θt+1) − L(Θt) = 1 2 � g∈G � � ˆQ2 g,t+1 − ˆQ2 g,t � + � ˆZ2 g,t+1 − ˆZ2 g,t � + � ˇQ2 g,t+1 − ˇQ2 g,t � + � ˇZ2 g,t+1 − ˇZ2 g,t � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (20) Use queue ˆQg,t as an example, based on the queue update equations (4), we have ˆQ2 g,t+1 = {max[ ˆQg,t − ˆxg,t, 0] + ˆag,t}2 ≤ ˆQ2 g,t + ˆa2 g,max + ˆx2 g,max + 2 ˆQg,t(ˆag,t − ˆxg,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (21) Thus, 1 2 � ˆQ2 g,t+1 − ˆQ2 g,t � ≤ 1 2 � ˆx2 g,max + ˆa2 g,max � + ˆQg,t (ˆag,t − ˆxg,t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (22) Similarly, for queue ˇQg,t, ˆZg,t, and ˇZg,t, we have 1 2 � ˇQ2 g,t+1 − ˇQ2 g,t � ≤ 1 2 � ˇx2 g,max + ˇa2 g,max � + ˇQg,t (ˇag,t − ˇxg,t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (23) 1 2[ ˆZ2 g,t+1 − ˆZ2 g,t] ≤ 1 2 max[( ηg Rg )2, ˆx2 g,max] + ˆZg,t[ ηg Rg − ˆxg,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (24) 1 2[ ˇZ2 g,t+1 − ˇZ2 g,t] ≤ 1 2 max[( ηg Rg )2, ˇx2 g,max] + ˇZg,t[ ηg Rg − ˇxg,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (25) We then substitute inequalities (22),(23), (24) and (25) into drift-plus-penalty term and yield ∆(Θt) + V E[−Ft|Θt] ≤ A + V E[−Ft|Θt] + � g∈G ˆQg,tE [ˆag,t − ˆxg,t|Θt] + � g∈G ˇQg,tE [ˇag,t − ˇxg,t|Θt] + � g∈G ˆZg,tE [−ˆxg,t|Θt] + � g∈G ˇZg,tE [−ˇxg,t|Θt] , (26) where A is a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', A = 1 2 � g∈G (ˆx2 g,max + ˆa2 g,max) + 1 2 � g∈G max[( ηg Rg )2, ˆx2 g,max] + 1 2 � g∈G (ˇx2 g,max + ˇa2 g,max) + 1 2 � g∈G max[( ηg Rg )2, ˇx2 g,max] + � g∈G [ ˆZg,max ηg Rg ] + � g∈G [ ˇZg,max ηg Rg ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By reorganizing the expression in (26) and ignoring the con- stant terms, we can obtain the following online optimization problem, P3 : min ˆxg,t,ˇxg,t,∀g,∀t � g∈G (−V πt − ˆQg,t − ˆZg,t)ˆxg,t + � g∈G (V πt − ˇQg,t − ˇZg,t)ˇxg,t, (27) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (9d) − (9f), where ˆQg,t, ˇQg,t, ˇZg,t, and ˇZg,t are first updated based on (4),(5),(12), and (13) before solving P3 in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In each time slot t, given the current system queue state Θt, the proposed method determines the current upper and lower aggregate EV power flexibility bounds ˆxg,t and ˇxg,t by solving problem P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the original offline optimization problem P1 has been decoupled into simple online (real-time) problems, which can be executed in each time slot without requiring prior knowledge of future uncertain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since the modified problem P3 is slightly different from the offline one P1, an important issue we care about is: what’s the gap between the optimal solutions of the online problem P3 the and offline problem P1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Proposition 3: Denote the obtained long-term time-average aggregate EV power flexibility value of P1 and P3 by F ∗ and F pro, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We have 0 ≤ −F pro + F ∗ ≤ 1 V A, (28) where A is a constant defined in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of Proposition 3 can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The optimality gap can be controlled by the parameter V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A bigger V value leads to a smaller optimality gap but increased queue sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In contrast, a smaller V value makes the queues more stable but results in a larger optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' DISAGGREGATION AND REAL-TIME FEEDBACK DESIGN In each time slot t, given the aggregate EV power flexibility region [� g ˇx∗ g,t, � g ˆx∗ g,t], the distribution system operator (DSO) can determine the optimal aggregate dispatch strategy for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This aggregate dispatch strategy should be further disaggregated to obtain the control strategy for each EV, which is studied in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Moreover, considering that the current dispatch strategy will influence the aggregate EV power flexibility in future time slots, real-time feedback is designed and integrated with the proposed online flexibility characterization method in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Disaggregation Suppose the dispatch strategy in time slot t is pdisp agg,t ∈ [� g ˇx∗ g,t, � g ˆx∗ g,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This can be determined by the DSO by solving an economic dispatch problem based on the up-to- date information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', the electricity price πt, the grid-side renewable generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since this paper focuses on the online characterization of aggregate EV power flexibility, the eco- nomic dispatch problem by the DSO is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Interested readers can refer to [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let the dispatch ratio αt be αt = pdisp agg,t − � g ˇx∗ g,t � g ˆx∗ g,t − � g ˇx∗ g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (29) Then, the dispatched power pdisp g,t for each group g can be determined according to the ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', pdisp g,t = (1 − αt)ˇx∗ g,t + αtˆx∗ g,t, (30) which satisfies pdisp agg,t = � g pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The next step is to allocate pdisp g,t to the EVs in the group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' All EVs in the group g are sorted according to their arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we follow the first-in-first-service principle to allocate the energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' namely, the EV that comes earlier will be charged with the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We have pdisp v,t = min � pdisp g,t , pmax v , emax v − ev,t ∆t � , ∀v ∈ Vg, (31) where Vg refers to the set of EVs in group g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The third term on the right side is used to ensure that the EV will not exceed its allowable maximum energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' After this charging assignment for an earlier EV is com- pleted, the following update procedures will execute pdisp g,t ← (pdisp g,t − pdisp v,t ), (32) which means deducting pdisp v,t from the total remaining dis- patched power pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, the pdisp g,t is allocated to the next earlier arrival EVs until the aggregate EV charging power is completely assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' At this time, the disaggregation is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The dispatched power disaggregation algorithm is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' State Update to Improve Power Flexibility Region Following the disaggregation procedures in Algorithm 1, we can get the actual EV dispatched charging power pdisp v,t , ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By now, we can move on to the next time slot t + 1 and evaluate the EV power flexibility by solving problem P3, determine the dispatch strategy, disaggregate the dispatched power, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' But considering that the current actual dispatched EV charging power can affect the future aggregate EV power flexibility, which is ignored in the aforementioned processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, we propose a real-time feedback method to integrate the current actual EV dispatched charging power into the future aggregate EV power flexibility characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Algorithm 1 EV Dispatched Power Disaggregation 1: Initialization: aggregate EV dispatched power pdisp agg,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2: Calculate the dispatched aggregate charging power pdisp g,t for each group g using (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3: for Each group g ∈ G do 4: for Each EV v in group g do 5: if the EV is not available for charging then 6: Let EV v’s charging power pdisp v,t = 0, ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7: else 8: Calculate pdisp v,t according to (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9: Update the remaining aggregate power via (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10: if If the updated pdisp g,t = 0 then 11: Break and return to Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12: end if 13: end if 14: end for 15: end for Disaggregation State update Dispatch Aggregate power flexibility region State feedback t=t+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Overall procedure of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The overall procedure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2 with the right-hand side blue box showing the real-time feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To be specific, we change the constraints (9d)-(9e) for time slot t into ˆxg,t = pdisp g,t , ˇxg,t = pdisp g,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (33) Since pdisp g,t ∈ [ˇx∗ g,t, ˆx∗ g,t], after replacing (9d)-(9e) with (33), the problem P3 is still solvable and the optimal solution is ˆxupdate∗ g,t = pdisp g,t , ˇxupdate∗ g,t = pdisp g,t , ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' With these updated lower and upper bounds, we update the queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1 according to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we move on the solve problem P3 for time slot t + 1 using the updated ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, ˇZg,t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' So far, we have developed a real-time feedback based online aggregate EV power flexibility characterization method as well as the EV dispatched charging power disaggregation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A completed description of the proposed method is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' SIMULATION RESULTS AND DISCUSSIONS In this section, we evaluate the performance of the proposed online algorithm and compare it with other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' System Setup The time resolution is set as 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The entire sim- ulation duration considered is 24 hours, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', 144 time slots JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 7 Algorithm 2 Real-time Feedback Based Online Aggregate EV Power Flexibility Characterization and Disaggregation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Aggregation 1: Aggregator classifies the arriving EVs and pushes them into different queues ˆQg, ˇQg, ˆZg and ˇZg according to their declared charging delay time Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2: Solve problem P3 and obtain the aggregate EV power flexibility region [ˇx∗ g,t, ˆx∗ g,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- cording to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Dispatch and Disaggregation 4: Receive the dispatch decision from the DSO, and decom- pose it to each group according to (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5: Perform EV dispatched power disaggregation according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Real-Time Feedback and Update 6: Update the lower and upper bounds ˆxupdate∗ g,t , ˇxupdate∗ g,t , ∀g of EV aggregate power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7: Update queues ˆQg,t+1, ˇQg,t+1, ˆZg,t+1, and ˇZg,t+1 ac- cording to (4), (5), (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8: Move to the next time slot t = t+1, and repeat the above steps I-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' each with a time interval of 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To reflect the actual fluctuations in the electricity prices, we use the real-time electricity price data obtained from the PJM market [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The dynamic electricity price data profile is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For the setting of EVs, we consider 30 EVs that are divided into three groups with different allowable charging time delays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', G = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Each group has 10 EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The EVs in the same group have identical allowable time delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Particularlly, R1 = 8 hours, R2 = 6 hours, and R3 = 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, for EV battery parameters, we refer to the Nissan Leaf EV model with a battery pack of 40 kWh and a maximum charging power of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='6 kW [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Considering that the EV charging behavior is uncertain, the EVs’ arriving times are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The initial battery energy level of each EV is selected from a uniform distribution in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5] × 40 kWh randomly [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We set the required state-of-charge (SOC) 1 upon departure as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 and the maximum SOC upon departure as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The weight parameter value of V is chosen as 1000, and the value of ηg is set as 648, 540, and 756 for the three groups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Real-time electricity price profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1The SOC of an EV is the ratio between the battery energy level and the battery capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Effectiveness of the proposed method We first show how the obtained aggregate EV power flexibility region looks like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Since the power grid dispatch determined by the DSO is beyond the scope of this paper, here the dispatch ratio αt in (29) is assumed to be randomly generated within the range of [0, 1] in each time slot, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Based on the generated dispatch ratio, we apply the proposed online flexibility characterization method and real-time feedback in turns (as in Algorithm 2) to obtain the aggregate EV power flexibility region (grey area) over time for each group and the charging station as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, the power flexibility region varies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because EVs dynamically arrive and leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Randomly generated dispatch ratio αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The obtained aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' To validate the effectiveness of the proposed algorithm, dis- aggregation of the dispatched EV charging power is performed and we check if the SOC curves of EVs satisfy the charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, if the final EV SOC value can reach or exceed the EV owner’s requirement (SOC ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) upon leaving, then it means that the proposed method is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6 shows the actual EV charging SOC curves under the randomly generated dispatch ratio αt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Each curve represents an EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As we can see from the figure, all EVs’ final SOC is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='7, greater than the required value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 and less than the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The right-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6 shows the number of delayed time slots (NDTS) to reach the requirement SOC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We can find that the maximum NDST is 10 for group 1, 7 for group 2, and 12 for group 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' All of them are within their respective declared allowable charging delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This validates the proposed algorithm in providing maximum power flexibility while meeting the charging requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7 shows the queue backlog evolution of the three groups over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Taking group 1 for example, the lower and upper bound queues ˇQ1 and ˆQ1 first increase be- cause EVs continue to arrive with their charging tasks pushing JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 8 Rg=8h=48 time slots Rg=6h=36 time slots Rg=7h=42 time slots Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' EV charging SOC of each group and the number of delayed time slots needed to reach SOC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' into the queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then as time moves on, through the EV charging dispatch pdisp v,t , ∀v, ∀t determined by disaggregation, the EV SOC gradually increases and reaches the minimum charging requirement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Hence, the lower bound queue ˇQ1 becomes zero because the ˇag,t, ∀g, ∀t is set using the charging as soon as possible method to meet the minimum charging requirement as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The upper bound queue ˆQ1 is still larger than zero since the EV SOC has not reach the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 6), so there is charging flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' At the same time, since ˆQ1 is nonnegative, the delay aware upper bound queue ˆZ1 keeps growing, aiming to increase the charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The queue evolution in groups 2 and 3 can be analyzed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Queue backlog of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Performance Evaluation To show the advantage of the proposed online algorithm, two widely used benchmarks in the literature are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Benchmark 1 (B1): This is a greedy algorithm that EVs start charging at the maximum charging power upon arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Let us denote the arrival time as t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' When the EV SOC reaches the minimum charging requirement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5, the lower bound of charging power ˇpd,t turns to be zero (time: t1), and the upper bound of charging power ˆpd,t remains the maximum charging power until the EV SOC reaches the maximum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='9 (time: t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The aggregate power flexibility region for [t0, t1] is empty and for t ∈ [t1, t2] is the region between 0 and the maximum charging power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Benchmark 2 (B2): This is the offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It directly solves P1 to obtain the aggregate EV power flexibility regions over the whole time horizon by assuming known future information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though not realistic, it provides a theoretical benchmark to verify the performance of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' But it is worth noting that since it does not take into account the real-time actual dispatch strategy when calculating the aggregate flexibility, its performance may be worse than the proposed real-time feedback based method even though it is an offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8 shows the accumulated flexibility values (�t τ=1 Fτ) under the three different methods, and TABLE I summarizes the total flexibility value (�T t=1 Ft) under different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', greedy algorithm, has the worst performance and the lowest total flexibility value due to the myopic strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For B2, since it has complete future knowledge of EV behaviors and real-time electricity prices, it outperforms B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, this method is usually impossible in practice since the accurate future information is hardly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Though predictions on future uncertainty realizations may be obtained, the potential prediction errors limit B2’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In contrast, the proposed online algorithm achieves the best performance with the highest total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is owing to the fact that it runs in a online manner with real-time feedback that allows it to utilize the most recent dispatch information to update its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, compared to the offline method B2, it does not require prior knowledge of future information or forecasts, which is more practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Accumulated flexibility value under different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' TABLE I TOTAL FLEXIBILITY VALUE COMPARISON BETWEEN B1, B2, AND THE PROPOSED ALGORITHM (UNIT: USD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Methods B1 B2 Proposed Value 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='69 586 647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='21 Improvement 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2% 25% The above result is obtained under the random dispatch ratio αt (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In fact, the dispatch ratio can affect the actual charging power of each EV and further affect their aggregate power flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, it is interesting to investigate the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 9 impact of αt on the aggregate EV power flexibility (or the total power flexibility value (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, we use a uniform α over time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', αt = α, ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We change α from 0 to 1 and record the total power flexibility value in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, the total power flexibility value depends on the dispatch ratio α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Generally, a larger α leads to a larger power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' However, this increase is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' When the dispatch ratio α exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5, the total power flexibility value no longer increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In the extreme case when α = 0, the total power flexibility value is 520 USD, which is still larger than the greedy algorithm B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' In addition, it can be concluded that if the average dispatch ratio α is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2, then the proposed online algorithm more likely outperforms the offline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This demonstrates the advantage of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We also present the aggregate EV power flexibility region under different α in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As α decreases, the aggregate EV power flexibility region gradually narrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because under a low dispatch ratio, the EVs are charged at a low charging rate and more likely to fail to meet the charging requirement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' hence, the lower bound of aggregate EV power flexibility region is raised to ensure the EVs can meet the charging requirement in the remaining time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of α on total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 0 50 Power [kW] ub lb =0 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4 0 20 40 60 80 100 120 140 Time [10 min] 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 0 50 Power [kW] ub lb =0 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4 0 20 40 60 80 100 120 140 Time [10 min] 0 50 Power [kW] ub lb =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The aggregate EV power flexibility under different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Impact of Parameters According to (19), the parameter V controls the trade- off between stabilizing the queues and maximizing the total power flexibility value in the objective function (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Here, we change the value of V to investigate its impact on the total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 11, the total power flexibility value becomes larger with an increasing V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of V on the total power flexibility value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of V on the average time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 12 depicts the impact of V on the number of time slots needed for EVs to meet their required battery energy level ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We calculate the maximum/minimum/average number of time slots for the EVs in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' As seen, with the growth of V , the number of delayed time slots slightly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because a larger V means putting more emphasis on maximizing the total power flexibility, which may result in a reduced lower bound ˇxg,tof the aggregate EV power flexibility region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Consequently, the charging time needed to reach the required energy level becomes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Comparing the three groups, we can find that the time delays of groups 1 and 3 are generally longer than that of group 2, which is owing to the shorter allowable time delay Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13 shows the impact of ηg on the total flexibility value and the number of time slots needed for EVs to meet their required battery energy level ed v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We can find that a larger ηg results in a lower total power flexibility value and less number of delayed time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This is because a larger ηg forces the virtual delay-aware queue ˇZg to grow rapidly, allowing EVs to get charged quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Meanwhile, the power flexibility is sacrificed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' CONCLUSION With the proliferation of EVs, it is necessary to better utilize their charging power flexibility, making them valuable resources rather than burdens on the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' This paper proposes a real-time feedback based online aggregate EV power flexibility characterization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It can output the aggregate flexibility region for each time slot in an online manner, with a total flexibility value over time similar to the offline counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that by choosing an aggregate JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The impact of ηg on the total flexibility value and average time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' dispatch strategy within the obtained flexibility region for each time slot, the corresponding disaggregated EV control strategies allow all EVs to satisfy their charging requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Simulations demonstrate the effectiveness and benefits of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' It is worth noting that the proposed method can even outperform the offline method in some cases since it can utilize up-to-date dispatch information via real-time feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Future research may take into account the conflicting interests between the operator, aggregator, and EVs when deriving the flexibility region.' metadata={'source': 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“Optimal transactive energy trading of electric vehicle charging stations with on-site PV generation in constrained power distribution networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Smart Grid, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 1427–1440, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' APPENDIX A PROOF OF PROPOSITION 1 Let {pd,t, ∀t} be the aggregate power trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' For each time slot t ∈ T , since pd,t ∈ [ˇp∗ d,t, ˆp∗ d,t], we can define an JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 11 auxiliary coefficient: βt := ˆp∗ d,t − pd,t ˆp∗ d,t − ˇp∗ d,t ∈ [0, 1] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) so that pd,t = βtˇp∗ d,t + (1 − βt)ˆp∗ d,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, we can construct a feasible EV dispatch strategy by letting pc v,t = βtˇpc∗ v,t + (1 − βt)ˆpc∗ v,t, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2a) ev,t = βtˇec∗ v,t + (1 − βt)ˆec∗ v,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2b) for all time slots t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' We prove that it is a feasible EV dispatch strategy as follows, pd,t = βtˇp∗ d,t + (1 − βt)ˆp∗ d,t = βt � v∈V ˇpc∗ v,t + (1 − βt) � v∈V ˆpc∗ v,t = � v∈V � βtˇpc∗ v,t + (1 − βt)ˆpc∗ v,t � = � v∈V pc v,t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) Hence, constraint (1c) holds for pd,t and pc v,t, ∀v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Similarly, we can prove that constraints (1d)-(1g) are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, we have constructed a feasible EV dispatch strategy, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■ APPENDIX B PROOF OF PROPOSITION 2 Here, we use the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' If a charging request ˆag,t arrives in time slot t cannot be fulfilled on or before time slot t + ˆδg,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Then, queue ˆQg,τ > 0always holds for τ ∈ [t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t + ˆδg,max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Thus, we have I ˆ Qg,t>0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' According to delay virtual queue dynamics (12), for all τ ∈ [t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t+ ˆδg,max], we have ˆZg,τ+1 ≥ ˆZg,τ + ηg Rg − ˆxg,τ, ∀g, ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) By summing the above inequalities over τ ∈ [t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=', t + ˆδg,max], we have ˆZg,t+ˆδg,max+1 − ˆZg,t+1 ≥ ηg Rg ˆδg,max + t+ˆδg,max � τ=t+1 (−ˆxg,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2) Since ˆZg,t+ˆδg,max+1 ≤ ˆZg,max and ˆZg,t+1 ≥ 0, we have ˆZg,max ≥ ηg Rg ˆδg,max + t+ˆδg,max � τ=t+1 (−ˆxg,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) Since the charging tasks are processed in a first-in-first-out manner, and the charging request is not fulfilled by t+ˆδg,max, we have t+ˆδg,max � τ=t+1 (ˆxg,τ) < ˆQg,max (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4) Combining the above two inequalities, we obtain ˆZg,max > ηg Rg ˆδg,max − ˆQg,max, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) which implies ˆδg,max < ( ˆQg,max + ˆZg,max)Rg ηg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='6) However, this result contradicts the definition of ˆδg,max in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' Therefore, the worst case delay should be less than or equal to ˆδg,max as defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' The proof of (15) follows a similar procedure, and we omit it here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■ APPENDIX C PROOF OF PROPOSITION 3 Denote the solution of P3 by the proposed algorithm by ˆxpro g,t and ˇxpro g,t , and the optimal solution of P1 by ˆx∗ g,t and ˇx∗ g,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' According to (26), we have ∆(Θt) + V E[−F pro t |Θt] ≤ A + V E[−F pro t |Θt] + � g∈G ˆQg,tE � ˆag,t − ˆxpro g,t |Θt � + � g∈G ˇQg,tE � ˇag,t − ˇxpro g,t |Θt � + � g∈G ˆZg,tE � −ˆxpro g,t |Θt � + � g∈G ˇZg,tE � −ˇxpro g,t |Θt � , ≤ A + V E[−F ∗ t |Θt] + � g∈G ˆQg,tE � ˆag,t − ˆx∗ g,t|Θt � + � g∈G ˇQg,tE � ˇag,t − ˇx∗ g,t|Θt � + � g∈G ˆZg,tE � −ˆx∗ g,t|Θt � + � g∈G ˇZg,tE � −ˇx∗ g,t|Θt � , ≤ A + V E[−F ∗ t |Θt] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) The result is based on the fact that lim T →∞ 1 T T � t=1 E [ˆag,t − ˆxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='2) lim T →∞ 1 T T � t=1 E [ˇag,t − ˇxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='3) lim T →∞ 1 T T � t=1 E [−ˆxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='4) lim T →∞ 1 T T � t=1 E [−ˇxg,t|Θt] ≤ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='5) which is due to constraints (9b)-(9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' By summing the above inequality (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content='1) over time slots t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' , T}, we have T � t=1 V E[−F pro t ] ≤ AT + V T � t=1 E[−F ∗ t ] − E[L(ΘT +1)] + E[L(Θ1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' X, FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' 2019 12 Based on the fact that L(ΘT +1) and L(Θ1) are finite, we divide both sides of the above inequalities by V T and let T → ∞, then we have lim T →∞ 1 T T � t=1 E(−F pro t ) ≤ A V + lim T →∞ 1 T T � t=1 E(−F ∗ t ), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'} +page_content=' ■' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfqQXv/content/2301.03342v1.pdf'}