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Performance Optimization of VPP in Fast Frequency Control Ancillary Service Provision
虚拟电厂在快速频率控制辅助服务中的性能优化

Chengrong Lin , Bo Hu , Heng-Ming Tai , Changzheng Shao ,
林成荣 ，胡波 ，戴恒明 ，邵长征 ，Kaigui Xie , Yu Wang
谢开贵 , 王宇 State Key Laboratory of Power Transmission Equipment and System Security at
国家电力输变电设备与系统安全重点实验室Chongqing University, Chongqing 400044, China
重庆大学，重庆 400044，中国 Department of Electrical and Computer Engineering, University of Tulsa, Tulsa, OK
塔尔萨大学电气与计算机工程系，塔尔萨，俄克拉荷马州

Abstract 摘要

This paper proposes a computation offloading method for performance optimization in the fast frequency control ancillary service (FFCAS) provision of a virtual power plant (VPP). VPP aggregates massive demand-side resources to provide power systems with the FFCAS. It faces excessive communication and computation tasks. Edge computing addresses these issues through computation offloading. Heavy tasks can be partially offloaded to the edge to relieve the burden of the central server. Existing offloading methods, however, are not specific for the VPP. To fill the gap, first, an age of information (AoI) model is introduced to characterize the data flow of an edge-enabled VPP. Next, an AoI-based FFCAS performance evaluation model is proposed considering the impacts of communication delay, communication failures, and computational delay. Then an FFCAS performance optimization model is
本文提出了一种针对虚拟电厂（VPP）快速频率控制辅助服务（FFCAS）提供的性能优化计算卸载方法。VPP 通过聚合大量需求侧资源为电力系统提供 FFCAS，面临通信与计算任务过重的挑战。边缘计算通过计算卸载解决这些问题，可将繁重任务部分卸载至边缘以减轻中心服务器负担。然而，现有卸载方法并不针对 VPP 特性。为此，首先引入信息年龄（AoI）模型来描述边缘赋能 VPP 的数据流特征。其次，提出一个考虑通信延迟、通信失败及计算延迟影响的基于 AoI 的 FFCAS 性能评估模型。进而构建一个 FFCAS 性能优化模型，

formulated. It aims to maximize VPP's profit through communication offloading and the DSR portfolio determination. Simulation results show that the proposed method can efficiently improve VPP's profit.
制定。其旨在通过通信卸载和 DSR 组合决策来最大化 VPP 的利润。仿真结果表明，所提出的方法能有效提升 VPP 的利润。
Keywords: Virtual power plant, demand-side resources, frequency control ancillary service, edge computing.
关键词：虚拟电厂、需求侧资源、频率控制辅助服务、边缘计算。

1. Introduction 1. 引言

With the increasing share of renewable energy in power systems, system inertia decreases and operation uncertainty increases [1]. Power system inertia is related to the system frequency stability [2], while fluctuations of the renewables are subject to uncertainty. Thus, the power system frequency deviation or the rate of change of frequency (RoCoF) becomes severe when the contingency event occurs. Moreover, the frequency protective relays of synchronous generators may be triggered and a cascade failure occurs [3].
随着可再生能源在电力系统中占比的增加，系统惯性降低，运行不确定性增强[1]。电力系统惯性与系统频率稳定性密切相关[2]，而可再生能源的波动具有不确定性。因此，在发生紧急事件时，电力系统频率偏差或频率变化率（RoCoF）会变得严重。此外，同步发电机的频率保护继电器可能被触发，导致连锁故障发生[3]。
A virtual power plant (VPP) is an aggregator of demand-side resources (DSRs), such as flexible loads, energy storage, and so on [4]. A VPP can provide the fast frequency control ancillary service (FFCAS) for power systems to reduce the RoCoF and frequency deviation. The FFCAS is positioned between the system inertia response and the primary frequency control. Specifically, the EirGrid Group, a transmission system operator, requires the FFACS to be activated within 2 seconds [5]. In addition, it is reported by Guidehouse Insights [6] that the capacity of VPP is forecasted to increase from 2.8 GW in 2020 to 36.9 GW by 2029. FFCAS by VPP has become an increasingly important service for low-inertia power systems.
虚拟电厂（VPP）是需求侧资源（DSRs）的聚合体，如灵活负荷、储能等[4]。VPP 能为电力系统提供快速频率控制辅助服务（FFCAS），以降低 RoCoF 和频率偏差。FFCAS 介于系统惯性响应和一次频率控制之间。具体而言，输电系统运营商 EirGrid 集团要求 FFACS 在 2 秒内激活[5]。此外，据 Guidehouse Insights[6]报道，VPP 的容量预计将从 2020 年的 2.8 GW 增长至 2029 年的 36.9 GW。VPP 提供的 FFCAS 已成为低惯性电力系统中日益重要的服务。
VPP earns a decent profit in the ancillary service market by providing
VPP 通过提供辅助服务在市场中获得了可观的利润
high-quality FFCAS. However, the performance of FFCAS is highly dependent on information and communication technologies (ICTs) [7]. The private and public cellular networks provide communication solutions for VPP. Note that the private cellular network has limited communication bandwidth and expensive construction costs, whereas the public one is more flexible, scalable, and economic. Thus, the public cellular network is a likely choice for VPP communication , considering the geographysically widespread DSRs. In addition, FFCAS is a data and computation-intensive function for VPPs. The public cellular network may get congested due to massive DSRs' communications, and may even cause communication failures [9]. Furthermore, VPP's control center needs time to solve the DSRs portfolio optimization problem, which causes the computational delay [10]. The communication uncertainties, including communication delay, communication failures, and computational delay, could lower the response speed of VPP and devalue the FFCAS .
高质量的 FFCAS。然而，FFCAS 的性能高度依赖于信息和通信技术（ICTs）[7]。私有和公共蜂窝网络为 VPP 提供了通信解决方案。需要注意的是，私有蜂窝网络通信带宽有限且建设成本高昂，而公共蜂窝网络则更为灵活、可扩展且经济。因此，考虑到地理上广泛分布的 DSRs，公共蜂窝网络是 VPP 通信的潜在选择 。此外，FFCAS 对于 VPP 而言是一个数据和计算密集型功能。由于大量 DSRs 的通信，公共蜂窝网络可能会出现拥堵，甚至可能导致通信失败[9]。此外，VPP 的控制中心需要时间来解决 DSRs 组合优化问题，这会导致计算延迟[10]。包括通信延迟、通信失败和计算延迟在内的通信不确定性可能会降低 VPP 的响应速度，并削弱 FFCAS 的价值 。
There are two approaches to cope with the cyber uncertainties of VPP for FFCAS. The first approach includes the adaptive control method [11], linear quadratic regulator method [12], primal-dual subgradient algorithm [13], and the fuzzy-PI method [14]. These methods could enhance the resilience of VPP, but would not improve the response speed of VPP because the delay or packet loss could not be reduced. The second approach aims for efficient communication resource management of VPP and includes the methods of 5G RAN slicing [7], dynamic communication spectrum allocation [15], and interference management [16]. Since the communication resources are limited, these methods may not support the growing communication re-
应对虚拟电厂(VPP)在快速频率响应辅助服务(FFCAS)中的网络不确定性问题，存在两种策略。第一种策略涉及自适应控制方法[11]、线性二次调节器方法[12]、原始-对偶次梯度算法[13]以及模糊-PI 方法[14]。这些方法虽能增强 VPP 的韧性，但无法提升其响应速度，因为它们无法减少延迟或数据包丢失。第二种策略则聚焦于 VPP 通信资源的有效管理，涵盖了 5G 无线接入网切片技术[7]、动态通信频谱分配[15]及干扰管理[16]等方法。然而，由于通信资源有限，这些方法可能难以支撑日益增长的通信需求。
quirements for VPP. Moreover, communication resource management does not improve the computational delay.
对 VPP 的要求。此外，通信资源管理并未改善计算延迟。
Edge computing has been considered as a promising technology for VPP to improve the FFCAS performance [17]. It allows DSRs to partially offload the heavy tasks of data processing, analysis, and storage to the edge servers, which are closer to the data source instead of the remote VPP's cloud control center. Edge computing enables VPP rapid and near real-time FFCAS response. Thus, more revenue will be gained for VPP.
边缘计算已被视为提升虚拟电厂（VPP）快速频率响应与辅助服务（FFCAS）性能的有前景技术[17]。它使得分布式电源（DSR）能够将繁重的数据处理、分析及存储任务部分卸载至更接近数据源的边缘服务器，而非远程的 VPP 云控制中心。边缘计算支持 VPP 实现快速且近乎实时的 FFCAS 响应，从而为 VPP 带来更多收益。
The application of edge computing to VPP has been addressed. The cloud-edge-end collaboration architecture was used for risk-averse optimal scheduling [18] and flexibility evaluation [19]. Both reports transformed the centralized optimization problems into distributed ones. Edge computing is also used to provide timely communication for energy trading [20]. However, these studies only discussed the role of edge computing in VPP in a qualitative manner, without considering the computation offloading for edgeenabled VPP.
边缘计算在虚拟电厂（VPP）中的应用已得到探讨。采用云-边-端协同架构进行风险规避型优化调度[18]及灵活性评估[19]，两项研究均将集中式优化问题转化为分布式问题。边缘计算还被用于为能源交易提供及时通信[20]。然而，这些研究仅定性地讨论了边缘计算在 VPP 中的作用，未涉及边缘赋能 VPP 的计算卸载问题。
This paper intends to investigate the efficient computation offloading method for edge-enabled VPP to gain more profit. How to tailor computation offloading to the requirements of VPP will be discussed quantitatively. The tradeoff between the economic benefits and the operating cost of edge computing will also be studied.
本文旨在探讨边缘计算支持下的虚拟电厂高效计算卸载方法，以实现更大收益。将定量讨论如何针对虚拟电厂需求定制计算卸载策略。同时，研究经济效益与边缘计算运营成本之间的权衡关系。
Contributions of this paper are
本文的贡献在于
  1. Develop an age of information (AoI) model to characterize the data flow of an edge-enabled VPP. AoI has been widely discussed in the communication field in recent years [21, 22], but few works have applied AoI
    构建一个信息年龄（Age of Information, AoI）模型，以表征边缘计算支持下的虚拟电厂数据流。近年来，AoI 在通信领域得到了广泛讨论[21, 22]，但将其应用于实际工作中的研究尚不多见。
    to VPP. This paper explores the use of AoI to establish the relationship between the computation offloading decisions and the communication uncertainties.
    本文探讨了利用时效性信息（AoI）来建立计算卸载决策与通信不确定性之间关系的方法。
  2. Propose an AoI-based FFCAS performance evaluation model considering the impacts of communication delay, communication failures, and computational delay. The benefits of the computation offloading are quantified as the improvement of FFCAS performance.
    提出一种基于年龄信息（AoI）的 FFCAS 性能评估模型，综合考虑通信延迟、通信故障及计算延迟的影响。计算卸载的益处被量化为 FFCAS 性能的提升。
  3. Formulate an FFCAS performance optimization model. It aims to maximize VPP's profit through communication offloading and the DSR portfolio determination.
    制定一个 FFCAS 性能优化模型。其目标是通过通信卸载和 DSR 组合决策来最大化 VPP 的利润。
The remainder of this paper is organized as follows. Section 2 introduces the edge-enabled VPP FFCAS control architecture. Section 3 formulates the AoI-based FFCAS performance evaluation model. The FFCAS performance optimization model is presented in Section 4. Section 5 presents the solution algorithm. Case studies are given in Section 6 to discuss the benefits of edge computing. Section 7 concludes this paper.
本文其余部分的组织结构如下。第 2 节介绍边缘赋能的 VPP FFCAS 控制架构。第 3 节构建基于 AoI 的 FFCAS 性能评估模型。第 4 节提出 FFCAS 性能优化模型。第 5 节展示求解算法。第 6 节通过案例研究探讨边缘计算的益处。第 7 节总结全文。

2. Edge Enabled VPP FFCAS Architecture
2. 边缘计算支持的虚拟电厂 FFCAS 架构

VPP is generally composed of two fundamental components: DSRs and ICTs, as shown in Figure 1. DSRs refer to the flexible loads and energy storage [1]. Flexible loads provide FFCAS in the on-off status. Energy storage such as the batteries of electric vehicles, flywheel energy storage, and battery energy storage, can support power system frequency continuously. While distributed generations can respond fast, FFCAS provided by the distributed
VPP 通常由两个基本组成部分构成：DSR 和 ICT，如图 1 所示。DSR 指的是灵活负荷和能量存储[1]。灵活负荷在开关状态下提供 FFCAS。能量存储，如电动汽车的电池、飞轮能量存储和电池能量存储，能够持续支持电力系统频率。尽管分布式发电能快速响应，但由分布式提供的 FFCAS
Figure 1: VPP components and control architecture for FFCAS.
图 1：FFCAS 的 VPP 组件及控制架构。
generation may instigate the required energy curtailment, which brings economic concerns for the power system operators and the VPP owner [23, 24].
发电量可能引发所需的能源削减，这对电力系统运营商和虚拟电厂所有者带来经济顾虑[23, 24]。
ICTs mainly include the communication network and edge servers. The communication network connects DSRs and coordinates the energy flow in a bidirectional way. Edge servers are geographically close to the DSRs and handle the heavy tasks of data processing, analysis, and storage. This improves the response speed of FFCAS for more VPP revenue gain.
信息通信技术主要包括通信网络和边缘服务器。通信网络连接分布式储能资源，并以双向方式协调能量流动。边缘服务器地理位置靠近分布式储能资源，负责处理数据处理、分析和存储等繁重任务。这提升了虚拟电厂辅助服务市场的响应速度，从而增加更多收益。
The workflow of FFCAS is also depicted in Figure 1. When a VPP receives the regulation instructions from a power system operator, it needs to collect DSRs' real-time status information and produces the DSR portfolio. Specifically, the cloud server and the edge servers receive the DSRs' status messages, such as the current power availability of energy storage and electric vehicles, the operation status of flexible loads, etc. Then, the VPP control center conducts an optimization problem, which aims to follow the regulation
FFCAS 的工作流程也在图 1 中展示。当虚拟电厂（VPP）接收到电力系统运营商的调节指令时，它需要收集需求侧响应（DSR）的实时状态信息并生成 DSR 组合。具体而言，云服务器和边缘服务器接收 DSR 的状态消息，例如当前储能和电动汽车的电力可用性、灵活负荷的运行状态等。随后，VPP 控制中心进行优化问题处理，旨在遵循调节指令。
instructions with a small operating cost and produce the control signals for the DSRs. The optimization problem can be solved by the VPP cloud server and the edge servers.
具有低操作成本的指令，并生成 DSR 的控制信号。优化问题可由 VPP 云服务器和边缘服务器解决。
VPP can provide the up and down regulation services. Up-regulation service supports the power system frequency level by turning off the flexible load, decreasing the charging power, or increasing the discharging power of the energy storage. Down-regulation service is for turning on the load, decreasing the discharging power, or increasing the charging power of the energy storage.
VPP 能够提供上调和下调调节服务。上调服务通过关闭灵活负荷、减少充电功率或增加储能放电功率来支撑电力系统频率水平。下调服务则是通过开启负荷、减少储能放电功率或增加充电功率来实现。

3. Performance Evaluation for DSR
3. 动态源路由协议性能评估

The performance score evaluates the response precision and the response speed of a VPP under scenario such that
性能评分 评估了虚拟电厂在场景 下的响应精度和响应速度，以达到 的效果
where is the regulation signal for the DSR under scenario at time ,
其中 是场景 下 DSR 在时间 的调节信号，
is the actual power output of the is the set of DSRs, is the settlement period, and represents the mathematical expectation over communication uncertainties. It is clear from (1) that the performance score ranges in . A large performance score means the FFCAS is delivered with a good performance.
表示实际功率输出， 是 DSR 集合， 代表结算周期，而 表示通信不确定性的数学期望。从（1）中可以明显看出，性能评分范围在 内。较高的性能评分意味着 FFCAS 的执行表现良好。

3.1. Computation Offloading and Communication Quality
3.1. 计算卸载与通信质量

Figure 2 illustrates the role of edge computing in a VPP. It can be seen from Figure 2 that AoI is observed by the VPP cloud control center. Smart
图 2 展示了边缘计算在虚拟电厂（VPP）中的作用。从图 2 中可以看出，VPP 云控制中心对活动状态指数（AoI）进行监测。
Figure 2: Computation offloading for an edge-assisted VPP. refers to the computational ratio and is the total data size related to the edge server. Computational tasks with a data size of can be offloaded to the edge to relieve the burden of the VPP cloud control center.
图 2：边缘辅助 VPP 的计算卸载。 表示计算比率， 是与 边缘服务器相关的总数据大小。数据大小为 的计算任务可以卸载到边缘，以减轻 VPP 云控制中心的负担。
sensors sample every DSR status and send it to edge servers. Suppose that the edge server manages DSRs and each DSR produces bits raw status data. Then the total data size reaches bits. Let the computation offloading ratio be . The edge server can process bits of raw data and the rest of the raw data will be sent to the cloud control center. Then the VPP cloud control center will cooperate with the edge servers to determine the DSR portfolio. This process is called computation offloading.
传感器对每个 DSR 状态进行采样，并将其发送至边缘服务器。假设 边缘服务器管理着 个 DSR，每个 DSR 产生 比特的原始状态数据。那么，总数据量达到 比特。设计算卸载比率为 。 边缘服务器能处理 比特的原始数据，剩余的原始数据将被发送到云控制中心。随后，VPP 云控制中心将与边缘服务器协同，确定 DSR 组合。这一过程称为计算卸载。
AoI is the time elapsed from the latest DSRs' status data received by a VPP's control center. It is used to characterize the decision-making process of a VPP. The AoI at the control center, denoted as , can be expressed as
AoI 是指从虚拟电厂（VPP）控制中心接收到最新分布式状态报告（DSRs）状态数据所经过的时间。它用于表征 VPP 的决策过程。控制中心的 AoI，记作 ，可以表示为 。
where is the time that smart sensors sample DSRs' status, is the index of the control periods, and is the time that the DSR portfolio is determined.
其中， 表示智能传感器采集 DSR 状态的时间， 为控制周期的索引， 则是确定 DSR 组合的时间。
Suppose that AoI is observed at the start of a control period and the initial value of AoI is set as . Figure 3 shows the evolution of AoI and Table I presents the nomenclature of the related variables. The first control period in Figure 3 shows the decision-making process without communication failures and the second control period involves the situation with those.
假设在控制周期开始时观测到时效性信息（AoI），并将 AoI 的初始值设定为 。图 3 展示了 AoI 的演变过程，表 I 列出了相关变量的命名。图 3 中的第一个控制周期展示了无通信故障情况下的决策过程，而第二个控制周期则涉及存在通信故障的情形。
It is clear from Figure 3 that the AoI is a piecewise function of time. AoI will not stop rising until the DSR portfolio is determined at time , . When one DSR's status data fails to be decoded by the VPP cloud control center at , the status data will be retransmitted based on the standard automatic repeat request policy [26]. If the VPP cloud control center can correctly decode the data and make the regulation signals, then AoI falls to from .
从图 3 可以明显看出，AoI 是时间的分段函数。AoI 将持续上升，直至 DSR 组合在时间 ， 确定。当某 DSR 的状态数据在 时刻未能被 VPP 云控制中心解码时，将依据标准自动重传请求策略[26]重新传输该状态数据。若 VPP 云控制中心能正确解码数据并生成调控信号，则 AoI 将从 降至 。
The response time of a VPP in the control period can be expressed as
在 控制周期内，VPP 的响应时间可表示为
where is the peak AoI such that
其中 为峰值年龄信息，使得

3.2. Impact of Communication Uncertainties
3.2. 沟通不确定性影响

The performance score (1) can be rewritten as
性能评分（1）可重写为
where is the control number in one settlement period.
其中 表示一个结算周期内的控制编号。
In practice, the settlement period is far greater than the control period . For example, the settlement period was in the range of an hour and the
实际上，结算周期 远大于控制周期 。例如，结算周期通常在小时范围内，而
Figure 3: Evolution of the AoI for an edge-assisted VPP.
图 3：边缘辅助虚拟电厂的 AoI 演化图
Table 1: Nomenclature of variables shown in Figure 3
表 1：图 3 中变量的命名规则
The time that smart sensors sample DSRs' status in the control period
智能传感器在控制周期内对 DSR 状态的采样时间
The time that the DSR portfolio is determined
DSR 投资组合确定的时间
The time that DSRs' status data arrives at the edge server
DSRs 状态数据到达边缘服务器的时间
The time that DSRs' status data arrives at the VPP cloud control center
DSRs 状态数据首次到达 VPP 云控制中心的时间
Waiting time for the next round of DSRs' status updating
下一轮 DSR 状态更新的等待时间
Transmission time of a DSR's status data from the smart sensor to the edge server
智能传感器向边缘服务器传输 DSR 状态数据的传输时间
Computational time for the DSR portfolio determination
DSR 投资组合确定的计算时间
The transmission time of DSRs' status from the edge server to the VPP cloud control center
从边缘服务器到 VPP 云控制中心的 DSR 状态传输时间
Response time of a VPP
虚拟电厂的响应时间
Control period 控制周期
control period was two seconds as considered in the PJM market in 2020 [27]. Thus, the performance score can be estimated as
控制周期为两秒，这是 2020 年 PJM 市场所考虑的设定[27]。因此，性能评分可据此进行估算。
It is clear from (6) that the FFCAS performance of a VPP depends on the response accuracy and response speed. Thus, according to (3), the average peak AoI, denoted as , should be proposed first.
从（6）中可以明显看出，虚拟电厂（VPP）的快速频率响应能力（FFCAS）取决于响应的准确性和速度。因此，根据（3），应首先提出平均峰值年龄信息（AoI），记为 。
This paper considers the VPP implemented in the urban grid [28]. Commercial and industrial buildings generally make the wireless communication signal suffer from small-scale fade and large-scale loss. Assume that the communication channels are independent and identically distributed, and they are constant for each information update. Then, the quasi-static Rayleigh channel model is a plausible model for a VPP. The channel gain from the edge server to the central server, denoted as , satisfies that [26]
本文考虑了城市电网中实施的虚拟电厂（VPP）[28]。商业和工业建筑通常会导致无线通信信号遭受小尺度衰落和大尺度损耗。假设通信信道独立同分布，并且对于每次信息更新保持恒定。那么，准静态瑞利信道模型是 VPP 的一个合理模型。从边缘服务器到中央服务器的信道增益，记为 ，满足[26]
where is the distance between the transmitter and the receiver, is the path loss exponent, and is the channel power gain at the reference distance. denotes the short-term fading. It is a random variable governed by the exponential distribution with unit mean. Only statistical channel state information is known at the smart sensor.
其中， 表示发射机与接收机之间的距离， 是路径损耗指数， 为参考距离处的信道功率增益。 表示短期衰落，它是一个服从均值为 1 的指数分布的随机变量。智能传感器仅知晓信道的统计状态信息。
The communication failure rate (which is also called the packet decoding error rate) is associated with the quasi-static Rayleigh channel model. It can
通信失败率 （亦称为数据包解码错误率）与准静态瑞利信道模型相关。
be estimated by
通过 进行估计
where is a function such that denotes the packet block length and is the signal-to-noise ratio (SNR) such that is the communication power of the transmitter. is the receiver noise power.
其中 是一个函数，使得 表示数据包块长度，而 是信噪比（SNR），其中 表示发射机的通信功率。 是接收机的噪声功率。
The computational delay for DSR portfolio determination can be expressed as
DSR 投资组合确定的计算延迟可以表示为
where is the complexity coefficient of the DSR portfolio determination problem, represents the CPU processing speed of the edge server, is the CPU processing speed of the central server.
其中， 表示 DSR 投资组合确定问题的复杂度系数， 代表边缘服务器的 CPU 处理速度， 为中央服务器的 CPU 处理速度。
The transmission time from a DSR to the edge server can be expressed as
从 DSR 到边缘服务器的传输时间 可以表示为
where is the channel bandwidth between the DSR and the edge server.
其中， 表示 DSR 与边缘服务器之间的信道带宽。
The transmission time from the edge server to the VPP cloud control center can be expressed as
边缘服务器到 VPP 云控制中心的传输时间 可以表示为
where is the channel bandwidth between the edge server and the VPP cloud control center.
其中， 表示边缘服务器与 VPP 云控制中心之间的信道带宽。
Theorem 1 estimates the average peak AoI for an edge-enabled VPP.
定理 1 估计了边缘计算支持的虚拟电厂的平均峰值时效性。
Theorem 1. Consider the edge-enabled VPP with the quasi-static Rayleigh channel. The average peak AoI is given as
定理 1. 考虑具有准静态瑞利信道的边缘赋能虚拟电厂。平均峰值时效性表示为
where is the expected number of the information transmission such that
其中 表示预期信息传输次数，使得
Proof. See Appendix A.
证明。见附录 A。
It can be seen from (12) that the average peak AoI is associated with the offloading ratio . When the offloading ratio gets close to one, the expected information transmission number decreases to one, as . Then, the average peak AoI declines, and the performance score rises.
从(12)可以看出，平均峰值 AoI 与卸载比 相关。当卸载比 接近 1 时，预期信息传输次数降至 1， 随着 。随后，平均峰值 AoI 下降，性能得分上升。

4. Performance Optimization in FFCAS Provision
4. FFCAS 供应中的性能优化

4.1. Performance-Dependent Profit Maximation
4.1. 基于绩效的利润最大化

A VPP's profit is equal to the difference between the revenue gained from the system operator and the operating cost of a VPP. The objective function of the profit maximation problem can be expressed as
VPP 的利润等于从系统运营商获得的收入与 VPP 运营成本之间的差额。利润最大化问题的目标函数可以表示为
where is the revenue of a VPP under scenario at time is the associated cost related to a VPP's reserve and edging computing, and is the occurrence probability of scenario . is the set of all scenarios, and
其中， 表示在情景 下 VPP 在时间 的收益， 是与 VPP 备用和边缘计算相关的成本， 是情景 发生的概率。 是所有情景的集合，
is the set of regulation time. The scenario denotes the frequency regulation instructions sent by the power system operator. The scenario-based analysis method is employed to estimate the revenue and the cost.
是规定的调节时间集合。该场景表示电力系统运营商发送的频率调节指令。采用基于场景的分析方法来估算收益和成本。
A VPP's revenue is the product of the performance score and capacity payment such that
VPP 的收入是性能评分 与容量支付 的乘积，以此构成
where is the revenue coefficient under scenario and is regulation signal for the DSR under scenario at time . The operating cost of a VPP is the sum of the DSR-related cost and the edge computing cost such that
其中， 表示情景 下的收益系数， 为情景 下 DSR 在时刻 的调控信号。VPP 的运行成本 是 DSR 相关成本 与边缘计算成本 之和，使得
The operating cost of the flexible load and the energy storage is expressed by (21) and (30). The cost of edge computing involves many factors, such as the data, location, and expertise. This paper employs the Platform-asa-Service option of edge computing and the pricing option provided by Microsoft Azure. The operating cost of the edge server mainly depends on the amount of data such that [30]
柔性负荷与储能的运行成本分别由(21)和(30)表示。边缘计算的成本涉及众多因素，如数据量、地理位置及专业技能等。本文采用边缘计算的 Platform-as-a-Service 选项及微软 Azure 提供的定价方案。 边缘服务器的运行成本主要取决于数据量，如[30]所示。
where is the cost coefficient for edge computing.
其中 表示边缘计算的成本系数。

4.2. Flexible Load Model
4.2. 灵活负荷模型

Consider the aggregated flexible load for providing up-regulation service. The down-regulation process is similar and can be easily extended. Suppose
考虑用于提供上调服务的聚合柔性负荷。下调过程类似，可轻松扩展。假设
the submitted up-regulation reserve of the aggregated flexible load under scenario is . Note that only the load which is turned on first can be turned down for providing the up-regulation service. Then, the reserve for up-regulation service can be expressed as
在场景 下，聚合灵活负荷的提交上调备用为 。需要注意的是，仅当负荷首先被开启时，才能通过下调来提供上调服务。因此，上调服务的备用可以表示为
where is the operation status variable and is binary. means that the load is turned on at time ; otherwise, is the rated power of load . is the set of the flexible loads.
其中， 表示操作状态变量，为二进制形式。 意味着在时间 时负载 被开启；否则， 为负载 的额定功率。 是柔性负载的集合。
If load is off at the initial time, i.e., , it cannot provide the up-regulation at any time . Thus, constraint (19) holds.
若负载 在初始时刻即关闭，即 ，则它无法在任何时刻 提供上调功率。因此，约束(19)成立。
To avoid the over-frequent on-off switches, suppose that each load can only be turned on once in one settlement period. Then, we have
为避免过于频繁的开关操作，假设每个负载在一个结算周期内只能开启一次。于是，我们有
The unit regulation cost of the flexible load, which is usually associated with the response power and the response holding time , can be expressed as
柔性负荷的单位调节成本，通常与响应功率 及响应保持时间 相关，可表示为
where is the cost coefficient of the flexible load and is the response holding time of load such that
其中， 表示灵活负荷 的成本系数，而 则为负荷 的响应保持时间，使得

4.3. Energy Storage Model
4.3. 能量存储模型

The up-regulation reserve of the energy storage can be obtained by increasing the discharging power or decreasing the charging power of energy storage. The down-regulation reserve can be determined similarly. Denote the base charging and discharging powers as and , respectively. The available regulation reserve is constrained by the energy storage's charging/discharging limits, which can be expressed as
能量存储的上调备用 可以通过增加放电功率或减少充电功率来获得。下调备用 可以类似地确定。分别将基础充电和放电功率记为 和 。可用的调节备用受限于能量存储的充放电极限，这可以表示为
where and are the rated discharging and charging power, respectively. and are the charging and discharging state variables of energy storage satisfying
其中， 和 分别为额定放电和充电功率， 和 为储能 的充电和放电状态变量，满足
Energy storage is also constrained by its state of charging. The real-time state of charging of the energy storage is defined as the ratio between its real-time energy and its rated energy capacity ; that is,
储能状态也受其充电状态的限制。储能设备 的实时充电状态 定义为其实时能量 与其额定能量容量 之比；即，
To avoid the over-discharge or over-charge behavior and improve the life cycle, each energy storage should operate in an allowable state of the charging interval , where and are the lower bound and the upper bound of the state of charging, respectively,
为避免过度放电或过度充电行为并延长使用寿命，每个储能单元 应在充电区间的允许状态下运行，其中 和 分别为充电状态的下限和上限
The real-time energy of energy storage is determined by its power and its energy level before regulation. The charging and discharging efficiencies of the energy storage is and . When energy storage provides the down-regulation service by either decreasing the charging power or increasing the discharging power, its energy can be expressed as
储能 的实时能量由其功率及其调控前的能量水平 决定。储能 的充放电效率分别为 和 。当储能 通过降低充电功率或增加放电功率提供下调服务时，其能量 可表示为
or 或者
Consider the aggregation of electric vehicles as a special type of energy storage when the electric vehicle stays at the charge station longer than one FFCAS market settlement period . Then, the constraints for the electric vehicle are similar to (23)-(29) with the following parameter settings.
考虑电动汽车在充电站停留时间超过一个 FFCAS 市场结算周期时，将其聚合视为一种特殊类型的储能。此时，电动汽车的约束条件类似于(23)-(29)，并采用以下参数设置。
The electric vehicle is generally charged at the rated power. This means that the electric vehicle's charging power in (23)-(29) is equal to its rated power. Then, the electric vehicle's down-regulation reserve is set as 0 when the electric vehicle is charging. On the other hand, the electric vehicle's owner may not be willing to, or the charge station may not be able to, discharge the vehicle's energy to the power system. Then, both the discharge power limit and the actual discharge power for the electric vehicle are set as 0 . Under this circumstance, electric vehicles provide FFCAS by starting or interrupting charging.
电动汽车通常以额定功率进行充电，这意味着在(23)-(29)中，电动汽车的充电功率等于其额定功率。因此，当电动汽车充电时，其下调储备设为 0。另一方面，电动汽车车主可能不愿意，或者充电站可能无法将车辆能量释放到电力系统中。因此，电动汽车的放电功率限制和实际放电功率均设为 0。在这种情况下，电动汽车通过启动或中断充电来提供 FFCAS。
The operating cost of energy storage for FFCAS mainly contains the life expectancy loss and the reserve capacity cost [31]
FFCAS 储能的运行成本主要包括寿命预期损失和备用容量成本[31]
where and are the cost coefficients of energy storage.
其中， 和 分别是储能的成本系数。

5. FFCAS Performance Optimization Algorithm
5. FFCAS 性能优化算法

This study considers the VPP to provide the FFCAS for the power system to gain profit from the system operator. Edge computing is used to improve the FFCAS performance and the VPP revenue, but it introduces additional operating costs. This problem can be formulated as an FFCAS performance optimization problem for VPP profit maximation:
本研究认为，虚拟电厂（VPP）为电力系统提供辅助服务（FFCAS），以从系统运营商处获得收益。采用边缘计算技术以提升 FFCAS 性能及 VPP 收益，但同时也引入了额外的运营成本。此问题可被构建为针对 VPP 利润最大化的 FFCAS 性能优化问题：
The decision variables of the optimization problem include the computation offloading ratio , the operation status variable of the flexible load , and the dispatched power of energy storage . This optimization problem is considered as a large-scale mixed-integer nonlinear programming problem with many scenarios. The number of DSRs is in the order of . It is very expensive to solve this problem directly through conventional optimization algorithms.
优化问题的决策变量包括计算卸载比率 、柔性负荷的运行状态变量 以及储能的调度功率 。该优化问题被视为具有多场景的大规模混合整数非线性规划问题。DSR 的数量级为 。直接通过常规优化算法求解此问题成本极高。
The problem formulated by is composed of two parts: the DSR regulation performance evaluation model (1) - (13) and the VPP FFCAS model (14) - (30). The peak average AoI calculation model gives the average peak AoI based on a given offloading ratio . The computation offloading ratio can be regarded as the input of the DSR portfolio determination model and is used to estimate a VPP's performance score and determine the DSR portfolio.
提出的问题由两部分组成：DSR 调控性能评估模型（1）-（13）与 VPP FFCAS 模型（14）-（30）。峰值平均 AoI 计算模型根据给定的卸载比率 得出平均峰值 AoI 。计算卸载比率可视为 DSR 组合确定模型的输入，用于评估 VPP 性能得分 并确定 DSR 组合。
Figure 4: Flowchart of the proposed algorithm.
图 4：所提算法的流程图。
Figure 4 shows the flowchart of the proposed algorithm, which contains three parts. The first part is to calculate the peak average AoI. The second part is to determine the DSR portfolio. The third part is to set the computation offloading ratio to the one with the largest VPP profit.
图 4 展示了所提算法的流程图，包含三个部分。第一部分是计算峰值平均 AoI。第二部分是确定 DSR 组合。第三部分是将计算卸载比率设定为能带来最大 VPP 利润的那个。
Step 2.1 is to calculate the dispatchable sets of flexible loads and energy storages for providing the up-regulation service, which can be
步骤 2.1 是计算灵活负荷 和储能 的可调度集合，以提供上调服务，这可以
defined as 定义为
It can be seen from (31) and (32) that DSR can provide up-regulation service if and only if . If the power system operator requires the up-regulation service, only the dispatchable set associated with the up-regulation service is needed. The holding time of the flexible load can be obtained according to the statistical data based on the first-on-first-off rule [7]. This rule ensures that the first turned-on flexible load should be turned off first. The holding time is the average value of the time interval between the up-regulation instruction and the down-regulation instruction.
从(31)和(32)可以看出，DSR 仅当 时能提供上调服务。若电力系统运营商需要上调服务，仅需调度与上调服务相关联的可调度集合。根据先开先关原则[7]的统计数据，可获得灵活负荷的持续时间。该原则确保最先开启的灵活负荷应优先关闭。持续时间是上调指令与下调指令之间时间间隔的平均值。
Step 2.2 calculates the VPP performance score and DSRs' profit coefficients. The energy storage needs to support the flexible load so that the total output can follow the regulation instructions. Thus, the profit coefficient for the flexible load is defined as the average profit gained from the system operator in unit time. Similarly, the profit coefficient for the energy storage is defined as the expected optimal profit in unit time.
步骤 2.2 计算 VPP 性能评分及 DSRs 的利润系数。储能需支撑灵活负荷，以使总输出能遵循调控指令。因此，灵活负荷 的利润系数定义为单位时间内从系统运营商处获得的平均利润。同理，储能 的利润系数定义为单位时间内的预期最优利润。
The profit coefficient is obtained by Theorem 2 .
利润系数由定理 2 得出。
Theorem 2. Consider the DSR model described by (14)-(30). The profit coefficient for the flexible load and the energy storage can be estimated by
定理 2. 考虑由(14)-(30)描述的 DSR 模型。灵活负荷与储能的利润系数可通过以下方法估算：
where is the dispatched power of the energy storage,
其中 表示储能的调度功率，
where is the rated power.
其中 为额定功率。
Proof. See Appendix B.
证明。见附录 B。
Steps 2.3 and 2.4 select the DSR portfolio and modify the output of DSRs. Step 6 computes the VPP's operating cost (16) and the revenue (15). Step 2.5 finds the optimal computation offloading ratio. Finally, output the optimal computation offloading ratio and the DSR portfolio.
步骤 2.3 和 2.4 选择 DSR 投资组合并调整 DSR 的输出。步骤 6 计算 VPP 的运营成本（16）和收入（15）。步骤 2.5 确定最优计算卸载比率。最终，输出最优计算卸载比率和 DSR 投资组合。

6. Case Studies 6. 案例研究

This section evaluates the economic benefits of edge-enabled VPP for FFCAS under various operating conditions such as the offloading ratios, cost coefficients, settlement periods, computing power, control periods, and message sizes. The average peak AoI and performance scores are also analyzed. The case studies consider two types of VPPs: the PJM and the Fingrid cases. The PJM case is a conventional power system with low integration of the renewables and the Fingrid case is with high penetration of renewables.
本节评估了在不同运行条件下，边缘计算支持的虚拟电厂（VPP）对前瞻性频率和辅助服务市场（FFCAS）的经济效益，这些条件包括卸载比率、成本系数、结算周期、计算能力、控制周期及消息大小。同时，还分析了平均峰值年龄指标（AoI）和性能评分。案例研究涉及两种类型的 VPP：PJM 案例和 Fingrid 案例。PJM 案例代表传统电力系统，可再生能源整合程度较低；而 Fingrid 案例则展示了高比例可再生能源渗透的情况。

6.1. System and Scenario Setting
6.1. 系统与场景设定

The communication parameters of VPP used in this study are adopted from [32] and [33] and listed in Table 2. For example, the path-loss exponent is set as 2 , the channel power gain as , the channel bandwidth as 20 MHz ,
本研究采用的 VPP 通信参数取自[32]和[33]，并列于表 2 中。例如，路径损耗指数设为 2，信道功率增益为 ，信道带宽为 20 MHz，
Table 2: Communication Parameters for VPP
表 2：VPP 通信参数
Communication Parameters
通信参数		 Value 价值
Path-loss exponent  路径损耗指数 2
Channel power gain at reference distance
参考距离处的信道功率增益
Channel bandwidth  信道带宽 20 MHz 20 兆赫
Noise power  噪声功率
Distance between transmitter and the receiver
发射机与接收机之间的距离 		100 m 100 米
Processing speed of the edge server
边缘服务器 的处理速度		 cycles/bit  周期/比特
Processing speed of the central server
中央服务器的处理速度 		cycles   循环
Packet blocklength  数据包块长度 500 channel 500 频道
Transmit power  传输功率 0.02 W 0.02 瓦
Number of CPU circles
CPU 周期数
Packet size 数据包大小 0.01 MB
and the noise power as . The processing speed of the edge server and that of the central server are cycles/bit and cycles/bit, respectively. The transmit power is assumed to be , the CPU runs cycles bit and the packet size is . The edge computing's operation cost is set as message.
噪声功率为 。边缘服务器和中心服务器的处理速度分别为 周期/比特和 周期/比特。假设传输功率为 ，CPU 运行 周期 比特，数据包大小为 。边缘计算的操作成本设定为 消息。
The DSR parameters are given in Table 3. The VPP contains 8612 DSRs, including 4000 residential users, 40 commercial buildings, 10 industrial customers, 100 small energy storage, 12 utility energy storage, and 4000 electric vehicles. Suppose that each residential user provides 1 electric devices for VPP, each commercial building has 10 floors, and every industrial customer
DSR 参数列于表 3 中。VPP 包含 8612 个 DSR，其中包括 4000 户居民用户、40 座商业建筑、10 个工业客户、100 个小规模储能设备、12 个公用事业储能设施及 4000 辆电动汽车。假设每个居民用户为 VPP 提供 1 台电气设备，每座商业建筑有 10 层，而每个工业客户
has 10 workshops for VPP.
拥有 10 个 VPP 车间。
The DSRs' power and energy operation ranges are as below. The rated power for residential load is determined in , the one for commercial load is , and the one for industrial load is . Small ES's rated power falls in and utility energy storage's rated power is in . The electric vehicle's rated power is larger than 5 kW and not more than 20 kW . The rated energy of the small energy storage is evenly distributed in the range of , that of the utility energy storage in , and the electric vehicle in . The lower bound of real-time energy of energy storage and electric vehicle is one in ten of their rated energy. The initial power of energy storage is evenly distributed in . The initial energy of energy storage and electric vehicles varies in their allowable energy interval. Both the charging and discharging coefficients and of the energy storage are set at . Under the setting, VPP's initial up-regulation reserve is 141.28 MW and the initial down-regulation reserve is 72.5 MW.
DSR 的功率与能量操作范围如下。居民负荷的额定功率在 确定，商业负荷为 ，工业负荷为 。小型 ES 的额定功率位于 ，公用事业储能的额定功率在 。电动汽车的额定功率大于 5 kW 且不超过 20 kW。小型储能的额定能量均匀分布在 范围内，公用事业储能的额定能量在 ，电动汽车则在 。储能与电动汽车的实时能量下限为其额定能量的十分之一。储能的初始功率均匀分布于 。储能与电动汽车的初始能量在其允许能量区间内变化。储能的充放电系数 和 均设定为 。在此设定下，VPP 的初始上调备用容量为 141.28 MW，初始下调备用容量为 72.5 MW。
Now consider the cost coefficients of DSRs. Flexible load's cost coefficient is set in the range of . Small energy storage's cost coefficients are set as and . The cost coefficients of utility energy storage are and . For the electric vehicle, the cost coefficients are and .
现在考虑需求侧响应的成本系数。灵活负荷的成本系数 设定在 范围内。小型储能的成本系数设定为 和 。公用事业储能的成本系数为 和 。对于电动汽车，成本系数为 和 。
Suppose that the ancillary service market settlement period is one hour, i.e., . The control period is set as 2 s . The representative frequency regulation instruction of the power system operator is obtained
假设辅助服务市场结算周期为 1 小时，即 。控制周期 设定为 2 秒。获取电力系统运营商的代表性频率调节指令。
Table 3: DSR Parameters for VPP
表 3：VPP 的 DSR 参数
DSR Type DSR 类型 Number 数字
 源文本：Power 翻译文本：功率
Power
Cost coefficient 成本系数
or   或
Residential load 居民负荷
Commercial load 商业负荷
Industrial load 工业负荷
Small energy storage 微型储能 100
Utility energy storage 实用能源储存 12
Electric vehicle 电动汽车 4000
from the historical Reg-D regulation data of PJM [27]. The Reg-D data from October to December 2020 is divided into one-hour pieces. Then there are scenarios. Each scenario contains regulation instructions. Note that the statistical value of the holding time only depends on the scenario. The average holding time is 432 s .
来自 PJM [27]的历史 Reg-D 监管数据。2020 年 10 月至 12 月的 Reg-D 数据被划分为每小时片段。随后存在 种情景。每种情景包含 条监管指令。需要注意的是，持有时间的统计值仅取决于情景。平均持有时间 为 432 秒。
The revenue coefficient is related to the share of renewables in the power system. In the PJM case, the renewables share is between and of the fossil fuels and the revenue coefficient is about MW. In the Fingrid case, the renewables share is about of the fossil fuels and the revenue coefficient is . Since the Fingrid power system has lower system inertia, FFCAS is more important. The revenue coefficient for the up-regulation is the same as that of the down-regulation.
收入系数与电力系统中可再生能源的份额相关。在 PJM 案例中，可再生能源占比介于 与 之间，收入系数约为 兆瓦。而在 Fingrid 案例中，可再生能源占比约为 ，收入系数为 。由于 Fingrid 电力系统的系统惯性较低，FFCAS 显得更为重要。上调与下调的收益系数相同。

6.2. Performance under Different Offloading Ratios
6.2. 不同卸载比例下的性能表现

Figure 5 shows the average peak AoI and the performance score with different offloading ratios. It can be seen from Figure 5(a) that the average
图 5 展示了不同卸载比率下的平均峰值 AoI 和性能得分。从图 5(a)可以看出，平均
Figure 5: Performance of edge computing enabled VPP under different offloading ratios. (a) Average peak AoI. (b) Performance score.
图 5：不同卸载比率下边缘计算支持的虚拟电厂性能。(a) 平均峰值时效性。(b) 性能评分。
peak AoI decreases rapidly when the offloading ratio rises to 0.67 and it increases slowly when the offloading ratio gets close to one. This may be explained as follows. The offloading ratio means some data are processed at the edge server, which supports VPP to make real-time decisions. However, the offloading ratio means too much data is processed by the edge servers. The performance score decreases because the edge server has a more limited processing speed than the VPP cloud control center.
峰值 AoI 在卸载比率上升至 0.67 时迅速下降，而当卸载比率接近 1 时则缓慢增加。这可以解释如下：卸载比率 意味着部分数据在边缘服务器上处理，这支持 VPP 进行实时决策。然而，卸载比率 表示过多数据由边缘服务器处理。性能评分 下降，因为边缘服务器的处理速度比 VPP 云控制中心更为有限。
Figure 6 shows the profit, cost, and revenue from edge-enabled platforms with different offloading ratios. Figure 6(a) shows these measures for the PJM case and Figure 6(b) for the Fingrid case. Edge computing's revenue is equal to the VPP's incremental revenue under a certain level of edge computing. Edge computing's profit is equal to the difference between edge computing's revenue and its operation cost.
图 6 展示了不同卸载比下边缘计算平台的利润、成本和收入。图 6(a)为 PJM 案例的各项指标，图 6(b)则为 Fingrid 案例。边缘计算的收入等于在一定边缘计算水平下 VPP 的增量收入。边缘计算的利润等于其收入与运营成本之差。
It can be seen from Figure 6(a) that edge computing's revenue increases as the computation offloading ratio falls within while the cost of edge
从图 6(a)可以看出，当计算卸载比率落在 范围内时，边缘计算的收入随着计算卸载比率的降低而增加，而边缘的成本
(a)
(b)
Figure 6: Profit, cost, and revenue of edge computing under different offloading ratios. (a) PJM case. (b) Fingrid case.
图 6：不同卸载比率下边缘计算的利润、成本和收入。(a) PJM 案例。(b) Fingrid 案例。
computing also rises. For edge computing's profit, a turning point exists at the offloading ratio of . For , the revenue and profit increase rapidly while the cost is small. For , the profit is lowered because the incremental revenue decreases and the cost continues to increase. Thus, a suitable offloading ratio for the PJM case is 0.67 .
计算成本也随之上升。对于边缘计算的收益，存在一个卸载比率的转折点，即 。在 时，收入和利润迅速增长，而成本较小。到了 ，利润下降，因为收入增量减少，成本持续增加。因此，对于 PJM 案例，合适的卸载比率为 0.67。
In the absence of edge computing , the VPP's revenue and profit are and for the PJM case, respectively. However, the implementation of edge computing demonstrably increases VPP revenue to . This translates to an increase in revenue. Furthermore, the analysis reveals that VPP's profit increases to . This represents a significant rise of in VPP's profit. A similar discussion can be extended to the Fingrid case.
在未采用边缘计算的情况下，VPP 在 PJM 案例中的收入和利润分别为 和 。然而，边缘计算的实施显著提升了 VPP 的收入至 ，这意味着收入增加了 。此外，分析显示 VPP 的利润提升至 ，这标志着 VPP 利润显著增长了 。类似讨论同样适用于 Fingrid 案例。
Let the cost coefficients of edge computing range from message to message. Figure 7 (a) shows the profit of edge computing in the PJM case and Figure 7(b) shows the results of the Fingrid case. It is clear from Figure 7(a) that edge computing's profit decreases when its cost coefficient rises. Specifically, VPP's profit increases by a range of to when the cost coefficient falls within message.
令边缘计算的成本系数从 消息到 消息变化。图 7(a)展示了在 PJM 案例中边缘计算的利润情况，而图 7(b)则显示了 Fingrid 案例的结果。从图 7(a)可以明显看出，当边缘计算的成本系数上升时，其利润会下降。具体而言，当成本系数落在 消息范围内时，VPP 的利润会增加 至 的范围。
(a)
(b)
Figure 7: Profit of edge computing under different edge computing's cost coefficients. (a) PJM case. (b) Fingrid case.
图 7：不同边缘计算成本系数下的边缘计算利润。(a) PJM 案例。(b) Fingrid 案例。
(a)
(b)
Figure 8: Profit of edge computing under different evaluation periods. (a) PJM case. (b) Fingrid case.
图 8：不同评估周期下边缘计算的利润。(a) PJM 案例。(b) Fingrid 案例。
This represents a significant rise of to in VPP's profit.
这标志着 VPP 利润从 显著上升至 。
Let the evaluation period range from one settlement period to four settlement periods . One settlement period refers to one hour. Figure 8(a) shows the profit of edge computing in the PJM case and Figure 8(b) shows the results of the Fingrid case. As depicted in Figure 8, VPP profitability exhibits a positive correlation with the evaluation period. In the absence of edge computing, the VPP's revenue is , and , respectively, for evaluation periods of , and . The implemen-
设定评估周期范围为一个结算周期 至四个结算周期 。一个结算周期指一小时。图 8(a)展示了 PJM 案例中边缘计算的利润，而图 8(b)则展示了 Fingrid 案例的结果。如图 8 所示，VPP 的盈利能力与评估周期呈正相关。在没有边缘计算的情况下，VPP 的收入分别为 和 ，对应评估周期为 和 。实施...
Figure 9: Profit of edge computing under different computing power of the VPP cloud control center. (a) PJM case. (b) Fingrid case.
图 9：在不同 VPP 云控制中心计算能力下的边缘计算利润。(a) PJM 案例。(b) Fingrid 案例。
tation of edge computing increases VPP's revenue by , and . This translates to a growth of revenue. Furthermore, the analysis reveals that VPP's profit increases by .
边缘计算的实施使 VPP 的收入增加了 ，并且 。这转化为收入 的增长。此外，分析显示 VPP 的利润增加了 。
Set the value of the computing power of the VPP cloud control center as and , where is cycles/bit. Figure 9(a) shows the profit of edge computing in the PJM case and Figure 9(b) shows the results of the Fingrid case. It is clear from Figure 9(a) that VPP's profit exhibits a negative correlation with the computing power of the VPP cloud control center. As the VPP cloud control center's computing power increases, the benefits of edge computing become quite limited. However, edge computing remains advantageous regardless of the computing power of the VPP cloud control center. When the computing power is set to , the profit of edge computing reaches , representing a increase in VPP's profit.
将 VPP 云控制中心的计算能力值设为 和 ，其中 为 周期/比特。图 9(a)展示了 PJM 案例中边缘计算的利润，而图 9(b)则显示了 Fingrid 案例的结果。从图 9(a)可以明显看出，VPP 的利润与 VPP 云控制中心的计算能力呈负相关。随着 VPP 云控制中心计算能力的增强，边缘计算的效益变得相当有限。然而，无论 VPP 云控制中心的计算能力如何，边缘计算始终具有优势。当计算能力设定为 时，边缘计算的利润达到 ，使 VPP 的利润增加了 。
Figure 10: Performance of edge computing enabled VPP under different control periods. (a) Average peak AoI. (b) Performance score. (c) VPP Profit in the PJM case.
图 10：不同控制周期下边缘计算支持的虚拟电厂性能。(a) 平均峰值时效性指数。(b) 性能评分。(c) PJM 案例中的虚拟电厂利润。

6.3. Performance under Different Control Periods
6.3. 不同控制周期下的性能表现

Next, we evaluate the impact of the control period on the FFCAS performance. Figure 10 illustrates the average peak AoI, the performance score, and the VPP profit under the PJM case with different control periods. It can be seen from Figure 10(a) that the average peak AoI decreases rapidly when the offloading ratio ranges in . When the computation offloading ratio is larger than 0.67 , the edge servers become overloaded, and the average peak AoI increases slowly.
接下来，我们评估控制周期对 FFCAS 性能的影响。图 10 展示了在 PJM 案例下，不同控制周期内的平均峰值 AoI、性能评分及 VPP 利润。从图 10(a)可以看出，当卸载比率在 范围内时，平均峰值 AoI 迅速下降。而当计算卸载比率超过 0.67 时，边缘服务器过载，平均峰值 AoI 缓慢上升。
For a fixed offloading ratio, Figure 10(b) shows that a higher performance score is obtained as the control period increases. A small control period enables VPP to provide timely frequency regulation service while it increases communication and computation pressure. As a result, VPP's performance score is reduced and its profit decreases.
对于固定的卸载比率，图 10(b)显示随着控制周期的增加，性能得分随之提高。较小的控制周期使得虚拟电厂能够及时提供频率调节服务，但同时增加了通信和计算压力。因此，虚拟电厂的性能得分降低，其利润也随之减少。
It is observed from Figure 10(b) that, without edge computing, the performance score with is only 0.77 , which is much smaller than the performance scores for . Nonetheless, with suitable edge
从图 10(b)中可以看出，在没有边缘计算的情况下， 的性能得分仅为 0.77，远低于 的 性能得分。然而，在适当的边缘计算条件下，
(a)
(c)
Figure 11: Performance of edge computing enabled VPP under different message sizes. (a) Average peak AoI. (b) Performance score. (c) VPP profit in the PJM case.
图 11：不同消息大小下边缘计算支持的虚拟电厂性能。(a) 平均峰值时效性指数。(b) 性能评分。(c) PJM 案例中的虚拟电厂利润。
computing involved by larger offloading ratios , the FFCAS with a small control period also performs well as the performance score is larger than 0.92. Edge computing supports VPP to provide timely frequency regulation service. This phenomenon is also evidenced by the profit improvement in the PJM case shown in Figure 10(c). It can be seen from Figure 10(c) that edge computing helps improve the profit. Figure 10(c) also shows that small control periods imply more profits with more edge computing involvement.
随着卸载比率的增大，具有较小控制周期的 FFCAS 同样表现出色，其性能评分高于 0.92。边缘计算支持 VPP 提供及时的频率调节服务。这一现象在图 10(c)所示的 PJM 案例中利润提升也得到了证实。从图 10(c)可以看出，边缘计算有助于提高利润。图 10(c)还表明，较小的控制周期意味着随着边缘计算参与度的增加，利润也会更多。

6.4. Performance under Different Message Sizes
6.4. 不同消息大小下的性能表现

This section evaluates the impact of message size on VPP's FFCAS performance. The control period is set at 2 s and three offloading ratios are considered: , and 0.6. Figure 11 displays the average peak AoI, performance score, and VPP profit versus message size up to 0.04 MB . It can be seen from Figure 11(a) that the average peak AoI increases linearly as the message size rises. Figure 11(b) shows that the performance score is lowered for the increasing message size. Figure 11 reflects the phenomenon that the DSRs' status data with smaller message size requires less time to
本节评估消息大小对 VPP 的 FFCAS 性能的影响。控制周期设定为 2 秒，并考虑了三种卸载比率： 、0.4 和 0.6。图 11 展示了平均峰值 AoI、性能评分及 VPP 利润随消息大小（最高至 0.04 MB）的变化关系。从图 11(a)可以看出，随着消息尺寸的增加，平均峰值 AoI 呈线性增长。图 11(b)显示，随着消息尺寸的增大，性能评分有所下降。图 11 反映了这样一个现象：消息尺寸较小的 DSRs 状态数据所需传输时间更少。
be processed and VPP's performance score is high.
经过处理，VPP 的性能评分较高。
Figure 11(c) illustrates the VPP's profit with different message sizes. It can be seen from Figure 11(c) that, without edge computing ( ), VPP's profit is reduced by , from to , when the message size increases from 0.01 MB to 0.04 MB . With the edge computing of , the profit reduction is only . In other words, the profit of VPP increases with edge computing of compared with the case without edge computing. This demonstrates the significant advantage of using edge computing on VPP.
图 11(c)展示了不同消息大小下 VPP 的利润情况。从图 11(c)可以看出，在没有边缘计算（ ）的情况下，当消息大小从 0.01 MB 增加到 0.04 MB 时，VPP 的利润减少了 ，从 降至 。而通过 的边缘计算，利润减少仅为 。换言之，与无边缘计算的情况相比，采用 的边缘计算使 VPP 的利润增加了 。这充分体现了边缘计算在 VPP 中的显著优势。

7. Conclusion 7. 结论

This paper presents a computation offloading method for edge-enabled VPP FFCAS performance improvement and profit growth. This paper explores the use of AoI to characterize the decision-making process of a VPP. Then, an AoI-based FFCAS performance evaluation model is established considering the impacts of communication delay, communication failures, and computational delay. Furthermore, the FFCAS performance optimization model is presented. It aims to maximize VPP's profit through communication offloading and the DSR portfolio determination.
本文提出了一种边缘计算卸载方法，旨在提升边缘赋能的 VPP FFCAS 性能并促进利润增长。本文探讨了利用 AoI（新鲜度指数）来表征 VPP 决策过程的方法。随后，建立了一个基于 AoI 的 FFCAS 性能评估模型，该模型综合考虑了通信延迟、通信故障及计算延迟的影响。此外，还提出了 FFCAS 性能优化模型，旨在通过通信卸载和 DSR 组合决策来最大化 VPP 的利润。
Case studies considered diverse operating conditions, including variations in offloading ratios, cost coefficients of edge computing resources, and network parameters like computing power, control periods, and message sizes. Results demonstrated that edge computing could improve VPP's profit in many cases. However, the benefits of edge computing become limited when the related operating cost is too high or the computing power of the VPP
考虑了包括卸载比率变化、边缘计算资源成本系数以及网络参数如计算能力、控制周期和消息大小在内的多种运行条件下的案例研究。结果表明，在许多情况下，边缘计算能够提升虚拟电厂（VPP）的利润。然而，当相关运营成本过高或 VPP 的计算能力有限时，边缘计算带来的效益将受到限制。
cloud control center is strong enough. The proposed method can serve as a tool for the VPP to determine the implementation of edge computing whether or not.
云控制中心足够强大。所提出的方法可作为虚拟电厂判断是否实施边缘计算的工具。

Author Statement 作者声明Chengrong Lin: Methodology, Writing, Formal analysis, Revision. Bo
林成荣：方法论、撰写、形式分析、修订。Hu: Validation, Supervision. Heng-Ming Tai: Data curation. Changzheng
胡：验证，监督。恒明泰：数据管理。长征Shao: Visualization. Kaigui Xie: Funding acquisition. Yu Wang: Conceptu-
邵：可视化。谢开贵：资金获取。王宇：概念构思。alization. 源文本：alization。 翻译文本：alization。

Declaration  声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明，他们不存在已知的竞争性财务利益或可能影响本论文所报告工作的个人关系。

Acknowledgements 致谢

This work was supported by the Fundamental Research Funds for the Central Universities of China (2023CDJYXTD-004).
本研究得到了中国中央高校基本科研业务费专项资金（2023CDJYXTD-004）的资助。

Data Statement 数据声明

The data that support the findings of this study are available on request from the corresponding author.
本研究结果所依据的数据，可应要求从通讯作者处获取。
    • Corresponding author 通讯作者
    Email addresses: chengronglin@u.nus.edu.com (Chengrong Lin), hboy8361@163.com (Bo Hu), tai@utulsa.edu (Heng-Ming Tai), cshao@cqu.edu.cn (Changzheng Shao), kaiguixie@vip.163.com (Kaigui Xie), wangyulz@hotmail.com (Yu Wang)
    电子邮件地址：chengronglin@u.nus.edu.com（林承荣），hboy8361@163.com（胡博），tai@utulsa.edu（戴恒明），cshao@cqu.edu.cn（邵长征），kaiguixie@vip.163.com（解开贵），wangyulz@hotmail.com（王宇）