## 1 Introduction
The difference between the actual load and the CBL is thought to be a reduction in the number of customers [39]. Large-scale DR also requires solving these two issues; thus, this paper focuses on how to operate DR resources on an aggregated level and how to measure each customer's performance. A greater number of customers and high DR event frequency are the main challenges in large-scale DR. A distributed DR scheme will have an easier and instant access in large-scale DR due to less communication and computation pressure on the DR center. Additionally, the privacy of customers can be protected in the distributed scheme [40]. On the individual level, there are many distributed DR schemes that have been proposed [2; 33]. In regard to the aggregated level, pre-determined fixed incentives can be thought of as a distributed approach; however, it cannot reflect real-time supply-demand status and accurately regulate demand-side resources, while CIDR or other similar better approaches are designed to be a centralized framework. More importantly, the CBL-based DR performance measurement method determines the whole DR process to be centralized because the CBL of each customer has to be established by a centralized DR aggregator. The CBL is a critical part of calculating the incentive payment; therefore, it is unfair to establish CBL by customers individually. For example, the Federal Energy Regulatory Commission (FERC) 745 order requires independent system operators (ISOs) or regional transmission operators (RTOs) to establish the CBL for all DR participants [41]. Nevertheless, in large-scale DR, establishing CBL for numerous customers every day will become a very big challenge for ISOs or RTOs. In fact, concerns about CBL in DR have been attended to in recent studies. Except for being an obstacle to distributed DR schemes, there still are other problems, as follows:  
1. The bias of CBL cannot be avoided, and it will result in welfare losses to customers or LSEs [41]. In addition, DR event frequency will be much higher in large-scale DR to integrate a higher proportion of RES, which causes obstacles to establishing accurate CBL, as it is calculated by historical load data in the absence of DR events. 2. The customers will be capable of strategically increasing CBL and make more load reductions in DR to get more incentives. However, such a reduction is illusory and cannot help regulate the grid. This is called a moral hazard problem [42; 43]. In general, models that allocate DR resources on aggregated levels aim to maximize LSE's welfare or social welfare from an ISO/RTO's viewpoint [29]. According to [29; 30; 31], this paper also considers LSEs, who can participate in electricity market as strategic bidders, as the DR aggregators. Considering that integrating and consuming more RES can bring much welfare to the whole society, this paper considers a single LSE in a certain region and promoting RES consumption as the goal of large-scale DR. In a real-time economic dispatch, due to the intermittent and stochastic characteristics of RES, the reserve capacity of conventional generators may not be enough and DR events will occur. With this background, our work focuses on designing a novel scheme to allocate DR resources and DR rebates to numerous customers. This paper's contributions can be summarized as follows:  
1. This paper proposes a novel DR performance measurement method called customer directrix load (CDL) to replace CBL. Due to the global uniqueness of CDL, it is more suitable for distributed schemes and the moral hazard problem can be avoided. 2. The process of allocating DR resources in this paper is designed as a distributed method, which is similar to the red envelop allocation in many social networks. The LSE only needs to publish a total DR rebate and corresponding DR regulation goal to the demand side through social networks, and customer claims for an ideal rebate are performed in a distributed manner. This process is modeled as a non-cooperative game, and the Nash equilibrium is proven to exist. 3.