## 1 Introduction
On the one hand, centralized optimization requires getting system-wide data, which makes the size of the optimization model increase dramatically and leads to a large computational cost. On the other hand, the operators cannot share all the information in the dispatch process due to the requirement of information privacy. Therefore, how to realize distributed cooperative dispatch among DN, DHN, and BESS needs to be solved urgently. The augmented Lagrangian relaxation (ALR) method is a more mature distributed algorithm with current applications, which decouples the multi-region optimization problem by introducing quadratic term relaxation coupling constraints in the objective function and iteratively updating the Lagrangian multipliers. Distributed algorithms based on ALR can be categorized into the fully distributed algorithms represented by the auxiliary problem principle [16], the alternating direction multiplier method (ADMM) [17], and the hierarchical distributed algorithms represented by analytical target cascading (ATC) [18]. Among the above distributed methods, since ADMM is a combination of the dyadic decomposition method and the ALR method, ADMM has the robustness of the multiplier method and the distributed computational ability of the dyadic decomposition and is also widely used in power system control and optimization for its excellent convergence, fast iteration speed, and protection of the subject's privacy. References [19; 20; 21] have investigated ADMM-based dispatch methods for distribution networks. In [19], a coordinated day-ahead dispatch method of multiple power distribution grids hosting stochastic resources based on ADMM is proposed to aggregate the power and energy flexibilities in an interconnected power distribution system. In [20], a fully decentralized hierarchical transactive energy framework for charging electric vehicles (EV), with local distributed energy resources (DER) in a power distribution system based on ADMM is used to ensure that the privacy of market participants is well preservedsince the bid data of each participant are not exposed to others. In [21], a decentralized distributed convex optimal power flow model for power distribution systems based on ADMM is proposed, which is based on dividing the power grid network into subproblems representing individual areas by interchanging minimum network information. Currently, only a few references have investigated the distributed cooperative dispatch of distribution networks with district heat networks. In [22], a transactive energy-supported economic operation method for multi-energy complementary microgrids (MECM) is proposed to coordinate interconnected MECM in a regional integrated energy system (RIES). In [23], a decentralized demand management method based on the alternating direction method of multipliers algorithm for an industrial park with CHP units and thermal storage, which can protect private data of all participants while achieving solutions with high quality. However, the above references do not establish the accurate model of DHN that includes rate and temperature of mass flow and therefore do not allow the flexible regulation capability of DHN to be utilized. Moreover, the above references do not consider the distributed cooperative dispatch among the three main actors: DN, DHN, and BESS. In conclusion, there is still a lack of research on distributed cooperative dispatch for DN, DHN, and BESS. In response to the above problems, a distributed cooperative dispatch method for DN with DHN and BESS that considers the flexible regulation capability is proposed. The main contributions are as follows:  
1. A distributed cooperative dispatch framework for DN-DHN-BESS is constructed, which transforms the cooperative dispatch problem into the distributed optimization problem by decoupling the system into three subsystems at the boundary power. 2.