## 1 Introduction  
Economic dispatch, as one of fundamental and key problem in power system, is formulated to an optimization problem that aims at minimizing the total operating cost while satisfying the supply-demand balance constraint and some capacity limit. With pressures of energy crisis [1] and environment pollution [2; 3], there is an increasing interest in utilizing combined heat and power (CHP) into the power system, to decrease carbon emission, increase energy efficiency, and save operational costs in recent years, which brings many challenges for the economic dispatch problem (EDP). During these decades, there have been plentiful literature studies concerned with solving the combined heat and power economic dispatch problem (CHPEDP) in a centralized manner, such as the lambda iteration method [4], the gradient method [5], multi-team particle swarm optimization method [6], evolutionary method [7], etc. However, with the rapid development of the CHP system, centralized method has become the bottleneck of solving CHPEDP and cannot satisfy the requirement of practical applications. To be specific, the centralized methods have to rely on a central controller to communication with all the other system components to collect global information, resulting in great implementation cost and high sensitivity to single-point failures [8; 9; 10]. Moreover, both the physical and communication topology of the future CHP system are prone to have a huge and variable topology with partially-unknown characteristic, which may seriously degrade the effectiveness of centralized methods. In addition, each component in CHP system should possess the plug-and-play feature, which is one of the key characteristics of the future CHP system [11; 12; 13]. Thus, the centralized method is not very suitable to address the distributed features of the future CHP system. The distributed methods featuring fast computation, high flexibility and reliability do not relay on the central controller and can provide cost-effective distributed solutions to meet the requirements of real-time application in modern power systems [14; 15; 16]. In fact, various distributed methods proposed only solve the EDP on the power grid side. For example, an incremental cost consensus method was proposed in [8] to solve the EDP in a distributed fashion, where a leader agent is needed to satisfy the electrical power balance constraint. In [17], a fully distributed method was proposed, in which the communication among generators are strongly connected and the estimation of power mismatch does not require a central controller or leader. Furthermore, the transmission losses, demand response and ramping rate limits have been taken into account and discussed in [18; 19; 20; 21; 22], respectively. In addition, there are also some researches analyzing the EDP from other perspectives, such as non-convex dispatch model [23], online generation adjustment [24], etc. It is worth noting that the aforementioned distributed methods are mainly focusing on the economic dispatch problem on the power side. However, there exist strong power-heat-coupling features both in the objective function and constraint limits in the CHP system scenario. Hence, the distributed methods above cannot apply to this kind of distributed coupling optimization problem. Based on above discussions, this paper presents a novel distributed method to solve the CHPEDP. The proposed method is implemented based on two consensus procedure modified their corresponding feedback parts where the one is used to get the optimal electrical incremental cost and the other is used to get the optimal heat incremental cost, resulting in achieving the optimal solutions. Meanwhile, the alternating iterative method is proposed and used to solve the electrical-heat-coupling problem. In addition, the estimation of the electrical and heat power are also through distributed fashion, which are then used as a feedback mechanism to meet the supply-demand balance.