Distributed Cooperative Dispatch Method of Distribution Network with District Heat Network and Battery Energy Storage System Considering Flexible Regulation Capability

Abstract

Flexible resources, including district heat networks (DHN) and battery energy storage systems (BESS), can provide flexible regulation capability for distribution networks (DN), thereby increasing the absorption capacity for renewable energy. In order to improve the operation economy of DN and ensure the information privacy of different operators, a distributed cooperative dispatch method of DN with DHN and BESS considering flexible regulation capability is proposed. First, a distributed cooperative dispatch framework of DN-DHN-BESS is constructed. Then, an optimal dispatch model of DHN under constant flow-variable temperature control strategy is established in order to utilize the heat storage capacity to provide flexible regulation capability for DN. Next, the optimal dispatch models of BESS and DN are established with the objective of minimizing the operation cost, respectively. Finally, a solution method based on the alternating direction multiplier method of distributed cooperative dispatch for DN-DHN-BESS is proposed. Case studies are performed on a system consisting of a 33-node DN and a 44-node DHN, and simulation results demonstrate that the proposed method differs from the centralized dispatch method by only 0.52% in the total system cost, and the proposed method reduces the total system cost by 34.5% compared to that of the independent dispatch method.

1. Introduction

With the increasing penetration of renewable energy sources, the need for flexibility in distribution networks (DN) over multiple time scales has increased [1]. In this context, flexible resources such as battery energy storage systems (BESS) and district heat networks (DHN) have become an important instrument for smoothing out power fluctuations of renewable energy sources and improving power quality due to their ability to transfer energy over time [2]. Therefore, it is important to study the cooperative operation method of DN with different types of flexible resources in order to improve the flexible regulation capability of the system and thus improve the operation economy of the system [3].

With the characteristics of fast power regulation and storage of electricity, battery energy storage systems play a large role in smoothing power fluctuations of intermittent energy sources, shaving peaks and filling valleys, improving voltage quality, and providing backup power sources, etc. It is the key to realizing flexible regulation of widely accessed distributed energy sources in DN. There have been more references on the cooperative dispatch of DN and BESS. In [4], an optimal dispatch method of BESS in unbalanced distribution networks is proposed so as to minimize power losses and simultaneously ensure that the network satisfies current and voltage constraints. In [5], an optimal dispatch method for BESS in DN considering peak-load transfer is proposed to improve the voltage distribution in DN. In order to mitigate the adverse effects of renewable energy sources in active distribution networks, energy management and the optimal dispatch method of BESS have been proposed in [6] to reduce operation costs, power losses, and voltage fluctuations in unbalanced distribution networks. In [7], a joint day-ahead and intraday optimal dispatch method for active distribution networks is proposed, which is capable of dispatching centralized and distributed energy storage based on their differences in capacity and responsiveness.

DHN is a potential flexible resource that can break the rigid constraints of real-time balance between heat supply and heat load by adjusting the water supply temperature [8,9,10]. This type of energy storage effect provides additional flexible regulation capability to DN to consume renewable energy. Several references have studied the cooperative dispatch of DN and DHN. In order to improve the absorption capacity of DN for wind power, an integrated electric-heat dispatch model considering the heat inertia of the DHN has been developed in [11], in which an integrated model for fully simulating the dynamic distribution of the district heat network has been proposed for the first time. In [12], a centralized optimal dispatch model for an integrated electric-heat energy system is developed by considering the use of electric boilers and heat storage tanks to improve the flexibility of combined heating and power (CHP) units for better consumption of wind power. In [13,14], pipelines in the DHN and buildings are further utilized as heat storage to provide flexibility in order to solve the problem of wind curtailment in the DN, and a dispatch model for CHP that takes into account the heat storage capacity of pipelines and buildings is developed. In [15], a coordinated operation strategy based on model predictive control for a multi-area integrated electric-heat energy system is proposed, which takes into account the heat storage capacity of DHN by taking into account the transmission delay of the pipelines and the heat inertia of the buildings.

All of the above references adopt the centralized cooperative dispatch methods, whereas the DN, DHN, and BESS are managed by the distribution network operator, the district heat network operator, and the energy storage aggregator, respectively, and the centralized cooperative dispatch of them is difficult to achieve in this context. 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 preserved since 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)

An optimal dispatch model for DHN under constant flow-variable temperature (CF-VT) control strategy is established, which takes into account the heat storage capacity by inscribing the temperature quasi-dynamics in pipeline and node.

(3)

An optimal dispatch model for DN is established, which is transformed into a second-order cone programming (SOCP) model based on the second-order cone relaxation and linearization method for quadratic constraint to improve the solution efficiency.

(4)

A solution method based on ADMM of distributed cooperative dispatch for DN-DHN-BESS is proposed, to realize the independent solution and cooperative dispatch of three subsystems.

The rest of the paper is organized as follows: In Section 2, the distributed cooperative dispatch framework for DN-DHN-BESS is constructed. In Section 3, the optimal dispatch model for DHN under CF-VT control strategy is established. In Section 4, the optimal dispatch model for BESS is established. In Section 5, the optimal dispatch model for DN is established. In Section 6, the solution method based on ADMM of distributed cooperative dispatch for DN-DHN-BESS is proposed. In Section 7, case studies are conducted to demonstrate the effectiveness and efficiency of the proposed method. Conclusions are given in Section 8.

2. Distributed Cooperative Dispatch Framework for DN-DHN-BESS

The structure of the DN-DHN-BESS system studied in this paper is shown in Figure 1. The DN includes distributed photovoltaic (PV), distributed wind turbine (WT), and load and is dispatched by the distribution network operator. The DHN includes CHP units, gas boilers (GB), electric heaters (EH), and loads and is dispatched by the district heat network operator. The charging and discharging of the BESS are dispatched by the energy storage aggregator. The difference in boundary power is to distinguish the boundary power information exchanged by different subjects; for example, P^D-E is used to represent the boundary power information exchanged by the distribution network, and P^E-D represents the boundary power information exchanged by the energy storage.

In order to follow the autonomous operation characteristics and the information privacy of different operators, a distributed cooperative dispatch framework for DN-DHN-BESS is constructed in this paper, as shown in Figure 2. The coupling variable between the DN, DHN, and BESS is the boundary power, so under the premise of satisfying the boundary power consistency constraint in (1), the system can be decoupled into three subsystems, and thus the cooperative dispatch problem can be transformed into a distributed optimization problem. By independently optimizing the three subsystems and exchanging the information of boundary power for coordination, accurate results can be obtained.

$$\{\begin{array}{c}{P}^{\mathrm{D}-\mathrm{E}}=P^{\mathrm{E}-\mathrm{D}}\\ P^{\mathrm{D}-\mathrm{H}}=P^{\mathrm{H}-\mathrm{D}}\end{array}$$

(1)

3. Optimal Dispatch Model for DHN under CF-VT Control Strategy

3.1. Objective of the Optimal Dispatch Model

The objective of the optimal dispatch model for DHN is to minimize the operation cost C^DHN, including the cost of purchased electricity $C_{\mathrm{e}}^{\mathrm{DHN}}$, the cost of purchased gas $C_{\mathrm{g}}^{\mathrm{DHN}}$, and the cost of pollutant emission penalties $C_{\mathrm{f}}^{\mathrm{DHN}}$, as shown in (2).

$$\min C^{\mathrm{DHN}}=C_{\mathrm{e}}^{\mathrm{DHN}}+C_{\mathrm{g}}^{\mathrm{DHN}}+C_{\mathrm{f}}^{\mathrm{DHN}}$$

(2)

$$C_{\mathrm{e}}^{\mathrm{DHN}}={\displaystyle \sum _t{\displaystyle \sum _{i\in N^{\mathrm{D}}}{P}_{i,t}^{\mathrm{H}-\mathrm{D}}{c}_t^{\mathrm{E},2}}}$$

(3)

$$C_{\mathrm{g}}^{\mathrm{DHN}}={\displaystyle \sum _t{\displaystyle \sum _{m\in N^{\mathrm{H}}}(G_{m,t}^{\mathrm{CHP}}+G_{m,t}^{\mathrm{GB}})c_t^{\mathrm{G}}}}$$

(4)

$$C_{\mathrm{f}}^{\mathrm{DHN}}={\displaystyle \sum _t{\displaystyle \sum _{m\in N^{\mathrm{H}}}{\displaystyle \sum _{k\in {\Omega }^{\mathrm{P}}}\left(P_{m,t}^{\mathrm{CHP}}{\lambda }_k^{\mathrm{CHP}}+H_{m,t}^{\mathrm{GB}}{\lambda }_k^{\mathrm{GB}}\right)c_k^{\mathrm{env}}}}}$$

(5)

where N^D is the set of nodes in DN; N^H is the set of nodes in DHN; Ω^P is the set of pollutants; $P_{i,t}^{\mathrm{H}-\mathrm{D}}$ is the power transmitted from the DN node i to the DHN at period t; $P_{m,t}^{\mathrm{CHP}}$ and $G_{m,t}^{\mathrm{CHP}}$ are the active power and gas consumption power of the CHP unit at DHN node m at period t, respectively; $G_{m,t}^{\mathrm{GB}}$ and $H_{m,t}^{\mathrm{GB}}$ are the gas consumption power and heat power of the GB at DHN node m at period t, respectively; $c_t^{\mathrm{E},2}$ is the price of electricity sold from the DN at period t; $c_t^{\mathrm{G}}$ is the gas price at period t; ${\lambda }_k^{\mathrm{CHP}}$ and ${\lambda }_k^{\mathrm{GB}}$ are the emission factors of category k pollutant of CHP and GB, respectively; $c_k^{\mathrm{env}}$ is the treatment cost factor of category k pollutant.

3.2. Constraints of the Optimal Dispatch Model

In order to take into account, the heat storage capacity, the temperature quasi-dynamics in pipeline and node are considered in the power flow constraints of DHN, and pipeline temperature decay and delay constraints and node temperature mixing constraints, are established, respectively [24]. Further combined with the heat power balance constraint, operation constraints of the energy device, and boundary power constraints between DHN and DN, the complete constraints are formed.

3.2.1. Pipeline Temperature Decay and Delay Constraints

Considering the transmission delay and heat loss of mass flow in the pipeline, the pipeline temperature decay and delay constraints in the supply and return networks are established as shown in (6) and (7), respectively [25]:

$$T_{mn,t}^{\mathrm{S},\mathrm{tail}}=T_t^{\mathrm{am}}+\left(T_{mn,t-{\tau }_{mn}}^{\mathrm{S},\mathrm{head}}-T_t^{\mathrm{am}}\right)e^{-{\displaystyle \frac{{\lambda }_{mn}{L}_{mn}}{f_{mn}^{\mathrm{S}}{A}_{mn}{\rho }_{\mathrm{w}}{c}_{\mathrm{w}}}}}, \forall mn\in L^{\mathrm{H}}$$

(6)

$$T_{mn,t}^{\mathrm{R},\mathrm{head}}=T_t^{\mathrm{am}}+\left(T_{mn,t-{\tau }_{mn}}^{\mathrm{R},\mathrm{tail}}-T_t^{\mathrm{am}}\right)e^{-{\displaystyle \frac{{\lambda }_{mn}{L}_{mn}}{f_{mn}^{\mathrm{R}}{A}_{mn}{\rho }_{\mathrm{w}}{c}_{\mathrm{w}}}}}, \forall mn\in L^{\mathrm{H}}$$

(7)

$${\tau }_{mn}={\displaystyle \frac{l_{mn}{A}_{mn}{\rho }_{\mathrm{w}}}{f_{mn}^{\mathrm{S}}}}, \forall mn\in L^{\mathrm{H}}$$

(8)

where L^H is the set of pipelines in DHN; $T_{mn,t}^{\mathrm{S},\mathrm{head}}$ and $T_{mn,t}^{\mathrm{S},\mathrm{tail}}$ are the temperature at the head and end of pipeline mn in supply network at period t, respectively; $T_{mn,t}^{\mathrm{R},\mathrm{head}}$ and $T_{mn,t}^{\mathrm{R},\mathrm{tail}}$ are the temperature at the head and end of pipeline mn in return network at period t; $T_t^{\mathrm{am}}$ is the outdoor temperature at period t; $f_{mn}^{\mathrm{S}}$ and $f_{mn}^{\mathrm{R}}$ are the mass flow rate of pipeline mn in supply and return network, respectively; τ_mn is the transmission delay of pipeline mn; λ_mn is the heat conductivity factor of pipeline mn; l_mn is the length of pipeline mn; A_mn is the cross-sectional area of pipeline mn; c_w is the specific heat capacity of water; ρ_w is the density of water.

3.2.2. Node Temperature Mixing Constraints

Considering that mass flow with different temperatures will mix at the nodes, the node temperature mixing constraints in the water supply and return networks are established as shown in (9) and (10) respectively.

$$T_{m,t}^{\mathrm{S}}({\displaystyle \sum _{lm\in L^{\mathrm{H}}}{f}_{lm}^{\mathrm{S}}}+f_m)=f_m{T}_{m,t}^{\mathrm{S},\mathrm{ex}}+{\displaystyle \sum _{lm\in L^{\mathrm{H}}}{f}_{lm}^{\mathrm{S}}{T}_{lm,t}^{\mathrm{S},\mathrm{tail}}}, \forall m\in N^{\mathrm{H}}$$

(9)

$$T_{m,t}^{\mathrm{R}}({\displaystyle \sum _{mn\in L^{\mathrm{H}}}{f}_{mn}^{\mathrm{R}}}-f_m)=-f_m{T}_{m,t}^{\mathrm{R},\mathrm{ex}}+{\displaystyle \sum _{mn\in L^{\mathrm{H}}}{f}_{mn}^{\mathrm{R}}{T}_{mn,t}^{\mathrm{R},\mathrm{head}}}, \forall m\in N^{\mathrm{H}}$$

(10)

where $T_{m,t}^{\mathrm{S}}$ and $T_{m,t}^{\mathrm{R}}$ are the temperature at node m in supply and return networks at period t, respectively; $T_{m,t}^{\mathrm{S},\mathrm{ex}}$ and $T_{m,t}^{\mathrm{R},\mathrm{ex}}$ are the supply and return temperatures of heat exchange station at node m at period t, respectively.

In addition, the temperature of the mass flow entering the pipeline from the nodes should be the same as the node temperature, thus the constraints shown in (11) and (12) are established. Moreover, there is a relation between the node temperature and the supply and return temperatures of the heat exchange station as (13) and (14) [26].

$$T_{m,t}^{\mathrm{S}}=T_{mn,t}^{\mathrm{S},\mathrm{head}}, \forall m\in N^{\mathrm{H}}, \forall mn\in L^{\mathrm{H}}$$

(11)

$$T_{m,t}^{\mathrm{R}}=T_{lm,t}^{\mathrm{R},\mathrm{tail}}, \forall m\in N^{\mathrm{H}}, \forall mn\in L^{\mathrm{H}}$$

(12)

$$T_{m,t}^{\mathrm{S}}=T_{m,t}^{\mathrm{S},\mathrm{ex}}, \forall m\in N^{\mathrm{H}}$$

(13)

$$T_{m,t}^{\mathrm{R}}=T_{m,t}^{\mathrm{R},\mathrm{ex}}, \forall m\in N_{\mathrm{S}}^{\mathrm{H}}$$

(14)

where $N_{\mathrm{S}}^{\mathrm{H}}$ is the set of source nodes in DHN.

3.2.3. Heat Power Balance Constraints

$$H_{m,t}^{\mathrm{N}}=c_{\mathrm{w}}{f}_m\left(T_{m,t}^{\mathrm{S},\mathrm{ex}}-T_{m,t}^{\mathrm{R},\mathrm{ex}}\right), \forall m\in N^{\mathrm{H}}$$

(15)

$$H_{m,t}^{\mathrm{N}}=H_{m,t}^{\mathrm{CHP}}+H_{m,t}^{\mathrm{GB}}+H_{m,t}^{\mathrm{EH}}-H_{m,t}^{\mathrm{load}}, \forall m\in N^{\mathrm{H}}$$

(16)

where $H_{m,t}^{\mathrm{N}}$ is the heat power of node m at period t; $H_{m,t}^{\mathrm{CHP}}$ is the heat power of CHP unit at node m at period t; $H_{m,t}^{\mathrm{EH}}$ is the heat power of EH at node m at period t; $H_{m,t}^{\mathrm{load}}$ is the heat power of the load at node m at period t [27].

3.2.4. Operation Constraints of Energy Devices

The operational constraints of energy equipment include the operational constraints of CHP units, GB, and EH [28,29].

$$H_{m,t}^{\mathrm{CHP}}=K_{\mathrm{e}}^{\mathrm{CHP}}{P}_{m,t}^{\mathrm{CHP}}, \forall m\in N^{\mathrm{H}}$$

(17)

$$H_{m,t}^{\mathrm{CHP}}=K_{\mathrm{h}}^{\mathrm{CHP}}{G}_{m,t}^{\mathrm{CHP}}, \forall m\in N^{\mathrm{H}}$$

(18)

$$P_{\min }^{\mathrm{CHP}}{S}_m^{\mathrm{CHP}}\le P_{m,t}^{\mathrm{CHP}}\le P_{\max }^{\mathrm{CHP}}{S}_m^{\mathrm{CHP}}, \forall m\in N^{\mathrm{H}}$$

(19)

where $K_{\mathrm{e}}^{\mathrm{CHP}}$ is the heat-to-electric ratio coefficient of the CHP unit; $K_{\mathrm{h}}^{\mathrm{CHP}}$ is the gas-heat conversion efficiency of the CHP unit; $S_m^{\mathrm{CHP}}$ is the capacity of the CHP unit at node m; $P_{\max }^{\mathrm{CHP}}$ and $P_{\min }^{\mathrm{CHP}}$ are the upper and lower limits of the generation power of the CHP unit, respectively.

$$H_{m,t}^{\mathrm{GB}}=K^{\mathrm{GB}}{G}_{m,t}^{\mathrm{GB}}, \forall m\in N^{\mathrm{H}}$$

(20)

$$H_{\min }^{\mathrm{GB}}{S}_m^{\mathrm{GB}}\le H_{m,t}^{\mathrm{GB}}\le H_{\max }^{\mathrm{GB}}{S}_m^{\mathrm{GB}}, \forall m\in N^{\mathrm{H}}$$

(21)

where K^GB is the heat-gas conversion factor of the GB; $S_m^{\mathrm{GB}}$ is the capacity of GB at node m; $H_{\max }^{\mathrm{GB}}$ and $H_{\min }^{\mathrm{GB}}$ are the upper and lower limits of the heat power of the GB, respectively.

$$H_{m,t}^{\mathrm{EH}}=K^{\mathrm{EH}}{P}_{m,t}^{\mathrm{EH}}, \forall m\in N^{\mathrm{H}}$$

(22)

$$H_{\min }^{\mathrm{EH}}{S}_m^{\mathrm{EH}}\le H_{m,t}^{\mathrm{EH}}\le H_{\max }^{\mathrm{EH}}{S}_m^{\mathrm{EH}}, \forall m\in N^{\mathrm{H}}$$

(23)

where $P_{m,t}^{\mathrm{EH}}$ is the active power of EH at node m at period t; K^EH is the heat-electric conversion factor of the EH; $S_m^{\mathrm{EH}}$ is the capacity of the EH at node m; $H_{\max }^{\mathrm{EH}}$ and $H_{\min }^{\mathrm{EH}}$ are the upper and lower limits of the heat power of EH, respectively.

3.2.5. Boundary Power Constraints between DHN and DN

In order to facilitate distributed cooperative dispatch, the boundary power between the DHN and the DN is calculated as follows:

$$P_{i,t}^{\mathrm{H}-\mathrm{D}}={\displaystyle \sum _{m\in N_i^{\mathrm{H}}}{P}_{m,t}^{\mathrm{EH}}}-{\displaystyle \sum _{m\in N_i^{\mathrm{H}}}{P}_{m,t}^{\mathrm{CHP}}}, \forall i\in N^{\mathrm{D}}$$

(24)

where $N_i^{\mathrm{H}}$ is the set of DHN nodes connected to DN node i.

4. Optimal Dispatch Model for BESS

4.1. Objective of the Optimal Dispatch Model

The objective of the optimal dispatch model for BESS is to minimize the operation cost of C^BESS, as shown in the following:

$$\min C^{\mathrm{BESS}}={\displaystyle \sum _t{\displaystyle \sum _{k\in N^{\mathrm{E}}}(P_{k,t}^{\mathrm{ES},\mathrm{c}}-P_{k,t}^{\mathrm{ES},\mathrm{f}})c_t^{\mathrm{E},2}}}$$

(25)

where N^E is the set of BESSs; $P_{k,t}^{\mathrm{ES},\mathrm{c}}$ and $P_{k,t}^{\mathrm{ES},\mathrm{f}}$ are the charging power and discharging power of BESS k at period t, respectively.

4.2. Constraints of the Optimal Dispatch Model

4.2.1. Operation Constraints of BESS

Referring to [30], the operation constraints of BESS are established as follows. Equation (26) is the mutual exclusion constraint of charging and discharging states. Equations (27) and (28) are the charging and discharging power constraints. Equation (29) is the total power constraint. Equations (30) and (31) are the energy constraints.

$$x_{k,t}^{\mathrm{ES},\mathrm{c}}+x_{k,t}^{\mathrm{ES},\mathrm{f}}\le 1, \forall k\in N^{\mathrm{E}}$$

(26)

$$0\le P_{k,t}^{\mathrm{ES},\mathrm{c}}\le x_{k,t}^{\mathrm{ES},\mathrm{c}}{S}_k^{\mathrm{ES}}, \forall k\in N^{\mathrm{E}}$$

(27)

$$0\le P_{k,t}^{\mathrm{ES},\mathrm{f}}\le x_{k,t}^{\mathrm{ES},\mathrm{f}}{S}_k^{\mathrm{ES}}, \forall k\in N^{\mathrm{E}}$$

(28)

$$P_{k,t}^{\mathrm{ES}}=P_{k,t}^{\mathrm{ES},\mathrm{c}}-P_{k,t}^{\mathrm{ES},\mathrm{f}}, \forall k\in N^{\mathrm{E}}$$

(29)

$$E_{k,t}^{\mathrm{ES}}=E_{k,t-1}^{\mathrm{ES}}+{\eta }^{\mathrm{ES},\mathrm{c}}{P}_{k,t}^{\mathrm{ES},\mathrm{c}}\Delta t-{\displaystyle \frac{P_{k,t}^{\mathrm{ES},\mathrm{f}}\Delta t}{{\eta }^{\mathrm{ES},\mathrm{f}}}}, \forall k\in N^{\mathrm{E}}$$

(30)

$$0\le E_{k,t}^{\mathrm{ES}}\le E_k^{\mathrm{ES}}, \forall k\in N^{\mathrm{E}}$$

(31)

where $x_{k,t}^{\mathrm{ES},\mathrm{c}}$ and $x_{k,t}^{\mathrm{ES},\mathrm{f}}$ are the charging and discharging states of the BESS k at period t, respectively; $P_{k,t}^{\mathrm{ES}}$ is the net power of the BESS k at period t; $S_k^{\mathrm{ES}}$ and $E_k^{\mathrm{ES}}$ are the power and capacity of the BESS k, respectively; $E_{k,t}^{\mathrm{ES}}$ is the capacity of the BESS k at period t; η^ES,c and η^ES,f are the charging and discharging efficiencies of the BESS, respectively.

4.2.2. Boundary Power Constraints between BESS and DN

In order to facilitate distributed cooperative dispatch, the boundary power between the BESS and the DN is calculated as follows:

$$P_{i,t}^{\mathrm{E}-\mathrm{D}}={\displaystyle \sum _{k\in N_i^{\mathrm{E}}}{P}_{k,t}^{\mathrm{ES}}}, \forall i\in N^{\mathrm{D}}$$

(32)

where $N_i^{\mathrm{E}}$ is the set of BESSs connected to DN node i.

5. Optimal Dispatch Model for DN

5.1. Objective of the Optimal Dispatch Model

The objective of the DN dispatch model is to minimize the operation cost C^DN, including the cost of purchased electricity $C_b^{DN}$, the penalty cost for curtailment for PV and WT $C_f^{DN}$, and the revenue from the sale of electricity $C_s^{DN}$, as shown in the following:

$$\min C^{\mathrm{DN}}=C_{\mathrm{b}}^{\mathrm{DN}}+C_{\mathrm{f}}^{\mathrm{DN}}-C_{\mathrm{s}}^{\mathrm{DN}}$$

(33)

$$C_{\mathrm{b}}^{\mathrm{DN}}={\displaystyle \sum _t{\displaystyle \sum _{i\in N^{\mathrm{D}}}{P}_{i,t}^{\mathrm{SG}}{c}_t^{\mathrm{E},1}}}$$

(34)

$${\mathrm{C}}_{\mathrm{f}}^{\mathrm{DN}}={\displaystyle \sum _t{\displaystyle \sum _{i\in N^{\mathrm{D}}}\left[y_{\mathrm{f}}^{\mathrm{PV}}\left({\tilde{P}}_{i,t}^{\mathrm{PV}}-P_{i,t}^{\mathrm{PV}}\right)+y_{\mathrm{f}}^{\mathrm{WT}}\left({\tilde{P}}_{i,t}^{\mathrm{WT}}-P_{i,t}^{\mathrm{WT}}\right)\right]}}$$

(35)

$$C_{\mathrm{s}}^{\mathrm{DN}}={\displaystyle \sum _t{\displaystyle \sum _{i\in N^{\mathrm{D}}}{c}_t^{\mathrm{E},2}(P_{i,t}^{\mathrm{D}-\mathrm{H}}+P_{i,t}^{\mathrm{D}-\mathrm{E}})}}$$

(36)

where $P_{i,t}^{\mathrm{SG}}$ is the active power transmitted from the superior grid to the DN node i at period t; $c_t^{\mathrm{E},1}$ is the price of electricity sold by the superior grid at period t; ${\tilde{P}}_{i,t}^{\mathrm{PV}}$ and $P_{i,t}^{\mathrm{PV}}$ are the predicted and actual output of PV at DN node i at period t, respectively; ${\tilde{P}}_{i,t}^{\mathrm{WT}}$ and $P_{i,t}^{\mathrm{WT}}$ are the predicted and actual output of WT at DN node i at period t, respectively; $y_{\mathrm{f}}^{\mathrm{PV}}$ and $y_{\mathrm{f}}^{\mathrm{WT}}$ are the penalty cost coefficients of curtailed power for PV and WT, respectively; $P_{i,t}^{\mathrm{D}-\mathrm{H}}$ is the active power transmitted from DN node i to DHN at period t; $P_{i,t}^{\mathrm{D}-\mathrm{E}}$ is the active power transmitted from DN node i to BESS at period t.

5.2. Constraints of the Optimal Dispatch Model

The DN optimal dispatch model contains the power flow constraint and output constraint of PV and WT.

5.2.1. Power Flow Constraint of DN

Based on the Distflow model, the power flow constraints of DN are established as follows [31].

$$\{\begin{array}{l}{P}_{i,t}^{\mathrm{N}}=P_{i,t}^{\mathrm{SG}}+P_{i,t}^{\mathrm{PV}}+P_{i,t}^{\mathrm{WT}}-P_{i,t}^{\mathrm{load}}-P_{i,t}^{\mathrm{D}-\mathrm{E}}-P_{i,t}^{\mathrm{D}-\mathrm{H}}\hfill \\ Q_{i,t}^{\mathrm{N}}=Q_{i,t}^{\mathrm{SG}}+Q_{i,t}^{\mathrm{PV}}+Q_{i,t}^{\mathrm{WT}}-Q_{i,t}^{\mathrm{load}}\hfill \end{array}, \forall i\in N^{\mathrm{D}}$$

(37)

$$\{\begin{array}{c}{\displaystyle \sum _{ki\in L^{\mathrm{D}}}(P_{ki,t}^{\mathrm{L}}-R_{ki}{I}_{ki,t}^2)}+P_{i,t}^{\mathrm{N}}={\displaystyle \sum _{ij\in L^{\mathrm{D}}}{P}_{ij,t}^{\mathrm{L}}}\\ {\displaystyle \sum _{ki\in L^{\mathrm{D}}}(Q_{ki,t}^{\mathrm{L}}-X_{ki}{I}_{ki,t}^2)}+Q_{i,t}^{\mathrm{N}}={\displaystyle \sum _{ij\in L^{\mathrm{D}}}{Q}_{ij,t}^{\mathrm{L}}}\end{array}, \forall i\in N^{\mathrm{D}}$$

(38)

$$U_{i,t}^2-U_{j,t}^2-I_{ij,t}^2(R_{ij}^2+X_{ij}^2)-2(P_{ij,t}^{\mathrm{L}}{R}_{ij}+Q_{ij,t}^{\mathrm{L}}{X}_{ij})=0, \forall i\in N^{\mathrm{D}}, \forall ij\in L^{\mathrm{D}}$$

(39)

$$U_{i,t}^2={\displaystyle \frac{{(P_{ij,t}^{\mathrm{L}})}^2+{(Q_{ij,t}^{\mathrm{L}})}^2}{I_{ij,t}^2}}, \forall i\in N^{\mathrm{D}}, \forall ij\in L^{\mathrm{D}}$$

(40)

$${(P_{ij,t}^{\mathrm{L}})}^2+{(Q_{ij,t}^{\mathrm{L}})}^2\le {(S_{ij})}^2, \forall ij\in L^{\mathrm{D}}$$

(41)

$${(U_{\min })}^2\le U_{i,t}^2\le {(U_{\max })}^2, \forall i\in N^{\mathrm{D}}$$

(42)

where L^D is the set of lines in DN; $P_{i,t}^{\mathrm{N}}$ and $Q_{i,t}^{\mathrm{N}}$ are the active and reactive power at node i at period t, respectively; $Q_{i,t}^{\mathrm{SG}}$ is the reactive power purchased by node i from the superior grid at period t; $P_{i,t}^{\mathrm{PV}}$ and $Q_{i,t}^{\mathrm{PV}}$ are the active and reactive outputs of the PV at node i at period t, respectively; $P_{i,t}^{\mathrm{WT}}$ and $Q_{i,t}^{\mathrm{WT}}$ are the active and reactive outputs of the WT at node i at period t, respectively; $P_{i,t}^{\mathrm{load}}$ and $Q_{i,t}^{\mathrm{load}}$ are the active and reactive power of the load at node i at period t, respectively; $P_{ij,t}^{\mathrm{L}}$ and $Q_{ij,t}^{\mathrm{L}}$ are the active and reactive power transmitted by line ij at period t, respectively; R_ij and X_ij are the resistance and reactance of line ij, respectively; $U_{i,t}^2$ is the squared value of the voltage at node i at period t; $I_{ij,t}^2$ is the squared value of the current of line ij at period t; S_ij is the capacity of line ij; U_min and U_max are the lower and upper limits of voltage, respectively.

5.2.2. Output Constraints of PV and WT

The output constraints of PV and WT are established as follows, where Equation (43) is the output constraint of PV and Equation (44) is the output constraint of WT.

$$0\le P_{i,t}^{\mathrm{PV}}\le {\tilde{P}}_{i,t}^{\mathrm{PV}}, \forall i\in N^{\mathrm{D}}$$

(43)

$$0\le P_{i,t}^{\mathrm{WT}}\le {\tilde{P}}_{i,t}^{\mathrm{WT}}, \forall i\in N^{\mathrm{D}}$$

(44)

5.2.3. Boundary Power Constraints between DN, DHN, and BESS

When DHN or BESS is connected to node i of DN, the boundary power is not 0, otherwise the boundary power is 0. Therefore, the constraints shown in (45) and (46) are established.

$$\{\begin{array}{cc}{P}_{i,t}^{\mathrm{D}-\mathrm{H}}=0& \mathrm{if} N_i^{\mathrm{HN}}=\varnothing \\ P_{i,t}^{\mathrm{D}-\mathrm{H}}\ne 0& \mathrm{otherwise}\end{array}, \forall i\in N^{\mathrm{D}}$$

(45)

$$\{\begin{array}{cc}{P}_{i,t}^{\mathrm{D}-\mathrm{E}}=0& \mathrm{if} N_i^{\mathrm{ES}}=\varnothing \\ P_{i,t}^{\mathrm{D}-\mathrm{E}}\ne 0& \mathrm{otherwise}\end{array}, \forall i\in N^{\mathrm{D}}$$

(46)

5.2.4. Second-Order Cone Relaxation and Linearization of Constraints

Equation (40) contains the product term of voltage and current. In order to improve the efficiency of the solution, (40) is first relaxed to the following form [32]:

$${(P_{ij,t}^{\mathrm{L}})}^2+{(Q_{ij,t}^{\mathrm{L}})}^2\le U_{i,t}^2{I}_{ij,t}^2, \forall i\in N^{\mathrm{D}}, \forall ij\in L^{\mathrm{D}}$$

(47)

Further, (47) is transformed into a second order cone constraint as shown in (48):

$$\Vert \begin{array}{ccc}[2P_{ij,t}^{\mathrm{L}}& 2Q_{ij,t}^{\mathrm{L}}& I_{ij,t}^2-U_{i,t}^2{]}^T\end{array}\Vert \le I_{ij,t}^2+U_{i,t}^2, \forall i\in N^{\mathrm{D}}, \forall ij\in L^{\mathrm{D}}$$

(48)

In addition, (41) contains the squared terms of active and reactive power; in order to improve the solution efficiency, (41) is linearized based on the quadratic constraint linearization method, and two square constraints (49) are employed to substitute for (41) [32].

$$\{\begin{array}{l}-S_{ij}\le P_{ij,t}^{\mathrm{L}}\le S_{ij}\hfill \\ -S_{ij}\le Q_{ij,t}^{\mathrm{L}}\le S_{ij}\hfill \\ -\sqrt{2}{S}_{ij}\le P_{ij,t}^{\mathrm{L}}+Q_{ij,t}^{\mathrm{L}}\le \sqrt{2}{S}_{ij}\hfill \\ -\sqrt{2}{S}_{ij}\le P_{ij,t}^{\mathrm{L}}-Q_{ij,t}^{\mathrm{L}}\le \sqrt{2}{S}_{ij}\hfill \end{array}, \forall ij\in L^{\mathrm{D}}$$

(49)

6. Solution Method of Distributed Cooperative Dispatch for DN-DHN-BESS

6.1. Modification of the Objective Function

Under the framework of ADMM, the boundary power consistency constraint needs to be relaxed and added as a penalty term to the objective functions of the optimal dispatch models of DN, DHN, and BESS so that the boundary power is as close as possible during the calculation process and finally achieves consistency [33]. The modified objective functions of the optimal dispatch models of DN, DHN, and BESS are shown in (50)–(52):

$$\begin{array}{r}\min F^{\mathrm{DN}}=C^{\mathrm{DN}}+{\displaystyle \sum _t{\displaystyle \sum _i\left\{{\lambda }_{i,t}^{\mathrm{D}-\mathrm{H}}\left(P_{i,t}^{\mathrm{D}-\mathrm{H}}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}\right)+{\left[{\rho }_{i,t}^{\mathrm{D}-\mathrm{H}}\left(P_{i,t}^{\mathrm{D}-\mathrm{H}}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}\right)\right]}^2\right\}}}+\\ {\displaystyle \sum _t{\displaystyle \sum _i\left\{{\lambda }_{i,t}^{\mathrm{D}-\mathrm{E}}\left(P_{i,t}^{\mathrm{D}-\mathrm{E}}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}\right)+{\left[{\rho }_{i,t}^{\mathrm{D}-\mathrm{E}}\left(P_{i,t}^{\mathrm{D}-\mathrm{E}}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}\right)\right]}^2\right\}}}\end{array}$$

(50)

$$\min F^{\mathrm{HN}}=C^{\mathrm{HN}}+{\displaystyle \sum _t{\displaystyle \sum _i\left\{{\lambda }_{i,t}^{\mathrm{D}-\mathrm{H}}\left({\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}-P_{i,t}^{\mathrm{H}-\mathrm{D}}\right)+{\left[{\rho }_{i,t}^{\mathrm{D}-\mathrm{H}}\left({\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}-P_{i,t}^{\mathrm{H}-\mathrm{D}}\right)\right]}^2\right\}}}$$

(51)

$$\min F^{\mathrm{ES}}=C^{\mathrm{ES}}+{\displaystyle \sum _t{\displaystyle \sum _i\left\{{\lambda }_{i,t}^{\mathrm{D}-\mathrm{E}}\left({\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}-P_{i,t}^{\mathrm{E}-\mathrm{D}}\right)+{\left[{\rho }_{i,t}^{\mathrm{D}-\mathrm{E}}\left({\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}-P_{i,t}^{\mathrm{E}-\mathrm{D}}\right)\right]}^2\right\}}}$$

(52)

where ${\lambda }_{i,t}^{\mathrm{D}-\mathrm{H}}$ and ${\lambda }_{i,t}^{\mathrm{D}-\mathrm{E}}$ are the Lagrange multipliers; ${\rho }_{i,t}^{\mathrm{D}-\mathrm{H}}$ and ${\rho }_{i,t}^{\mathrm{D}-\mathrm{E}}$ are the penalty parameters; ${\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}$ and ${\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}$ are the average value of boundary power, which can be calculated by (53).

$$\{\begin{array}{l}{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H}}={\displaystyle \frac{1}{2}}\left(P_{i,t}^{\mathrm{D}-\mathrm{H}}+P_{i,t}^{\mathrm{H}-\mathrm{D}}\right)\hfill \\ {\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E}}={\displaystyle \frac{1}{2}}\left(P_{i,t}^{\mathrm{D}-\mathrm{E}}+P_{i,t}^{\mathrm{E}-\mathrm{D}}\right)\hfill \end{array}$$

(53)

6.2. Solution Process

The specific solution process includes the following five steps:

Step 1: Initialize the Lagrange multipliers, penalty parameters, and boundary power, and set the iteration flag r = 0.

Step 2: Let r = r +1 and calculate the average value of the boundary power according to (53).

Step 3: The optimal dispatch models for DN, DHN, and BESS are solved separately to obtain the dispatch results and update the boundary power.

Step 4: Calculate the original residuals ($M_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}$, $M_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}$) and pairwise residuals ($D_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}$, $D_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}$) according to (54) and (55), and check whether the residuals satisfy the convergence condition in (56). If satisfied, output the calculation results; otherwise, continue with step 5.

$$\{\begin{array}{l}{M}_{i,t}^{\mathrm{D}-\mathrm{I},\left(r\right)}={\Vert P_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}\Vert }_2\hfill \\ M_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}={\Vert P_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\Vert }_2\hfill \end{array}$$

(54)

$$\{\begin{array}{l}{D}_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}={\Vert {\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H},\left(r-1\right)}\Vert }_2\hfill \\ D_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}={\Vert {\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E},\left(r-1\right)}\Vert }_2\hfill \end{array}$$

(55)

$$\{\begin{array}{l}{M}_{\max }^{\left(r\right)}=\max \left\{M_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)},M_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}\le {\epsilon }_{\mathrm{M}}\hfill \\ D_{\max }^{\left(r\right)}=\max \left\{D_{i,t}^{\mathrm{D}-\mathrm{H},\left(r\right)},D_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}\le {\epsilon }_{\mathrm{D}}\hfill \end{array}$$

(56)

where the superscript (r) is the value of the variable corresponding to the r-th iteration; ε_M and ε_D are the convergence margins of the original and pairwise residuals, respectively.

Step 5: Update the Lagrange multipliers and penalty parameters according to (57)–(60) and go to step 2.

$${\rho }_{i,t}^{\mathrm{D}-\mathrm{H},(r+1)}=\{\begin{array}{ll}{\xi }_1{\rho }_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\hfill & \max \left\{M_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\right\}>\nu \max \left\{D_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\right\}\hfill \\ {\displaystyle \frac{{\rho }_{i,t}^{\mathrm{D}-\mathrm{H},(r)}}{{\xi }_2}}\hfill & \max \left\{D_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\right\}>\nu \max \left\{M_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\right\}\hfill \\ {\rho }_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\hfill & \mathrm{other}\hfill \end{array}$$

(57)

$${\lambda }_{i,t}^{\mathrm{D}-\mathrm{H},(r+1)}={\lambda }_{i,t}^{\mathrm{D}-\mathrm{H},(r)}+{\rho }_{i,t}^{\mathrm{D}-\mathrm{H},(r+1)}\left(P_{i,t}^{\mathrm{D}-\mathrm{H},(r)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{H},(r)}\right)$$

(58)

$${\rho }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r+1\right)}=\{\begin{array}{ll}{\xi }_1{\rho }_{i,t}^{\mathrm{D}-\mathrm{E},\left(k\right)}\hfill & \max \left\{M_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}>\nu \max \left\{D_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}\hfill \\ {\displaystyle \frac{{\rho }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}}{{\xi }_2}}\hfill & \max \left\{D_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}>\nu \max \left\{M_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right\}\hfill \\ {\rho }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\hfill & \mathrm{other}\hfill \end{array}$$

(59)

$${\lambda }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r+1\right)}={\lambda }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}+{\rho }_{i,t}^{\mathrm{D}-\mathrm{E},\left(r+1\right)}\left(P_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}-{\overline{P}}_{i,t}^{\mathrm{D}-\mathrm{E},\left(r\right)}\right)$$

(60)

where ξ₁ and ξ₂ are the update coefficients of penalty parameters; υ is the update judgment coefficient.

7. Case Studies

The case studies are completed on a computer with an Intel i7-14700K CPU and 32 GB of RAM. The programs are developed using Matlab R2023a and are solved by Gurobi 11.0.0. The relative gap of the MILP solver is 0.1%.

7.1. Test System

The proposed method is tested using the topology shown in Figure 3, containing a 33-node DN and a 44-node DHN [34]. In the DN, nodes 13 and 28 are connected to PV with parameters of 1600 kW and 600 kW, nodes 6 and 24 are connected to WT with parameters of 600 kW and 300 kW, and nodes 6, 24, and 28 are connected to BESS with parameters of 150 kW/300 kWh, 100 kW/200 kWh, and 200 kW/400 kWh, respectively. In the DHN, node 1 is a heat station equipped with 1000 kW CHP, 2000 kW GB, and 2000 kW EH, where CHP and EH are connected to the DN node 13.

7.2. Analysis of the Effectiveness of the Proposed Distributed Cooperative Dispatch Method

In order to verify the effectiveness of the proposed distributed cooperative dispatch method, the centralized dispatch method is used as a comparison [34].

Table 1. Comparison of system cost of the proposed method and centralized dispatch method.

Cost (103 CNY)	Proposed Method	Centralized Dispatch Method
CDN	9.45	9.38
CDHN	13.76	13.72
CBESS	−0.105	−0.117
Total	23.10	22.98

shows the comparison of the system costs of the two methods.

Table 2. Comparison of computation time of the proposed method and centralized dispatch method.

	Proposed Method	Centralized Dispatch Method
Computation time	75.46 s	2.98 s

shows the comparison of the calculation times of the two methods.

The comparison in Table 1 shows that the operation costs of DN, DHN, and BESS of the two methods have a very small difference, and the proposed method differs from the centralized dispatch method by only 0.52% in the total system cost, which proves that the proposed distributed cooperative dispatch method does not fall into the local optimum during the solution process and has the same solution accuracy as the centralized dispatch method. The comparison in Table 2 shows that although the solution time of the proposed distributed cooperative dispatch method is much more than that of the centralized dispatch method, it is still within the acceptable range.

In summary, it can be concluded that the proposed distributed cooperative dispatch method can achieve the same solution accuracy as the centralized dispatch method.

7.3. Analysis of the Efficiency of the Proposed Distributed Cooperative Dispatch Method

In order to verify the efficiency of the proposed distributed cooperative dispatch method, the independent dispatch method is used as a comparison. The independent dispatch method indicates that the optimal dispatch models for DHN and BESS are first solved to obtain the boundary power, and then the optimal dispatch model for DN is solved to obtain the dispatch scheme.

Table 3. Comparison of system cost of the proposed method and independent dispatch method.

Cost (103 CNY)	Proposed Method	Independent Dispatch Method
CDN	9.45	24.33
CDHN	13.76	11.23
CBESS	−0.105	−0.307
Total	23.10	35.26

shows the comparison of the system costs of the two methods.

Table 4. Comparison of curtailed energy for PV and WT of the proposed method and independent dispatch method.

	Curtailed Energy for PV (kWh)	Curtailed Energy for WT (kWh)
Proposed method	1185.3	0
Independent dispatch method	8578.5	0

shows the comparison of curtailed energy for PV and WT of the two methods. Figure 4 shows the comparison of curtailed power for PV and WT of the two methods. Figure 5 shows the comparison of the power purchased from the superior grid by DN of the two methods.

As shown in Table 3, compared to the independent dispatch method, the operation cost of DN is reduced by 61.2%, the operation cost of DHN is increased by 22.5%, the profitability of BESS is reduced by 65.8%, and the total cost of the system is reduced by 34.5% of the proposed method. This is because the proposed method achieves efficient synergy among DN, DHN, and BESS, which cannot be achieved by an independent dispatch method. As shown in Table 4, compared with the independent dispatch method, the curtailed energy for PV of the proposed method is reduced by 7393.2 kWh. This is because in 7:00–18:00, the curtailed power for PV and WT of the proposed method is significantly smaller than that of the independent dispatch method, as shown in Figure 4. Further, the penalty cost for curtailment for PV and WT of the proposed method is reduced by 86.2% compared with the independent dispatch method. Due to the higher capacity of PV and WT power consumption of the proposed method, less power is purchased from the superior grid compared to the independent dispatch method, as shown in Figure 5. As a result, the cost of purchased electricity from the superior grid of the proposed method is reduced by 7.0% compared to the independent dispatch method. Ultimately, the total system cost of the proposed method is reduced by 34.5% compared to the independent dispatch method.

In summary, compared with the independent dispatch method, the proposed distributed cooperative dispatch method realizes the efficient synergy between DN and the two types of flexible resources, DHN and BESS, which can effectively improve the absorption capacity of DN for PV and WT and then reduce the operation cost.

7.4. Analysis of the Impact of Heat Storage Capacity on Cooperative Dispatch

In order to analyze the impact of heat storage capacity on cooperative dispatch, the following case is set up for comparison with the proposed method.

Case 1-1: adopt the exiting distributed cooperative dispatch method in [22], i.e., the heat storage capacity of DHN is not taken into account.

Table 5. Comparison of system cost of the proposed method and Case 1-1.

Cost (103 CNY)	Proposed Method	Case 1-1
CDN	9.45	11.54
CHN	13.76	14.41
CES	−0.105	−0.221
Total	23.10	25.73

shows the comparison of the system cost of the proposed method and Case 1-1.

Table 6. Comparison of curtailed energy for PV and WT of the proposed method and Case 1-1.

	Curtailed Energy for PV (kWh)	Curtailed Energy for WT (kWh)
Proposed method	1185.3	0
Case 1-1	2349.6	0

shows the comparison of curtailed energy for PV and WT of the proposed method and Case 1-1. Figure 6 shows the comparison of the water supply temperature of DHN of the proposed method and Case 1-1. Figure 7 shows the comparison of the heat generation power of DHN for the proposed method and Case 1-1. Figure 8 shows the comparison of curtailed power for PV and WT of the proposed method and Case 1-1.

As seen in Table 5, compared to Case 1-1, the operation cost of DN of the proposed method is reduced by 18.1%, the operation cost of DHN is reduced by 2.0%, and the profitability of BESS is reduced by 52.5%, which ultimately results in a reduction of the total system cost by 10.2%. This is because the proposed method takes into account the heat storage capacity of DHN, which significantly increases the operational flexibility of the system. As shown in Figure 6 and Figure 7, under the proposed method, DHN can flexibly adjust the water supply temperature to change the heat generation power. In the period when the output of PV and WT is large, DHN raises the water supply temperature to increase the heat generation power, which converts the electric energy of PV and WT that cannot be absorbed by DN into the heat energy of water and increases the heat storage capacity. In the period when the output of PV and WT is small, DHN reduces the heat generation power and utilizes the heat storage to provide the auxiliary power support, which achieves the effect of increasing the absorption capacity of DN for PV and WT. As shown in Figure 8, the curtailed power for PV and WT of the proposed method is significantly smaller than that of Case 1-1 in several periods. So as shown in Table 6, the curtailed energy for PV of the proposed method is reduced by 1164.3 kWh compared to Case 1-1, and consequently the penalty cost for curtailment for PV and WT of the proposed method is reduced by 49.6% compared with that of Case 1-1.

From the above analysis, it can be concluded that the proposed method considering the heat storage capacity of DHN can provide additional regulation capacity for DN by flexibly adjusting the water supply temperature of DHN, which in turn improves the absorption capacity of DN for PV and WT and then reduces the system operation cost.

7.5. Analysis of the Impact of BESS Capacity on Cooperative Dispatch

In order to analyze the impact of BESS capacity on cooperative dispatch, the following four cases are set up for comparison.

Case 2-1: Reduce the capacity of the energy storage in Section 7.1 by 50%.

Case 2-2: The capacity of the energy storage in Section 7.1 is unchanged.

Case 2-3: Increase the capacity of the energy storage in Section 7.1 by 50%.

Case 2-4: Increase the capacity of the energy storage in Section 7.1 by 100%.

Table 7. Comparison of system costs of the four cases.

Cost (103 CNY)	Case 2-1	Case 2-2	Case 2-2	Case 2-3
CDN	9.82	9.45	9.42	9.41
CHN	14.27	13.76	13.66	13.65
CES	−0.087	−0.105	−0.205	−0.217
Total	24.00	23.10	22.88	22.84

shows the comparison of system costs for the four cases.

Table 8. Comparison of curtailed energy for PV and WT of the four cases.

	Curtailed Energy for PV (kWh)	Curtailed Energy for WT (kWh)
Case 2-1	1222.9	0
Case 2-2	1185.3	0
Case 2-3	1146.3	0
Case 2-4	1135.1	0

shows the comparison of curtailed energy for PV and WT of the four cases. Figure 9 shows the trend in total system cost and curtailed energy for PV and WT of the four cases.

As seen in Table 7, compared to Case 2-2, the operation cost of DN in Case 2-1 is increased by 3.9%, the operation cost of DHN is increased by 3.7%, and the profitability of BESS is reduced by 17.1%, which ultimately results in an increase of the total system cost by 3.9%. This is because when the BESS capacity is reduced, the regulation capacity it can provide to the DN is weakened, and therefore the absorption capacity of DN for PV and WT is reduced. As shown in Table 8, the curtailed energy for PV and WT of Case 2-1 is increased by 3.2% compared to that of Case 2-2. As seen in Table 7, compared to Case 2-2, the operation costs of DN of Case 2-3 and Case 2-4 are reduced by 0.32% and 0.42%, respectively; the operation costs of DHN of Case 2-3 and Case 2-4 are reduced by 0.73% and 0.80%, respectively; and the profitability of BESS of Case 2-3 and Case 2-4 is increased by 95.2% and 106.7%, respectively, which ultimately results in a reduction of the total system cost by 0.95% and 1.1%, respectively. This is because when the BESS capacity is increased, the regulation capacity it can provide to the DN is increased, and therefore the absorption capacity of DN for PV and WT is increased. As shown in Table 8, the curtailed energy for PV and WT of Cases 2-3 and 2-4 is reduced by 3.3% and 4.2% compared to that of Case 2-2.

In addition, as seen in Figure 9, with the increase of BESS capacity, the reduction of the total system cost and the curtailed energy for PV and WT gradually become slower. Therefore, increasing the BESS capacity within a certain range can reduce the total system cost and improve the dispatch economy.

8. Conclusions

In this paper, a distributed cooperative dispatch method for DN with DHN and BESS considering flexible regulation capability is proposed. According to the simulation results, the following conclusions are drawn:

(1) The proposed distributed cooperative dispatch method does not fall into the local optimum during the solution process and differs from the centralized dispatch method by only 0.52% in the total system cost, which indicates that it can achieve the same solution accuracy as the centralized dispatch method.

(2) Compared with the independent dispatch method, the proposed distributed cooperative dispatch method realizes the efficient synergy between DN and the two types of flexible resources, DHN and BESS, which can effectively improve the absorption capacity of DN for PV and WT and then reduce the operation cost of the system by 35.6%.

(3) The proposed method, considering the heat storage capacity of DHN, can provide additional regulation capacity for DN by flexibly adjusting the water supply temperature of DHN, which in turn improves the absorption capacity of DN for PV and WT and then reduces the system operation cost.

In the future, the authors will further consider flexible resources such as electric vehicles in distributed cooperative dispatch to provide more regulation capability for the distribution network and further investigate how to take into account the impact of uncertainty of renewable energy in the distribution network in cooperative dispatch. In addition, the authors will apply the proposed method to a specific case in Wuxi City in the future to further test the performance of the proposed methods in practical applications.