Research on Integrated Energy Distributed Sharing in Distribution Network Considering AC Power Flow and Demand Response

Abstract

Under the constraint of the AC power flow architecture considering reactive power regulation, the examination of integrated energy distributed transactions among multiple subsystems can promote the research in the field of energy sharing. It is difficult to fully cover the consideration of AC power flow, demand response, integrated energy, and other factors in traditional related research. In response, a study is therefore conducted in this paper on integrated energy sharing in the distribution network. First, this paper introduces the system operation framework of the proposed distribution network model, and explains the interaction between all the players. Secondly, a distribution network power flow model and an integrated energy subsystem model are respectively. In particular, the subsystem model specifically considers new energy, demand response, integrated energy, and other factors. Then, a cooperative game model is constructed based on the cooperative relationship among subsystems in the distribution network system, followed by the analysis of the benefits brought by cooperation to the distribution network and the subsystems themselves. Finally, a distributed solution flow is established for the model based on the Alternating Direction Method of Multipliers (ADMM) algorithm. The results of the example analysis reveal the effectiveness of the model proposed in increasing the degree of energy utilization and further absorbing new energy in the distribution network system, each subsystem can generate up to 12% more absorption capacity than it would otherwise operate separately to accommodate more renewable energy in the distribution system.

1. Introduction

Environmental pollution and energy shortage have accelerated the global trend of pursuing low-carbon and clean energy development. The energy field is now undergoing and will continue to undergo extensive and profound changes. The integrated energy system (IES), which can enable multiple energy sources to engage in complementation and coordination and ensure the cascade usage of energy [1], offers an inevitable way to deal with a series of issues such as traditional energy depletion, environmental pollution, and global climate degradation.

The active distribution network (ADN) re-lies on the autonomous control of the IES downward, and dynamically interacts with the main network upward. The connection of more IESs and distributed generation (DG) to the distribution network may give rise to some problems such as bidirectional energy flow in the distribution network, increased loss of the system, and complex dispatching management [2]. The optimal operation of the IES and ADN has been widely studied. In Refs. [3,4,5], the operation strategy of regional IESs is optimized through applying intelligent optimization algorithms and planning methods with the goal of improving economic performance, promoting energy conservation, and achieving emission reduction. However, these studies focus on a single IES or multiple IESs, but ignore the optimal dispatching of the upper-layer ADN in the IES. In Refs. [6,7,8], the hierarchical optimization of the ADN with multiple IESs is studied to improve economic benefits, environmental benefits, and power quality. Nevertheless, only the power interaction between IESs is considered, without the involvement of more energy forms. When it comes to the research on the incorporation of integrated energy into the ADN, (1) no consideration is given to the impact of the interaction of multiple integrated energy sources on the coordinated operation of the ADN and multiple IESs and to the comprehensive exploration of the characteristics of Energy internet. (2) As for the complex model to optimize ADN-IES cooperative operation, the major problem lies in the insufficient optimization ability of the existing intelligent algorithms.

The IES-ADN coordinated planning can improve social and economic benefits to a large extent [9]. Nonetheless, among the large number of devices in the IES, those devices that consume electric energy to produce other energy sources consume reactive power in addition to active power. When coupled with the ADN, the node voltage of the ADN would be affected. Therefore, it is necessary to adopt AC power flow to consider the balance of both active power and reactive power when seeking the coordinated planning of the ADN substation that includes integrated energy production components. Meanwhile, because the integrated energy production component model and multi-energy network model are often complex, it is necessary to find a suitable balance between modeling accuracy and solution accuracy. The difficulty of unified planning would be further increased if the random fluctuations of new energy and load are also taken into consideration.

At present, the methods for multi-IES energy management are mainly divided into centralized and distributed ones. In centralized energy management methods, power generation and consumption information are collectively collected and processed by the administrator. After the formation of the cost function, it is solved by particle swarm optimization algorithm [10,11,12,13], genetic algorithm [14], etc. to obtain the best operation scheme. However, the adoption of a single-point to multi-point communication mode in centralized methods [15] may lead to communication overload, privacy exposure, and other problems. In order to better protect user privacy, extensive re-searches have been carried out in academic on distributed methods [16]. In Ref. [17], a dis-tributed hierarchical scheme is employed to manage energy in multiple IESs with star topology, but this method cannot be well applicable to island multiple IESs. In Ref. [18], a distributed control algorithm for island multiple IESs is proposed, in which the multi-agent system (MAS) technology is used to optimize the communication process and control process. Nevertheless, the methods above neither thoroughly consider the self-interest of each IES as a stakeholder nor fully protect private information. Alternating Direction Multiplier Method (ADMM) is quite suitable for large-scale distributed computing systems [19,20], since there is no need to centralize the computing of IES information. Instead, each IES solves the objective function by itself and then updates and iterates the multipliers ac-cording to the constraint at the multi-IES syetem, thus minimizing the operating cost of each IES while achieving energy balance in the whole system.

With the development of the IES, there are inevitable conflicts of interest arising from the participation of multiple players in the market competition, gradually highlighting the applicability of game theory to IES-related optimization problems. The difference between relevant studies lies in the focus of the construction and expansion of system lines and devices in the planning model, with the former paying more attention to supply-demand interaction and the optimization and allocation of resources. Nevertheless, they have some similarities in the solution methods due to their common involvement of optimization issues. The IES game optimization models proposed upon a non-cooperative game in Refs. [21,22] for solving the competition of multiple parks in electricity purchase under the premise of ensuring privacy can better balance the interests of multiple parks. Ref. [23], guided by cooperative game theory, examines different integrated energy investment players and their capacity planning under the mode of cooperation and uses the Shapley value method to distribute benefits among players, so as to improve the enthusiasm of multiple IES players to engage in collaborative operation. Non-cooperative game discusses the existence of a Nash equilibrium, focusing on the competition between multiple players. Different from the non-cooperative game, a cooperative game pays more attention to the promotion of collective interests and whether cooperative surplus is distributed fairly and reasonably. The emphasis of this paper is put on a cooperative game.

Intelligent algorithms assist in optimizing the dispatch of integrated energy systems, improving overall efficiency and resource utilization. In Ref. [24], a two-stage optimization strategy for electric vehicle charging and discharging was developed using particle swarm optimization, considering elasticity demand response. The proposed strategy not only reduces charging costs by 40~110% and distribution network load variance by 19~100%, but also demonstrates that the battery wear cost from cyclic charging and discharging is minimal compared to the benefits of discharging. In Ref. [25], this paper introduces a new microgrid framework that utilizes a hybrid of parallel connected inductive-coupling inverters and capacitive-coupling inverters, with the latter being more cost-effective for reactive power conditioning. An adaptive power sharing method, supported by a back-propagation neural network control layer, was developed to reduce total rated power and losses, and its efficacy was confirmed through simulations and experiments. In Ref. [26], this study presents an optimal energy management system for integrating photovoltaic systems, energy storage systems, and electric vehicles within a university campus, aiming to enhance sustainability. Simulations reveal that the proposed energy management system can reduce energy consumption costs by approximately 45%, with electric vehicles contributing significantly as both energy sources and loads, and the system’s resilience during power outages is also examined. In Ref. [27], this study introduces a novel hybrid heap-based and jellyfish search algorithm that merges the strengths of both the heap-based algorithm and the jellyfish search algorithm to address the combined heat and power economic dispatch problem. Experimental results on various system sizes indicate that the hybrid heap-based and jellyfish search algorithm consistently outperforms the standard heap-based algorithm, jellyfish search algorithm, and other techniques, showcasing its superior stability and efficiency in minimizing the whole fuel cost of power and heat generation. In Ref. [28], this study employs the Manta Ray Foraging Optimization Algorithm, a recent meta-heuristic optimization method, to address the Economic Power and Heat Dispatch problem in Cogeneration Energy Systems, aiming to minimize the total fuel cost while adhering to operational constraints. Testing on three systems (5 units, 7 units, and 48 units) showcases the Manta Ray Foraging Optimization Algorithm’s superior performance in achieving lower total fuel cost compared to other optimization techniques, highlighting its potential for Economic Power and Heat Dispatch challenges in Cogeneration Energy Systems. In Ref. [29], this research introduces an Improved version of the Heap-based Technique with enhanced discriminatory attributes and adaptive parameters, designed to optimize the Combined Heat and Power Economic Dispatch by minimizing fuel costs while meeting heat and power demands. Tested on various scales and benchmark functions, the Improved version of the Heap-based Technique demonstrates superior performance and resilience in finding optimal solutions for Combined Heat and Power Economic Dispatch compared to traditional methods. In Ref. [30], this study introduces a Modified Artificial Ecosystem Algorithm incorporating a Fitness Distance Balance Model to enhance the Economic Dispatch of Combined Heat and Power Units, demonstrating significant improvements in robustness, convergence, and solution quality compared to traditional methods and other reported results. In Ref. [31], this study introduces a multi-objective teaching-learning studying-based algorithm enhanced with Pareto archiving and a fuzzy decision-making approach for the Combined Heat and Power Economic Environmental Dispatch problem, demonstrating superior performance in minimizing fuel costs and emissions across multiple test systems compared to established algorithms like NSGA-II and SPEA 2.

In summary, this paper examines the cooperative operation of the ADN and IES: firstly, an IES scheduling model is established for the simultaneous supply of cooling, heating, and electricity, in which energy storage device, renewable energy, and demand response are also considered. Then, the scale model is added with AC power flow to model the distribution network, through which the impact of reactive power consumed by various devices in the IES on the voltage stability of the ADN can be considered. The cooperative relationship among subsystems in the ADN system is extracted and analyzed to build a cooperative game model, followed by the analysis of the benefits of cooperation for the ADN and subsystems. Finally, the distributed solution work-flow is established for the model based on ADMM algorithm. The IEEE33 node distribution system is used in example analysis to verify that the proposed model can ensure promising economic benefits and high practical application capability under the premise of protecting user privacy and meeting the constraint on system power flow.

The coupling and uncertainty within integrated energy systems are further deepened with the development of distributed resources and increase in the installed capacity of renewable energy. When it comes to energy revolution, one of the inevitable choices is the creation of integrated energy systems that can meet diversified demands (cooling, heating, electricity, gas, etc.) by making it possible for distributed energy and new technologies to realize deep integration.

The integrated energy system built herein aims to meet this need, and its energy sharing framework is shown in Figure 1. Different subsystems relate to each other through power lines and heat pipes. Due to the differences between systems in new energy curve and load curve, adjacent subsystems can form a cooperative alliance to further absorb new energy in the distribution network system. The multi-subsystem group has the following relationship with the distribution network: at the upper layer, energy management is performed by the distribution network operator which provides the lower layer with an energy sharing platform by maintaining AC power flow constraints. At the lower layer, multiple subsystems engage in energy trading with the help of the plat-form, and cooperate with the upper layer in implementing the reactive power optimization strategy. The distribution network operator and multiple sub-systems are equipped with communication devices, covering price information, energy sharing information, etc. All the concerned stages of the work in Figure A1.

Figure 2 shows the park IES integrated energy subsystem examined herein. Specifically, the sub-system mainly includes new energy, heat-electric cogeneration, electric boiler, energy storage, electric load, thermal load, etc. For the subsystem, there are two states: grid-connected operation and isolated operation. During grid-connected operation, the subsystem can access the distribution network through power lines to participate in system energy sharing and trade with the distribution network; during isolated operation, it is self-reliant in balancing power balance. In this system, for the supply side, the renewable energy sources including the wind power device and the photovoltaic power generation device give rise to power using natural resources. The gas-fired turbine shows strong coupling characteristics in that it generates electric energy and thermal energy. The electric boiler can use electric energy to generate heat, thus delivering a certain amount of heat energy to users. With the timely transfer of energy by the energy storage device, the flexibility of system energy can be improved effectively. For the load side, users mainly need electric and thermal loads, and the daily flexible adjustment of different energy loads. All parameters and abbreviations of IES are presented in Appendix A (

Table A1. All parameters of the IES.

Symbol	Description
$i$	energy storage device
$t$	period of time
$E_{\mathrm{i},\mathrm{t}}$	storage capacity of $i$ during $t$
$P_{i,char,t}$	energy storage and charging power of $i$ during $t$
$P_{i,dis,t}$	energy storage and discharging power of $i$ during $t$
${\eta }^{loss}$	energy storage loss rate of $i$
${\eta }^{char}$	charging efficiency of $i$
${\eta }^{dis}$	discharging efficiency of $i$
$E_{i,min}$	lower limits of $E_{\mathrm{i},\mathrm{t}}$
$E_{i,max}$	upper and lower limits of $E_{\mathrm{i},\mathrm{t}}$
$P_{i,char,max}$	largest charging power
$P_{i,dis,max}$	largest discharging power
$P_{i,s}^t$	electricity sold by the upper-layer operator during $t$
$P_{i,b}^t$	electricity bought by lower-layer subsystems during $t$
$P_{i,Y,min}^t$	lower limit on the electricity traded between the lower layer and the upper layer
$P_{i,Y,max}^t$	upper limit on the electricity traded between the lower layer and the upper layer
${L_e}^t$	post-participation electric load
${L_g}^t$	post-participation thermal load
${L_{\mathrm{e},\mathrm{o}}}^t$	initially predicted pre-participation electric load
${L_{\mathrm{g},\mathrm{o}}}^t$	initially predicted pre-participation thermal load
$\mathsf{\Delta }{L_{\mathrm{e}}}^{\mathrm{t}}$	changes in electric load response
$\mathsf{\Delta }{L_{\mathrm{g}}}^{\mathrm{t}}$	changes in thermal load response
$V_{\mathrm{in}}$	cut-in wind speed
$V_r$	rated wind speed
$V_{\mathrm{out}}$	cut-out wind speed
$r$	actual light intensity in the current period
$r_{max}$	maximum light intensity in the current period
$\alpha$/$\beta$	Beta distribution shape parameters
$p_{\mathrm{rated}}^{\mathrm{p}}$	rated light intensity of the rated output power
$r_{rated}$	rated light intensity of the PV module
$P_{PV,t}$	maximum of the predicted output of the photovoltaic device during the period t
$P_{FJ,t}$	maximum of the predicted output of the fan during the period t
$P_{i-j,e}^t$	interaction between the $i$th subsystem and the $j\mathrm{t}\mathrm{h}$ subsystem in terms of electric power within a day
$C_{i,MT}$	the cost incurred by the gas-fired unit in the subsystem $\mathrm{i}$
$p_{i-j,g}^t$	thermal power interaction between the $i\mathrm{t}\mathrm{h}$ subsystem and the jth subsystem within a day
$C_{i,Y}$	the cost incurred from the purchasing of electric power from the distribution network operator
$C_{i,ES}$	the cost incurred from the storage of energy
$C_i^{MG}$	objective function of the $\mathrm{i}$th subsystem
$C_{\mathrm{i}}^{\mathrm{P}2\mathrm{P}}$	the cost of interaction with the distribution network
$r_{\mathrm{j}-\mathrm{l}}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}$/$r_{\mathrm{j}-\mathrm{l}}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}$	represent dual variables (Lagrange multiplier)
${\rho }_1$/${\rho }_2$	penalty parameters

and

Table A2. All abbreviations of IES.

Symbol	Abbreviation
AC	Alternating Current
ADMM	Alternating Direction Method of Multipliers
ADN	Active Distribution Networks
IES	Integrated Energy Systems
DG	Distributed Generation
DC	Direct Current
PV	Photovoltaic
MAS	Multi-Agent System

).

2. Mathematical Model of AC Optimal Power Flow

The AC power flow model is a cornerstone in power system analysis. It’s used to describe and calculate the voltage magnitude, voltage phase angles, and power flow in transmission lines within a power system. It’s why establishing an AC power flow model is essential: Compared to the DC power flow model, the AC power flow model provides a more accurate representation of the actual operating conditions of a power system. It takes into account both active and reactive power, as well as variations in voltage magnitude and phase angles. The AC power flow model is crucial for assessing the stability of a power system, especially in response to significant disturbances or unexpected events. Power system operators rely on the AC power flow model for daily system dispatch to ensure both the reliability and economic efficiency of power delivery. In a power market environment, the AC power flow model underpins price formation and power trading decisions.

The power flow model examined is a multi-period dynamic optimization power flow model. Based on the dynamics of second-order cone relaxation, the problem is made convex to facilitate solving it with a solver. The AC optimal power flow model simplified for the convenience of expression is as follows:

(1)

Constraint on node power balance

$$\sum _{i\in {\Psi }_n^G}{g}_{it}+\sum _{r\in {\Psi }_n^{RE}}{p}_{rt}+\sum _{s\in {\Psi }_n^S}{p}_{st}-\sum _{l:s\left(l\right)=n}{f}_{lt}+\sum _{l:r\left(l\right)=n}{f}_{lt}=\sum _{j\in {\Psi }_n^D}{d}_{jt} \forall n,\forall t$$

(1)

where $r$ represents the new energy unit; $t$ represents the period of time; $s$ represents the energy storage device; $p_{rt}$ stands for the generating power of $r$ during $t$; $p_{st}$ represents the power of $s$ during $t$; $f_{lt}$ represents the active power flow of the transmission line $l$, to which the node $n$ is attached; $d_{jt}$ represents the load power; ${\Psi }_n^G,{\Psi }_n^{RE},{\Psi }_n^S,{ \Psi }_n^D$ are the sets corresponding to the electric generator, new energy unit, energy storage, and load on the node n, respectively; $r\left(l\right),s\left(l\right)$ represent the head and end nodes of the line $l$, respectively.

(2)

Constraint on line power flow

$$\left\{\begin{array}{ccc}-f_l^{max}\le f_{lt}\le f_l^{max}& l\in {\Omega }^{L_0}& \forall t\\ f_{lt}-\frac{\left[{\theta }_{s\left(l\right)t}-{\theta }_{r\left(l\right)t}\right]}{X_l}& l\in {\Omega }^{L_0}& \forall t\end{array}\right.$$

(2)

where $X_l$ represents the line reactance; $f_l^{max}$ represents the line capacity; ${\theta }_{s\left(l\right)t},{\theta }_{r\left(l\right)t}$ represent the voltage phase angle of the head and end nodes of the line l, respectively; ${\mathsf{\Omega }}^{L_0}$ represents the set of existing lines.

(3)

Constraint on node voltage phase angle

$$\left\{\begin{array}{c}-\pi \le {\theta }_{s\left(l\right),t}\le \pi \\ -\pi \le {\theta }_{r\left(l\right),t}\le \pi \end{array}\right.$$

(3)

where the node voltage phase angle shall be guar-anteed to be within the constraint range.

(4)

Objective function:

In the comprehensive consideration of factors such as renewable energy absorption and voltage excursion, the normalized objective function of the system is as follows:

$$\mathit{min}{C}_P={\sum }_{t\in {\Omega }^T}{\sum }_{m\in {\Omega }^{RE}}{P}_{mt}d{u}_t{C}_{ab}+\sum _{t\in {\Omega }^T}\sum _{i\in {\Omega }^G}{\kappa }_t{g}_{it}{G}_i+\sum _{t\in {\Omega }^T}(-p_{gs}^t{P}_{gs}^t+p_{gb}^t{P}_{gb}^t)+\sum _{t\in {\Omega }^T}{C}_A+\sum _{i\in {\Omega }^I}\sum _{t\in {\Omega }^T}{C}_J\left|U_i-U_N\right|$$

(4)

$${\kappa }_t=\frac{1}{(1+r{)}^{t-1}}$$

(5)

where $C_A$ stands for the cost arising from penalty for light and wind curtailment; $P_{mt}$ represents the wind and light curtailment power of the node m during $t$; $C_{ab}$ represents the penalty coefficient of wind and light curtailment; $g_{it}$ stands for the generating power of $i$ during $t$; $G_i$ stands for the operating cost coefficient of the unit ($/MWh). $p_{gs}^t$ stands for the purchasing price set by the large power grid at the upper layer during $t$, while $p_{gb}^t$ stands for the corresponding selling price. $P_{gs}^t$ and $P_{gb}^t$ stand for the amount of electricity bought and sold by the distribution network and users, respectively, during $t$; $C_J$ is the unit voltage excursion cost; ${\kappa }_t$ is the present value coefficient; r is the discount rate.

3. Mathematical Model of the Integrated Energy Subsystem

3.1. Heat-Electric Cogeneration

The heat-electric cogeneration unit of the subsystem $i$ gives rise to the following:

$$C_{i,MT}=\sum _{t\in T}\frac{P_{i,MT,t}}{{\eta }_{i,t}}\frac{C_{g,t}}{LH{V}_{gas}}$$

(6)

where $C_{g,t}$ is the price of gas bought during $t$; ${\eta }_{i,t}$ stands for the rate of conversion by the ith gas-fired unit during $t$; $LH{V}_{gas}$ is the low calorific value of natural gas; $P_{i,MT,t}$ stands for the electric power generated by the subsystem $i$ during $t$.

It should be noted that there is the following operational constraint:

$$P_{i,min}\le P_{i,MT,t}\le P_{i,max}$$

(7)

where $P_{i,max}$ and $P_{i,min}$ stand for the upper and lower limits of $P_{i,MT,t}$.

The electric and thermal coupling constraint is as follows:

$$P_{i,MT,t}=\delta P_{i,H,t}$$

(8)

where $P_{i,H,t}$ is the thermal power generated by the subsystem $i$ during $t$; $\delta$ is the heat-electric ratio of the heat-electric cogeneration unit.

3.2. Electric Boiler

As a typical electric-heat coupling device, the electric boiler consumes electric energy to generate heat energy ultimately for the purpose of meeting the need of thermal load and the heat storage tank. Under the guidance of time-sharing electricity prices, the electric boiler cooperates with the heat-electric cogeneration system to meet the demand of thermal load and increase the amount of electricity consumed during the valley period.

It should be noted that there is the following operational constraint:

$$P_{i,EH,min}\le P_{i,EH,t}\le P_{i,EH,max}$$

(9)

where $P_{i,EH,t}$ is the electric boiler’s thermal output; $P_{i,EH,min}$ and $P_{i,EH,max}$ stand for the upper and lower limits of $P_{i,EH,t}$.

The electric and thermal coupling constraint is as follows:

$$P_{i,EH,t}=\gamma P_{i,E,t}$$

(10)

where $P_{i,E,t}$ is the electric power consumed during $t$; $\gamma$ is the efficiency of the electric boiler.

3.3. Energy Storage Device

In the system, the flexibility of energy could be improved significantly in virtue of the timely transfer of energy by the energy storage device. In the access system, the following operating cost would be incurred when the energy storage device participates in coordinating and optimizing operation:

$$C_{i,ES}=\sum _{t\in T}{p}_{ES}(P_{i,char,t}+P_{i,dis,t})$$

(11)

The operational constraint is as follows:

$$E_{i,t}=E_{i,t-1}(1-{\eta }^{loss})+({\eta }^{char}{P}_{i,char,t}-\frac{P_{i,dis,t}}{{\eta }^{dis}})\mathsf{\Delta }t$$

(12)

$$E_{i,min}\le E_{i,t}\le E_{i,max}$$

(13)

$$0\le P_{i,char,t}\le {\alpha }_{i,char,t}{P}_{i,char,max}$$

(14)

$$0\le P_{i,dis,t}\le {\alpha }_{i,dis,t}{P}_{i,dis,max}$$

(15)

$${\alpha }_{i,char,t}+{\alpha }_{i,dis,t}\le 1$$

(16)

$$E_{i,1}=E_{i,T+1}$$

(17)

where $i$ represents the energy storage device; $t$ represents the period of time; $E_{i,t}$ represents the storage capacity of $i$ during $t$; $P_{i,\mathrm{char},t}$ stands for the energy storage and charging power of $i$ during $t$; $P_{i,dis,t}$ stands for the energy storage and discharging power of $i$ during $t$; ${\eta }^{loss}$ stands for the energy storage loss rate of $i$; ${\eta }^{char}$ stands for the charging efficiency of $i$; ${\eta }^{dis}$ stands for the discharging efficiency of $i$; $E_{i,min}$ and $E_{i,max}$ stand for the upper and lower limits of $E_{i,t}$; $P_{i,char,max}$ and $P_{i,dis,max}$ stand for the largest charging and discharging power; ${\alpha }_{i,char,t}$/${\alpha }_{i,dis,t}$, which is either 0 or 1, stand for the charging/discharging state, with 0 hinting at the suspension of the charging/discharging state, and 1 hinting at the proceeding of charging/discharging state.

3.4. Transaction Cost

The subsystem $i$ is assumed to incur the following transaction cost:

$$C_{i,Y}=\sum _{t\in T}(-p_{nb}^t{P}_{i,b}^t+p_{ns}^t{P}_{i,s}^t)$$

(18)

where $P_{i,s}^t$ stands for the electricity sold by the upper-layer operator during $t$; $P_{i,b}^t$ stands for the electricity bought by lower-layer subsystems during $t$.

It should be noted that there is the following operational constraint:

$$P_{i,Y,min}^t\le P_{i,b}^t P_{i,s}^t\le P_{i,Y,max}^t$$

(19)

where $P_{i,Y,min}^t$/$P_{i,Y,max}^t$ stands for the upper/lower limit on the electricity traded between the lower layer and the upper layer.

3.5. Demand Response

Load taking part in demand response is known as flexible load, while its counterpart absent in demand response is called rigid load. These two types of load are the main components of the integrated energy demand load. After taking part in demand response, there are the following electric and thermal loads:

$${L_e}^t={L_{\mathrm{e},\mathrm{o}}}^t+\mathsf{\Delta }{L_e}^t$$

(20)

$${L_g}^t={L_{\mathrm{g},\mathrm{o}}}^t+\mathsf{\Delta }{L_g}^t$$

(21)

where ${L_e}^t$ represents the post-participation electric load, and ${L_g}^t$ stands for the post-participation thermal load; ${L_{\mathrm{e},\mathrm{o}}}^t$ stands for the initially predicted pre-participation electric load, while ${L_{\mathrm{g},\mathrm{o}}}^t$ stands for the initially predicted pre-participation thermal load; $\mathsf{\Delta }{L_e}^t$ and $\mathsf{\Delta }{L_g}^t$, respectively, represents the changes of electric and thermal load response.

Throughout the demand response process, there is constantly no change in total flexible load:

$$\sum _{t=1}^T\mathsf{\Delta }{L_e}^t=\sum _{t=1}^T\mathsf{\Delta }{L_g}^t=0$$

(22)

The following constraint should be met when it comes to the percentage of fall/rise in electric and thermal loads:

$$-\mathsf{\Delta }{L_e}^{max}\le \mathsf{\Delta }{L_e}^t\le \mathsf{\Delta }{L_e}^{max}$$

(23)

$$-\mathsf{\Delta }{L_g}^{max}\le \mathsf{\Delta }{L_g}^t\le \mathsf{\Delta }{L_g}^{max}$$

(24)

where $\mathsf{\Delta }{L_e}^{max}$ and $\mathsf{\Delta }{L_g}^{max}$ are, respectively, the allowable upper limits of $\mathsf{\Delta }{L_e}^t$ and $\mathsf{\Delta }{L_g}^t$.

3.6. Renewable Energy

Mathematical model of renewable energy:

In the case that the wind speed exceeds the cut-out wind speed or is below the cut-in wind speed, the fan has no output; the fan output is found to have a nearly linear relationship with the wind speed that ranges from the cut-in value to the rate value; it equals to the rated output in the case that there is a wind speed ranging from the rated to the cut-out values. The following formula can be used to approximately express the relationship between the fan output and the wind speed:

$$P_w(V)=\left\{\begin{array}{cc}0& 0\le V\le V_{\mathrm{in}}\\ P_r\frac{V-V_{\mathrm{in}}}{V_r-V_{\mathrm{in}}}& V_{\mathrm{in}}<V\le V_r\\ P_r& V_r<V\le V_{\mathrm{out}}\\ 0& V>V_{\mathrm{out}}\end{array}\right.$$

(25)

where $V_{\mathrm{in}}$ represents the cut-in wind speed; $V_r$ represents the rated wind speed; $V_{\mathrm{out}}$ represents the cut-out wind speed.

The solar illumination intensity is the most important factor affecting the actual output power of PV. In the planning study, the long-term light intensity can generally be considered to conform to Beta distribution, and its probability density function is as follows:

$$f(r)=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}({\frac{r}{r_{max}})}^{\alpha -1}\left[(1-\frac{r}{r_{max}}{)}^{\beta -1}\right]$$

(26)

where r and $r_{max}$ are the actual light intensity and maximum light intensity in the current period respectively; $\alpha$ and $\beta$ are Beta distribution shape parameters.

If the light intensity r at the location is known, the available output power $p^p$ of PV in the ideal state is a piecewise function: when r is lower than the rated light intensity $r_{rated}$ of the PV module, its output power increases linearly with the light intensity; when r reaches or exceeds $r_{rated}$, the output of PV is maintained at the level of its rated output power and expressed as follows:

$$p^p(r)=\left\{\begin{array}{cc}{p}_{\mathrm{rated}}^p\left(r/r_{\mathrm{rated}}\right)& r\le r_{\mathrm{rated}}\\ p_{\mathrm{rated}}^p& r>r_{\mathrm{rated}}\end{array}\right.$$

(27)

where $r_{rated}$ and $p_{\mathrm{rated}}^p$ are respectively the rated light intensity of the PV module and its corresponding rated output power.

The renewable energy output constraint is as follows:

$$0\le P_{pv,t}\le P_{PV,t}$$

(28)

$$0\le P_{\mathrm{fj},t}\le P_{FJ,t}$$

(29)

where $P_{PV,t}$ is the maximum of the predicted output of the photovoltaic device in the system during the period t; $P_{FJ,t}$ is the maximum of the predicted output of the fan in the system during the period t.

3.7. Interaction Coupling Constraint

$$P_{i-j,e}^{min}\le P_{i-j,e}^t\le P_{i-j,e}^{max}$$

(30)

where $P_{i-j,e}^t$ represents the interaction between the $i$th subsystem and the $j\mathrm{t}\mathrm{h}$ subsystem in terms of electric power within a day; $P_{i-j,e}^{min}$ and $P_{i-j,e}^{max}$, respectively, represent the largest and lowest value of $P_{i-j,e}^t$.

$$P_{i-j,g}^{min}\le p_{i-j,g}^t\le P_{i-j,g}^{max}$$

(31)

where $p_{i-j,g}^t$ is the thermal power interaction between the ith subsystem and the jth subsystem within a day; $P_{i-j,g}^{min}$ and $P_{i-j,g}^{max}$ respectively represent the largest and lowest value of $p_{i-j,g}^t$.

In terms of the general subsystem, the detail above gives description of all factors involved. Moreover, the need to satisfy the balance of energy in each subsystem by all energy sources is the same as what has been previously described.

4. Model of Cooperative Operation and Distributed Workflow

4.1. Model of Cooperative Operation

With the help of energy pipelines or power lines, there are interconnections among multiple integrated energy subsystems, so that different subsystems are allowed to absorb energy from others and transfer energy to each other during different periods. For the system group, the overall operating cost is obtained through the objective function. The following formula is used to express this function:

$$C_{MMGs}=\sum _{i\in I}(C_{i,MT}+C_{i,Y}+C_{i,ES})$$

(32)

where $C_{i,MT}$ is used to denote the cost incurred by the gas-fired unit in the subsystem $i$; $C_{i,Y}$ is used to denote the cost incurred from the purchasing of electric power from the distribution network operator; $C_{i,ES}$ is used to denote the cost incurred from the storage of energy.

According to cooperative game theory and the cooperative dispatching model of multiple subsystems designed in this paper, each subsystem is a participant of Nash bargaining, and the cooperative alliance is targeted at the optimization objective of maximizing the utility product of all participants, that is:

$$\left\{\begin{array}{c}{R}_i^{\mathrm{MG}}=\mathit{max}{\displaystyle \prod _{i\in N}^N}[C_i^0-(C_i^{\mathrm{MG}}+C_i^{\mathrm{P}2\mathrm{P}}\left)\right]\\ st.C_i^0-(C_i^{\mathrm{MG}}+C_i^{\mathrm{P}2\mathrm{P}})\ge 0\end{array}\right.$$

(33)

where $C_i^{MG}$ represents the objective function of the $i$th subsystem, while $C_i^{\mathrm{P}2\mathrm{P}}$ is used to denote the cost of interaction with the distribution network.

In terms of their own utility expectations, different players in the cooperative alliance obey the rule of diminishing income. The logarithmic function is used to describe the decision-making preferences and income expectations of different players. Therefore, the logarithm of Equation (33) is taken, and the product problem is converted into a sum problem, as shown in Equation (34).

$$\left\{\begin{array}{c}{R}_i^{\mathrm{MG}}=\mathit{min}{\displaystyle \prod _{i\in N}^N}-\mathit{ln}[C_i^0-(C_i^{\mathrm{MG}}+C_i^{\mathrm{P}2\mathrm{P}}\left)\right]\\ st.C_i^0-(C_i^{\mathrm{MG}}+C_i^{\mathrm{P}2\mathrm{P}})\ge 0\end{array}\right.$$

(34)

4.2. Distributed Workflow

In the process of Nash bargaining, each participant in the alliance takes the energy interaction between each other, namely the amount of electricity ${\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}$ and the amount of heat ${\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}$, as the bargaining strategy and meets $\left\{\begin{array}{c}{\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}={\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}\\ {\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}={\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}\end{array}\right.$. Then, the augmented Lagrangian function of the modified Nash bargaining model is obtained as follows:

$$\begin{gathered} L_j=\mathit{min}\sum _{j=1}^N-d_j\mathit{ln}[C_j^0-(C_j^{\mathrm{MG}}+C_j^{\mathrm{P}2\mathrm{P}})]+\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}})+\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_1}{2}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}{)}^2 \\ +\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}})+\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_2}{2}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}{)}^2 \end{gathered}$$

(35)

where $r_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}$ and $r_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}$ represent dual variables (Lagrange multiplier); ${\rho }_1$ and ${\rho }_2$ represent penalty parameters.

ADMM can decompose large-scale problems into smaller subproblems that can be solved in parallel. This makes it particularly suitable for distributed optimization scenarios. It can handle a variety of constraints, including equality and inequality constraints. Under certain conditions, ADMM is guaranteed to converge to the optimal solution.

ADMM algorithm is used to process the formula above hierarchically

(1)

Optimization model for lower-layer sub-problems

Based on fixed dual variables $r_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}$, $r_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}$ and coupling variables the amount of transaction is locally optimized by ${\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}$ and ${\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}$.

$$\begin{gathered} L_j=\mathit{min}\sum _{j=1}^N-d_j\mathit{ln}[C_j^0-(C_j^{\mathrm{MG}}+C_j^{\mathrm{P}2\mathrm{P}})]+\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}({\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}})+\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_1}{2}({\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}{)}^2 \\ +\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}({\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}})+\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_2}{2}({\lambda }_{j-l,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j,t}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}{)}^2 \end{gathered}$$

(36)

(2)

Model for upper-layer sub-problems:

To minimize the deviation from the “optimal amount of transaction”, dual variables and auxiliary variables are calculated and updated. The objective function is as follows:

$$\begin{gathered} \mathit{min}(\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}})+\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_1}{2}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{e}}{)}^2+\sum _{i=1,i\ne j}^N\sum _{t=1}^T{r}_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}) \\ +\sum _{i=1,i\ne j}^N\sum _{t=1}^T\frac{{\rho }_2}{2}({\lambda }_{j-l}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}-{\lambda }_{l-j}^{\mathrm{P}2\mathrm{P}\_\mathrm{g}}{)}^2) \end{gathered}$$

(37)

5. Example Analysis

5.1. Example Setting

One AC distribution network system and two electric-heat integrated energy subsystems are involved in the scenarios considered. The standard IEEE33 node system is selected for the AC distribution network. The specific parameters of the electric-heat integrated energy subsystem are within a specific scope, it is permitted to adjust the thermal and electric loads, without any change before and after adjustment in the sum of electric and gas loads; 3% and 5% are the upper limits of the proportion of permitted adjustment within a day in thermal and electric loads respectively [5]. When it comes to the energy storage battery, its rated capacity is set to be 300 kWh, while its largest charging and discharging power is 60 kW; for the state of charge, the initial value is 0.2, the lowest value is 0.1, and the largest value is 0.9; 0.95 is the value of both its coefficients of charging and discharging efficiency. As for the heat storage tank, its rated capacity is 300 kWh, while its largest heat storage and release power is 60 kW; for the state of charge, the initial value is 0.5, the lowest value is 0.1, and the largest value is 0.9; 0.95 is the value of both its coefficients of charging and discharging efficiency. When it comes to the heat-electric cogeneration unit, its heat-electric ratio equals to 1.2; the efficiency of the electric boiler is 90%; the largest and lowest electric outputs of the heat-electric cogeneration unit are 500 kW and 0 kW separately; the climbing rate has upper and lower limits which are both 200 kW/h. When examining the three subsystems, their load forecast values and renewable energy output are modified with reference to the data of general industrial electric-heat users.

5.2. Example Results

To call CPLEX for solution, Yalmip language is adopted herein under the MATLAB 2021a compiling environment.

The following four scenarios are established for the purpose of verifying the model proposed:

Scenario 1: Each subsystem operates independently and trades directly with the large power grid, without the need to consider demand response within the subsystem.

Scenario 2: Each subsystem operates cooperatively and trades directly with the large power grid, with the need to consider demand response within the subsystem.

Scenario 3: All subsystems operate cooperatively without the need to consider demand response within the subsystem.

Scenario 4: All subsystems operate cooperatively, with the need to consider demand response within the subsystem.

Table 1. Operating cost of each stakeholder under Scenario 2 and Scenario 4.

Computing Mode	Distribution Network/yuan	IES 1/yuan	IES 2/yuan	Total Cost/yuan
Independent operation	195.36	7266.34	5635.23	13,096.93
Cooperative operation (centralized)	12.95	7202.00	5680.10	12,895.05
Cooperative operation (distributed)	12.36	7225.05	5639.38	12,876.79

and

Table 2. Operating cost of each stakeholder under Scenario 1 and Scenario 3.

Computing Mode	Distribution Network/yuan	IES 1/yuan	IES 2/yuan	Total Cost/yuan
Independent operation	195.36	7266.34	5635.23	13,096.93
Cooperative operation (centralized)	12.95	7333.51	5736.42	13,087.30
Cooperative operation (distributed)	12.61	7314.29	5740.41	13,067.31

show the cost results of optimization within the system where demand response is considered (Scenario 2 and Scenario 4) and demand response is not considered (Scenario 1 and Scenario 3).

From the analysis of data in Table 1, it can be concluded from the perspective whether demand response is considered in the subsystem that the consideration of demand response can reduce the cost in the subsystem, because demand response can realize the flexible adjustment of multiple energy loads in the subsystem. If the subsystem group is directly connected to the distribution network, the distribution network will receive further impact on its operating cost. The reason is that when the cooperation mode allows the subsystem group to interact with the distribution network in energy, so that the advantages of demand response can be extended to the distribution network. From the perspective whether the participation of subsystems in the cooperative operation is considered, the cooperation mode allows energy to be flexibly used between the subsystem group and the distribution network, further absorbs surplus new energy in the distribution network, releases the pressure of load shedding in each subsystem, and thus achieves the overall economic improvement of the subsystem group and the distribution network. In addition, a distributed solution process is designed based on ADMM algorithm. The operating costs obtained by distributed solution and centralized solution are similar, according to the table, which further explains the correctness of the distributed process proposed, showing that the method and model proposed in this paper meet corresponding engineering application standards.

Figure 3 shows the convergence of total operating cost, original residual, and dual residual in the distributed computing process under scenario 4.

In distributed optimization algorithms, the primal (original) residual and the dual residual are two commonly used indicators for convergence. They both serve to measure the discrepancy between the current solution and the optimal solution, but they arise from different considerations and have distinct implications:

Original Residual: It measures the discrepancy in the constraints of the original problem. Specifically, it quantifies how far the current solution is from satisfying the constraints of the primal problem. A high primal residual indicates that the current solution is far from feasible, i.e., it doesn’t satisfy the problem’s constraints well. As the algorithm progresses, the primal residual should decrease, indicating that the solution is becoming more feasible.

Dual Residual: It measures the discrepancy in the dual problem, which is derived from the original problem by swapping the roles of the objective function and the constraints. Specifically, the dual residual quantifies how much the dual variables change from one iteration to the next. A high dual residual indicates that the algorithm is making significant updates to the dual variables, suggesting that the solution might be far from optimal. As the algorithm progresses, the dual residual should decrease, indicating that the solution is approaching optimality.

Difference: While both residuals provide insights into the algorithm’s progress towards an optimal solution, the primal residual focuses on feasibility (satisfying constraints), whereas the dual residual focuses on optimality (minimizing or maximizing the objective function). In many algorithms, both residuals are used in tandem to ensure that the solution is both feasible and optimal.

As shown in Figure 3, after 16 times of distributed computing, the total operating cost, original residual, and dual residual in the system all guarantee convergence, which shows that the method designed has better convergence performance and does not require many tedious iterative calculations. Figure 4 shows the reactive power, active power, and voltage distribution of the node system under Scenario 4.

Synergistic cooperation between active distribution networks and integrated energy systems can lead to significant results in reactive power optimization. The following are some of the ways in which this synergy can affect reactive power optimization:

Voltage stability: Integrated energy systems can help maintain grid voltage stability by adjusting reactive power injection. When voltage levels need to be adjusted in the distribution network, the integrated energy system can provide reactive power on demand to ensure that the voltage remains within an appropriate range, thus preventing voltage fluctuations and voltage dips.

Current balancing: In distributed and integrated energy systems, various energy sources and loads can provide or absorb reactive power. By working together, these systems can balance currents and prevent current imbalances. This is essential to prevent overloading of equipment and to improve the stability of the grid.

Reactive power exchange: Collaboration enables the exchange and sharing of reactive power. A part of an integrated energy system can provide excess reactive power to meet the demands of the grid or other systems. This synergy reduces reactive power waste and improves resource utilization.

Power Factor Improvement: Integrated energy systems can improve the power factor, bringing it closer to 1. This helps to reduce the extra cost of electricity in the grid and improves the efficiency of energy transmission.

Taken together, synergies between distribution grids and integrated energy systems can effectively optimize reactive power, improve grid stability, lower energy costs, reduce energy waste, and facilitate the integration of clean and sustainable energy sources. This is important for realizing sustainable energy systems and improving grid reliability.

The active power of the node system under Scenario 3 is shown in Figure 5.

From a meticulous examination of Figure 4 and Figure 5, several pivotal insights emerge regarding the dynamics of the distribution network and its subsystems. When demand response is factored into the scenarios, the cooperative interplay between the distribution network and its associated subsystems becomes evident. This collaboration ensures that both reactive and active power, as well as voltage distribution, consistently remain within the stipulated parameters set by the distribution network. Such adherence to these parameters is paramount, as it guarantees the reliable and safe operation of the distribution network, a cornerstone for ensuring energy security and system stability.

Furthermore, a side-by-side comparison of Figure 4b and Figure 5 unveils another intriguing facet: the cooperative operation significantly enhances the distribution of active power throughout the system. This not only optimizes energy flow but also ensures a more balanced and efficient power distribution.

But the revelations don’t stop there. The intricate coupling relationship between active power, reactive power, and voltage plays a pivotal role in enhancing the overall system’s performance. This relationship, when harnessed effectively, can lead to improved voltage distribution. Such an improvement has a cascading benefit: it negates the need for deploying costly reactive power voltage regulating devices within the distribution network system. The result is a win-win: the system achieves voltage stability without necessitating alterations in the reactive power distribution. Moreover, this approach effectively minimizes voltage deviations, ensuring that the system operates at its optimal efficiency while reducing potential operational costs. The comparison of new energy absorption by the node system between Scenario 3 and Scenario 4 is shown in Figure 6.

According to what is presented in Figure 6, the cooperative operation allows the distribution network system to further absorb new energy. Specifically, the PV of the distribution network system decreases in varying degrees during 9:00~17:00 of the day. Due to the mutual aid between subsystems, each subsystem can generate more absorption capacity to accommodate more renewable energy in the distribution network system.

In Scenario 4, all subsystems are designed to operate in a cooperative manner, with an emphasis on incorporating demand response within each subsystem. This scenario showcases the synergy between the distribution network and the subsystems, ensuring a harmonious balance of electric and thermal energy supply and demand.

Figure 7 and Figure 8 delve into the electric-heat supply and demand dynamics of two distinct electric-heat integrated energy subsystems under this scenario. The visual representations in these figures underscore the equilibrium achieved in the system, highlighting the intricate interplay of energy flows.

From the data presented in these figures, it’s evident that the cooperative operation facilitates the distribution network and its subsystems to maintain the reactive power, active power, and voltage distribution within the stipulated range. This ensures the reliability and safety of the distribution network’s operations.

Figure 9 shows the curve of changes of subsystem 1 before and after demand response in under Scenario 4 in terms of electric and thermal loads.

According to Figure 9, at different times of the day, there can be transfers of electric and thermal loads in subsystem 1. Particularly, the electric load is mainly reduced during 0:00~9:00 and 18:00~24:00 of the day, and increased during 9:00~18:00 of the day; meanwhile, the thermal load is mainly increased during 0:00~9:00 and 16:00~24:00, and is reduced during 9:00~16:00. This trend is explained as follows: subsystem 1 is required to transfer the periods incurring a high total cost from unit operation, trading of electricity with the distribution network, trading with other subsystems, new energy absorption, and other activities. The subsystem should achieve balance between the cost benefit and loss benefit of response in all periods of the day for the purpose of improving the profitability of the system.

6. Conclusions

In consideration of the constraint of the AC power flow architecture, this paper designs a distributed transaction process based on ADMM algorithm, and examines the distributed transaction of multiple integrated energy sources among multiple subsystems. There are the following conclusions drawn from the analysis and discussion above:

(1)

We’ve developed a distributed transaction process using the ADMM algorithm, tailored for the efficient handling of multiple integrated energy sources across various subsystems.

(2)

Our cooperative game model, set within a distribution network framework, facilitates efficient energy sharing among subsystems, ensuring optimal resource utilization.

(3)

The model effectively manages reactive power deviations, minimizing the operational costs associated with reactive power devices in the upper-layer distribution network.

(4)

By incorporating factors like demand response, new energy absorption, and integrated energy, our approach maximizes the utilization of new energy within the distribution network.

(5)

Our distributed algorithm not only streamlines energy sharing but also reduces computational demands and safeguards user privacy during interactions.

As the landscape of integrated energy systems (IES) evolves and technological advancements continue, we foresee several key directions that will likely be the focus of research in the coming years:

Advanced Control Strategies: To further optimize the performance of IES, there will be a need to develop more advanced control strategies, especially leveraging artificial intelligence and machine learning techniques for real-time optimization and adaptive control.

Energy Storage Advancements: As the importance of energy storage devices has been highlighted, future research will likely delve into the development of more efficient, longer-lasting, and environmentally friendly storage solutions.

Integration of Renewable Resources: With the global push towards sustainability, integrating a higher percentage of renewable energy sources into IES will be crucial. Research will focus on addressing the challenges of variability and intermittency associated with renewables.

Grid Resilience and Security: As IES become more complex, ensuring the resilience of the system against external threats and disturbances will be paramount. This includes both physical security and cybersecurity measures.

Economic and Policy Implications: As technologies mature, there will be a need to study the economic viability of different IES configurations and the policy frameworks that can support their widespread adoption.

In conclusion, the future of IES is promising, with numerous opportunities for research, development, and real-world implementation. As this paper has laid the groundwork for understanding the current state of IES, we are optimistic about the innovations and advancements the future holds.