Decentralized Optimization of Electricity-Natural Gas Flow Considering Dynamic Characteristics of Networks

: The interconnection of power and natural gas systems can improve the ﬂexibility of system operation and the capacity of renewable energy consumption. It is necessary to consider the interaction between both, and carry out collaborative optimization of energy ﬂow. For space-time related line packs, this paper studies the optimal multi-energy ﬂow (OMEF) model of an integrated electricity-gas system, taking into account the dynamic characteristics of a natural gas system. Besides, in order to avoid the problem of large data collection in centralized algorithms and consider the characteristics of decentralized autonomous decision-making for each subsystem, this paper proposes a decentralized algorithm for the OMEF problem. This algorithm transforms the original non-convex OMEF problem into an iterative convex programming problem through penalty convex-concave procedure (PCCP), and then, uses the alternating direction method of multipliers (ADMM) algorithm at each iteration of PCCP to develop a decentralized collaborative optimization of power ﬂow and natural gas ﬂow. Finally, numerical simulations verify the e ﬀ ectiveness and accuracy of the algorithm proposed in this paper, and analyze the e ﬀ ects of dynamic characteristics of networks on system operation.


Introduction
With the energy crisis and global environmental issues becoming increasingly prominent, it is necessary to develop multi-energy complementarity and optimization of an integrated energy system to achieve low-carbon environmental operations [1]. The development of gas turbines [2] and power to gas (P2G) [3] has enabled a bidirectional energy flow between the power system and the natural gas system, which promotes the formation of integrated electricity-gas system (IEGS). As an important part of Energy Internet, IEGS is also the development direction of Smart Grid [4][5][6].
Recently, a considerable amount of research has been made on the optimization of IEGS [7][8][9][10][11][12][13][14][15]. A unified modeling framework for IEGS is proposed in [7]. Based on this model, the effect of a natural gas system on the operation of a power system in accordance with the security-constrained unit commitment is discussed. Based on mixed integer linear programming, paper [8] presents an integrated electricity-gas system model that can be used to research interdependencies and identify critical vulnerabilities. Paper [9] studies the electricity and steady-state natural gas flow. The above studies are directed at steady-state models. However, for the network characteristics of IEGS, different systems have different dynamic characteristics. Natural gas flow operates at slower rates than electric power. Some natural gas is stored in the pipeline, which is called the "line pack". In order to describe the natural gas network model more accurately, it is necessary to consider the line pack. Papers [10][11][12][13] have proved the improvement effect of the management of IEGS on the flexibility

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The non-convex OMEF problem is transformed into a convex optimization problem by PCCP. For the power system model, based on the paper [25], this paper studies the method of convexity of the second-order power flow equation in the multi-period OMEF model, which effectively retains all the electrical quantities and is suitable for any topology. For the natural gas system model considering the dynamic characteristics, the positive and negative natural gas flow are introduced into the model, and the PCCP method is used to convexize the flow equation, which can efficiently solve the natural gas system model.

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The decentralized optimization process is based on the obtained convex OMEF model. At each iteration of the PCCP, ADMM is used to establish and solve the decentralized OMEF problem of IEGS. During the solution process, each system only needs to exchange a small amount of boundary information, which achieves independent optimization and partition coordination. The solution process ensures the privacy and security of each system information. Finally, testing IEGS combined with an IEEE 39-bus power system and a Belgian 20-bus natural gas system is used to verify the proposed algorithm and can effectively solve the decentralized OMEF problem.

Modeling for OMEF
Due to the non-convexity of the OMEF model, the OMEF problem of IEGS is a difficult non-convex optimization problem. In this part, the OMEF is modeled and the non-convex term is effectively Appl. Sci. 2020, 10, 3348 3 of 18 processed with PCCP. The OMEF problem is transformed into a convex optimization problem. The IEGS studied in this paper consists of a power network and a natural gas network. The power and natural gas systems are interconnected through P2Gs and gas turbines, as shown in Figure 1.

Modeling for OMEF
Due to the non-convexity of the OMEF model, the OMEF problem of IEGS is a difficult nonconvex optimization problem. In this part, the OMEF is modeled and the non-convex term is effectively processed with PCCP. The OMEF problem is transformed into a convex optimization problem. The IEGS studied in this paper consists of a power network and a natural gas network. The power and natural gas systems are interconnected through P2Gs and gas turbines, as shown in Figure  1. Wind where PG,i,t is the active output of generator at bus i; PP2G,i,t is the power input of P2G at bus i; QG,i,t is the reactive output of generator at bus i; PD,i,t and QD,i,t are the active power demand and reactive power demand at bus i; Vi,t is the voltage magnitude of bus i; θij,t is the phase angle difference of branch ij; Pij,t and Qij,t are the active power flow and reactive power flow from bus i to bus j; gii and bii are the shunt conductance and shunt susceptance at bus i; gij and bij are the conductance and susceptance of branch ij.
(2) Power output limits (4) Ramp rate limits where P G,i,t is the active output of generator at bus i; P P2G,i,t is the power input of P2G at bus i; Q G,i,t is the reactive output of generator at bus i; P D,i,t and Q D,i,t are the active power demand and reactive power demand at bus i; V i,t is the voltage magnitude of bus i; θ ij,t is the phase angle difference of branch ij; P ij,t and Q ij,t are the active power flow and reactive power flow from bus i to bus j; g ii and b ii are the shunt conductance and shunt susceptance at bus i; g ij and b ij are the conductance and susceptance of branch ij.
(2) Power output limits (4) Ramp rate limits where DR i and UR i are the down and up ramp rate limits. The classic power system model is non-convex, and its non-convexity comes from Equations (3) and (4). In order to ensure the efficiency and quality of model solving, it is necessary to carry out convex processing for that equations.
First, equivalent substitutions are made to the variables in Equations (3) and (4), as follows: Then, Equations (3) and (4) can be equivalent to the following equations: For general mesh networks, the cycle conditions (for each loop in the network, the values of the voltage phase angle difference accumulated cyclically in a certain direction is 0 (modulo 2π)) [26] need to be considered. That is, it is necessary to consider Equation (13).
In the equivalent equations, Equations (12) and (13) are non-convex. Aiming at Equations (12) and (13), paper [25] proposed a convex relaxation method for single-period economic dispatch of power system. Based on this, this paper studies the convex processing method of Equations (12) and (13) in the multi-period OMEF model.
For Equation (13), the equivalent transformation is performed as follows: C ij,t = cos θ ij,t According to the voltage phase angle difference between the two ends of the branch in the actual power system is generally not more than 10 • [27], by using the Taylor expansion formula of the trigonometric function and ignoring the higher-order terms, Equations (14) and (15) can be transformed into the following equations: Through the above transformations, Equations (12), (16), (17), and (19) are still non-convex, and further transformations are needed. Among them, Equation (12) can be equivalent to the following two inequalities: Equation (20) is a standard second-order cone constraint and has convexity. Equation (21) is a difference-of-convex constraint, which is non-convex and can be accurately convex relaxed by using Appl. Sci. 2020, 10, 3348 5 of 18 the PCCP method [25]. The idea is to deal with the difference-of-convex constraint through the Taylor expansion formula, and add relaxation variables to make the constraint hold, as follows: where M k ij,t , W k ij,t and U k i,t are the value obtained by the kth iteration, and are constant; ε 1,ij,t is a relaxation variable and is a positive number.
In the same way, Equations (16), (17), and (19) can be transformed into two inequalities, and then transformed as above.
In order to ensure the accuracy of relaxation, a penalty term of relaxation variables is added to the objective function, and the relaxation field is continuously tightened by continuously punishing the relaxation variables until the accuracy is satisfied. The penalty term ∆e can be expressed as: where γ k e is the dynamic penalty coefficient of the power system penalty term; ε i,ij,t (i = 1, 2, . . . , 5) are relaxation variables of Equations (16), (17), (19) and (22), Therefore, the power system model is first transformed into convex-concave programming (CCP), and then, the PCCP method is used to solve the difference-of-convex constraint. The power system model is eventually transformed into convex programming, each of which has convexity, and the global optimal solution of the optimal energy flow problem can be obtained efficiently.
In addition, in order to obtain good initial iteration values, the solution obtained by the QC relaxed power flow model [28] is selected as the initial iteration values in the PCCP method.
The PCCP method can be expressed as follows: Step 1 Initialization: Set initial penalty coefficient γ 0 e > 0, maximum penalty coefficient γ e,max , penalty adjustment coefficient µ e > 0, number of iterations k = 0, convergence accuracy ω e .
Step 2 Initial iteration value: Solve the initial iteration value based on the QC relaxed power flow model.
Step 3 Solve the iterative PCCP.

Modeling for Natural Gas System
(1) Line pack Considering the dynamic characteristics of natural gas flow in time and space, natural gas flow has a transmission delay in the pipeline transmission process and the pipeline has a certain buffer capacity. The natural gas inflow at the beginning of the pipeline is different from the outflow at the end. Some natural gas is stored in the pipeline, which is called the "line pack", which can be expressed as [16]: where H ij,t is the line pack of pipeline ij; L ij and D ij are the pipe length and diameter; R, T, Z and ρ 0 are respectively the gas constant, natural gas temperature, natural gas compression factor, and natural gas Appl. Sci. 2020, 10, 3348 6 of 18 density under standard conditions; p ij,t = p i,t + p j,t /2 is the average pressure of pipeline; q in ij,t and q out ij,t are the gas inflow and outflow of pipeline ij.
In order to use the line pack reasonably and ensure a certain adjustment margin in the next scheduling cycle, the amount of the total line pack after running one cycle should not be lower than the initial value, as follows: (2) Natural gas flow equation According to the Weymouth equation, the natural gas flow equation can be expressed as: where q ij,t = q in ij,t + q out ij,t /2 is the average gas flow of pipeline ij; K ij is the transmission parameter of pipeline ij, which is positive; F ij is the friction coefficient of pipeline ij.
(3) Node gas flow balance equation where A s , A P2G , A R , A g and A GT are respectively the association matrix of node-natural gas source, node-P2G, node-gas storage tank, node-natural gas pipeline, node-gas turbine; q s,t , q P2G,t , q out R,t , q in R,t , q out g,t , q in g,t , q D,t and q GT,t are respectively the matrix of gas source output, P2G gas outflow, gas storage tank output, gas storage tank input, pipeline gas outflow, pipeline gas inflow, nodal load, gas turbine gas inflow.

(6) Compressors limits
In the long-distance transmission of natural gas, a natural gas compressor must be set up at a distance to pressurize the natural gas in the pipeline to offset the pressure consumed along the way. Because the energy consumed by the compressors is less, this article simplifies the compressor model and only considers the pressure-boosting relationship between the inlet and outlet ends of the compressor, as follows: where k max com and k min com are the upper and lower limits of the gas pressure boost ratio. (7) Gas storage tanks limits where S R,i,t is the gas storage capacity of the ith gas storage tank. Additionally, the volume of gas storage capacity, gas storage tank input and gas storage tank output meet the capacity constraints.

Natural Gas Flow Equation Convex Relaxation
In the natural gas system model, the gas flow Equation (27) is non-convex, which increases the difficulty of solving the model. It is necessary to convexize the Equation (27). First, introduce the positive and negative gas flow and Equation (34), as follows: where q + ij,t and q − ij,t are the positive and negative gas flow of pipeline ij. Then, assuming Equation (34) holds, there is the following relationship: Equation (27) is equivalent to the following inequality: Therefore, Equation (27) can be equivalent to Equations (34) and (37). Among them, Equation (37) can be equivalent to: According to the PCCP method described above, non-convex Equations (34), (38), and (39) are subjected to convex relaxation processing, and Equation (34) is taken as an example for description.
Equation (34) can be equivalent to: According to the PCCP method, Equation (40) can be expanded as follows: Similarly, Equations (38) and (39) can be transformed as above by PCCP. The penalty term ∆g in the natural gas system can be expressed as: where γ k g is the dynamic penalty coefficient of the natural gas system penalty term; ε i,ij,t (i = 1, 2, 3, 4) are relaxation variables of Equations (38), (39) and (41).
In this paper, the initial iteration value is obtained by solving the natural gas system model that ignores Equation (27), and the initial values of positive and negative gas flow are as follows: Appl. Sci. 2020, 10, 3348 8 of 18

Modeling for Coupling Elements
The power system and the natural gas system are connected by coupling elements. Common coupling elements have P2Gs and gas turbines. This paper uses the energy conversion equation and the gas turbines' consumption characteristic equation to represent the models of P2Gs and gas turbines, respectively.
(1) Energy conversion equation where q P2G,i,t is the gas outflow of ith P2G; α P2G,i and C g,i are the conversion factor and calorific value of natural gas.
(2) Gas turbines' consumption characteristic equation where q GT,i,t is the gas flow consumed by ith gas turbine; P GT,i,t is the power output of ith gas turbine; z 2,I , z 1,i and z 0,i are the consumption characteristic constants of ith gas turbine. Consumption characteristic Equation (45) is non-convex. Since unnecessary consumption will increase the cost of energy supply, Equation (45) can be relaxed into the following form: If the system optimization goal is the cost of energy supply, Equation (46) is always tight.

Objectives of OMEF
The optimization objectives of the multi-period OMEF model of the integrated electricity-gas system include the cost of generating electricity from coal-fired units, the cost of generating electricity from gas turbines, the cost of gas sources, and the penalty cost of wind power curtailment. The expression is as follows: where P GM,i,t is the power output of ith coal-fired unit; q GT,i,t is the gas flow consumed by ith gas turbine; q s,i,t is the ith gas source output; P wf WT,i,t and P WT,i,t is the predicted power and actual output of ith wind turbine; a 2,i , a 1,i and a 0,i are respectively the cost factor of ith coal-fired unit; b i is the cost factor of ith gas turbine; c i is the cost factor of ith gas source; d i is the cost factor of wind power curtailment; Ω GM , Ω GT , Ω GS and Ω WT are respectively the sets of coal-fired units, gas turbines, gas sources and wind turbines; ∆e and ∆g are the penalty terms for relaxation variables in power and natural gas systems to ensure the accuracy of convex relaxation models. The specific forms of ∆e and ∆g are respectively (23) and (42).

Decentralized Optimal Scheduling of OMEF for IEGS
Because the power system and the natural gas system belong to different stakeholders, and their respective information cannot be fully shared, it is necessary to study the decentralized optimal scheduling method.

Decentralized Framework
There are two common methods for regional topology separation: (1) Node tearing method. By node tearing, the node is divided into two to form two boundary nodes. (2) Branch tearing method. Introduce a virtual node at the midpoint of the interconnection branch, and divide the virtual node.
Both topological separation methods must satisfy the boundary coupling condition, that is, the boundary nodes of the sub-regions must meet the consistency of variables.
In the IEGS, the power system and the natural gas system are connected together through P2Gs and gas turbines. This article uses the node tearing method to decouple the entire system, as shown in Figure 2. The consistency of boundary variables can be described as follows: where x e and x g are the boundary variable vectors of power system and natural gas system.

Decentralized Algorithm
The ADMM algorithm uses the ideas of dual decomposition and decoupling. It has the advantages of fast convergence and strong robustness. It is suitable for solving decentralized optimization problems, and the convex programming problem can be guaranteed to converge to the optimal solution through the ADMM algorithm.
Through the PCCP method, the power system and natural gas system have established corresponding convex models. In a decentralized framework, in order to ensure the convexity of the model, the ADMM algorithm is used to solve the decentralized OMEF problem at each iteration of PCCP. First where y is the Lagrange multiplier; ρ is the step size, which is constant. Then, this problem is transformed into two independent sub-optimization problems according to the ADMM method. Its iterative format is as follows: where Z e,m+1 and Z g,m+1 are the iterative formats for power system optimization and natural gas system optimization problems; F e is the objective function of power system optimization, including

Decentralized Algorithm
The ADMM algorithm uses the ideas of dual decomposition and decoupling. It has the advantages of fast convergence and strong robustness. It is suitable for solving decentralized optimization problems, and the convex programming problem can be guaranteed to converge to the optimal solution through the ADMM algorithm.
Through the PCCP method, the power system and natural gas system have established corresponding convex models. In a decentralized framework, in order to ensure the convexity of the model, the ADMM algorithm is used to solve the decentralized OMEF problem at each iteration of PCCP.
First, according to the augmented Lagrange Equation, the coupling constraint (48) is relaxed into the objective function F, and the augmented Lagrange Equation is established as follows: where y is the Lagrange multiplier; ρ is the step size, which is constant.
Then, this problem is transformed into two independent sub-optimization problems according to the ADMM method. Its iterative format is as follows: where Z e,m+1 and Z g,m+1 are the iterative formats for power system optimization and natural gas system optimization problems; F e is the objective function of power system optimization, including the cost of generating electricity from coal-fired units and penalty cost of wind power curtailment; F g is the objective function of natural gas system optimization, including the cost of generating electricity from gas turbines and the cost of gas sources; y m+1 is the iterative format of Lagrange multiplier; m is the number of iterations.
The convergence criteria for decentralized optimization are as follows: where r m+1 is the original residual, which reflects the infeasibleness of the decentralized model; s e,m+1 and s g,m+1 are the dual residuals of the power system and the natural gas system, reflecting whether the iterative calculations have converged to the optimal solution; τ is the convergence accuracy. The decentralized solution can be described as: The power system calculates the optimal energy flow value according to the iterative format Z e,m+1 , and passes the value of the boundary variable x e,m+1 to the natural gas system. The natural gas system calculates the optimal energy flow value according to the iterative format Z g,m+1 , and passes the value of the boundary variable x g,m+1 to the power system for the next iterative operation. During the system optimization process, the network topology, operating conditions, and calculation optimization processes of the power system and natural gas system are independent. The power and natural gas system make global adjustments by exchanging a small amount of boundary information to obtain a global optimal solution.

Flowchart of Decentralized OMEF Optimization Algorithm
The algorithm proposed in this paper first transforms the non-convex OMEF problem into a convex programming problem through PCCP, and then, performs a decentralized solution at each iteration of PCCP according to ADMM. The algorithm flowchart is shown in Figure 3.
The detailed solution steps of the algorithm are as follows: Step 1: System partition. According to the node tearing method, the IEGS is divided into two regions: power system and natural gas system.
Step 2: Initialization. Set PCCP algorithm parameters and ADMM algorithm parameters, and use ADMM to solve the OMEF problem modeled with QC relaxed power flow model and the natural gas system model that ignores Equation (27) to obtain the initial value of the iteration in the PCCP process.
Step 3: Power system and natural gas system establish corresponding convex models according to PCCP method.
Step 4: According to the model obtained in step 3, the ADMM method is used to establish and solve the decentralized OMEF problem at each iteration of the PCCP. Determine whether the ADMM convergence criterion is satisfied, that is, whether Equation (51) is satisfied. If it is satisfied, the optimization result is output, and the process proceeds to step 5; otherwise, update the number of iterations m and the Lagrange multiplier y for the next optimization iteration.
Step 5: According to the partial output of step 4, that is, the value of the relaxation variables of the power system and the natural gas system determines whether all meet the PCCP convergence conditions (max ε i,ij,t < ω). If both systems meet the convergence conditions, output of the optimization results; otherwise, update the penalty coefficients γ k e and γ k g , the number of iterations k, and skip to step 3 to continue the optimization. To speed up the convergence, the value obtained at the kth iteration is set as the initial iteration value of k+1th iteration of PCCP and ADMM.
The decentralized solution can be described as: The power system calculates the optimal energy flow value according to the iterative format Z e,m+1 , and passes the value of the boundary variable x e,m+1 to the natural gas system. The natural gas system calculates the optimal energy flow value according to the iterative format Z g,m+1 , and passes the value of the boundary variable x g,m+1 to the power system for the next iterative operation. During the system optimization process, the network topology, operating conditions, and calculation optimization processes of the power system and natural gas system are independent. The power and natural gas system make global adjustments by exchanging a small amount of boundary information to obtain a global optimal solution.

Flowchart of Decentralized OMEF Optimization Algorithm
The algorithm proposed in this paper first transforms the non-convex OMEF problem into a convex programming problem through PCCP, and then, performs a decentralized solution at each iteration of PCCP according to ADMM. The algorithm flowchart is shown in Figure 3. The detailed solution steps of the algorithm are as follows: Step 1: System partition. According to the node tearing method, the IEGS is divided into two regions: power system and natural gas system.
Step 2: Initialization. Set PCCP algorithm parameters and ADMM algorithm parameters, and use ADMM to solve the OMEF problem modeled with QC relaxed power flow model and the natural gas system model that ignores Equation (27) to obtain the initial value of the iteration in the PCCP process.
Step 3: Power system and natural gas system establish corresponding convex models according to PCCP method.
Step 4: According to the model obtained in step 3, the ADMM method is used to establish and solve the decentralized OMEF problem at each iteration of the PCCP. Determine whether the ADMM convergence criterion is satisfied, that is, whether Equation (51) is satisfied. If it is satisfied, the optimization result is output, and the process proceeds to step 5; otherwise, update the number of iterations m and the Lagrange multiplier y for the next optimization iteration.
Step 5: According to the partial output of step 4, that is, the value of the relaxation variables of the power system and the natural gas system determines whether all meet the PCCP convergence conditions (max , , < ). If both systems meet the convergence conditions, output of the

Case Studies
In order to verify the accuracy of the algorithm proposed in this paper, case studies are carried out on a testing IEGS combined with an IEEE 39-bus power system and a Belgian 20-bus natural gas system. The topology of testing IEGS is shown in Appendix A, Figure A1. The proposed model and algorithm are implemented with GAMS and Matlab2019a, where GAMS is to as an optimization solver to solve the problem, and Matlab2019a is to call and analyze the calculation results from GAMS.

Simulation Model Description
In this simulation model, there are five coal-fired units, three gas turbines, and two wind turbines in the IEEE 39-bus power system. There are two gas sources, four gas storage tanks, and three natural gas compressors in the Belgian 20-bus natural gas system. The gas turbines (GT 1 , GT 2 , GT 3 ) are connected to gas nodes 4, 10, 12 respectively. The input terminals of P2Gs are connected to the electricity nodes 33, 34, respectively, and the corresponding output terminals are connected to the gas nodes 13, 14. The initial total line pack is 5.86 × 10 6 m 3 . The cost factor of wind power curtailment is set as d i = 50$/MW·h. Other main parameters of the simulation model are shown in the Appendix A Tables A1-A4. The curve of electric load and gas load is shown in Figure 4. This paper takes one day with 24 h as a dispatch cycle. the electricity nodes 33, 34, respectively, and the corresponding output terminals are connected to the gas nodes 13, 14. The initial total line pack is 5.86 × 10 6 m 3 . The cost factor of wind power curtailment is set as di = 50$/MW•h. Other main parameters of the simulation model are shown in the Appendix A Tables A1-A4. The curve of electric load and gas load is shown in Figure 4. This paper takes one day with 24h as a dispatch cycle.

Analysis of the Algorithm
In order to verify the accuracy of the decentralized framework and algorithm for the OMEF problem proposed in this paper, the algorithm solving process and optimization results are analyzed in detail. The key parameters of the algorithm in this paper are shown in Table 1.

Analysis of the Algorithm
In order to verify the accuracy of the decentralized framework and algorithm for the OMEF problem proposed in this paper, the algorithm solving process and optimization results are analyzed in detail. The key parameters of the algorithm in this paper are shown in Table 1. In order to verify the accuracy and effectiveness of the decentralized algorithm proposed in this paper, the decentralized optimization results are compared with the centralized algorithm.
As shown in Table 2, the NLP algorithm is to solve the OMEF problem of IEGS without processing non-convex equations. Meanwhile, the PCCP algorithm is to use the PCCP method proposed in this paper to model for the power system and the natural gas system and solve it centrally. Comparing the two results, although NLP can directly use the IPOPT solver to solve the OMEF problem, the results obtained are not optimal solutions, and the non-convex nature makes the decentralized algorithm unable to converge or obtain ideal results. The PCCP algorithm used to solve the OMEF problem can converge to an optimal solution. PCCP-ADMM algorithm is the decentralized algorithm proposed in this paper. Compared with the PCCP centralized algorithm, the results obtained by the decentralized algorithm are similar to the centralized algorithm, which verifies the feasibility of the decentralized algorithm. The difference between the two results is 0.4%, which is acceptable. For optimization time, the decentralized algorithm takes more time than the centralized algorithm. However, the decentralized algorithm only needs to exchange a small amount of boundary information in each area to coordinate the global optimal scheduling, which guarantees the privacy of the information in each area. The iterative process of PCCP-ADMM algorithm is shown in Figure 5. It can be seen from (a) that the optimization process performed 7 PCCP iterations and 22 ADMM iterations. As shown in (b), at the boundary of each iteration of PCCP, the ADMM residual will basically increase, because the algorithm's solution mechanism is that the ADMM algorithm is performed at each iteration of PCCP, but the overall trend is declining. In addition, in the second iteration of PCCP and later, the number of ADMM iterations is small. The reason is that the energy flow results obtained by the PCCP-ADMM in these steps are close to the optimal solution, and the ADMM algorithm in k+1th iteration of PCCP uses the Lagrange multiplier result in kth iteration as the initial iteration value. Therefore, the proposed algorithm can not only guarantee convergence to the optimal solution, but also speed up the convergence speed. algorithm proposed in this paper. Compared with the PCCP centralized algorithm, the results obtained by the decentralized algorithm are similar to the centralized algorithm, which verifies the feasibility of the decentralized algorithm. The difference between the two results is 0.4%, which is acceptable. For optimization time, the decentralized algorithm takes more time than the centralized algorithm. However, the decentralized algorithm only needs to exchange a small amount of boundary information in each area to coordinate the global optimal scheduling, which guarantees the privacy of the information in each area. The iterative process of PCCP-ADMM algorithm is shown in Figure 5. It can be seen from (a) that the optimization process performed 7 PCCP iterations and 22 ADMM iterations. As shown in (b), at the boundary of each iteration of PCCP, the ADMM residual will basically increase, because the algorithm's solution mechanism is that the ADMM algorithm is performed at each iteration of PCCP, but the overall trend is declining. In addition, in the second iteration of PCCP and later, the number of ADMM iterations is small. The reason is that the energy flow results obtained by the PCCP-ADMM in these steps are close to the optimal solution, and the ADMM algorithm in k+1th iteration of PCCP uses the Lagrange multiplier result in kth iteration as the initial iteration value. Therefore, the proposed algorithm can not only guarantee convergence to the optimal solution, but also speed up the convergence speed.

Analysis of Optimization Results
The PCCP-ADMM algorithm proposed in this paper is used to solve and the OMEF problem. In order to analyze the impact of the dynamic characteristics of the natural gas system on the optimal scheduling of the system, this paper analyzes the following two cases: (1) Case 1: Regardless of the dynamic characteristics of the natural gas system (i.e., removing the Equations (24)-(26)), the objective is to optimize the cost of energy supply. (2) Case 2: Considering the dynamic characteristics of the natural gas system, the objective is to optimize the cost of energy supply.
This paper uses PCCP-ADMM algorithm to optimize two cases and analyzes the results to research the impact of the line pack on the optimal operation of the system, as shown in Figures 6 and 7. (2) Case 2: Considering the dynamic characteristics of the natural gas system, the objective is to optimize the cost of energy supply. This paper uses PCCP-ADMM algorithm to optimize two cases and analyzes the results to research the impact of the line pack on the optimal operation of the system, as shown in Figures 6  and 7.   optimize the cost of energy supply. This paper uses PCCP-ADMM algorithm to optimize two cases and analyzes the results to research the impact of the line pack on the optimal operation of the system, as shown in Figures 6  and 7.   According to the optimization results, during the period of 1-6 h, the system's electric and gas loads are in the trough period. Compared with Case 1, Case 2 has more gas sources output, and part of the gas is stored in the natural gas pipeline. As shown in Figure 6, the value of the total line pack rises. During the period of 8-12 h, the system load is at its peak. The load in the natural gas system in Case 1 is mainly satisfied by the output of the gas sources; in Case 2, during this period, there is a certain amount of the line pack to release, and the output of the gas sources is less. Therefore, considering the dynamic characteristics of natural gas systems, IEGS can buffer the gas inflow and gas outflow. The gas network has a certain energy storage capacity, which can effectively buffer load fluctuations. It improves the flexibility of the system operation while ensuring the quality of energy supply.
The cost of energy supply in each case is shown in Table 3. In Case 2, the cost is reduced due to the release of the line pack during the peak load period and used for system scheduling, with a reduction of 0.77%. This paper focuses on Case 2. The output of some devices in the IEGS is shown in Figure 8.
gas outflow. The gas network has a certain energy storage capacity, which can effectively buffer load fluctuations. It improves the flexibility of the system operation while ensuring the quality of energy supply. The cost of energy supply in each case is shown in Table 3. In Case 2, the cost is reduced due to the release of the line pack during the peak load period and used for system scheduling, with a reduction of 0.77%. This paper focuses on Case 2. The output of some devices in the IEGS is shown in Figure 8.  For the power system, according to Figure 8b,d, it can be obtained that during the periods of 1-6 h and 22-24 h (load trough periods), the wind power output is relatively high. In order to absorb wind power, the excess power is injected into the natural gas system through P2Gs. In this period, P2Gs are basically close to full load operation. As shown in Figure 8a, during the period of 8-21 h, For the power system, according to Figure 8b,d, it can be obtained that during the periods of 1-6 h and 22-24 h (load trough periods), the wind power output is relatively high. In order to absorb wind power, the excess power is injected into the natural gas system through P2Gs. In this period, P2Gs are basically close to full load operation. As shown in Figure 8a, during the period of 8-21 h, because the generating cost of coal-fired units is generally lower than that of gas turbines, coal-fired units are basically close to full power output; meanwhile the output of gas turbines is relatively flat, which is mainly output at peak loads.
For the natural gas system, as shown in the Figure 8c, it can be found that during the period of 2-6 h, the excess wind power is injected into the gas network through P2Gs and a part of it is stored in the gas storage tanks, and the gas inflow of the gas storage tanks is large. During the period of 8-12 h, due to the increase of system load, the gas storage tanks increase the gas outflow to meet the load demand.

Conclusions
Considering the dynamic characteristics of the natural gas system, the privacy of each system's information, and the non-convexity of the OMEF model, this paper proposes a PCCP-ADMM decentralized algorithm to solve the OMEF problem. The simulation analysis of the testing IEGS proves the effectiveness of the algorithm and reaches the following conclusions: (1) For the non-convex OMEF model, this paper introduces the PCCP method to transform the non-convex OMEF problem into a convex programming problem, and proves the feasibility of the method from theoretical and simulation results. (2) This article analyzes the impact of the line pack on system performance. The calculation results show that considering the dynamic characteristics of the natural gas system can improve the flexibility and economy of the system. (3) Considering the privacy of the information of each subsystem in IEGS, this paper proposes a decentralized optimization algorithm for the OMEF problem based on PCCP-ADMM. In decentralized optimization, each system only needs to exchange a small amount of information to perform global coordination and optimization.