Energy is the basis and important guarantee for human survival. There are many problems in traditional energy systems, such as independent energy supply, low cascade utilization level, energy waste, and environmental pollution (Zhou et al., 2013; Fan et al., 2021; Hu et al., 2022). The multi-energy flow coupling system (MEFCS) is an energy system form that integrates public cold, heat, electricity, and gas. Its purpose is to integrate multiple energy sources, such as electric energy, natural gas, and thermal energy in a certain area, so as to realize collaborative optimal operation, collaborative management, and complementary mutual assistance among various forms of energy subsystems (Zhao et al., 2018; Klyapovskiy et al., 2019). In addition, under the background of “double carbon”, the transformation of clean and low-carbon energy is an inevitable trend of global energy development.

Traditional energy systems are planned and operated independently, and only a single situation needs to be considered in their optimal scheduling. However, for a multi-energy flow coupling system, the correlation among energy subsystems should be considered in planning and operation (Sirvent et al., 2017). In Liu et al. (2019), considering the multi-timescale characteristics, an electrical and thermal energy sharing model of interconnected microgrids with combined heating and power (CHP) and photovoltaic systems was built, in which CHP could operate in a hybrid mode by selecting the operating point flexibly. In Wang et al. (2019), a multi-objective bi-level optimization model considering the total cost and carbon dioxide emission was built, while the energy efficiency of multi-energy flow coupling system was ignored. In Barati et al. (2015) and Clegg and Mancarella (2016), under the condition of meeting the basic needs of power, gas, and heat loads, the coordinated planning of multi-energy flow coupling system was considered, in order to reduce the construction cost of transmission lines, gas pipelines, and power plants as much as possible. In Koltsaklis and Knápek (2021), the authors presented an optimization framework for the optimal scheduling of a multi-energy microgrid, where a number of aggregated end-users were considered. In Nicolosi et al. (2021), a novel mixed integer linear programming optimization algorithm has been developed to compute the optimal management of a micro-energy grid, where the total cost, the NOx, and the CO₂ emissions of the system were taken into consideration. To meet the safety constraints, in Wang D. et al. (2018), an optimal coordination control strategy (OCCS) for a hybrid energy storage system was developed considering the state-space equation to describe the OCCS, the constraints of the OCCS, and the objective function to express the optimal coordination control performance. In Sun et al. (2020), the authors considered the day ahead optimal scheduling problem of electricity gas interconnected systems, where the two-way energy flow was taken as a non-convex nonlinear mixed integer linear programming problem, and a second-order cone programming (SOCP) method has been proposed. In Luo et al. (2018) and Zhang et al. (2021), the uncertainty caused by renewable energy and multi-energy load was considered, and the robust optimization and stochastic optimization methods were adopted to deal with it respectively, so as to ensure that the system can still maintain stable operation under the worst conditions. In Ghosh and Kamalasadan (2017), a grid-connected two mass DFIG and a grid-supportive single mass squirrel cage induction generator-based flywheel energy storage system model have been considered for controller design and proof-of-concept exploration. In Wang L. et al. (2021), the flexible resources (FRs) on both the energy supply and load sides were introduced into the optimal dispatch of the integrated electricity-heat energy system (IEHES) and further modeled to alleviate the renewable fluctuations, and the solution for FRs participating in IEHES dispatch was given, with goals of maximizing the renewable penetration ratio and lowering operation costs. It can be seen that most of the existing results consider optimization of the economic objectives of the multi-energy flow coupling system, where the index is relatively single, and less consideration is paid on the carbon emission level in the operation of the system. At the same time, the operation strategy is the lack of comprehensive comparison and verification.

In solving the MEFCS collaborative optimization model, when considering multiple optimization objectives including carbon emission, investment and operation cost, and energy utilization, the traditional single objective optimization algorithm may be difficult to ensure that the solution result is the optimal solution of the original problem. In Wang W. et al. (2021), the load characteristics and various constraints of the integrated community energy system were considered, and the operating model with the goal of minimizing operating costs was optimized. In Ma et al. (2018), the energy consumption cost and environmental cost of the multi-energy flow coupling system were considered comprehensively, the optimal scheduling model of multi-energy flow coupling system was proposed, and the optimal scheduling model was transformed into a mixed integer linear programming problem. In Xiao et al. (2018), the method of the probability scenario had been used to model the uncertainties of the distributed renewable energies (DREs) and loads, which could better characterize the impact of uncertainty on the planning and design of the MEFCS. In Yang et al. (2018), a two-stage robust generation scheduling model was proposed for the dynamic safety constraints of the natural gas pipeline network and the uncertainty of wind power, and a new solution method was developed to avoid the nonlinearity of gas flow constraints. In Wu et al. (2021), the multi-objective optimization model was transformed into a single objective optimization model through the multi-objective programming hierarchical solution method, and the primal dual interior point method was used to solve the model. Based on the fast particle swarm optimization algorithm, in Qu et al. (2021), a dual-decomposition-based distributed algorithm was designed to address the problem that the data and information of the EHs during the operation were confidential and should be kept by each owner, where the optimal consensus problem was used for the dual problem to update the multipliers, in Li et al. (2020), the proposed MEFCS planning model, formulated as a two-stage MILP problem, was solved by the Benders decomposition (BD) method to determine the optimal capacity of each component in MEFCS planning.

To be pointed out that, the research on the optimization of multi-energy flow coupling system at home and abroad mainly focuses on the simplification of the optimization model. However, on the one hand, it will lead to the reduction of solution accuracy, at the same time, because the models are more and more complex, which are difficult to be simplified. Therefore, the heuristic algorithm has become an important way to deal with optimization problems. However, the traditional heuristic algorithm has the problems of poor convergence and easy to fall into local optimization, and how to find a simplified and better algorithm is another motivation of this paper. Based on the above discussions, in this paper, the environmental protection goal is taken as the leading factor, the economic and energy efficiency goals are comprehensively considered, the multi-objective collaborative optimization model is developed for the multi-energy flow coupling system, which can be transformed into a single objective optimization model through the linear combination of the analytic hierarchy process and the improved entropy weight method, and then the model can be solved by using the simplified primal dual interior point method. The results avoid falling into local optimization and accelerate convergence. Case studies verify the effectiveness of the proposed algorithm.

A typical multi-energy flow coupling system structure is shown in Figure 1, which is internally connected through the power grid, thermal pipe network, and cooling transmission network. The equipment involved distribution power source includes a wind turbine (WT) and photovoltaic (PV). Cogeneration includes CHP, a gas turbine (GT), a waste heat boiler (WHB), a ground source heat pump (HP), an electric refrigerator (ER), an absorption refrigerator (AR), and other energy conversion equipment, as well as electric energy storage (EES), heat energy storage (HES), and other energy storage equipment.

In the multi-objective collaborative optimization of MEFCS considering carbon emissions, the optimization objectives considered in this paper include the environmental protection objective, economic objective, and energy efficiency objective.

Based on the above analysis, it can be found that the multi-objective collaborative optimization of the multi-energy flow coupling system considered in this paper is a complex nonlinear programming problem. In order to make the solution speed and convergence meet the requirements of practical problems, the simplified primal dual interior point algorithm is used in this paper. For the sake of brevity, first, the optimization model described above is transformed into the following general form: { max    F ( x ) , s .t .      h ( x ) = 0 , g min ≤ g ( x ) ≤ g max , (30) where x is the state variable, including the output power, external power purchase, gas purchase, etc., of each equipment, h ( x ) is the equality constraint, including the power balance constraint of the system, the energy balance constraint at the beginning and end of the scheduling cycle of energy storage equipment, etc.; g ( x ) is the inequality constraint, including the upper and lower limits of equipment output, energy storage charge and discharge constraints, etc.; and g max and g min represent the upper and lower bounds of the inequality, respectively.

When dealing with this optimization model with the traditional interior point algorithm, relaxation variables u = [ u 1 , ⋯ , u r ] T and l = [ l 1 , ⋯ , l r ] T are introduced first, where r represents the number of inequality constraints; thus, the original inequality constraints are transformed into the equality constraints. The resulting optimization model is formulated as { min    − F ( x ) , s .t .      h ( x ) = 0 , g ( x ) + u = g max , g x − l = g min , u > 0 , l > 0 . (31)

At the same time, the size of relaxation variables u and l should be restricted to ensure that the objective function F ( x ) is always far away from the solution boundary so that it can be solved in the feasible domain as follows: { min    − F ( x ) − μ ∑ i = 1 r log ( l i ) − μ ∑ i = 1 r log ( u i ) , s .t .      h ( x ) = 0 , g ( x ) + u = g max , g ( x ) − l = g min , (32) where μ represents the introduced disturbance factor.

At this point, the inequality constraints contained in the optimization model described in this paper have all been converted into the equality constraints, and the Lagrange function for this optimization problem can be expressed as L = − F ( x ) − y T h ( x ) − z T [ g ( x ) − l − g min ] − w T [ g ( x ) + u − g max ] − μ ∑ i = 1 r log ( l i ) − μ ∑ i = 1 r log ( u i ) , (33) where y , z , and w all represent the Lagrange operators, also known as the dual variables. By deriving this Lagrange function, the optimal solution to this optimization problem can be obtained.

In this paper, by simplifying the original dual interior point algorithm, the simplified original dual interior point method can be utilized to solve the optimization model, and the simplified process is to rewrite the inequality constraint to { g ^ ( x ) ≤ g ^ max , g ^ ( x ) = [ g ( x ) , − g ( x ) ] T , g ^ max = [ g max , − g max ] T , (34) where g ^ ( x ) and g ^ max represent the generalized inequality constraints and generalized upper bounds, respectively.

It can be found from the traditional interior point algorithm that in the process of dealing with inequality constraints, the upper and lower bounds of inequality constraints need to be relaxed, and then converted into equality constraints, respectively. At the same time, Lagrange operators are also introduced for equality constraints converted from upper-bound inequality constraints and lower-bound inequality constraints, respectively, which introduce more variables in the Lagrange function. This simplification algorithm greatly reduces the relaxation variables and corresponding Lagrange operators introduced in the optimization model, improves the convergence speed of the algorithm while guaranteeing the calculation accuracy, and reduces the amount of programming to a certain extent. The rest of the algorithm is handled similarly to the traditional algorithm, which are not discussed here. For ease of understanding, the multi-objective collaborative optimization calculation flow of the multi-energy flow coupling system based on the simplified primal dual interior point algorithm is shown in Figure 2, where r gap is the dual gap, ε represents the convergence accuracy, which is taken as 10–6 in this paper, and K max represents the maximum number of iterations, normally is set as 300.

Based on the above analysis, the collaborative optimization of MEFCS dominated by the carbon emission targets in this paper can be summarized as the following steps:

Step 1. Enter parameter information of the multi-energy flow coupling system, such as system load, rated capacity of each unit, equipment parameters, and time-of-use electricity price.

Step 2. Establish the steady-state operation model of each equipment, as shown in Eqs 1–7.

Step 3. Establish the objective functions and constraints of the MEFCS, as shown in Eqs 8–15.

Step 4. Weight each objective function using the analytic hierarchy process-improved entropy weight method, as shown in Eqs 18–28.

Step 5. Convert each objective function into a collaborative optimization objective through the membership function and the obtained weight information, as shown in Eqs 16, 17, 29.

Step 6. Solve the MEFCS collaborative optimization model by the primal dual interior point algorithm until the optimal solution is obtained or the algorithm does not converge.

The above steps can be clearly represented by the flowchart shown in Figure 3.

In this paper, the multi-objective collaborative optimization model of MEFCS has been developed considering each of the environmental protection, system economy, and energy efficiency as the objectives of this study, in which the carbon emission orientation goal could be achieved. Moreover, the simplified primal dual interior point method has been used to solve the constructed model. According to the obtained results, the following findings have been concluded: 1) The contradiction and conflict between the three objectives (CO2 emission, system operation cost, and primary energy utilization rate) were relatively balanced under the proposed collaborative optimization operation, which have clearly demonstrated that the satisfaction of collaborative optimization operation considering multiple objectives could be higher than that considering a single objective of the system. 2) Each equipment of the system could achieve stable energy supply as well as continuous output throughout the whole operation process, whereas the working state was difficult to fluctuate sorely. 3) At the same time, the operator could adjust the weight of the three objectives through his own will, so as to get the best operation results that meet his requirements. 4) In addition, the simulation results have illustrated that the simplified primal dual interior point method being adopted in this paper has better convergence and global optimization ability in multi-objective collaborative optimization. However, more investigations are needed regarding the sensitivity analysis on the critical parameters of the multi-energy flow coupling system, which will be considered in our future research work.