A collaborative approach to integrated energy systems that consider direct trading of multiple energy derivatives

0 Introduction

With the increasing environmental pollution and energy crisis, reforms in energy production, transmission,distribution, and utilization are gaining increasing attention. Regional integrated energy system (RIES) can improve energy efficiency and reduce carbon emissions[1], providing a feasible solution for the development of low-carbon cities [2, 3]. The diverse energy conversion equipment in RIES not only enables the system coupling of electricity, gas, and heat, but can also meet the needs of end users [4, 5]. In addition, the flexible operation of RIES can provide useful support and services for energy distribution, such as reducing operating costs [6]. Therefore,it is important to optimize integrated energy systems at the regional level by using flexible exchange of multiple energies.

Different units of RIES generally belong to different interest subjects, each of which only possesses information related to the subject and rationally pursues to maximize their own interests. However, the phenomenon of disorderly competition often occurs due to information asymmetry,which greatly reduces market efficiency. Cooperative game theory is often used to deal with the interests of multiple subjects [7, 8]. The cooperative game considers the unity of individual interests and overall interests and can achieve the global overall optimum [9]. For example, the model of joint venture between wind farms and electricity market (EM)to operate large hydrogen production plants was proposed based on the cooperative game to address the problems of high investment cost and low utilization of large hydrogen production plants [10]. As an important part of the cooperative game, Nash negotiation is often used in the multi-entity benefit distribution problem [11], and the idea is to transform the cooperative benefit distribution problem into the process of finding the Nash negotiation solution,and the economic significance of the Nash negotiation equilibrium solution is to maximize the total benefits of the alliance. The Nash negotiation method is used to establish the cooperation model of electric trading between multiple micro-networks, and the benefits of multi-micro-network cooperation are distributed by solving the Nash product[12, 14]. A Nash-negotiated cooperative operation model between wind and multiple hydrogen production stations is proposed, and the model is solved using the Benders decomposition method [15]. The literature [16] considers multiple types of subjects, such as distributed power sources, energy storage operators, and load aggregators and proposes a Nash bargaining method for optimal scheduling of multi-subject microgrid day-ahead. All the above studies actively explore the analysis of the interest relationship among multiple subjects; however, most of them focus on the traditional energy trading and cannot solve the problem of cooperative operation and benefit distribution among multiple subjects involved in the trading of new energy derivatives; in addition, some models violate the privacy of subjects by solving the benefit distribution among subjects through the analytical method. How to consider the individual and coalition rationality of multiple subjects and how to protect the privacy of each subject are the issues that must be focused on.

Distributed optimization methods are widely used for solving Nash equilibria [17], and their idea of decentralized optimization can effectively protect the privacy of each subject. Due to the natural decoupling property and stable convergence performance, alternating directed multiplier method (ADMM) has been widely used for distributed optimization of integrated energy systems. Coordination of electric and natural gas systems was studied through ADMM [18]. Day-ahead scheduling of electric and natural gas systems is proposed to minimize operating costs by using an improved ADMM [19]. Optimal distributed energy management is proposed for local energy agencies consisting of energy suppliers and consumers [20]. In the above references, the interactions between different energy systems are coupled and exchanged only through gas units or combined heat and power system (CHP) for natural gas and electrical and thermal energy. However, at the regional level, the multi-energy coupling and exchange may be more complex. RIES involves not only the coupling and exchange of traditional energy sources, but may also involve the exchange of new energy derivatives, such as green certificates (GC) [21, 22], CO2 [23], etc. Therefore, the difficulty of solving optimization problems has increased.Moreover, with the increasing number of diverse energy allocations and exchanges, the operating entities in RIES are becoming diversified. Therefore, the original ADMM cannot be directly applied to the optimal dispatch of RIES and needs to address the interactions between energy systems with the participation of multiple energy sources and their derivatives. Meanwhile, the computational cost of distributed optimization methods will increase with the system size. Therefore, it is necessary to propose a new optimization strategy in a distributed framework to improve the operational efficiency of RIES.

To address the above issues, a cooperative model that aggregates renewable energy generation, conventional energy generation, and integrated energy supply hub is studied. Among them, wind-photovoltaic generation system(WG) as a representative renewable energy generation;traditional energy generation is mainly CHP coupled with carbon capture system (CCS); and the integrated energy supply hub is designed as an electricity–carbon–hydrogen load (PCH) with power to gas (P2G) as the main equipment and coupled with hydrogen fuel cell (HFC) and CCS.

In this way, the cooperative model consisting of WG,CHP, and PCH not only has transactions of traditional energy sources such as electricity, but also considers transactions of new energy media such as GC and CO2.Simultaneously, considering the characteristics of the multisubject cooperation of WG–CHP–PCH, it is necessary to consider both individual rationality and coalition rationality,focus on two key problems of cooperative optimization operation and fair distribution of the benefits of WG, CHP,and PCH multi-subjects, and use ADMM to conduct the optimization process of each subsystem separately and in parallel to achieve an efficient solution. The contributions of this paper are summarized as follows:

(1) The trading mechanism of the WG–CHP–PCH cooperative model involving electricity, GC, CO2, and service power (SP) trading is analyzed.

(2) The WG, CHP, and PCH sub-models under a cooperative operation are established, respectively.

(3) The WG–CHP–PCH cooperative operation Nash negotiation model is established and equivalently transformed into the two sub-problems of cooperative benefit maximization and electric energy transaction payment negotiation; the former corresponds to the global optimal operation problem of cooperative alliance and the latter corresponds to the fair distribution problem of cooperative benefits.

(4) To protect the privacy of each subject, ADMM is used to solve both subproblems in a distributed manner,and the proposed method is verified through arithmetic examples.

The rest of this paper is organized as follows: Section 1 introduces the RIES model and performs the transaction mechanism analysis. Section 2 establishes the operation model of each subject. Section 3 establishes the WG–CHP–PCH multi-entity cooperative operation Nash negotiation model and the solution procedure. Section 4 presents the arithmetic analysis. The conclusion is given in Section 5.

1 Trading mechanisms under cooperative operation

A multi-species trading mechanism consisting of WG,CHP, and PCH is shown in Fig. 1. Four trading varieties are involved, namely, electricity, GC, CO2, and SP. It is important to explain that one GC represents 1 MW of nonwater renewable energy [24, 25]. In terms of generation, the purchase of one GC represents the notional generation of 1 MW of non-water renewable energy; for the consumption,the purchase of one GC represents the notional use of 1 MW of non-water renewable energy. WG mainly outputs electricity and GC, while WG’s output electricity has a forecast error; thus, WG needs to purchase SP to neutralize its forecast error. CHP can output electricity and CO2; PCH has the purchase demand of electricity and CO2, and PCH also has the purchase demand of GC because it has the task of renewable energy consumption.

Fig. 1 Multi-energy derivative trading framework of WG–CHP–PCH

In the non-cooperative model, electricity trading: WG and CHP sell electricity to the EM, and PCH purchases electricity from EM. GC trading: WG’s GC and electricity are bundled and sold to the green certificate market (GCM),and CHP and PCH purchase GC from GCM. CO2 trading:PCH purchases CO2 from the carbon market (CARM) to meet its own carbon demand. SP trading: WG buys capacity on the capacity market (CAPM) to purchase capacity to neutralize WG’s generation error.

The trading approach will be more flexible under the cooperative model.

Electricity trading: According to the distributed trading model [26]: distributed generation projects can be traded directly with electricity consumers. Assumably, in the cooperative model, WG and CHP can negotiate with PCH to determine the transaction and price, and then sell electricity directly to PCH through EM. Meanwhile, WG and CHP will sell surplus power to EM, and PCH can also purchase power from EM to meet the surplus load demand.

GC trading: According to the renewable energy quota system [27, 28], each coal-fired power producer should bear a quota (at least 15%) of non-water renewable energy generation to thermal power generation. Under the cooperative model, WG can negotiate with CHP and PCH to determine the transaction and price and sell in GCM. Moreover, CHP and PCH can choose to seek GC transactions from GCM and WG to meet their renewable energy quota mandate.

SP Trading: To avoid WG’s penalty due to power forecast error, WG will seek capacity services from CHP and CAPM to neutralize WG’s power forecast error and reduce forecast penalty.

CO2 trading: In the context of “dual carbon” [29], CHP will face the constraint of carbon emission quota. In the cooperative operation model, CHP can sell the absorbed CO2 to PCH or CARM, and PCH can buy CO2 from CHP and CARM.

In this paper, assumably, WG, CHP, and PCH belong to different subjects of interest. If WG, CHP, and PCH reach a direct transaction agreement through bargaining negotiation, the revenue of each subject will be enhanced,which will stimulate WG, CHP, and PCH to cooperate and trade electricity and energy derivatives and enhance the comprehensive revenue of each subject. In the following,we establish the optimization model of WG–CHP–PCH under the cooperative operation mode.

2 Optimization model for each subject

2.1 WG

The operational objective of the WG is to optimize the WG’s trading variables with other agents and markets to maximize the operational revenue Uwg. This mainly includes the WG’s trading revenue with the PCH , trading revenue with the CHP , trading revenue with the market Uwg-m, and WG’s maintenance cost . Among them:

where is the revenue of WG selling power to PCH; is the revenue of WG selling GC to PCH; is the revenue of WG selling GC to CHP; is the cost of WG purchasing SP from CHP; is the revenue of WG selling power to EM; is the revenue of WG selling GC to GCM; is the cost of WG purchasing SP from CAPM.

Trading variables include WG’s prices and volumes with other subjects and markets. Among them, the transaction price of WG with other subjects includes WG–PCH transaction priceand WG–CHP transaction price; the transaction volume with other subjects includes WG-PCH transaction volumeand WG–CHP transaction volumeThe transaction volume of WG with each market(the transaction price of WG with each marketis determined by the market and is regarded as a constant factor in this paper). Here, is the price of WG selling electricity to PCH at time t, RMB/kWh; , are the prices of WG selling GC to PCH and CHP at time t, RMB/units, respectively; is the price of WG buying SP from CHP at time t,RMB/kWh; is the electricity sold by WG to PCH at time t, kWh; , are the GC sold by WG to PCH and CHP at time t, units/MW; is the SP purchased by WG from CHP at time t, kWh; and are the electricity and GC sold by WG to EM and GCM at time t, respectively; is the SP purchased by WG from CAPM at time t; is the feed-in tariff at time t; is the market purchase price of GC at time t; is the market price of SP at time t.

The benefit-maximizing operating model of WG can be expressed as (4a) – (4j).

In the above model, (4a) is the objective function;(4b)–(4f) are the revenue expressions; (4g) is the power constraint; (4h) is the GC constraint; (4i) is the SP constraint. Where, is the price of the maintenance cost of WG, RMB/kWh; and are the forecast power and forecast error of WG at time t, kWh, respectively.

2.2 CHP

The combined operation model of CHP with CCS is shown in Fig. 2, which mainly consists of CHP, CCS, and CO2 storage tanks.

Fig. 2 CHP operation model

1) CHP

CHP is a “heat and power” model that generates both heat and power. Its operation model can be simplified as follows:

where Pet and Pht are the power generation and thermal power of CHP at time t, respectively, kW; Ph0 is the base thermal power of CHP in the minimum operation state, kW; hchp1 and hchp2 are the heat-to-electricity conversion efficiency of CHP in the minimum and maximum operation states,respectively; Pe,max and Pe,min are the upper and lower limits of CHP power generation, kW, respectively.

Meanwhile, CHP’s “heat and power” operation is also subject to power constraints and electrical power creep constraints.

Here, Ph,max and Ph,min are the upper and lower limits of CHP heating power, respectively, kW; is the maximum climbing power of CHP, kW.

In addition, the CO2 emissions from CHP can be expressed as:

where is the CO2 emission rate of CHP at time t,kg/s; aco2, bco2, and cco2 are the emission factors of CHP,respectively [30].

2) CCS

Under a steady-state operation, the carbon capture rate of CCS is approximately linearly related to the power consumption [30]; thus, the simplified model of CCS can be expressed as:

where is the CCS power consumption at time t, kW;is the CCS carbon capture rate at time t, kg/s; χ is the capture rate, kW/kg [31, 32].

CCS operation is subject to the following power constraints and power creep constraints, respectively.

where Pccs,max and Pccs,min are the maximum and minimum input power consumption of CCS, respectively, kW; is the maximum climbing/downhill power of CCS, kW.3) CO2 storage tanks (CST)

The CST is used to store compressed CO2, and its internal air pressure can indirectly represent the storage CO2 volume, which must be met by the internal air pressure of the CST [25].

where is the internal pressure of the CST at time t, bar;Tco2 is the internal temperature of CST, K; molco2 is the molar mass of CO2, kg/mol; is the output flow rate of CO2 at time t, kg/s; Pa,min and Pa,max are the minimum and maximum atmospheric pressures of CST, respectively.

4) CHP operation model

The operating revenue of CHP Uchp mainly includes the revenue from CHP’s transactions with PCH ,revenue from transactions with WG , revenue from transactions with the market Uchp-m, and the cost of CHP maintenance . Of these,

where is the proceeds from CHP’s sale of electricity to PCH; is the proceeds from CHP’s sale of CO2 to PCH; is the proceeds from CHP’s sale of SP to WG; is the cost of CHP’s purchase of GC from WG; is the proceeds from CHP’s sale of electricity to EM; is the proceeds from CHP’s sale of SP to CAPM; is the proceeds from CHP’s sale of CO2 to CARM;and is the cost of CHP’s purchase of GC from GCM.

Trading variables include CHP’s trading prices and volumes with other subjects and various markets.Among them, the transaction price of CHP with other subjects includes CHP–PCH transaction priceand CHP–WG transaction price; the transaction volume with other subjects includes CHP–PCH transaction volumeand CHP –W G t r a n s a c t i o n v o l u m eThe transaction volume of CHP with each market(the transaction price of WG with each marketis determined by the market and considered a constant factor in this paper).Where, is the feed-in tariff of CHP at time t, RMB/kWh; is the price of electricity sold by CHP to PCH at time t, RMB/kWh; are the electricity sold by CHP to EM and PCH at time t, kW, respectively; and is the price of SP sold by CHP to CAPM at time t, RMB/kWh; is the SP sold by WG to CAPM at time t, kW; are the price of electricity sold by CHP to CARM at time t, respectively and and are the prices of CO2 sold by CHP to CARM and PCH, respectively, RMB/kg; are the prices of CO2 sold by CHP to CARM and PCH at time t, respectively, kg/s;and is the price of GC purchased by CHP from GCM at time t, RMB/unit; is the number of GC purchased by CHP from GCM at time t, units/(MW).

The benefit-maximizing operational model of CHP can be expressed as (19a) – (19k).

In the above model, (19a) is the objective function;(19b)–(19f) are the revenue expressions; (19g) is the electric power balance constraint with non-negative constraint;(19h) is the GC equilibrium constraint with a non-negative constraint; (19i) is the CO2 balance constraint with nonnegative constraint; (19j) is the reserved capacity constraint,i.e., CHP sells less SP power than CHP’s reserved capacity;(19k) is the thermal power constraint. Here, , , are the CHP O&M cost coefficients, RMB/kWh; is CHP’s non-water, renewable energy generation quota ratio; is CHP’s standby capacity ratio.

2.3 PCH

PCH aims to minimize its operating costs to optimize the amount of electrical energy and green certificate interactions with WG and CHP to meet PCH’s needs. The structure of PCH is shown in Fig. 3.

Fig. 3 PCH system

1) Electrolyzer (EL) [33].

Here, ηH2 is the EL energy conversion rate; is the EL hydrogen production power at time t; is the EL power consumption power at time t; Pel,max is the upper limit of EL power consumption; is the upper limit of climbing power.

2) Hydrogen compressor (HC) [17].

where is the HC power consumption at time t; RH2 is the specific heat capacity constant of hydrogen; is the HC compressed hydrogen flow rate at time t; Tin is the HC input hydrogen temperature; ηcom is the compressor efficiency; K is the hydrogen isentropic index;Po ut/Pin is the compression ratio.

3) Methanation (MR).

where is the gas production power of MR at time t,kW; is the hydrogen consumption rate of MR at time t, kg/s; ηmr is the energy conversion efficiency of MR, kW/kg; Pmr,max and Pmr,min are the upper and lower limits of MR operation, respectively.

According to the chemical equation for methane synthesis, the CO2 demand of MR and the gas production approximately satisfy the following relationship.

where is the CO2 demand of MR at moment t, kW; ϖ is the carbon-to-hydrogen conversion coefficient.

4) Hydrogen storage tank (HST) [25].

where is the internal pressure of the HST at time t, bar;TH2 is the internal temperature of the HST, K; molH2 is the molar mass of hydrogen, kg/mol; mctom is the compressed hydrogen flow rate of the compressor at time t, kg/s;is the hydrogen load demand at time t, kg/s; Pa,H 2 ,min and Pa,H 2 ,max represent the minimum and maximum at mospheric pressures of HST, respectively.

5) Electricity storage (ES).

where is the stored energy at time t; Ebat,min and Ebat,max are the minimum and maximum stored energy of the ES, respectively, kWh; and are the charging and discharging power at time t, respectively, kW; ebat,c and ebat,d are the charging and discharging efficiency,respectively; Pbat,c,max and Pbat,d,max are the maximum charging and discharging powers, respectively, kW; is a binary variable to avoid the charging and discharging processes to be performed simultaneously.

6) PCH model.

The operating benefits of PCH Uchp include transaction costs with CHP , transaction costs with WG ,transaction costs with the market Cpch-m, and CHP maintenance costs

whereare the costs of electricity purchased by PCH from CHP, WG, and EM, respectively;are the costs of CO2 purchased by PCH from CHP and CARM, respectively;are the costs of GC purchased by PCH from WG and GCM,respectively.

Trading variables include PCH’s trading prices and volumes with other subjects and various markets.Among them, the transaction price of PCH with other subjects includes the transaction price of PCH–CHPand the transaction price of PCH–WG; the transaction volume with other subjects includes the transaction volume of PCH–CHPand the transaction volume of PCH–WGThe transaction volume of PCH with each market; the transaction pri ce of PCH with each marketis determined by the market and is regarded as a constant factor in this paper. Here, is the industrial electricity price at time t, RMB/kWh; is the electricity purchased by PCH from EM at time t, kWh; is the price of CO2 purchased by PCH from CARM at time t, RMB/kg; is the price of CO2 purchased by PCH from CARM at time t, RMB/kg; is the price of GC purchased by PCH from GCM at time t, RMB/unit; is the price of GC purchased by PCH from GCM at time t, units/MW.

The benefit-maximizing operating model of PCH can be expressed as (33a)–(33j).

In the above model, (33a) is the objective function;(33b)–(33f) are the cost expressions; (33g) is the electric power balance constraint and non-negative constraint; (33h)is the GC balance constraint and non-negative constraint,indicating that under the renewable energy consumption responsibility system, PCH needs to purchase GC to meet its renewable energy consumption tasks; (33i) is the CO2 balance constraint and non- negative constraint; (33j) is the hydrogen balance constraint; (33k) is the gas power constraint. is the combined O&M cost factor for PCH,RMB/kWh; is the percentage of renewable energy generation consumed by PCH.

3 The WG–CHP–PCH collaborative operating model

As independent individuals, each subject intends to reach a consensus through negotiation and seek a balanced strategy to maximize their respective benefits. Therefore,how to negotiate a fair and reasonable deal solution between subjects is the focus of all subjects. Using Nash negotiation theory to characterize the degree of the maximum gain enhancement of each subject under the cooperative model of WG–CHP–PCH, the model is obtained as shown in (34):

whereare the benefits of WG, CHP, and PCH when they are operated independently, which can be obtained by model (B1)–(B3) in Appendix B, respectively.The difference between the two returns indicates the degree of enhancement of the returns of each subject under the cooperative model. The solution that maximizes the product of the degree of revenue enhancement of each subject in (21)is the Nash equilibrium solution [34].

3.1 Equivalent transformation of models

The multi-entity cooperative operation model of WG–CHP–PCH is essentially a nonconvex nonlinear optimization problem, which is difficult to solve directly. In this paper,model (34) is equivalently transformed. It is converted into the following two easily solvable subproblems: the cooperative benefit maximization subproblem of WG–CHP–PCH (P1) and the transaction payment subproblem(P2). The optimal solution of the original problem (34) is obtained by solving the two subproblems sequentially.

P1: Maximizing the benefits of WG–CHP–PCH collaboration.

P2: Payment issues for WG–CHP–PCH transactions.

whereare the optimal solutions derived from P1. The proof process is detailed in Appendix A. In P1, the transaction amounts between subjects are cancelled out in the superposition process, and thus, the inter-subject transactions cannot be determined. Therefore, by solving for P2, the negotiated transaction price between subjects can be determined, and thus, the amount of inter-subject transactions can be determined.

3.2 Solving models using ADMM

To protect the privacy of each subject when participating in the negotiation, ADMM is used for the distributed solution of P1 and P2.

A. Solving for P1

Since the volume equation constraint of (37) can be seen such that it becomes more difficult to decouple the cooperative model considering multi-species trading, two pairs of Lagrange multipliers are no longer sufficient for the distributed solution in the typical three-subject cooperative model. Therefore, three pairs of Lagrange multipliers ,, and penalty factors ρwg, ρchp, ρpch are introduced:

whereare the transaction volume gaps among WG, CHP, PCH, and other subjects,respectively. When the decreasing process of transaction volume gap represents the negotiation process of intersubject transaction, when the transaction volume gap is equal to 0, the transaction is reached between subjects.

Considering the opposite of the objective function of P1 and transforming it into a minimization problem, the increasing Lagrange function of the objective function of P1 is obtained as follows:

According to the principle of the ADMM algorithm,the decomposition of (39) leads to the distributed optimal operation models of WG, CHP, and PCH, respectively.

1) WG distributed optimized operation model.

2) CHP distributed optimization operation model.

3) PCH distributed optimization operation model.

Update Lagrange Multiplier λ:

P1 iteration termination condition.

The distributed solution algorithm 1 for P1 is shown in Appendix D for the specific steps.

B. Solving for P2

By solving P1, one can simultaneously derive the optimal desired transaction volume between WG–CHP–PCH:, the intersubject transaction volume can then be expressed as

Substituting (45) into (36), we obtain the equivalent P2 model.

Based on the objective function of (46), the augmented Lagrange function is constructed by introducing three pairs of Lagrange multipliersand penalty factors ϕwg,ϕchp, ϕpch. Then the augmented Lagrange function of the objective function of (46) can be expressed as

whereare the transaction price gaps among WG, CHP, PCH, and other subjects, respectively.When the transaction price gap is equal to 0, the transaction amount is determined between subjects. They are expressed as in (48).

To visualize the price advantage of inter-subjective transactions, this paper assumes that the inter-subjective transaction price is greater than the market acquisition price and less than the market sale price, i.e.,

According to the decomposition idea of the ADMM algorithm, (47) is decomposed to obtain the distributed optimization models of transaction prices for WG, CHP,and PCH, respectively.

1) WG trading price distributed optimization model.

2) CHP trading price distributed optimization model.

3) PCH trading price distributed optimization model.

Update of Lagrange multipliers.

P2 Iteration termination condition.

Distributed Algorithm 2 for P2, see Appendix D for the specific steps.

C. On the update of the penalty factor

Literature [33] shows that the convergence rate of ADMM is strongly influenced by the value of the step size.The step size of the original ADMM is generally fixed, such as ρ in P1 and ϕ in P2. Therefore, the performance of the algorithm may degrade in the later stages of the iteration[35]. To solve this problem, a two-stage dynamic step correction method is improved and utilized in this paper. (P1 is used as an example below.)

Stage 1: Each iteration calculates the change in the value of the original residual and the pair-wise residual, as shown in (57). If the minimum value change of the original residuals and pair-wise residuals is greater than the set value Δ (e.g.: 0.1), the step size remains unchanged because the current step size moves toward the decrease in the original residuals and pair-wise residuals. Otherwise, the current step size may affect the convergence of the algorithm. In this case, the step size is updated, according to stage 2.

Stage 2: The step size is updated based on the current values of the original residuals and the pair-wise residuals,as shown in (58). If the original residuals are considerably larger than the pair-wise residuals, the step size is increased,resulting in an increased penalty for violating the original constraint. If the pair-wise residuals are substantially larger than the original residuals, the step size decreases to achieve convergence of the pair-wise residuals. In this case, the convergence of the original residuals and the pairwise residuals can be balanced alternatively.

The dynamic step update is shown in Algorithm 3, and the specific steps are shown in Appendix D.

4 Case analysis

The WG–CHP–PCH multi-subject energy system shown in Fig. 1 is used as an example to illustrate the application of the WG–CHP–PCH cooperative model proposed in this paper considering multi-energy derivatives trading and to verify its effectiveness. The parameters of WG, CHP, and PCH are shown in Table C1 in Appendix C, where the feed-in tariff of wind-PV is selected from the 2020 regional guideline price for Class II resources [36]. The WG power predicted values are shown in Fig. C2 of Appendix C,and the actual values are the sum of predicted values and prediction errors. The thermal load carried by CHP is shown in Fig. C3 of Appendix C. The combined load carried by PCH is shown in Fig. C4 of Appendix C. The industrial time-of-use tariff for EM is given in Table C5 [37].

4.1 Algorithm Convergence Analysis

Fig. 4 gives the convergence results of the objective function and cooperative efficiency function for each subject of P1, and the proposed algorithm achieves convergence after 46 iterations. Fig. 5 provides the convergence results of the original residuals of P2, which converge after the 10th iteration. It shows that this paper has good convergence characteristics for both P1 and P2 based on ADMM and can achieve distributed and efficient solutions for both subproblems while considering the protection of the privacy information of each subject.

Fig. 4 Cost-iterative convergence process of P1

Fig. 5 The original residual convergence process for problem 2

4.2 Inter-subject transaction analysis

A. Transaction Energy Analysis

Fig. 6(a) gives the results of electricity trading between PCH and WG, CHP, and EM under the WG–CHP–PCH cooperative model. PCH chooses to purchase electricity from WG, CHP, and EM in the period 01:00–06:00, when the base price of EM is low. In the period 08:00–22:00,when the base price is higher, PCH mainly purchases electricity from WG and CHP. During the periods 08:00–14:00 and 21:00–22:00, PCH’s power requirements are fully supplied by WG and CHP; during the period 15:00–21:00, PCH purchases a small amount of power from EM.By arranging the power purchase schedule in this way,the power consumption cost can be minimized. Figs. 6(b)and 6(c) provide the power trading scheme of WG and CHP, respectively, and the power generated by WG and CHP is mainly traded to PCH, which indicates that for the direct feed-in of power generated by WG and CHP, the direct power trading among WG, CHP, and PCH is more profitable.

Fig. 6 Energy transaction between subjects

Fig. 7 GC transactions between entities

Fig. 7 shows the results of GC transactions between WG and PCH, CHP, and GCM under the cooperative model.WG seeks direct transactions above the GC base price. In Fig. 7(a), during the 05:00–8:00 and 15:00–20:00 hours,WG prioritizes seeking direct transactions with CHP and PCH guided by the GC base price to maximize the revenue.In the other periods, WG sells the remaining GCs to GCM at the base price after satisfying the direct GC transactions,which agrees with the basic guideline of “scarcity is precious” in the market context.

Fig. 7(b) shows the GC purchase scheme of PCH.Combined with 6(b), the renewable energy consumption of PCH cannot meet its consumption task during 05:00–8:00 and 15:00–20:00 hours; thus, PCH must purchase additional GC to meet its consumption task. During other hours, both WG’s electricity and GC output are high, and the renewable energy power purchased by PCH from WG is sufficient to meet its renewable energy consumption task; thus, PCH does not need to purchase additional GCs.

Fig. 7(c) shows the GC purchase scheme of CHP. The renewable energy generation quota task of CHP will be accompanied by its electricity output curve all the time, and it keeps a synchronous trend with electricity in Fig. 6(c).During the 05:00–8:00 and 15:00–20:00 hours, CHP will choose the appropriate GC purchase option in competition with PCH’s purchase due to WG’s lower GC production and PCH’s GC demand. This is demonstrated by the fact that CHP’s GC demand rises, but its GC purchase from WG falls, and the shortage of GC will be purchased from GCM.This is the optimal GC trading scheme in view of the CHP subject. In other time periods, CHP only trades GCs directly with WG.

Fig. 8 shows the CO2 trading results among PCH,CHP, and CARM in the cooperative mode. The CO2 trading volume of CHP with PCH and CARM depends on the CO2 uptake by CHP. Combined with the analysis in Figs. 8(b) and 6(c), the power sold by CHP peaks during 17:00–19:00, while the CO2 absorption is almost 0. This is because CHP sells all the power generated at this time,and the carbon absorption unit of CHP is stopped, which is considered to be the revenue maximizing operation by the CHP subject. In other time periods, CHP sells less power,and the extra power sent out by CHP will be used for CO2 absorption to increase CHP’s revenue. In Fig. 8(a),as the CO2 available for sale from CHP slowly decreases throughout the day, PCH’s purchased CO2 is traded from CHP to CARM.

Fig. 8 Inter-subject carbon trading

Fig. 9 shows the results of SP trading between WG,CHP and CAPM under cooperative model. When the actual generation of WG is larger than the forecasted value, WG can buy the power sales from CHP, such as the 00:00–06:00 and 22:00–24:00 periods, and can also buy the power sales from EM, such as the 10:00–13:00 and 15:00–20:00 hours.When WG’s actual generation is smaller than forecasted generation, WG can purchase generation rights from CHP or CAPM, such as 09:00 and 14:00 hours.

Fig. 9 Service electricity transaction between subjects

B. Transaction Price Analysis

Fig. 10 shows the trading prices between subjects and the market, where Figs. 10a, 10b, 10c, and 10d show the trading prices of electric energy, GC, CO2, and SP,respectively. Under the price constraint of (49)–(51),each subject will seek a suitable trading price between the base prices to maximize their respective trading returns.For example, in Fig. 10(a), PCH’s purchase of renewable energy power from WG offsets PCH’s consumption tasks and reduces GC costs. Therefore, compared with CHP,PCH is more willing to trade electricity with WG; thus,the price of electricity from WG–PCH is higher than that of electricity from CHP–PCH. CHP is at a disadvantage in the direct trading of electricity. In Fig. 10(b), at around 06:00 and 18:00, the GC production of WG is smaller and the corresponding trading price will become expensive, and CHP and PCH will choose to seek a portion of GC from the market. Similarly, the direct trading prices of carbon and SP shown in Fig. 10(c) and (d) are inversely proportional to their internal spot quantity in the cooperative model, and each subject will seek to trade with the market when the internal spot quantity is insufficient.

Fig. 10 Inter-subject transaction prices

4.3 Operational benefit comparison

Table 1–3 show the operating income results of each subject before and after the cooperative operation,respectively. Positive and negative values in the table represent costs and benefits, respectively. After the cooperative operation, the income of WG and CHP increased by 1.615669 million RMB and 1.7056 million RMB, i.e., about 50.8% and 49.26%, respectively, while the cost of the PCH main body decreased by 3.389771 million RMB, a decrease of about 24.19%. This shows that individual interests have been significantly improved through tripartite cooperation.

Table 1 Comparison of WG’s revenue before and after cooperation

After cooperation(10e4＊RMB)Cwg WGBefore cooperation(10e4＊RMB)wh5.39335.3933 wg,sp53.28945.4802 Cchp2wg wg,p-245.5355-41.0436 Uwg2em wg,p0-357.9872 Uwg2pch wg,gcs-119.4497-55.392 Uwg2gcm wg,gcs0-57.2891 Uwg2chp wg,gcs0-7.0315 Uwg2pch Total cost-306.303-467.8699 the difference in the total cost161.5669

Table 2 Comparison of CHP’s revenue before and after cooperation

CHPBefore cooperation(10e4＊RMB)After cooperation(10e4＊RMB)Cchp wh27.520122.5188 chp,p-158.388-1.6699 Uchp2em chp,p0-382.8 Uchp2pch chp,gcs52.79611.8761 Cgcm2chp chp,gcs057.2891 Cwg2chp chp,e-226.8-18.9897 Uchp2carm chp,e0-132.238 Uchp2pch chp,sp0-15.5899 Total cost-289.0425-459.6033 the difference in the total cost170.56 Uchp2wg

Table 3 Comparison of PCH’s revenue before and after cooperation

PCHBefore cooperation(10e4＊RMB)After cooperation(10e4＊RMB)Cchp wh6.93666.9366 pch,p105384.9126 Cem2pch pch,p0357.9872 Cwg2pch pch,p0382.8003 Cchp2pch pch,gcs141.195623.1283 Cgcm2pch pch,gcs07.0315 Cwg2pch

continue

PCHBefore cooperation(10e4＊RMB)After cooperation(10e4＊RMB)Cchp wh6.93666.9366 pch,p105384.9126 Cem2pch pch,p0357.9872 Cwg2pch pch,p0382.8003 Cchp2pch pch,gcs141.195623.1283 Cgcm2pch pch,gcs07.0315 Cwg2pch

4.4 Influence of base price on the cooperative operation

This section will explore the impact of the base price on the revenue of the WG–CHP–PCH subject cooperation.Scenarios 1–3 are designed to compare the impact of different WG sales base prices on WG–CHP–PCH cooperative operation, as shown in Table 4. Scenario 1 is the basic scenario used in this paper; scenario 2 is the average renewable energy feed-in tariff in 2021; scenario 3 is the predicted feed-in tariff in 2022 based on the price reduction of the base price in the last two years. Comparing the calculation results of the three scenarios, the returns of each subject are improved substantially under different scenarios. In the three scenarios, the revenue of WG is increased by 50.81%, 63.70%, and 69.63% respectively;the revenue of CHP is increased by 49.26%, 50.26%, and 51.05% respectively; the revenue of PCH is increased by 24.19%, 24.65%, and 24.78% respectively.

Table 4 Effect of the feed-in tariff of wind-PV on cooperation

AgentUnit:10e4＊RMBScenario 1Scenario 2Scenario 3 WG wg,t0.370.320.3 Before 306.303273.123259.8503 After 467.8699447.0909440.7869 Increase rate52.75%63.70%69.63%pwg2em CHP wg,t0.370.320.3 Before 289.0425289.0425289.0425 After 459.6033462.3227466.5857 Increase rate59.01%59.95%61.42%pwg2em wg,t0.370.320.3 Before 1,401.2361,401.2361,401.236 After 1062.31055.91054 Increase rate24.19%24.65%24.78%pwg2em PCH

Similarly, scenarios 4–6 are designed to compare the effects of different GC base prices on WG–CHP–PCH,as in Table 5; scenarios 7–9 are designed to compare the effects of different CO2 base prices on WG–CHP–PCH,as in Table 6. By comparing the above scenarios, general conclusions can be summarized as follows. 1) With the reduction in various base prices, the total revenue of each subject is affected with or without cooperation, and all of them show a decreasing trend. 2) Through cooperation, the degree of revenue enhancement or cost reduction of each subject shows a gradual increase. This further indicates that the operating efficiency of each subject can be improved through cooperative operation. 3) Under cooperation, the lower the base price of each type, the higher the percentage of revenue improvement of each subject, and the more obvious the effect of improving the operating efficiency of each subject through cooperation.

Table 5 Effect of GC’s base price on cooperation

AgentUnit:10e4＊RMBScenario 4 Scenario 5Scenario 6 wg ,t0.190.180.17 Before312.9391306.303299.6669 After468.1553467.8699461.6614 Increase rate49.60%52.75%54.06%gwg2gcm WG wg ,t0.190.180.17 Before289.0425289.0425289.0425 After428.5404459.6033464.3826 Increase rate48.26%59.01%60.66%gwg2gcm CHP wg ,t0.190.180.17 Before1,401.2361,401.2361,401.236 After1062.61062.31062.1 Increase rate24.17%24.19%24.20%gwg2gcm PCH

Table 6 Effect of base price of carbon on cooperation

AgentUnit:10e4＊RMBScenario 7Scenario 8Scenario 9 chp ,t0.650.60.5 echp2carm WG Before306.30298306.303306.303 After458.341467.8699470.7433 Increase rate49.64%52.75%53.69%chp ,t0.650.60.5 echp2carm CHP Before307.9425289.0425251.2425 After434.5301459.6033425.1853 Increase rate41.11%59.01%69.23%

continue

AgentUnit:10e4＊RMBScenario 7Scenario 8Scenario 9 echp2carm PCH chp ,t0.650.60.5 Before1,401.21,401.2361,401.236 After1063.11062.31059.9 Increase rate24.13%24.19%24.36%

4.5 Convergence rate analysis

We introduce a penalty factor update algorithm to improve the convergence speed of the calculation process.The number of iterations (ITER) for ADMM with adaptive parameters (AP), fixed parameters (FP), and dynamic parameters (DP) to solve P1 are shown in Table 7.

Table 7 Convergence speed comparison

Initial0.01 0.05 0.01 DP AP FP DP AP FP DP AP FP P1 ITER 4681 205 215753444444 P2 ITER7--9--136767

First, we find that P1 requires a relatively large initial penalty parameter, while P2 requires a relatively small initial penalty parameter. This suggests that it is optimal to tune the parameters for enhanced performance. In addition, the number of iterations and computation time for convergence are acceptable in practical implementations.Second, adaptive and dynamic ADMMs do not always outperform standard ADMM in solving P1, but this still depends on the parameter choice. In Fig. 11–12, we further show the convergence of the original and dual residuals of the algorithm for solving P1. The results show that ADMM with dynamic penalty parameters can converge faster by coordinating the convergence of the original and dual residuals. Notably, for computing cooperative game problems of different agents in a distributed environment,dynamic ADMM provides excellent results for problems of different scales.

Fig. 11 Convergence comparison of original residuals

Fig. 12 Convergence comparison of dual residuals

5 Conclusion

A cooperative model of WG–CHP–PCH considering multi-energy derivatives trading is proposed. Only trading information (trading varieties, curves, and prices)is exchanged between the subjects, which protects the privacy and security of the subject models and provides a feasible and optimal operation scheme for the cooperative model considering multi-energy derivatives trading. The following main conclusions are drawn. 1) The distributed optimization algorithms based on the two subproblems of cooperation benefit maximization and transaction payment negotiation proposed by ADMM have good convergence properties and achieve an efficient solution of the WG–CHP–PCH cooperation problem while protecting the privacy information of each subject. 2) Simulation results show that the proposed WG–CHP–PCH cooperative model insignificantly increases the benefits of each subject compared to the non-cooperative case. 3) With the decrease in the base price, the benefits of each subject show a decreasing trend. Through the cooperative operation,the benefits of each subject tend to increase compared with the uncooperative case; the lower the base price, the higher the percentage of benefit increase, i.e., the more obvious the effect of enhancing the respective benefits through cooperation. Therefore, the decrease in the base price will encourage independent subjects to seek thirdparty cooperation to improve their benefits. 4) Dynamic adjustment of the penalty factor of ADMM can achieve convergence of the optimal solution with fewer iterations.Measurements show that with an initial penalty factor of 0.01, dynamic ADMM can save up to 43.2% and 77.56% of the computational cost compared with standard and adaptive ADMM. In future, further efforts will be made to improve the proposed decentralized scheduling framework.

Appendix A

Appendix B

1) WG operation model of the traditional trading mechanism.

2) CHP operation model of the traditional trading mechanism.

3) PCH revenue model for traditional trading mechanisms.

Appendix C

Fig. C1 Predicted power and prediction error of WG

Fig. C2 Heat load carried by CHP

Fig. C3 Electric load, natural gas load, and heat load carried by PCH

Table C1 Parameters of equipment

Note: The unit of price is: [RMB/kWh]

ParametersValues ParametersValues ParametersValue ParametersValues pwg2em wg,t0.37 pchp2em chp,t0.3 Ebat,max1800 Pccs,max7.5 gwg2gem wg,t0.18schp2capm chp,t0.4ebat,c0.95Pccs,min0 scapm2wg wg,t0.45echp2carm chp,t0.6ebat,d0.96Prp ccs1.5 awhwg0.008ggcm2chp chp,t0.2Pbat,c,max500Tco235 Ph05αchp wh13.29Pbat,d,max600ηmr0.6 hchp10.15bchp wh0.004ggcm2pch pct,t0.2Pmr,max25 pct,t0.7Pmr,min0 Pe,max50βchp hchp20.08cchp wh22egcm2pch gcs0.2sgcm2pch pct,t0.45 ω 1 Pe,min10βchp wh0.008molco244 Ph,max40ηH20.87βpch sp0.1αpch wh0.15Pa,min20 Ph,min0Pel,min0Pel,max72Pa,max35 Prp chp10Prpel14.4TH260 χ0.5 aco20.89RH214.304molH20.002Pout/Pin1.5 bco20.0017Tin40Pa,H2,min20Ebat,min200 cco226.15ηcom0.7Pa,H2,max35

Table C2 Time-of-use electricity price

Time intervalpem2pch pch ,e / (RMB/kWh)01:00—07:00、23:00—24:000.38 08:00—11:00、15:00—18:000.68 12:00—14:00、19:00—22:001.2

Appendix D

Algorithm 1−5−4 1: = 50, = 10 , 10 *ones(6,1), =1, (6,24),K max ξρλ==kzeros Initialize each transaction volume in the alliance to 0,wg, ,wg, ,t kt k transaction volumes = [; ;PPP chp, ,chp, ,pch, ,pch, ,t kt kt kt k wg-pchwg-chpchp-pch ;; ;];PPP chp-wgpch-chppch-wg 2: while 1 do wg,t 3: WG sends to PCH,P wg-pch CHP sends to PCH,P chp,t chp-pch PCH calculates the model(42), andsendsP pch, , 1pch, , 1 t kt k++pch-wg and ;Ppch-chp P pch, , 1 t k+4: PCH sends to CHP,pch-chp wg, ,t k WG sends to CHP,wg-chp CHP calculates the model(41), and sends and ;PP P chp, , 1chp, , 1 t kt k++P chp-pchchp-wg 5:pch, , 1 t k PCH sends to WG,pch-wg chp, , 1 P t k s++CHP sends to WG,chp-wg WG calculates the model(40), and wg, , 1wg, , 1 t kt k++sends and ;PP wg-pchwg-chp 6: Update according to formula (43);λ 7: 1;kk ← +8: Judging the convergence of the alg orithm:if the formula (44) is met, break;else, return to step 3;

Algorithm 2−2−8 1: = 50, = 10 , 10 *ones(6,1), =1, (6,24),K max ξψγ==kzeros Initialize each transaction volume in the alliance to 0,wg, ,wg, ,t kt k transaction volumes = [; ;ppp chp, ,chp, ,pch, ,pch, ,t kt kt kt k wg-pchwg-chpchp-pc;; ;];ppp hchp-wgpch-chppch-wg 2: while 1 do 3: WG sends to PCH,p wg,wg-pch t chp,t CHP sends to PCH,PCH calculates the model(54), andsendsp pch, , 1pch, , 1 chp-pch t kt k+and ;pch-wg+ppch-chp 4: PCH sends to CHP,pch, , 1 pch-chp t k+WG sends to CHP,wg, ,wg-chp t k CHP calculates the model(53), and pp pp ppp chp, , 1chp, , 1 chp-pchchp-wg t kt k++sends and ;5:pch, , 1 pch-wg t k PCH sends to WG,chp, , 1 t k s++CHP sends to WG,chp-wg WG calculates the model(52), and sends and ;t kt k+wg, , 1wg, , 1 wg-pchwg-chp+pp 6: Update according to formula (55);γ 7: 1;kk ← +8: Judging the convergence of the alg orithm:if the formula (56) is met, break;else,return to step 3;

Algorithm 3 1:Set the residual change tolerance .∆2:fork 3: Find the symbol with the largest o riginal residual according to (57a),and return the position label ;i 4: Find the symbol w ith the largest dual residual according to (57c),and return the position label ;j 5: ifi j =k+12 k+12 Calculate and according to∆r∆s ijij 2 2(57b), (57d) and ;i k+12 if min(∆rs,)≥∆k+12 ijij 2∆2 kk+1 ρρ=else k+1 ρ is updated accordi ng to (58).e d n e se l Calculate and accordin o∆r k+1 k+1 2 2 i 2∆s i 2 g t (5 7b) and ;i k+1 2 if min(,)≥∆∆∆rs 2 i 2 k+1 i 2 kk+1 ρρ=else k+1 ρ is updated accordi ng to (58).end Calcula e k+12 t d∆r k+12 j 2 an∆s j 2 according to (57d) and ;j k+12 k+12 if min(,)≥∆2 rs j j 2 kk+1 ρρ∆∆=e s l e ρ k+1 is updated accordingto(5).end end 6:k+1 ρ is send back t o the A lgor h 1 itm 8.7:end

Acknowledgements

This work was supported in part by the Science and Technology Project of State Grid Corporation of China under Grant (No. 52272220002T), and in part by the project supported by Sichuan Provincial key research and development program of China (No. 2022YFG0123), and in part by Central Government Funds for Guiding Local Scientific and Technological Development of China (No.2021ZYD0042).

Declaration of competing interest

We declare that we have no conflict of interest.