Introduction

The severity of energy consumption, environmental degradation, and climate change has escalated in tandem with the progress of the economy and society. CCS can separate \({\text{CO}}_2\) from the exhaust gas emitted by the energy industry, which is one of the key technologies to realize the low carbonization of IES1. Moreover, the \({\text{CO}}_2\) that is caught and stored can be utilized as a carbon source in the P2G reaction process. The integration of the two provides an opportunity to achieve deep decarbonization of energy systems. Yang et al. combined P2G with PIES to achieve electricity-heat-gas complementarity and improved the stability and economic benefits of the system2. Li et al. combined the P2G with the traditional combined CCHP microgrid, and enhanced economy of the system by improving the electricity-gas coupling3. Chen et al. introduced a model that optimizes the utilization of renewable energy sources and reduces carbon dioxide emissions by integrating P2G and CCS technologies4. Zhang et al. considered the coupling operation of CCS and P2G, which reduced the carbon emissions, gas purchase and operating cost5. The above-described references confirm that P2G–CCS can reduce system carbon emissions from the perspective of equipment coupling and low-carbon energy consumption, but do not fully consider environmental costs and detailed modeling analysis of EL.

The carbon trading mechanism is a viable approach to harmonize the environmental advantages and economic advantages of the system6. Wang et al.7 described carbon transaction costs with a carbon emission cost coefficient, and constructed a refined carbon emission model. Xiang et al. proposed a low-carbon economic dispatch model for an electricity-gas system that incorporates CCS, carbon emissions trading, and demand response. The findings indicate that the model effectively decreases operational expenses and minimizes carbon emissions from the system8. Xiang et al.9 used the low-carbon contribution of renewable energy as an evaluation index to construct and design a coupled market for energy and carbon trading, which reduced the environmental costs of the PIES. Wang et al. introduced a low-carbon operation model for PIES by incorporating CCS technology and integrating a LCPMRP. Their research demonstrates that a well-designed carbon trading market can effectively drive the low-carbon development of PIES7. Zhang et al.5 considered the carbon trading policy based on the CHP-P2G cooperative operation mode, which effectively reduced the environmental costs of the system. Sun et al.10 introduced an optimization model within the framework of IES carbon trading and demonstrated the benefits of carbon trading in decreasing operating costs. However, the majority of current studies on PIES optimization with CCS and P2G coupling focuses on equipment coupling and carbon trading mechanism constraints, and less research considers the uncertainty factors of source and load.

The generation of wind and solar power exhibits a high degree of randomness and intermittency, and user-side electricity consumption is highly subjective. At the same time, there are limitations in forecasting technology. Therefore, the uncertainty in the power system is gradually manifested as Knightian uncertainty. When considering the uncertainty of system source-load, the two generally employed approaches are robust optimization and fuzzy random optimization11. Sharma et al.12 introduced a two-stage robust optimization approach for enhancing the adaptability of building energy management systems in the face of uncertainties to improve the adaptability of system to uncertainties while ensuring user comfort and system reliability. Ju et al. established a robust optimization model for source-load uncertainty, thereby reducing the impact of microgrid uncertainty13. Zheng et al. introduced a data-driven stochastic collaborative model to integrate electricity and natural gas systems. This model enhanced the utilization of renewable energy and decreased carbon emissions in the system14. Yang et al. introduced a framework for integrating a combined CCHP system with P2G technology. An operation optimization model was developed using an improved risk explicit interval parameter programming (REIPP) method to enhance energy efficiency15. However, robust optimization does not consider the economics of the system, and random optimization is difficult to achieve under the risk that cannot be calculated and measured. In order to balance the benefits and risks in microgrid operation, Wu and Dai et al. introduced the IGDT to introduce RA and RS strategies based on the deterministic cost of system operation to provide flexible decision-making solutions for system operators when the system is uncertain16,17. Qu et al. successfully addressed the dual uncertainty of the source side and the load side of the system using IGDT. This approach enhanced the system’s economy and overall stability18. Najafi et al. proposed a hybrid optimization model of IGDT and RO for a multi-energy coupled energy center integrated system with CCHP. RO is used to deal with the uncertainty of the current electricity market price, and IGDT is used to deal with the uncertainty of wind power generation. The results show that this hybrid method can optimize operating costs under uncertain conditions19. Mobasseri et al. modeled electricity and heat demand, renewable energy generation uncertainty, and fuel cell electric vehicle hydrogenation demand uncertainty through IGDT and scenario analysis20. Currently, while examining the impact of uncertain parameters on the functioning and scheduling of a system, the uncertainty of multiple parameters such as wind power, solar energy and load is rarely involved, and the IGDT optimal scheduling of PIES with CES is not yet involved.

The emergence of the “sharing economy” paradigm provides a novel framework for the operational mechanisms of energy storage within PIES21. Li et al. introduced a business model for CES. The model aims to enhance the efficiency of energy storage resources and minimize the expenses associated with energy storage by consolidating and reusing resources22. Zhou et al.23 presented a novel approach by integrating Distributed Energy Systems (DES) with CES via a subscription model, significantly enhancing sustainability through optimizing economic, environmental, and flexibility performances, ultimately reducing storage costs by 13–53%. Yang et al.24 proposed an optimal planning strategy for energy storage system in CES mode considering inertia support and electro-thermal coordination. Li et al.25 presented a renewable energy sharing cloud mechanism composed of energy service providers and cloud users to balance the interests of multiple entities. He et al. constructed a dual-agent CES bi-level optimization model for service providers and users. The upper-level decision variables are CES power, CES capacity, and CES annual lease price. The lower-level decision variables are user’s rental CES capacity and grid interaction power26. However, the references described above rarely consider the PIES optimization strategy for the coordinated operation of P2G–CCS and CES.

Table 1 Comparison between this study and other major publication.
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Table1 listed the comparison between the study and the above reviewed major publications in detail. In view of the shortcomings of the existing research, the main contributions of this paper are in three aspects:

  1. (1)

    Based on the coupling of traditional two-stage P2G and CCS, a multi-condition electrolytic cell model is proposed. The multi-condition conversion characteristics and operation constraints of the alkaline electrolytic cell are analysed, and the mathematical description of the multi-condition and conversion relationship is established. In the real-time scheduling process of the electrolytic cell, the coordinated operation of multiple electrolytic cells and the rapid switching of multiple operating conditions are realised.

  2. (2)

    An LCPMRP with an incentive coefficient is established based on the carbon emissions of the equipment. A two-stage optimal scheduling model is formulated, where the upper model aims at minimising the cost for the PIES operator, while the lower model aims at minimising the operating cost for the CES operator. The sensitivity analysis of carbon trading price, incentive coefficient and energy storage capacity is carried out.

  3. (3)

    An entropy weight adaptive information gap decision theory method is proposed. In the case of the EAIGDT method dealing with multiple uncertain factors, the subjective factors are eliminated, the non-probabilistic characteristics are retained, and the optimal scheduling model is constructed considering the uncertainty of the source and load. According to the operator’s risk preference, the RA and RS strategies are constructed respectively. While effectively solving the source-load uncertainty problem, it satisfies the risk preference of decision makers and achieves a balance between risk management.

The remaining portion of the paper is structured in the following manner. Section “Structure of PIES and CES and mathematical modeling of components” provides an introduction to the operational business model of the CES system in PIES. Furthermore, the architecture of PIES considering CES is constructed. The Section “Ladder-type carbon price mechanism with reward and punishment”, construct a reward and punishment ladder carbon trading model. The Section “Bi-level optimization model of PIES” presents the bi-level optimal scheduling model, with the PIES operator representing the higher subject and the CES operator representing the lower subject. The paper in the fifth section employs the EAIGDT method to measure the level of uncertainty in the source-load of the system and develops RA and RS strategies. The efficacy of the proposed optimization strategy is validated in the sixth section.

Structure of PIES and CES and mathematical modeling of components

Cloud energy storage service model

The deep integration of cloud economy and power system has become a new trend in the development of power system. CES refers to the business model that the physical energy storage collected by the virtual cloud platform provides shared energy storage services for the integrated energy system28. Unlike traditional user-side energy storage, this mode involves third parties investing in and operating the energy storage. Paying a low service fee provides access to the energy storage, reducing its initial investment cost and the time and money required for user operation and maintenance. This also increases the rate at which energy storage resources are utilized. The basic architecture of CES is shown in Fig. 1.

Figure 1
figure 1

Basic architecture of CES.

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The system architecture has two parts: CES operator and PIES operator. The two realize two-way communication through real-time information interaction, and realize physical connection and energy transmission through energy network. The CES service center is supported by advanced monitoring, control and measurement technology, and the optimal energy storage charging and discharging schedule is determined according to the historical system load, wind, photovoltaic output and electricity price information. The operator signs a service contract with the PIES operator based on the optimization results, and provides services according to the content of the contract in actual operation. The charging and discharging services enjoyed by CES users are settled with the service provider in the form of electricity purchase and sale, and the service fee is paid for the right to use energy storage.

Basic structure of PIES

Figure 2
figure 2

Basic structure of PIES.

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The specific structure of PIES considering the coordination of P2G–CCS and CES is shown in Fig. 2, including energy supply, utilization, conversion and storage. Energy supply refers to wind power, photovoltaic, external power grid and natural gas grid. Energy utilization refers to regional loads of electricity, heat, gas and CCS. Energy conversion refers to CHP, GB, two-stage P2G. Energy storage refers to CES, HS and H2S devices. The two-stage P2G consists of EL, MR and HFC. Among them, the captured \({\text {CO}}_2\) is delivered by CCS to MR, which converts it to \({\text {CH}}_4\) via a Sabatier reaction to supply gas-fired CHP units and gas load. Due to the introduction of CES structure, distributed batteries are no longer built within the PIES. The PIES interacts with the power grid and CES through the power bus, and sends the charging and discharging information demand signal to the CES operator through the information connection line.

Mathematical model of components

CHP

In this paper, the feasible operation region (FOR) method is used to model the behavior of CHP. The model is as follows29:

$$\left\{ \begin{aligned}&{{P}_{\text {CHP,g}}}(t)=\frac{{{P}_{\text {CHP,e}}}(t)+{{P}_{\text {CHP,h}}}(t)}{{{\eta }_{\text {CHP}}}} \\&{{P}_{\text {CHP,e}}}(t)-P_{\text {CHP,e}}^{\text {A}}-\frac{P_{\text {CHP,e}}^{\text {A}}-P_{\text {CHP,e}}^{\text {B}}}{P_{\text {CHP,h}}^{\text {A}}-P_{\text {CHP,h}}^{\text {B}}}\left( {{P}_{\text {CHP,h}}}(t)-P_{\text {CHP,h}}^{\text {A}} \right) \le 0 \\&{{P}_{\text {CHP,e}}}(t)-P_{\text {CHP,e}}^{\text {B}}-\frac{P_{\text {CHP,e}}^{\text {B}}-P_{\text {CHP,e}}^{\text {C}}}{P_{\text {CHP,h}}^{\text {B}}-P_{\text {CHP,h}}^{\text {C}}}({{P}_{\text {CHP,h}}}(t)-P_{\text {CHP,h}}^{\text {B}}) \le ({{I}_{\text {CHP}}}(t)-1)M \\&{{P}_{\text {CHP,e}}}(t)-P_{\text {CHP,e}}^{\text {C}}-\frac{P_{\text {CHP,e}}^{\text {C}}-P_{\text {CHP,e}}^{\text {D}}}{P_{\text {CHP,h}}^{\text {C}}-P_{\text {CHP,h}}^{\text {D}}}({{P}_{\text {CHP,h}}}(t)-P_{\text {CHP,h}}^{\text {C}}) \le ({{I}_{\text {CHP}}}(t)-1)M \\&0\le {{P}_{\text {CHP,h}}}(t) \le P_{\text {CHP,h}}^{\text {B}}{{I}_{\text {CHP}}}(t) \\&0 \le {{P}_{\text {CHP,e}}}(t) \le P_{\text {CHP,e}}^{\text {A}}{{I}_{\text {CHP}}}(t) \\&-R_{\text {dn}}^{\text {CHP}}\le {{P}_{\text {CHP,e}}}(t)-{{P}_{\text {CHP,e}}}(t-1)\le R_{\text {up}}^{\text {CHP}} \\ \end{aligned} \right. $$
(1)

where \(P_{\text {CHP,e}}^{\text {A}}\), \(P_{\text {CHP,e}}^{\text {B}}\), \(P_{\text {CHP,e}}^{\text {C}}\), \(P_{\text {CHP,e}}^{\text {D}}\), \(P_{\text {CHP,h}}^{\text {A}}\), \(P_{\text {CHP,h}}^{\text {B}}\), \(P_{\text {CHP,h}}^{\text {C}}\), \(P_{\text {CHP,h}}^{\text {D}}\) are the power generation and heating power of the CHP unit at the four endpoints of the feasible region, respectively, as shown in Fig. 3M is a large enough constant; \({{I}_{\text {CHP}}}(t)\) is a 0–1 variable representing the start-stop state of CHP unit.

Figure 3
figure 3

Feasible operating region of CHP unit.

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GB

GB is a high carbon emission equipment that produces heat energy by burning natural gas. The model is as follows30:

$$ \left\{ \begin{array}{ll} {{P}_{\text {GB,h}}}(t)={{\eta }_{\text {GB}}}{{P}_{\text {GB,g}}}(t) \\ P_{\text {GB},\text {g}}^{\text {min}}\le {{P}_{\text {GB,g}}}(t)\le P_{\text {GB},\text {g}}^{\text {max}} \\ -\Delta{P}_{\text {GB},\text {g}}^{\text {max}}\le {{P}_{\text {GB,g}}}(t+1) -{{P}_{\text {GB,g}}}(t)\le \Delta{P}_{\text {GB},\text {g}}^{\text {max}} \\ \end{array} \right. $$
(2)

Multi-condition EL

The working state of a single EL can be roughly divided into: fluctuating operating conditions, overload operating conditions, low load operating conditions, standby operating conditions and shutdown state. The specific model is as follows31:

$$\left\{ \begin{array}{ll} &{} 0.3P_{\text {el}}^{e}\le {{P}_{\text {el,e},n}}(t)\le P_{\text {el}}^{e}(L_{n}^{t}=1) \\ &{} 0.1P_{\text {el}}^{e}\le {{P}_{\text {el,e},n}}(t)\le 0.3P_{\text {el}}^{e}(V_{n}^{t}=1) \\ &{} P_{\text {el}}^{e}\le {{P}_{\text {el,e},n}}(t)\le 1.5P_{\text {el}}^{e}(R_{n}^{t}=1) \\ &{} {{P}_{\text {el,H2}}}(t)=0\quad {{P}_{\text {el,e}}}(t)=0(I_{n}^{t}=1) \\ &{} {{P}_{\text {el,H2}}}(t)=0\quad {{P}_{\text {el,e}}}(t)=P_{\text {el}}^{b}(S_{n}^{t}=1) \\ \end{array} \right. $$
(3)

where \(L_{n}^{t}\), \(V_{n}^{t}\), \(R_{n}^{t}\), \(I_{n}^{t}\) and \(S_{n}^{t}\) represent the nth EL at time t in fluctuating production condition, low load production condition, overload production condition, shutdown condition and standby condition, respectively.

The specific procedure is as follows:

  1. 1.

    Shutdown state: EL is completely turned off in the shutdown state, and it usually takes 30–60 min to restart.

  2. 2.

    Standby state: EL is closed but not stopped, with low power standby to maintain control operation, in order to switch quickly, need 5–10 min to complete the start.

  3. 3.

    Fluctuating operating conditions, overload operating conditions and low load operating conditions: In order to ensure the safety of hydrogen production, EL runs in fluctuating operating conditions most of the time. At the same time, EL can run in overload and low load conditions for a short time, which makes EL flexible and can adjust the running state by changing the load.

Due to the loss of output hydrogen energy during the start-up process under standby conditions. Therefore, the relationship between EL electric power and hydrogen production power can be expressed as:

$${{P}_{\text {el,H2,}n}}(t)=\eta _{\text {el}}^{\text {H2}}\left( {{P}_{\text {el,e},n}}(t)-S_{n}^{t}P_{\text {el}}^{b} \right) -{{\zeta }_{\text {el}}}W_{n}^{t} $$
(4)

where \({{\zeta }_{\text {el}}}\) is the penalty coefficient of hydrogen production power during standby start-up. When the binary variable W is 1, it means that EL is converted from standby state to working state, which satisfies the following constraints:

$$ \left\{ \begin{array}{ll} W_{n}^{t}\le S_{n}^{t-1} \\ W_{n}^{t}\le L_{n}^{t}+R_{n}^{t}+V_{n}^{t} \\ W_{n}^{t}\ge S_{n}^{t-1}+L_{n}^{t}+R_{n}^{t}+V_{n}^{t}-1 \\ \end{array} \right. $$
(5)

In addition, the EL operating state and state transition must satisfy certain logical constraints, as shown below:

$$\begin{aligned}{} & {} P_{\text {el}}^{b}S_{n}^{t}+0.3L_{n}^{t}P_{\text {el}}^{e}+R_{n}^{t}P_{\text {el}}^{e}+0.1V_{n}^{t}P_{\text {el}}^{e}\le P_{\text {el,e},n}^{t}\le P_{\text {el}}^{b}S_{n}^{t}+L_{n}^{t}P_{\text {el}}^{e}+1.5R_{n}^{t}P_{\text {el}}^{e}+0.3V_{n}^{t}P_{\text {el}}^{e} \end{aligned}$$
(6)
$$\begin{aligned}{} & {} -I_{n}^{t-2}+I_{n}^{t-1}-I_{n}^{t}\le 0 \end{aligned}$$
(7)
$$\begin{aligned}{} & {} V_{n}^{t}+R_{n}^{t}+L_{n}^{t}+S_{n}^{t}+I_{n}^{t-1}-1\le Y_{n}^{t} \end{aligned}$$
(8)
$$\begin{aligned}{} & {} V_{n}^{t}+R_{n}^{t}+L_{n}^{t}+S_{n}^{t}+I_{n}^{t}=1 \end{aligned}$$
(9)
$$\begin{aligned}{} & {} \underset{i=t}{\overset{t+{{T}_{\max }}-1}{\mathop {\sum }}}\,R_{n}^{i}\le {{T}_{\max }} \nonumber \\{} & {} \underset{i=t}{\overset{t+T_{_{\max }}^{'}-1}{\mathop {\sum }}}\,V_{n}^{i}\le T_{_{\max }}^{'} \end{aligned}$$
(10)

Equation (6) represents the coordinated control constraints between the various working conditions of the EL; Eq. (7) is the start-stop constraint; Eq. (8) is the start-up interval constraint, and \(Y_{n}^{t}\) is the complete start-up interval of the nth EL at time t. When it is 1, it means that EL is completely started from the shutdown state; Eq. (9) is the mutual exclusion constraint of the running state; Eq. (10) is the maximum time limit constraint for overload and low load.

HFC

The HFC burns hydrogen for power and heating. By changing the cooling circulating water flow rate and hydrogen input rate, the fuel cell can adjust the thermoelectric ratio according to the real-time electric and heat load. The model is as follows32:

$$ \left\{ \begin{array}{ll} &{} {{P}_{\text {HFC,h}}}(t)+{{P}_{\text {HFC,e}}}(t)={{\eta }_{\text {HFC}}}{{P}_{\text {HFC,H2}}}(t) \\ &{} P_{\text {HFC,H2}}^{\text {min}}\le {{P}_{\text {HFC,H2}}}(t)\le P_{\text {HFC,H2}}^{\text {max}} \\ &{} \Delta P_{\text {HFC,H2}}^{\text {min}}\le {{P}_{\text {HFC,H2}}}(t+1)-{{P}_{\text {HFC,H2}}}(t)\le \Delta P_{\text {HFC,H2}}^{\text {max}} \\ &{} K_{\text {HFC}}^{\min }\le {{P}_{\text {HFC,h}}}(t)/{{P}_{\text {HFC,e}}}(t)\le K_{\text {HFC}}^{\text {max}} \\ \end{array} \right. $$
(11)

MR

MR can combine hydrogen with \({\text {CO}}_2\) to produce methane. The model is as follows32:

$$ \left\{ \begin{aligned}&{{P}_{\text {MR,g}}}(t)={{\eta }_{\text {MR}}}{{P}_{\text {MR,H2}}}(t) \\&P_{\text {MR,H2}}^{\text {min}}\le {{P}_{\text {MR,H2}}}(t)\le P_{\text {MR,H2}}^{\text {max}} \\&{{C}_{\text {MR}}}(t)=\varsigma {{P}_{\text {MR,g}}}(t) \\&\Delta P_{\text {MR,H2}}^{\text {min}}\le {{P}_{\text {MR,H2}}}(t+1)-{{P}_{\text {MR,H2}}}(t)\le \Delta P_{\text {MR,H2}}^{\text {max}} \\ \end{aligned} \right. $$
(12)

where \(\varsigma \) is the calculation coefficient of \({\text {CO}}_2\).

CCS

CCS consists of two parts: carbon capture and carbon sequestration. One part of the captured \({\text {CO}}_2\) is fed into the P2G equipment for recycling and the other part is sealed by the \({\text {CO}}_2\) compressor. The model is as follows30:

$$ \left\{ \begin{array}{ll} {{P}_{\text {CCS}}}(t)=P_{\text {CCS}}^{\text {B}}(t)+P_{\text {CCS}}^{\text {Y}}(t) \\ P_{\text {CCS}}^{\text {Y}}(t)={{\lambda }_{\text {c}}}{{C}_{\text {C}{{\text {O}}_{2}}}}(t) \\ {{C}_{\text {C}{{\text {O}}_{2}}}}(t)={{C}_{\text {MR}}}(t)+{{C}_{\text {F}}}(t) \\ {{C}_{\text {C}{{\text {O}}_{2}}}}(t)={{\eta }_{\text {CCS}}}{{C}_{\text {CC}{{\text {O}}_{2}}}}(t) \\ \end{array} \right. $$
(13)

Where, \({{\lambda }_{\text {c}}}\) is the power consumed per unit of \({\text{CO}}_2\) captured.

Energy storage

The energy storage devices of this system mainly include CES device, H2S, and HS. The model is as follows33:

$$\left\{ \begin{array}{ll} {{P}_{s}}\left( t \right) =\eta _{s}^{\text {cha}}P_{s}^{\text {cha}}\left( t \right) -P_{s}^{\text {dis}}\left( t \right) /\eta _{s}^{\text {dis}}\\ {{S}_{s}}\left( t \right) ={{S}_{s}}\left( t-1 \right) +{{P}_{s}}\left( t \right) /{{P}_{sn}} \\ {{S}_{s}}\left( 1 \right) ={{S}_{s}}\left( T \right) \\ 0\le P_{s}^{\text {cha}}\left( t \right) \le U_{s}^{\text {cha}}\left( t \right) P_{s}^{\text {cha},\max } \\ 0\le P_{s}^{\text {dis}}\left( t \right) \le U_{s}^{\text {dis}}\left( t \right) P_{s}^{\text {dis},\max } \\ U_{s}^{\text {cha}}\left( t \right) +U_{s}^{\text {dis}}\left( t \right) =1 \\ S_{s}^{\min }\le {{S}_{s}}\left( t \right) \le S_{s}^{\max } \\ \end{array} \right. $$
(14)

\(s\in \{CES,HS,H2S\}\). \(\eta _{s}^{\text {cha}}\), \(\eta _{s}^{\text {dis}}\) are the charging and discharging efficiencies respectively.

Ladder-type carbon price mechanism with reward and punishment

Initial carbon allowance model

The baseline technique is a widely employed method in the electric power business for calculating the carbon emission allowance free34. This exemption is often separated into two categories: heating and power supply. The model is presented below35:

$$ \left\{ \begin{aligned}&{{E}_{\text {p}}}=\sum \limits _{t=1}^{T}{({{E}_{\text {el}}}(t)+{{E}_{\text {h}}}(t)}) \\&{{E}_{\text {el}}}(t)={{P}_{\text {CHP,e}}}(t){{B}_{e}}{{\delta }_{1}}{{\delta }_{2}}+\left( {{P}_{\text {WT}}}(t)+{{P}_{\text {PV}}}(t) \right) {{B}_{e}}{{\delta }_{2}} \\&{{E}_{\text {h}}}(t)=({{P}_{\text {CHP,h}}}(t)+{{P}_{\text {GB,h}}}(t)){{B}_{h}} \\ \end{aligned} \right. $$
(15)

Where, \({{E}_{\text {el}}}(t)\) and \({{E}_{\text {h}}}(t)\) are the carbon quotas for powering and heating the unit; \({{B}_{e}}\) is the reference value of unit power supply, which is 1.003 kg/kWh; \({{\delta }_{1}}\) is the unit cooling correction coefficient, which is 1; \({{\delta }_{2}}\) is the correction coefficient of unit output, which is 1.087; \({{B}_{h}}\) is the heating reference value of the unit, which is 0.735 kg/kWh35.

Actual carbon emission model

Considering that a large amount of \({\text {CO}}_2\) will be consumed by the P2G–CCS coupling, the actual carbon emission is modeled in the following equation:

$$ \left\{ \begin{array}{ll} &{} {{E}_{\text {e}}}=E_{\text {e,buy}}^{\prime}+E_{g}^{'}-E{{}_{\text {CS}}} \\ &{} E_{\text {e,buy}}^{\prime}={{\sigma }_{\text {e}}}\sum \limits _{t=1}^{T}{{{P}_{\text {e,buy}}}} \\ &{} E_{\text {g}}^{\prime}={{\sigma }_{\text {g}}}\sum \limits _{t=1}^{T}{({{\eta }_{\text {e,CHP}}}{{P}_{\text {CHP,e}}}(t)+{{\eta }_{\text {h,CHP}}}{{P}_{\text {CHP,h}}}(t)+{{\eta }_{\text {h,GB}}}{{P}_{\text {GB}}}(t))} \\ &{} {{E}_{\text {CS}}}=\sum \limits _{t=1}^{T}{{C}_{\text {C}{{\text {O}}_{2}}}}(t) \\ \end{array} \right.$$
(16)

where \({{E}_{\text {e}}}\) is actual total carbon emissions; \(E_{\text {e,buy}}^{\prime}\) and \(E_{\text {g}}^{\prime}\) are the actual carbon emissions of the electricity and gas units purchased from the grid; \({{\sigma }_{\text {e}}}\) and \({{\sigma }_{\text {g}}}\) are the unit carbon emission coefficients of power and gas units, which are 0.788 kg/kWh and 0.358 kg/kWh, respectively.

Carbon transaction cost model

The specific functions are as follows36:

$$ {{E}_{\text {co2}}}=\left\{ \begin{aligned}&-{{c}_{\text {e}}}\left( 1+2\mu \right) ({{E}_{\text {p}}}-h-{{E}_{\text {e}}}),{{E}_{\text {e}}}\le {{E}_{\text {p}}}-h \\&-{{c}_{\text {e}}}\left( 1+2\mu \right) h-{{c}_{\text {e}}}(1+\mu )({{E}_{\text {p}}}-{{E}_{\text {e}}}),{{E}_{\text {p}}}-h<{{E}_{\text {e}}}\le {{E}_{\text {p}}} \\&{{c}_{\text {e}}}\left( {{E}_{\text {p}}}-{{E}_{\text {e}}} \right) ,{{E}_{\text {p}}}<{{E}_{\text {e}}}\le {{E}_{\text {p}}}+h \\&{{c}_{\text {e}}}h+{{c}_{\text {e}}}\left( 1+\lambda \right) ({{E}_{\text {e}}}-h-{{E}_{\text {p}}}),{{E}_{\text {p}}}+h<{{E}_{\text {e}}}\le {{E}_{\text {p}}}+2h \\&{{c}_{\text {e}}}\left( 2+\lambda \right) h+{{c}_{\text {e}}}\left( 1+2\lambda \right) ({{E}_{\text {e}}}-2h-{{E}_{\text {p}}}),{{E}_{\text {p}}}+2h<{{E}_{\text {e}}}\le {{E}_{\text {p}}}+3h \\&{{c}_{\text {e}}}\left( 3+\lambda \right) h+{{c}_{\text {e}}}\left( 1+3\lambda \right) ({{E}_{\text {e}}}-3h-{{E}_{\text {p}}}),{{E}_{\text {p}}}+3h<{{E}_{\text {e}}}\\ \end{aligned} \right. $$
(17)

where \({{E}_{\text {co2}}}\) is the PIES carbon transaction cost; \({{c}_{\text {e}}}\) is the carbon emission trading price, which is 100 yuan/t; \(\mu \) and \(\lambda \) are the reward and punishment coefficients, which are 0.2 and 0.15, respectively; h is the value interval length.

Bi-level optimization model of PIES

In the two-tier structure, PIES operator and CES operator have their own decision makers and influence each other. Each decision maker tries to achieve the best interest, but cannot make the final decision unilaterally. Considering the competition and cooperation between the two, it is necessary to ensure the relative autonomy of the two as a premise to achieve the best benefit of the whole system.

The upper-level PIES operator optimization model

Upper model objective function

The PIES operator simultaneously considers the cost of energy purchase \({{f}_{\text {buy}}}\), penalty cost of abandoning renewable energy \({{f}_{\text {ab}}}\), carbon transaction cost \({{f}_{\text {C}{{\text {O}}_{\text {2}}}}}\), carbon sequestration cost \({{f}_{\text {F}}}\) and CES power interaction cost \({{f}_{\text {CES}}}\), and aims for the lowest operating cost, which can be described as:

$$\left\{ \begin{array}{ll} &{} \min {{f}_{\text {PIES}}}={{f}_{\text {buy}}}+{{f}_{\text {ab}}}+{{f}_{\text {C}{{\text {O}}_{\text {2}}}}}+{{f}_{\text {F}}}+{{f}_{\text {CES}}}\\ &{} {{f}_{\text {buy}}}=\sum \limits _{t=1}^{\text {T}}{{{c}_{\text {e}}}\left( t \right) {{P}_{\text {e,buy}}}\left( t \right) +\sum \limits _{t=1}^{\text {T}}{{{c}_{\text {g}}}\left( t \right) {{P}_{\text {g,buy}}}\left( t \right) }} \\ &{} {{f}_{\text {ab}}}=\sum \limits _{t=1}^{\text {T}}{{{\zeta }_{\text {ab}}}\left( {{P}_{\text {ab,WT}}}(t)+{{P}_{\text {ab,PV}}}(t) \right) } \\ &{} {{f}_{\text {CES}}}=\sum \limits _{t=1}^{\text {T}}{\left( -\alpha (t)P_{\text {CES}}^{\text {cha}}(t)+\beta (t)P_{\text {CES}}^{\text {dis}}(t) \right) +}\sum \limits _{t=1}^{\text {T}}{\phi (t)\left( P_{\text {CES}}^{\text {cha}}(t)+P_{\text {CES}}^{\text {dis}}(t) \right) } \\ &{} {{f}_{\text {C}{{\text {O}}_{\text {2}}}}}=\sum \limits _{t=1}^{\text {T}}{{{E}_{\text {co2}}}(t)} \\ &{} {{f}_{\text {F}}}=\sum \limits _{t=1}^{T}{{{f}_{\text {C}{{\text {O}}_{\text {2}}}}}{{C}_{\text {F}}}(t)} \\ \end{array} \right.$$
(18)

where \({{\zeta }_{\text {ab}}}\) is the penalty cost coefficient of abandoning wind power and photovoltaic, which is 0.12 yuan/kW. \({{f}_{\text {C}{{\text {O}}_{\text {2}}}}}\) is the unit price of carbon sequestration.

Upper model constraints

  1. (1)

    Electric energy balance: without considering the PIES to sell electricity to the power grid. The constraints are shown below:

    $$ \left\{ \begin{aligned}&{{P}_{~\text {e.buy}}}(t)+{{P}_{\text {CHP,e}}}(t)+{{P}_{\text {wt}}}(t)+{{P}_{\text {pv}}}(t) +{{P}_{\text {HFC,e}}}(t)\\&={{P}_{\text {E}}}(t)\text {+}\sum \limits _{n=1}^{N}{{{P}_{\text {el,e},n}}(t)} +{{P}_{\text {CCS}}}(t)+P_{\text {CES}}^{\text {cha}}(t)-P_{\text {CES}}^{\text {dis}}(t)\\&0\le {{P}_{~\text {e.buy}}}(t)\le P_{\text {e.buy}}^{\text {max}} \\&0\le {{P}_{\text {wt}}}(t)\le P_{\text {wt}}^{\text {max}} \\&0\le {{P}_{\text {pv}}}(t)\le P_{\text {pv}}^{\text {max}} \\ \end{aligned} \right. $$
    (19)
  2. (2)

    Heat energy balance: the heat supply equipment in PIES includes CHP, GB and HFC, which are mainly supplied to heat load and heat storage equipment. The constraints are shown below:

    $${{P}_{\text {CHP,h}}}(t)+{{P}_{\text {GB,h}}}(t)+{{P}_{\text {HFC,h}}}(t) ={{P}_{\text {H}}}(t)+P_{\text {HS}}^{\text {cha}}(t)-P_{\text {HS}}^{\text {dis}}(t) $$
    (20)
  3. (3)

    Gas energy balance: regardless of PIES selling gas to the gas network. The constraints are shown below:

    $$\left\{ \begin{array}{ll} {{P}_{\text {g,buy}}}(t)+{{P}_{\text {MR,g}}}(t)={{P}_{\text {G}}}(t) +{{P}_{\text {GB,g}}}(t)+{{P}_{\text {CHP,g}}}(t)\\ 0\le {{P}_{\text {g,buy}}}(t)\le P_{\text {g,buy}}^{\max } \\ \end{array} \right.$$
    (21)
  4. (4)

    Hydrogen energy balance: The hydrogen supply equipment in PIES is EL, which is mainly supplied to MR, HFC and hydrogen storage. The hydrogen energy balance is as follows:

    $$ \sum \limits _{n=1}^{N}{{{P}_{\text {el,H2},n}}(t)}\begin{array}{ll} ={{P}_{\text {HFC,H2}}}(t)+{{P}_{\text {MR,H2}}}(t)+P_{\text {H2S}}^{\text {cha}}(t)-P_{\text {H2S}}^{\text {dis}}(t) \\ \end{array} $$
    (22)

The lower-level CES operator optimization model

Lower model objective function

The objective function is defined as the lowest operating cost for the CES operator, as shown below:

$$ \left\{ \begin{aligned}&{\min F={{F}_{1}}+{{F}_{2}}+{{F}_{3}}} \\&{{F}_{1}}=\sum \limits _{t=1}^{T}{{{\mu }_{\text {w}}} \left( P_{\text {CES}}^{\text {cha}}(t)+P_{\text {CES}}^{\text {dis}}(t) \right) } \\&{{F}_{2}}=\sum \limits _{t=1}^{T}{\left( \alpha (t)P_{\text {CES}}^{\text {cha}}(t) -\beta (t)P_{\text {CES}}^{\text {dis}}(t) \right) } \\&{{F}_{3}}=\sum \limits _{t=1}^{T}{-\phi (t)\left( P_{\text {CES}}^{\text {cha}}(t) +P_{\text {CES}}^{\text {dis}}(t) \right) } \\ \end{aligned} \right.$$
(23)

Where, \({F}_{1}\) is the loss cost of energy storage charge and discharge; \({F}_{2}\) is the interaction cost of CES and PIES power; \({F}_{3}\) is the revenue component of CES service.

Lower model constraints

The constraints on the lower CES operator are the same as the generalized energy storage constraints, which are the same as the energy storage constraints above.

Source-load uncertainty scheduling model based on IGDT

The IGDT theory describes the range in which the factual data differs from its predicted data by constructing the interval error (information gap area) between the actual data and the predicted data. IGDT has two strategies: RA and RS. The purpose of the former is to find the largest set of uncertain variables that can meet the given objective, thus avoiding the risk of uncertainty; the purpose of the latter is to use uncertain factors to gain advantage and find the smallest set of uncertain variables that can bring advantage27.

The model for optimization with uncertain variables in the parameters can be expressed as:

$$ \left\{ \begin{array}{ll} \min f(X,u) \\ \text {s}.\text {t}.h(X,u)=0,g(X,u)\le 0 \\ \end{array} \right. $$
(24)

where f is the optimization goal; h and g are model constraints; X is the decision variable; u is an uncertain variable.

To solve the problem with multiple uncertain variables, the weighting method is usually used to add the uncertainty of each uncertain variable as an optimisation objective. However, to eliminate subjectivity arising from the weight of uncertainty, this paper proposes the use of the entropy weight approach to determine the weight assigned to each uncertainty component and assess its significance37. To determine the weight of each index, the reference data sequence is normalized and the information entropy is determined. The model is as follows:

$$\left\{ \begin{aligned}{}&{{X}_{k}}=\left\{ {{x}_{k,1}},{{x}_{k,2}},\ldots ,{{x}_{k,T}} \right\} \\&{{{\hat{X}}}_{k,t}}=\frac{{{{\hat{X}}}_{k,t}}-\min {{X}_{k}}}{\max {{X}_{k}}-\min {{X}_{k}}} \\&{{\nu }_{k,t}}=\frac{{{{\hat{X}}}_{k,t}}}{\underset{t=1}{\overset{T}{\mathop {\sum }}}\,{{{\hat{X}}}_{k,t}}} \\&{{H}_{k}}=-\ln {{(K)}^{-1}}\underset{t=1}{\overset{T}{\mathop {\sum }}}\,{{\nu }_{k,t}}\ln {{\nu }_{k,t}} \\&{{w}_{k}}=\frac{1-{{H}_{k}}}{k-\underset{k=1}{\overset{K}{\mathop {\sum }}}\,{{H}_{k}}} \\&{{\alpha }_{k}}={{w}_{k}}\alpha \\&\underset{k=1}{\overset{K}{\mathop {\sum }}}\,{{w}_{k}}=1 \\ \end{aligned} \right. $$
(25)

where \({{X}_{k}}\) is the evaluation index; \({{x}_{k,t}}\) and \({{v}_{k,t}}\) are positive indicators and their ratios; \({{H}_{k}}\) is information entropy; \({{w}_{k}}\) is the weight of each index; \({{\alpha }_{k}}\) is the proportion of uncertain variables in the total uncertainty. The uncertainty index is expressed by the predicted value of each uncertainty variable, and the corresponding entropy weight is calculated to obtain the part of the corresponding uncertainty variable in the total uncertainty38.

The risk-averse IGDT model based on entropy weight is as follows39:

$$ \begin{aligned} \left\{ \begin{array}{ll} \max \alpha \\ \text {s}.\text {t}.f(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})\le (1+\delta ){{f}_{\text {cr}}} \\ h(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})=0 \\ g(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})\le 0 \\ \forall {{u}_{k}}\in U({{u}_{k}},{{\alpha }_{k}}),k\in \{1,2,\ldots ,k\} \\ \end{array} \right. \end{aligned} $$
(26)

where \(\delta \) is the robust deviation factor of the objective function; \({{f_{\mathrm{{cr}}}}}\) is the objective function under the deterministic model.

The risk-seeking IGDT model based on entropy weight is as follows39:

$$\begin{aligned}\left\{ \begin{array}{ll} \min \alpha \\ \text {s}.\text {t}.f(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})\le (1-\theta ){{f}_{\text {cr}}} \\ h(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})=0 \\ g(X,{{u}_{1}},{{u}_{2}},\ldots ,{{u}_{k}})\le 0 \\ \forall {{u}_{k}}\in U({{u}_{k}},{{\alpha }_{k}}),k\in \{1,2,\ldots ,k\} \\ \end{array} \right. \end{aligned} $$
(27)

where \(\theta \) is the opportunity deviation factor of the objective function.

The above method eliminates the subjectivity of weight and target deviation factor, and retains the non-probabilistic characteristics of IGDT method.

Scheduling model solving method

Figure 4
figure 4

Schematic diagram of the dual-layer optimization model.

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The relationship between the two layers is shown in Fig. 4. As shown in Fig. 4, the upper layer is optimised to minimise the operating costs of the PIES operator, optimising the output of the equipment and the power interacting with the outside. The lowest layer is designed to minimize the operational expenses of the CES operator and optimize the charging and discharging power of the CES. The upper layer initially conducts scheduling by utilizing historical factors and the equipment model, and subsequently transmits the operational outcomes to the lower layer. The lower layer model returns the CES charging and discharging results to the upper layer, and the optimal scheduling of the whole system is finally achieved by coupling information and repeating iterations.

During the scheduling process, the calculation iteration involves the transmission of upper and lower variables due to the coupling of the two-layer model, making the calculation method more complex. Therefore, the KKT condition and the Big-M method are utilised to reduce the dimension and linearise the two-layer model. The steps for model transformation using the KKT-condition and the BIG-M method are shown below:

  1. (1)

    Construct the Lagrangian function of the lower layer model:

    $$ \begin{aligned}&L(x,\mu ,\lambda )=\sum \limits _{t=1}^{T}{{{\mu }_{\text {w}}}(P_{\text {CES}}^{\text {cha}}(t) +P_{\text {CES}}^{\text {dis}}(t))+}\sum \limits _{t=1}^{T}{(\alpha (t)P_{\text {CES}}^{\text {cha}}(t) -\beta (t)P_{\text {CES}}^{\text {dis}}(t))}+\sum \limits _{t=1}^{T}{-\phi (t) \left( P_{\text {CES}}^{\text {cha}}(t)+P_{\text {CES}}^{\text {dis}}(t) \right) } \\&+{{\mu }_{1}}({{S}_{\text {CES}}}(t)-S_{\text {CES}}^{\max })+{{\mu }_{2}}(S_{\text {CES}}^{\min } -{{S}_{\text {CES}}}(t))+{{\mu }_{3}}(P_{\text {CES}}^{\text {cha}}(t)-{{U}_{\text {cha}}}(t) P_{\text {CES}}^{\text {cha,max}})+{{\mu }_{4}}(P_{\text {CES}}^{\text {dis}}(t)-{{U}_{\text {dis}}} (t)P_{\text {CES}}^{\text {dis,max}}) \\&+{{\mu }_{5}}({{U}_{\text {cha}}}(t)+{{U}_{\text {dis}}}(t)-1 )+{{\lambda }_{1}} ({{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t)-\eta _{\text {CES}}^{\text {CH}}P_{\text {CES}}^{\text {cha}}(t) \Delta{t})+{{\lambda }_{2}}({{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t) +P_{\text {CES}}^{\text {dis}}(t)\Delta{t}/\eta _{\text {CES}}^{\text {DC}}) \\&+{{\lambda }_{3}}({{S}_{\text {CES}}}(1)-{{S}_{\text {CES}}}(T))\text {+}{{\lambda }_{\text {4}}} ({{S}_{\text {CES}}}(1)-\text {0.3}S_{\text {CES}}^{\max }) \\&s.t.\left\{ \begin{aligned}&{{\mu }_{i}}\ge 0,{{\lambda }_{i}}\ge 0 \\&{{S}_{\text {CES}}}(t)-S_{\text {CES}}^{\max }\le 0 \\&S_{\text {CES}}^{\min }-{{S}_{\text {CES}}}(t)\le 0 \\&P_{\text {CES}}^{\text {cha}}(t)-{{U}_{\text {cha}}}(t)P_{\text {CES}}^{\text {cha,max}}\le 0 \\&P_{\text {CES}}^{\text {dis}}(t)-{{U}_{\text {dis}}}(t)P_{\text {CES}}^{\text {dis,max}}\le 0 \\&{{U}_{\text {cha}}}(t)+{{U}_{\text {dis}}}(t)-1 \le \text {0} \\&{{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t)-\eta _{\text {CES}}^{\text {CH}}P_{\text {CES}}^{\text {cha}}(t) \Delta{t}=0 \\&{{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t)+P_{\text {CES}}^{\text {dis}}(t)\Delta{t}/ \eta _{\text {CES}}^{\text {DC}}=0 \\&{{S}_{\text {CES}}}(1)-{{S}_{\text {CES}}}(T)=0 \\&{{S}_{\text {CES}}}(1)-\text {0.3}S_{\text {CES}}^{\max }=0 \\ \end{aligned} \right. \\ \end{aligned}$$
    (28)
  2. (2)

    The KKT condition is employed to incorporate Eq. (28) as a constraint into the upper layer model. The objective function of the two-layer model, following the dimensionality reduction procedure, aims to minimize the operational expenses of the PIES operator, as depicted in Eq. (29):

    $$ \begin{aligned}&\min {{f}_{\text {PIES}}}={{f}_{\text {buy}}}+{{f}_{\text {ab}}} +{{f}_{\text {C}{{\text {O}}_{\text {2}}}}}+{{f}_{\text {F}}}+{{f}_{\text {CES}}}\\&\text {s.t.}\left\{ \begin{aligned}&\text {Eq.}\,\text{19}{-}\text{Eq.}\,\text{22} \\&\frac{\partial L(x,\mu ,\lambda )}{\partial x}=0 \\&{{\mu }_{i}}\ge 0,{{\lambda }_{i}}\ge 0 \\&{{S}_{\text {CES}}}(t)-S_{\text {CES}}^{\max }\le 0 \\&S_{\text {CES}}^{\min }-{{S}_{\text {CES}}}(t)\le 0 \\&P_{\text {CES}}^{\text {cha}}(t)-{{U}_{\text {cha}}}(t)P_{\text {CES}}^{\text {cha,max}}\le 0 \\&P_{\text {CES}}^{\text {dis}}(t)-{{U}_{\text {dis}}}(t)P_{\text {CES}}^{\text {dis,max}}\le 0 \\&{{U}_{\text {cha}}}(t)+{{U}_{\text {dis}}}(t) -1\le \text {0} \\&{{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t)-\eta _{\text {CES}}^{\text {CH}}P_{\text {CES}}^{\text {cha}}(t)\Delta{t}=0 \\&{{S}_{\text {CES}}}(t+1)-{{S}_{\text {CES}}}(t)+P_{\text {CES}}^{\text {dis}}(t)\Delta{t}/\eta _{\text {CES}}^{\text {DC}}=0 \\&{{S}_{\text {CES}}}(1)-{{S}_{\text {CES}}}(T)=0 \\&{{S}_{\text {CES}}}(1)-\text {0.3}S_{\text {CES}}^{\max }=0 \\ \end{aligned} \right. \\ \end{aligned} $$
    (29)
  3. (3)

    Due to the presence of complementary constraints in Eq. (29), it is nonlinear and nonconvex in nature. The nonlinear constraints are linearized by introducing binary variables using the BIG-M method, which can be described as:

    $$ \left\{ \begin{aligned}&-M{{U}_{1,t}}\le {{S}_{\text {CES}}}(t)-S_{\text {CES}}^{\max }\le 0 \\&-M(1-{{U}_{1,t}})\le {{\mu }_{1}}\le 0 \\&-M{{U}_{2,t}}\le S_{\text {CES}}^{\min }-{{S}_{\text {CES}}}(t)\le 0 \\&-M(1-{{U}_{2,t}})\le {{\mu }_{2}}\le 0 \\&-M{{U}_{3,t}}\le P_{\text {CES}}^{\text {cha}}(t)-{{U}_{\text {cha}}}(t)P_{\text {CES}}^{\text {cha,max}}\le 0 \\&-M(1-{{U}_{3,t}})\le {{\mu }_{3}}\le 0 \\&-M{{U}_{4,t}}\le P_{\text {CES}}^{\text {dis}}(t)-{{U}_{\text {dis}}}(t)P_{\text {CES}}^{\text {dis,max}}\le 0 \\&-M(1-{{U}_{4,t}})\le {{\mu }_{4}}\le 0 \\&-M{{U}_{5,t}}\le {{U}_{\text {cha}}}(t)+{{U}_{\text {dis}}}(t) -1\le 0 \\&-M(1-{{U}_{5,t}})\le {{\mu }_{5}}\le 0 \\&{{U}_{i,t}}\in [0,1] \\ \end{aligned} \right.$$
    (30)

    The specific procedure is shown in the Fig. 5.

    Figure 5
    figure 5

    Flow chart of optimization scheduling solution.

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Case study

In this paper, the actual engineering data of an PIES in Xinjiang is selected, and the simulation is carried out with 24 h a day as the scheduling cycle and 1 hour as the step size. Predicted statistics for wind power, photovoltaic, electric, heat, and gas loads are displayed in Fig. 6. Figure 7 shows the power purchase price, gas purchase price, power sale price from PIES to CES, and power purchase price from CES. The upper limit for PIES operator to purchase electricity and gas energy from the grid is 1500 kW. The data of carbon trading model are shown in Ref.35. The EL array consists of four ELs connected in parallel. The operating coefficient of the unit in this paper, as shown in Tables 2 and 3.

Figure 6
figure 6

Forecast value of WT and PV output Power and load demand.

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Figure 7
figure 7

Time-of-use pricing for the grid and CES.

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Table 2 Basic equipment parameters.
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Table 3 Parameters of energy storage equipment.
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Verification of the P2G–CCS and CES coupling model

In order to confirm the economic and low-carbon properties of the coupling between P2G–CCS and CES, three operational scenarios have been created:

  • Scenario 1: Not considered P2G–CCS devices and cloud energy storage;

  • Scenario 2: Consider P2G–CCS devices but not cloud energy storage;

  • Scenario 3: Consider P2G–CCS devices and cloud energy storage coupling.

Table 4 The specific cost of three scenarios.
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From Table 4, compared to Scenario 1, Scenario 2 shows a 30% decrease in the total operating cost of PIES. Additionally, there is a reduction of 1247.49 yuan in energy abandonment cost and a decrease of 1444.35 yuan in carbon transaction cost. The primary reason for this is that the two-stage P2G equipment has a high utilization rate of hydrogen energy and is flexible. This promotes the CCS equipment to supply the carbon dioxide consumption of MR. Additionally, the hydrogen fuel cell uses hydrogen energy to achieve cogeneration, which reduces the output of high-yield carbon equipment, and also lowers the energy purchase cost. In Scenario 3, PIES experienced a 2% reduction in its overall operating expenses compared to Scenario 2. Furthermore, there was a reduction of 297.04 yuan in the cost of carbon storage, and a decrease of 1396.49 yuan in the cost of energy purchasing. In addition, the flexible adjustment capability of CES can fully accommodate renewable energy generation, and when wind power generation is low, the adjustment output can achieve a more economical power supply, which can alleviate the phenomenon of external grid energy purchase due to lower wind power generation.

According to the lithium iron phosphate battery for the energy storage project has an average bid price of 1897 yuan/(kW h), the power cost is 1000 yuan/(kW h), and the operation cost is 72 yuan/(kW h)40. The energy storage power station has a lifespan of 10 years and incurs a total setup cost of 2.397 million yuan, with an annual operation and maintenance cost of 36,000 yuan. The daily operational income of the CES operator is 1483.31 yuan, while the energy storage power station has a static investment recovery period of 4.8 years. It can be seen that the CES operator has considerable profit space. Investing in the building of CES is financially advantageous, and the joint service mode of P2G–CCS and CES is theoretically feasible.

Analysis of deterministic scenes

The system equipment outputs in an orderly manner within a 24-h scheduling cycle, which greatly improves energy utilization and operation economy. By solving the two-layer model, the operation plans of the system power supply, heating, gas supply and hydrogen energy system are obtained respectively, as shown in Figs. 8, 9, 10 and 11.

Figure 8 shows that wind power resources are abundant from 1:00–8:00, and the electrical load is provided by the output power of WT and HFC, supplemented by CHP. EL, as the only source of hydrogen, converts the enriched electric energy into hydrogen through high-power operation. This improves the wind power consumption rate and provides hydrogen energy supply for HFC cogeneration and MR hydrogen to methane. During periods of abundant renewable energy output, CES is charged to absorb excess renewable energy. From 9:00–19:00, when the electricity price is high and renewable energy output is insufficient, EL hydrogen production is reduced, the HFC does not work, and power output and CES output are increased through the CHP unit to meet the load demand. The CCS device absorbs the \({\text{CO}}_2\) emitted by CHP and GB in real-time throughout the scheduling cycle. This provides the MR with raw materials for hydrogen methanation. From 20:00 to 24:00, wind power gradually increases, P2G equipment consumes energy, and HFC output rises.

From Fig. 9, it is evident that the heat load demand is higher from 1:00 to 8:00, during which CHP and HFC supply the heat energy, supplemented by GB and heat storage tanks. HFC, with its zero carbon emissions, is the primary supplier of heat energy during this period. The heat load demand is lower from 9:00 to 18:00. As a result of the EL shutdown in the system, there is no hydrogen source available. Therefore, all the heat load is provided by CHP. The CHP stores the excess heat energy from the cogeneration in a heat storage tank. At 19:00–24:00, the output of renewable energy increases, and HFC starts to realize cogeneration and take over thermal energy supply.

Upon comparing Figs. 10 and 11, it is evident that the demand for hydrogen in HFC and MR comes entirely from EL electrolysis. Hydrogen is produced by EL during the hours of 1:00–10:00 and 21:00–24:00. Some of it is directly supplied to HFC for cogeneration, while the rest is produced by MR to create natural gas. The remaining hydrogen is stored in tanks and released during the high energy demand period of 19–20 h to provide HFC for cogeneration. This achieves the effect of peak shaving and valley filling, reducing the cost of energy purchase. The conversion of hydrogen to methane by MR reduces the gas purchase cost of PIES. During the period of 1:00–5:00, hydrogen energy is primarily supplied to HFC for cogeneration, and the remaining portion is used by MR and hydrogen storage tanks.

Figure 8
figure 8

Electric power balance figure.

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Figure 9
figure 9

Heat power balance figure.

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Figure 10
figure 10

Gas power balance figure.

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Figure 11
figure 11

Hydrogen power balance figure.

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Analysis of the effect of carbon trading mechanism on PIES

In order to verify the benefits of the carbon trading model proposed in this paper, three scenarios are set up:

  • Scenario 4: do not consider carbon transaction cost;

  • Scenario 5: consider traditional ladder carbon transaction cost;

  • Scenario 6: consider the LCPMRP.

Table 5 Comparison of scheduling results under different scenarios.
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Table 5 shows that scenario 5, which considers traditional carbon transaction costs, allows the system to obtain free carbon quotas. This results in a reduction of part of the transaction costs and a decrease of 4.54% in the system’s total PIES cost and 5.8% in system carbon emissions compared to scenario 4.

On the basis of scenario 6, a reward coefficient is introduced to give some economic compensation when the initial quota is surplus. In order to prevent excessive environmental cost, PIES operator have increased environmental cost constraints. Compared to Scenario 5, Scenario 6 reduces PIES operating costs by 1.33 % and carbon emissions by 3.29 %. Increasing the reward coefficient to reduce the carbon transaction cost can fully mobilise the enthusiasm of PIES towards carbon reduction.

Sensitivity analysis

Figure 12 shows the trend of carbon emissions and carbon transaction costs with the change of carbon price. In the range of small carbon prices, carbon price changes have little effect on unit output, and carbon emissions remain stable. However, as the cost of carbon rises, so does the cost of carbon trading. Carbon emissions are anticipated to decrease significantly as the price of carbon increases to approximately 70 yuan/t. Carbon emissions exhibit stability when the price reaches 200 yuan/t. Carbon transaction costs increase initially and subsequently decrease due to the combined impact of carbon pricing and carbon emissions. As the cost of carbon emissions increases, there is a corresponding decrease in carbon emissions.

By changing the reward and punishment coefficients, the scheduling models under different carbon prices are solved, respectively, and the results of Figure 13 are obtained. It is clear from Fig. 13 that as the carbon price increases and the incentive and punishment coefficients become larger, the cost of carbon transactions falls at a faster rate. When \(\omega \) is 0.2 and the carbon transaction price is approximately 300 yuan, the carbon transaction cost tends to remain stable. This shows that the system unit output is gradually stabilising.

Figure 12
figure 12

Impact of carbon trading price on system carbon emissions.

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Figure 13
figure 13

Cost of carbon trading with different incentive factors.

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Analysis of operation characteristics of CES

Figure 14 shows that CES is charged during periods of low electricity prices when power load levels are low and there are abundant wind power resources. Using CES to store excess electric energy can reduce system energy purchase costs while consuming surplus renewable energy. CES discharges during periods of high electricity prices when electric load is high, fluctuation is strong, and wind power resources are scarce. CES effectively uses the peak-valley price difference to achieve ’high incidence and low storage’ cashing. It balances the benefits of multiple stakeholders and relieves pressure on the power grid. In addition, the discharge period of CES is not only influenced by the time-of-use price of electricity, but is also related to the distribution of energy and load. It is only when the system energy supply is insufficient that CES can benefit by filling the energy shortfall.

Figure 14
figure 14

CES operation plan.

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Figure 15
figure 15

Influence of CES capacity on PIES cost.

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To evaluate the impact of CES operating characteristics on the economics of each subject, the sensitivity analysis of PIES operator cost and CES operator revenue with different capacities under the deterministic model is shown in Fig. 15. The capacity of CES is adjusted to 0%, 20%, 40%,... 140% of the preset value, respectively.

Figure 15 shows that the operating cost of PIES operator shows a decreasing trend with the increase of CES capacity, because CES can effectively improve the charging enthusiasm of renewable energy surplus period. Moreover, the profit of CES shows an increasing trend, which is due to the increase of power interaction between PIES and CES. In addition, due to the limitation of CES capacity, the cost of PIES operator and the revenue of CES operator will not change after reaching a certain fixed value (minimum cost), and the CES capacity is optimal (1188 kWh). The analysis shows that an appropriate increase in CES capacity can guarantee the lowest cost for the PIES operator while increasing the revenue of the CES operator, thus achieving mutual benefits and win-win results.

Analysis of EL operating characteristics under multiple operating conditions

The production schedule of the EL array in the deterministic model is shown in Fig. 16. By analyzing the variation trend of the electric power of each EL over time, it can be seen that during the period of 1:00–6:00,22:00–24:00, when the wind power is high, most of the EL works under fluctuating production conditions, but it is allowed to work under overload production conditions for a short time, switching between overload conditions and fluctuating conditions, and absorbing the excess wind power as much as possible. This reflects that the scheduling strategy maintains the working state of the EL as much as possible on the basis of meeting energy and price fluctuations; at 8:00, in response to changes in energy prices and fluctuations in renewable energy output. The No.1 EL runs in a cold standby state for a short time, thus avoiding economic losses caused by downtime. This reflects that the model proposed in this paper takes into account the operating characteristics and start-up costs of EL, effectively reduces the number of startup and shutdown of the EL, and the operating state is more flexible.

Figure 16
figure 16

Resolving flow chart.

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Verification and analysis of uncertainty scheduling model based on IGDT

In this paper, other parameters are kept unchanged on the basis of the deterministic model. According to the predicted values of wind power, photovoltaic power, electric load, heat load and gas load, the entropy weights are 0.385,0.221,0.167,0.120 and 0.107 respectively. The deviation factor of the objective function is set to 0.02 steps. As the length increases from 0 to 0.20, the IGDT models based on RA and RS respectively are established.

It can be seen from Table 6 that under the RA strategy, uncertain factors have a negative impact on the PIES operator cost. As the deviation factor grows, the acceptable uncertainty of PIES gradually increases, the risk of robust optimisation decisions by operators gradually decreases, and the cost gradually increases. Under the RS strategy, uncertain factors will have a positive impact on the PIES operator cost. As the deviation factor grows, the acceptable uncertainty of PIES continues to increase, the system risk of opportunity seeking behaviour continues to increase, the system cost gradually decreases, and operators can release more operating costs. Therefore, decision makers need to set an appropriate cost deviation factor by balancing system economics and uncertainty risk.

Table 6 Sensitivity analysis of risk preference.
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Figures 17 and 18 show the scheduling scheme under the RA strategy with \(\delta \) = 0.1 as an example. Figures 17 and 18 show that the IGDT robustness schedule is more conservative than the initial power schedule. The output of renewable energy is below the projected level, while the load exceeds the projected level. This can increase the pressure on the supply during periods of high system load fluctuations, resulting in insufficient supply capacity. To manage load fluctuations, the system buys electricity from the external power grid to compensate for any energy shortages. Additionally, the system increases the output of EL hydrogen production, prioritising the supply of hydrogen to HFC for cogeneration. This results in a reduction of natural gas supplied to the methane reactor and an increase in gas load. The system needs to purchase more gas energy, resulting in an increase in scheduling cost. Additionally, the reduction of renewable energy output leads to a decrease in the charging power of CES during the electricity price trough, which reduces the revenue of CES operator.

Figures 19 and 20 show the scheduling scheme under the RS strategy with \(\theta \) = 0.1 as an example. Figures 19 and 20 demonstrate that the RS method is more extreme than the output plan in the deterministic case. Because of the reduction of the electric load and the increase of the renewable energy output, the PIES operator reduces the output of the traditional unit CHP and the gas boiler. While reducing the carbon transaction cost, the consumption of natural gas is reduced, thereby reducing the gas purchase cost and the operating cost of PIES.

Figure 17
figure 17

Scheduling results of power flow from risk-averse strategy.

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Figure 18
figure 18

Scheduling results of gas flow from risk-averse strategy.

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Figure 19
figure 19

Scheduling results of power flow from risk-seeking strategy.

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Figure 20
figure 20

Scheduling results of gas flow from risk-seeking strategy.

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Discussion

In this paper proposes a two-tier optimal scheduling model for PIES that considers the co-operation of P2G–CCS and CES. The proposed model aims to balance and enhancement of the income of each individual while reducing carbon emissions and system costs. The simulation results of different scenarios are compared, and the main conclusions are as follows:

  1. (1)

    After the implementation of cooperation of P2G–CCS and CES, the daily carbon transaction cost of PIES decreased by 1411.02 yuan, and the cost of purchasing energy dropped by 4084.36 yuan. Additionally, the overall operation cost decreased by 3974.96 yuan; The coordinated operation of P2G–CCS and CES is significant for the low-carbon economic operation and energy cascade utilization of the system. Among them, the proposed multi-condition EL model adjusts the output of the devices according to the load in real time, which effectively improves the flexibility of the electro-hydrogen coupling. In addition, an appropriate increase in CES capacity can increase the revenue of CES operator while reducing operating cost, and achieve mutual benefits and win-win results for two different stakeholders.

  2. (2)

    Set the reward coefficient for the carbon trading mechanism. A higher reward coefficient leads to greater benefits from carbon trading. Comparative analysis has verified that the ladder-type carbon trading mechanism with rewards and punishments can better constrain system carbon emissions, reduce emission reduction costs, and promote coordinated utilization of energy cascades.

  3. (3)

    The subjective factors in the IGDT method will affect the results. The EAIGDT method proposed in this paper effectively eliminates the subjective factors in the IGDT method and retains the non-probabilistic characteristics of the IGDT method. EAIGDT is used to effectively describe the uncertainty of source-load prediction, which helps scheduling managers to formulate scheduling schemes under RA and RS strategies according to risk preference, so as to achieve the balance between prediction error and system operation economy.