Low-carbon collaborative dual-layer optimization for energy station considering joint electricity and heat demand response

Abstract

In the park-level integrated energy system (PIES) trading market involving various heterogeneous energy sources, the traditional vertically integrated market trading structure struggles to reveal the interactions and collaborative relationships between energy stations and users, posing challenges to the economic and low-carbon operation of the system. To address this issue, a dual-layer optimization strategy for energy station-user, taking into account the demand response for electricity and thermal, is proposed in this paper. The upper layer, represented by energy stations, makes decisions on variables such as the electricity and heat prices sold to users, as well as the output plans of energy supply equipment and the operational status of battery energy storage. The lower layer, comprising users, determines their own electricity and heat demand through demand response. Subsequently, a combination of differential evolution and quadratic programming (DE-QP) is employed to solve the interactive strategies between energy stations and users. The simulation results indicate that, compared to the traditional vertically integrated structure, the strategy proposed in this paper increases the revenue of energy stations and the consumer surplus of users by 5.09% and 2.46%, respectively.

Graphical abstract

Keywords

demand response / dual-layer optimization / energy station / integrated energy system

1 Introduction

With the implementation of “Carbon Peaking and Carbon Neutrality” goals initiative, there is a strong impetus to develop the renewable energy industry, with wind and solar power as representatives, and to establish park-level integrated energy system (PIES) with renewable energy sources (RESs) as the main component [1,2]. This serves as the fundamental support to promote the coordinated complementarity of various types of energy and reduce carbon emissions. However, the inherent characteristics of RES, such as strong randomness [3], low reliability [4], and weak controllability [5], pose higher demands on the operation and scheduling of PIES.

At present, combined heating and power (CHP) is considered the core energy supply for PIES [6–8]. CHP with rapid response capabilities is adopted to enhance the flexibility of the energy system, as seen in Refs. [9,10]. Most of the natural gas required for CHP comes from the natural gas pipeline network [11]. However, the natural gas supplied by the natural gas pipeline is non-renewable fossil fuel, which does not align with the current goals of sustainability and environmental protection. Additionally, influenced by factors such as market transaction and geopolitical considerations, natural gas supplied through the pipeline may experience significant volatility, consequently affecting the stability of electricity production [12]. Moreover, relying on natural gas pipeline exposes CHP to the risk of energy supply interruptions due to disruptions in the supply chain, pipeline damage, and other factors [13,14].

It is worth mentioning that biogas, as one of the representatives of renewable natural gas, provides a feasible solution to address the aforementioned issues [15]. Generated through the anaerobic digestion process of organic waste (such as sludge, agricultural residues, and food waste), biogas not only contributes to waste disposal but also reduces carbon emissions [16]. Unlike traditional natural gas pipelines, the renewable nature of biogas allows it to be less constrained by geopolitical factors and market transition fluctuations, enhancing the stability of natural gas supply and electricity production in PIES [17]. This, in turn, reduces the likelihood of facing the risk of energy supply interruptions. Importantly, the production process of biogas is relatively flexible [18], making it better adaptable to the strong stochastic characteristics of RES. Thus, the introduction of biogas into comprehensive energy stations as a source of natural gas supply is recognized as one effective means of mitigating pressure on external natural gas pipelines, while concurrently boosting the economic efficiency of energy station scheduling.

With the reform of the energy market, the proportion of energy sources such as electricity and heat traded in the market has gradually increased [19–21]. Jung et al. [22] and Faraji & Hemmati [23] engage in power and heat transactions through the signing of fixed-time trading contracts. A reference framework for a transactive energy market based on distributed ledger technology, designed in Gourisetti et al. [24], is employed to support various energy transactions in the energy market. However, the aforementioned market transactions follow the traditional vertical integrated trading structure. In a PIES trading market involving multiple heterogeneous energy sources, various energy conversions and storage are involved. Real-time considerations of supply-demand relationships [25–27], prices, and output plans [28] need to be taken into account during the trading process. Nevertheless, under the traditional vertical integrated trading structure, users’ electricity demand, heat demand, and regulation capabilities cannot be transmitted in real-time to the higher level [29]. Meanwhile, PIES cannot instantly access information about users’ demands, regulation capabilities, or their participation in the regulation process. PIES often rely on predicting users’ electricity consumption behaviors to make decisions on prices, energy devices output, and other information. In such situations, optimization strategies often lack customized multi-energy coupling requirements [30].

In summary, as a nonlinear, strongly coupled, multi-agent complex heterogeneous system, the IPES exhibits diversity in energy sources and unpredictability in the coupling characteristics among various energy sources. Leveraging an integrated energy market for energy transactions is one effective means to facilitate complex interactions among multiple energy sources. However, the trading process typically involves multiple stakeholders, including energy station and users, with their decision-making behaviors influencing each other. Currently, in the traditional vertically integrated market, the energy demands and flexibility of users cannot be communicated upstream, and energy stations cannot access information regarding users’ demands and flexibility, nor can they determine whether users are participating in the flexibility process. Consequently, energy stations primarily rely on predicting users’ behavior to make decisions regarding prices and energy allocation. This approach to scheduling strategies fails to reveal the interactions and cooperation between energy station and users, often lacking customization for specific demand characteristics [31]. Moreover, in trading markets of IPES involving multiple heterogeneous energy sources, the processes of production, consumption, conversion, and storage of various energy sources are complex. Balancing the revenues of energy station and users, establishing a multi-node, multi-agent-based trading strategy for multi-energy sources present an extremely challenging task.

Thus, a heat-electricity co-optimization strategy considering demand response is proposed for PIES comprised of a series of supplying units such as photovoltaic (PV), wind turbine (WT), CHP, and battery energy storage (BES). In this strategy, the supplying side (energy station) is positioned at the upper level, while the consuming side (users) is at the lower level. Through the interaction between the energy station and users, the energy output of the station and the energy demands of users are optimized. By substituting internal biogas supply for natural gas pipeline procurement, the stability of natural gas supply to the energy station and electricity production is enhanced. Additionally, to support low-carbon energy transformation, a stepped carbon trading strategy is introduced in this paper, thereby promoting carbon reduction in the PIES. This paper is contributive because, to address the issues of the traditional vertically integrated trading markets, which fail to reveal the interactions and cooperation between energy station and users and often lack customized demand characteristics, it considers the complex conversion relationships among electricity, gas, and heat, and electricity-heat integrated demand response. Based on this, it proposes a thermal–electric co-optimization strategy based on a dual-layer interactive framework. In this strategy, energy stations are seen as leaders at the upper level, while users are seen as followers at the lower level, achieving mutual benefit through the coordinated operation of energy stations and users. Moreover, in the context of a PIES considering carbon trading, it introduces, for the first time, an energy station with biogas power generation capabilities. It integrates biogas into the energy station, enhancing the stability of natural gas supply and electricity production. Thus, it alleviates the pressure on the external natural gas network, and enhances the economic efficiency of the energy station.

2 System description

The PIES studied in this paper integrates a variety of RES and load types, structure and energy flow relationship are shown in Fig.1.

The devices in the green block diagram are the main devices of the energy station, which mainly include the PV, a WT, an anaerobic digestion reactor (ADR), CHP units, gas boilers (GBs), heat exchangers (HEs), BES, and thermal energy storage (TES), where the CHP units include gas turbines (GTs) and waste heat boilers (WHBs). The methane generated from the ADR is used as fuel and fed into the GT and GB, respectively. The WHB collects the waste heat from the operation of the GT, which is then used together with the heat from the GB to supply heat to the users, and the TES is used to regulate the imbalance between the heat supply and demand of the system. The PV, WT, and GT serve as the primary power generation device within the energy station. When the electricity generated by the units exceeds the electricity demand from users, the surplus electricity can be stored in the BES within the energy station. When there is insufficient electricity supply, the stored electricity is released, mitigating the imbalance between electricity supply and demand in the system. The cold water at the inlet of the HE is heated to a specified temperature, providing heat to users. Additionally, the waste heat generated by the GT during operation can be collected by the WHB to supply heat to users, or be stored in the HES. The CHP units and GB within the energy station serve as the primary energy supply device, with their fuel sourced from the ADR. Therefore, it is essential to ensure the continuous operation of the ADR.

The PIES proposed in this paper not only considers the interaction between the energy station and users but also addresses the energy exchange between the external grid and the energy station. These interactions are facilitated through energy transactions. Both electricity and heat are directly sold to users by the energy station as the energy station for energy retailers. In cases where the electrical supply falls short of meeting user demands, the energy station procures additional electricity from the external grid. Conversely, when there is excess electricity, the surplus is sold back to the external grid by the energy station. Meanwhile, the heating load for users is entirely supplied by the WHB and GB of the energy station.

The energy transaction process in PIES is illustrated in Fig.2, involving participants such as the energy station, users, and the external grid. Throughout the energy transaction process, there is continuous information exchange and iteration of electricity and heat prices between the energy station and users. The transaction process also encompasses the exchange of energy, including electric power flow and heat power flow. In terms of electricity trading, the energy station establishes electricity prices based on information from the external grid and then communicates price signals to users. Users, in response to real-time electricity prices, adjust their demand. For example, they may reduce electrical load during periods of higher prices or modify their usage times. The adjusted electricity demand from users is then communicated back to the energy station. Simultaneously, upon receiving the user’s expected electricity demand, the energy station modifies the output plan for the GT, the operational status of the BES, and the plan for purchasing electricity from the external grid. The iteration of electricity prices is conducted with the aim of maximizing the revenue for the energy station. The iteration process for heat prices is similar, and further details are not reiterated here.

The external grid is assumed to be exclusively based on thermal power generation, while the energy station is assumed to employ a combination of thermal power generation and clean energy sources (wind and solar power), such that purchasing electricity from the energy station by users not only ensures a reliable energy supply but also promotes carbon reduction. To prevent users from directly trading with the external grid, this paper assumes that the electricity prices set by the energy station for user purchases are lower than the prices users would incur by buying directly from the external grid. This ensures priority for internal energy transactions within the PIES. Furthermore, to incentivize user participation in carbon reduction within the PIES, this paper introduces a stepped carbon trading cost. To encourage participants to reduce carbon emissions, carbon emissions are divided into multiple intervals. If the actual carbon emissions fall below the free carbon emissions quota allocated, the energy station can sell excess carbon emissions quotas in the carbon trading market. Conversely, if the emissions exceed the quota allotted, participants are required to pay penalty fees.

In light of the scenario described above, user demand response and carbon trading are incorporated into the energy transaction process. Electricity and heat pricing strategies are formulated by the energy station, while the output of the CHP units and the GB is optimized. The goals of thermal-electric co-optimization and carbon reduction in the PIES are accomplished.

3 System model

3.1 Model of stepped carbon trading cost

A method of free allocation to distribute carbon emission quotas to the energy station, users, and the external grid within the PIES is adopted in this paper. Carbon emissions primarily originate from the CHP units and the GB of the energy station, and the external grid, primarily powered by coal-fired power plants. The carbon emission quotas for the GB, the CHP units, and the external grid are defined as E_GB, E_CHP, and E_Grid, respectively. The total carbon emission quota is defined as E_{\mathrm{I}\mathrm{E}\mathrm{S}}, whose expressions are

(1)E_{\mathrm{G}\mathrm{B}}= {\delta }_{\mathrm{g}}\sum _{t=1}^T{Q}_{\mathrm{G}\mathrm{B}}^t,

(2)E_{\mathrm{C}\mathrm{H}\mathrm{P}}={\delta }_{\mathrm{g}}\sum _{t=1}^T\left(\beta P_{\mathrm{G}\mathrm{T}}^{\mathrm{t}}+Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t\right),

(3)E_{\mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d}}= {\delta }_{\mathrm{e}}\sum _{t=1}^T{P}_{\mathrm{b}\mathrm{u}\mathrm{y}}^t,

(4)\begin{array}{c}{E}_{\mathrm{I}\mathrm{E}\mathrm{S}}=E_{\mathrm{G}\mathrm{B}}+E_{\mathrm{C}\mathrm{H}\mathrm{P}}+E_{\mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d}},\end{array}

where δ_g represents the carbon emission quota per unit of gas supply, δ_e represents the carbon emission quota per unit of electricity supply, Q_{\mathrm{G}\mathrm{B}}^t denotes the thermal power output of the GB at time t, P_{\mathrm{G}\mathrm{T}}^t represents the electric power output of the GT at time t, Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t represents the WHB from the CHP units at time t, β is the conversion factor for converting electricity into heat, and P_{\mathrm{b}\mathrm{u}\mathrm{y}}^t represents the electricity purchased by the energy station from the external grid at time t.

Based on the relationship between each carbon emission quota of the participant and their actual carbon emissions, users face penalties or rewards in the carbon trading process. The traditional model for actual carbon emissions is linear. To better fit the actual carbon emission data, the quadratic term coefficients of the model are adjusted, resulting in the actual carbon emissions for each participant, represented as [32,33]

(5)E_{\mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d}}^{\mathrm{a}\mathrm{c}\mathrm{t}}={\sum }_{t=1}^T\left[a_1+b_1{P}_{\mathrm{b}\mathrm{u}\mathrm{y}}^t+c_1{{(P}_{\mathrm{b}\mathrm{u}\mathrm{y}}^t)}^2\right],

(6)E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{a}\mathrm{c}\mathrm{t}}={\sum }_{t=1}^T\left[a_2+b_2{P}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^t+c_2({P_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^t)}^2\right],

(7)\begin{array}{c}{P}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^t=P_{\mathrm{G}\mathrm{T}}^t+Q_{\mathrm{G}\mathrm{B}}^t+Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t,\end{array}

(8)E_{\mathrm{I}\mathrm{E}\mathrm{S}}^{\mathrm{a}\mathrm{c}\mathrm{t}}= E_{\mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d}}^{\mathrm{a}\mathrm{c}\mathrm{t}}+E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{a}\mathrm{c}\mathrm{t}},

where E_{\mathrm{G}\mathrm{r}\mathrm{i}\mathrm{d}}^{\mathrm{a}\mathrm{c}\mathrm{t}} represents the actual carbon emissions from the external grid; E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{a}\mathrm{c}\mathrm{t}} represents the actual carbon emissions from the energy station; a₁, b₁, c₁, a₂, b₂, and c₂ are the relevant parameters for the external grid and the energy station; P_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^t is the sum of the electricity and heat output from the energy station at time t; and E_{\mathrm{I}\mathrm{E}\mathrm{S}}^{\mathrm{a}\mathrm{c}\mathrm{t}} represents the actual total carbon emissions from the PIES.

Carbon trading settlement strategies are divided into multiple gradient levels, each assigned different reward and penalty coefficients. The carbon trading cost is

(9)

where F_c represents the carbon trading cost, p is the carbon trading price, μ is the reward factor in the carbon trading strategy, λ is the penalty factor in the carbon trading strategy, and h is the carbon emission range.

3.2 Model of energy supply devices

To ensure a continuous energy supply and reduce the reliance of PIES on external natural gas, an ADR is introduced into the energy station to produce the methane fuel required. Given the small scale of the ADR in this paper and for the sake of simplifying the model, the surface of the anaerobic digestion process is assumed to be adiabatic. The heat released by biomass feedstock to the surrounding soil through walls and lids during the anaerobic digestion process is neglected. To ensure the normal daily operation of the ADR, biomass feedstock (i.e., slurry, straw, and animal manure) is fed into it. The production process of biogas is shown in Fig.3.

The production of biogas is constrained by the quantity of feedstock, whose constraints are

(10)m_{\mathrm{b}\mathrm{i}\mathrm{o}}^t⩾{\eta }_{\mathrm{b}\mathrm{i}\mathrm{o}}{G}_{\mathrm{b}\mathrm{i}\mathrm{o}}^t,

where m_{\mathrm{b}\mathrm{i}\mathrm{o}}^t represents the mass of feedstock input into the ADR at time t, G_{\mathrm{b}\mathrm{i}\mathrm{o}}^t represents the amount of methane produced during anaerobic digestion at time t, and η_bio is the efficiency of converting biomass feedstock into biogas.

The methane production is influenced by the temperature in the ADR, whose expression is [15]

(11)G_{\mathrm{b}\mathrm{i}\mathrm{o}}^t=\frac{B_0{S}_0{V}_{\mathrm{A}\mathrm{D}}}{24\cdot \theta }\left(1-\frac{K}{\theta {\cdot \mu }_{\mathrm{m}}^t-1+K}\right),

where B₀ is the biochemical methane potential, which is the amount of methane gas produced per unit of volatile solids, S₀ is the concentration of volatile solids in the incoming biomass feedstock, θ is the hydraulic retention time, representing the time the biomass feedstock stays in the anaerobic digestion pool, V_AD is the volume of the ADR, {\mu }_{\mathrm{m}}^t is the maximum specific growth rate of microorganisms at time t, and K is a dimensionless kinetic parameter related to the rate and stability of an aerobic digestion process. The expressions for {\mu }_{\mathrm{m}}^t and K are [15]

(12){\mu }_{\mathrm{m}}^t= {\alpha }_1{\mathrm{e}}^{{\alpha }_2{T}_{\mathrm{d}}^t},

(13)K={\beta }_1{\mathrm{e}}^{{\beta }_2{S}_0}+{\beta }_3,

where α₁, α₂, β₁, β₂, and β₃ are a series of parameters related to anaerobic digestion observed in full-scale experiments, and T_d is the temperature of the anaerobic digestion process. The temperature evolution in the digestion pool is

(14)T_{\mathrm{d}}^t= T_{\mathrm{d}}^{t-1}+\frac{Q_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}}^t}{c_{\mathrm{m}}{\rho }_{\mathrm{m}}{V}_{\mathrm{A}\mathrm{D}}}.

where T_d is the temperature of the anaerobic digestion process, Q_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}}^t is the thermal power input into the biogas digester at time t, c_m is the heat capacity of the biomass feedstock, and ρ_m is the density of the biomass feedstock.

The biomass feedstock undergoes chemical reactions in the anaerobic digestion pool, and the resulting biogas G_{\mathrm{b}\mathrm{i}\mathrm{o}}^t is split into two streams at the outlet. One part, G_{\mathrm{G}\mathrm{T}}^t, is fed into the gas turbine, and the other part, G_{\mathrm{G}\mathrm{B}}^t, is directed into the gas boiler. The relationship and constraints are

(15)G_{\mathrm{b}\mathrm{i}\mathrm{o}}^t=G_{\mathrm{G}\mathrm{T}}^t+G_{\mathrm{G}\mathrm{B}}^t,

(16)

(17)

where G_{\mathrm{G}\mathrm{T}}^t is the amount of biogas input into the CHP unit at time t, and G_{\mathrm{G}\mathrm{B}}^t is the amount of biogas input into the GB at time t, Q_{\mathrm{C}{\mathrm{H}}_4} is the heating value of methane; and {\eta }_{\mathrm{G}\mathrm{T}}^{\mathrm{m}\mathrm{a}\mathrm{x}} and {\eta }_{\mathrm{G}\mathrm{B}}^{\mathrm{m}\mathrm{a}\mathrm{x}} are the maximum efficiencies of the GT and GB, respectively.

In this paper, the energy station serves both as an energy supplier and as an energy retailer. The energy station earns profits by selling electricity and heat to users, while bearing the fuel cost C_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}^t and carbon trading cost F_c. Additionally, when there is excess electricity, the energy station could sell the surplus to the external grid for profit, and conversely, it needs to purchase electricity from the external grid when there is a shortfall. The optimization objective of this paper is to maximize the profit R_SUPL of the energy station, whose model is

(18)R_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}={\mathrm{\Sigma }}_{t=1}^T(P_{\mathrm{e}}^t{c}_{\mathrm{e}}^t+ Q_{\mathrm{h}}^t{c}_{\mathrm{h}}^t- C_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}^t\mathrm{\Delta }t- F_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}^t)- F_{\mathrm{c}},

where c_{\mathrm{e}}^t and c_{\mathrm{h}}^t are the electricity and heat prices, respectively, charged by the energy station to users at time t, P_{\mathrm{e}}^t and Q_{\mathrm{h}}^t are the electric power and thermal power output from the energy supply system at time t, C_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}^t represents the fuel cost, and F_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}^t is the cost associated with the interaction between the energy station and the external grid, whose expression is

(19)F_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}^t={\mathrm{\Sigma }}_{t=1}^T\left(P_{\mathrm{b}\mathrm{u}\mathrm{y}}^t{c}_{\mathrm{g}\mathrm{b}}^t-P_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}}^t{c}_{\mathrm{g}\mathrm{s}}^t\right),

where P_{\mathrm{b}\mathrm{u}\mathrm{y}}^t is the amount of electricity purchased by the energy station from the external grid at time t, and P_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}}^t is the amount of electricity sold by the energy station to the external grid at time t, and c_{\mathrm{g}\mathrm{b}}^t and c_{\mathrm{g}\mathrm{s}}^t represent the grid purchase and grid sale electricity prices to the external grid at time t, respectively.

The relationship between fuel cost and the output of the GT and GB is expressed as

(20)\begin{array}{rl}{C}_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}^t=& a_{\mathrm{e}}{\left(P_{\mathrm{G}\mathrm{T}}^t\right)}^2+b_{\mathrm{e}}{P}_{\mathrm{G}\mathrm{T}}^t+c_{\mathrm{e}}+a_{\mathrm{h}}{\left(Q_{\mathrm{G}\mathrm{B}}^t\right)}^2+\\ & b_{\mathrm{h}}{Q}_{\mathrm{G}\mathrm{B}}^t+c_{\mathrm{h}}+C_{\mathrm{b}\mathrm{i}\mathrm{o}}{m}_{\mathrm{b}\mathrm{i}\mathrm{o}}^t,\end{array}

where C_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}}^t is the fuel cost for the energy supply at time t; P_{\mathrm{G}\mathrm{T}}^t and Q_{\mathrm{G}\mathrm{B}}^t are the electric power output of the GT and the thermal power output of the GB at time t, respectively; a_e, b_e, c_e, a_h, b_h, and c_h are the cost coefficients for the gas turbine and gas boiler; and C_bio is the material and maintenance cost for the input biomass feedstock.

The model for the electric power and thermal power output from the energy station is

(21)P_{\mathrm{e}}^t= P_{\mathrm{P}\mathrm{V}}^t+ P_{\mathrm{W}\mathrm{T}}^t+ P_{\mathrm{G}\mathrm{T}}^t,

(22)Q_{\mathrm{h}}^t= \left(Q_{\mathrm{G}\mathrm{B}}^t+Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t\right){\eta }_{\mathrm{h}},

where η_h is the efficiency of the heat exchanger under winter conditions, P_{\mathrm{P}\mathrm{V}}^t and P_{\mathrm{W}\mathrm{T}}^t represent the output electric power of the PV and the WT at time t, and the predictive curves for the PV and WT are provided in the case analysis. The relationship between the thermal power output of the WHB and the electric power output of the GT is expressed as

(23)Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t=\frac{P_{\mathrm{G}\mathrm{T}}^t\left(1-{\eta }_{\mathrm{G}\mathrm{T}}\right)}{{\eta }_{\mathrm{G}\mathrm{T}}}{\eta }_{\mathrm{W}\mathrm{H}\mathrm{B}},

where Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t is the waste heat recovery from the WHB at time t, {\eta }_{\mathrm{G}\mathrm{T}} is the efficiency of the GT, and {\eta }_{\mathrm{W}\mathrm{H}\mathrm{B}} is the efficiency of the WHB.

3.3 Model of battery energy storage

To ensure the safe and reliable operation of the BES and extend their lifespan, the model must satisfy certain charging and discharging constraints, as well as the state of charge (SOC) [34]. The SOC and constraints for the BES are expressed as [35,36]

(24)S_t= \left(1-{\eta }_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\right)S_{t-1}+{\eta }_{\mathrm{c}\mathrm{h}}{P}_{\mathrm{c}\mathrm{h}}^t\mathrm{\Delta }t- \frac{P_{\mathrm{d}\mathrm{c}\mathrm{h}}^t\mathrm{\Delta }t}{{\eta }_{\mathrm{d}\mathrm{c}\mathrm{h}}},

(25)0⩽P_{\mathrm{c}\mathrm{h}}^t⩽\mu P_{\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{a}\mathrm{x}},

(26)0⩽P_{\mathrm{d}\mathrm{c}\mathrm{h}}^t⩽\left(1-{\mu }_{\mathrm{c}}\right)P_{\mathrm{d}\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{a}\mathrm{x}},

(27)S_{\mathrm{m}\mathrm{i}\mathrm{n}}⩽S_t⩽S_{\mathrm{m}\mathrm{a}\mathrm{x}},

(28)E_{24}= E_0,

where S_t represents the SOC of the BES at time t, directly reflecting the remaining battery capacity, η_loss is the energy loss factor, η_ch and η_dch are the charging and discharging efficiencies of the BES, respectively, P_{\mathrm{c}\mathrm{h}}^t and P_{\mathrm{d}\mathrm{c}\mathrm{h}}^t are the charging and discharging power of the BES at time t, P_{\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{a}\mathrm{x}} and P_{\mathrm{d}\mathrm{c}\mathrm{h}}^{\mathrm{m}\mathrm{a}\mathrm{x}} are the maximum charging and discharging power of the BES, and S_max and S_min are the maximum and minimum values of the SOC. A binary variable μ_c ϵ {0, 1} is introduced to prevent the BES from charging and discharging simultaneously. Equation (28) is the daily cycling balance constraint for the BES, and E₂₄ represents the energy stored in the battery energy storage system (BESS) capacity at the end of the day, and E₀ represents the energy stored in the BESS capacity at the beginning of the day.

When there is excess heat supply, the TES could store the surplus heat and release the energy stored during the periods of insufficient heat or when the cost of generating heat is high. This process is expressed as [37]

(29)H_t= H_{t-1}+{\eta }_{\mathrm{i}\mathrm{n}}{H}_{\mathrm{i}\mathrm{n}}^t\mathrm{\Delta }t- \frac{H_{\mathrm{o}\mathrm{u}\mathrm{t}}^t\mathrm{\Delta }t}{{\eta }_{\mathrm{o}\mathrm{u}\mathrm{t}}},

where H_t represents the heat storage degree (HSD) of the TES at time t, directly reflecting the remaining heat in the TES, η_in and η_out are the efficiency of heat storage and heat release of the TES, respectively, and H_{\mathrm{i}\mathrm{n}}^t and H_{\mathrm{o}\mathrm{u}\mathrm{t}}^t represent the heat stored and released by the TES at time t.

Similar to the BES, to ensure the safe operation and extend the lifespan of the TES, the following constraints must be satisfied [37].

(30)0⩽H_{\mathrm{i}\mathrm{n}}^t⩽\mu H_{\mathrm{i}\mathrm{n}}^{\mathrm{m}\mathrm{a}\mathrm{x}},

(31)0⩽H_{\mathrm{o}\mathrm{u}\mathrm{t}}^t⩽\left(1-{\mu }_{\mathrm{c}}\right)H_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}},

(32)H_{\mathrm{m}\mathrm{i}\mathrm{n}}⩽H_t⩽H_{\mathrm{m}\mathrm{a}\mathrm{x}},

where H_{\mathrm{i}\mathrm{n}}^{\mathrm{m}\mathrm{a}\mathrm{x}} and H_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}} are the maximum capacities of the TES for storing and releasing heat at time t, H_{\mathrm{i}\mathrm{n}}^t and H_{\mathrm{o}\mathrm{u}\mathrm{t}}^t represent the heat stored and released by TES at time t, and H_max and H_min are the maximum and minimum values of the HSD.

3.4 Model of users

The users participating in the trading process in this paper have flexible and adjustable electrical and thermal loads. The electrical and thermal loads of a user at time t are expressed as

(33)P_{\mathrm{e}\mathrm{l}}^t= P_{\mathrm{f}\mathrm{e}\mathrm{l}}^t+P_{\mathrm{t}\mathrm{r}\mathrm{s}}^t+P_{\mathrm{c}\mathrm{u}\mathrm{t}}^t,

(34)Q_{\mathrm{h}\mathrm{l}}^t= Q_{\mathrm{f}\mathrm{h}\mathrm{l}}^t+Q_{\mathrm{c}\mathrm{h}\mathrm{l}}^t,

where P_{\mathrm{e}\mathrm{l}}^t and Q_{\mathrm{h}\mathrm{l}}^t represent the electrical and thermal loads of a user at time t, respectively; P_{\mathrm{f}\mathrm{e}\mathrm{l}}^t is the fixed electrical load at time t, representing the part of the load with a fixed and less adjustable usage time, reflecting the user’s basic electricity demand; P_{\mathrm{t}\mathrm{r}\mathrm{s}}^t is the translatable electrical load at time t, and P_{\mathrm{c}\mathrm{u}\mathrm{t}}^t is the reducible electrical load at time t; and Q_{\mathrm{f}\mathrm{h}\mathrm{l}}^t and Q_{\mathrm{c}\mathrm{h}\mathrm{l}}^t represent the fixed and reducible heating loads of users at time t. The constraints for the proportions of translatable and reducible electrical loads to the total electrical load are

(35)\{\begin{array}{c}{W}_{\mathrm{t}\mathrm{r}\mathrm{s}}={\mathrm{\Sigma }}_{t=1}^T{P}_{\mathrm{t}\mathrm{r}\mathrm{s}}^t\mathrm{\Delta }t,\\ \frac{\left|P_{\mathrm{t}\mathrm{r}\mathrm{s}}^t\right|}{P_{\mathrm{e}\mathrm{l}}^t}⩽{\theta }_{\mathrm{t}\mathrm{r}\mathrm{s}},\\ \frac{\left|P_{\mathrm{c}\mathrm{u}\mathrm{t}}^t\right|}{P_{\mathrm{e}\mathrm{l}}^t}⩽{\theta }_{\mathrm{c}\mathrm{u}\mathrm{t}},\end{array}

(36)0⩽Q_{\mathrm{c}\mathrm{h}\mathrm{l}}^t⩽Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t,

where W_trs is the total translatable electrical load during time period T, θ_trs and θ_cut are the upper limits of the translatable rate and reducible rate, and Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t is the upper limit of the reducible thermal load at time t.

The difference between the utility function and the cost for users is defined as consumer surplus, whose relationship is expressed as

(37)R_{\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{r}}={\sum }_{t=1}^T\left[C_i-\left(P_{\mathrm{e}\mathrm{l}}^t{c}_{\mathrm{e}}^t+Q_{\mathrm{h}\mathrm{l}}^t{c}_{\mathrm{h}}^t\right)\right],

(38)C_i= {\alpha }_{\mathrm{e}}{P}_{\mathrm{e}\mathrm{l}}^t- \frac{1}{2}{\beta }_{\mathrm{e}}{P_{\mathrm{e}\mathrm{l}}^t}^2+{\alpha }_{\mathrm{h}}{Q}_{\mathrm{h}\mathrm{l}}^t- \frac{1}{2}{\beta }_{\mathrm{h}}{Q_{\mathrm{h}\mathrm{l}}^t}^2.

In Eq. (37), R_user represents the user’s revenue, C_i represents the user’s utility function, which quantifies the user’s preferences and satisfaction with electricity or heat; α_e, β_e, α_h, and β_h represent the preference coefficients for the user’s purchase of electricity and heat.

To ensure the safety and reliability of PIES, certain constraints need to be satisfied. The supply–demand balance of the PIES is expressed as

(39)P_{\mathrm{W}\mathrm{T}}^t+ P_{\mathrm{P}\mathrm{V}}^t+ P_{\mathrm{G}\mathrm{T}}^t+ P_{\mathrm{d}\mathrm{c}\mathrm{h}}^t+P_{\mathrm{b}\mathrm{u}\mathrm{y}}^t=P_{\mathrm{c}\mathrm{h}}^t+ P_{\mathrm{f}\mathrm{e}\mathrm{l}}^t+P_{\mathrm{t}\mathrm{r}\mathrm{s}}^t+P_{\mathrm{c}\mathrm{u}\mathrm{t}}^t+P_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}}^t,

(40)Q_{\mathrm{G}\mathrm{B}}^t+ Q_{\mathrm{W}\mathrm{H}\mathrm{B}}^t+H_{\mathrm{o}\mathrm{u}\mathrm{t}}^t=H_{\mathrm{i}\mathrm{n}}^t+ Q_{\mathrm{f}\mathrm{h}\mathrm{l}}^t+Q_{\mathrm{c}\mathrm{h}\mathrm{l}}^t.

4 DE-QP iterative optimization strategy based on a two-layer optimization

To achieve the optimal energy pricing strategy for the energy station under the low-carbon constraints, ensuring a relative balance of benefits between the energy station and users, the energy station needs to simultaneously consider the impact of carbon trading costs and user behavior in each time period. In an interactive competitive electricity market, there is continuous information exchange (such as electricity and heat pricing information) between users and the energy station. The selling prices for electricity and heat are continuously adjusted by the energy station and eventually determined. However, the combination of electricity prices, heat prices, and output information sold to users by the energy station is too complex to achieve optimal revenue. Thus, the multi-energy trading process is transferred to a two-layer interactive framework. Through an iterative optimization process, the interaction and optimal decisions between the upper-layer energy station and the lower-layer users are realized. The optimization iteration process is illustrated in Fig.3.

In Fig.4, the energy station is considered the upper-layer decision-maker, and the users are considered the lower-layer decision-makers. At time t–1, the energy station first formulates the electricity and heat prices to be sold to users for each time period within a day based on the energy transaction prices from the external grid. The price information is then transmitted to the users. Upon receiving the price signal, users adjust their energy consumption behavior according to their own circumstances, forming the expected values of electricity and heat demand, which are then fed back to the energy station. The feedback results are received by the energy station and used to adjust the optimal combination of electricity prices, heat prices, and output plans. At time , the energy station sends the optimized electricity and heat prices to the users, and one iteration cycle is completed. This process is repeated continuously until the revenues of both the energy station and users converge, ending the iteration. During the iteration process, maximizing the revenue of the energy station is considered as the optimization objective, whose model is

(41)\{\begin{array}{c}\mathrm{m}\mathrm{a}\mathrm{x}{R}_{\mathrm{S}\mathrm{U}\mathrm{P}\mathrm{L}},\\ \mathrm{s}.\mathrm{t}.\mathrm{E}\mathrm{q}\mathrm{s}.\left(1\right)-\left(17\right),\left(19\right)-\left(40\right).\end{array}

In the solving process, the maximization of energy station revenue is considered as the optimization objective of the upper level, while the maximization of consumer surplus is considered as the optimization objective of the lower level. The energy station and users pursue their respective optimal values while their optimization decisions are interrelated and mutually influential. Additionally, due to the lack of transparency in information among participants in the electricity market, each participant needs to be optimized individually. The algorithm selected should maintain diversity while exhibiting high computational efficiency and convergence. Thus, this paper adopts a distributed equilibrium-solving approach that combines the differential evolution (DE) algorithm with quadratic programming (QP). It utilizes the DE algorithm for global search and then transforms the search results into a quadratic programming problem, which is solved using quadratic programming algorithms to obtain more accurate optimal solutions. This algorithm combines the advantages of global search and precise solving, making it suitable for handling optimization problems with complex constraints and non-convexity.

The equilibrium of strategy-solving results below has been demonstrated.

According to Hua et al. [11], when the following conditions are met by the model, a unique equilibrium solution exists.

Condition (1): The revenue functions of the participants are non-empty, continuous functions with respect to the proposed set of strategies.

Condition (2): The revenue functions of the upper and lower levels are continuous concave or convex functions with respect to their respective sets of strategies.

The strategy set of the upper-level energy station satisfies Eqs. (1)–(38), while the strategy set of the lower-level users satisfies Eqs. (33)–(38). It is evident that both strategy sets are non-empty and continuous, fulfilling condition (1).

The revenue function of the lower-level users is analyzed by taking the second partial derivatives of the function R_user with respect to P_{\mathrm{e}\mathrm{l}}^t and Q_{\mathrm{h}\mathrm{l}}^t, whose results are

(42)\{\begin{array}{l}{\displaystyle \frac{{\mathrm{\partial }}^2{R}_{\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{r}}}{[{\mathrm{\partial }\left(P_{\mathrm{e}\mathrm{l}}^t\right)]}^2}}=-{\beta }_{\mathrm{e}},\\ {\displaystyle \frac{{\mathrm{\partial }}^2{R}_{\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{r}}}{[{\mathrm{\partial }\left(Q_{\mathrm{h}\mathrm{l}}^t\right)]}^2}}=-{\beta }_{\mathrm{h}}.\end{array}

Since β_e and β_h are all positive values, the second partial derivatives of the equation are all less than 0. Additionally, the cross-partial derivatives of the function R_user are equal to 0. Therefore, the Hessian matrix of R_user is negative definite, indicating that R_user has a unique maximum point, satisfying condition (2).

The payoff function R_SUPL of the upper-level energy station has been analyzed. Clearly, R_SUPL is a linear function of multiple decision variables. According to the definition of concavity and convexity, a linear function is both concave and convex.

In conclusion, the dual-level model proposed in this paper exhibits a unique equilibrium.

The electricity and heat prices that the energy station sells to users in each hour of the day are set as variables in the population. Based on this, the solution approach of the double-layer optimization algorithm is employed to solve this model, where the upper layer uses the DE-QP algorithm, and the lower layer uses the CPLEX solver for solving. The solution process is illustrated in Fig.5, and the specific steps are as follows.

(1) Running parameters, such as population size, maximum iteration times, crossover factor, mutation factor, and deviation amplification coefficient, are input. Subsequently, the population is initialized, i.e., the electricity and heat prices that the energy station sells to users in each hour of the day are randomly initialized based on external electricity and heat price information.

(2) Price signals from each population are sent by the upper-layer energy station to the lower-layer users. Upon receiving the price signals, users optimize and adjust their electricity consumption behavior, and then feedback the expected electricity and heat demand values for each time period to the upper-layer energy station.

(3) Based on the feedback information, the fitness function of each parent population in the population (i.e., the revenue of the energy station) is calculated by the upper layer.

(4) Mutation operations are applied to the parent price population, and mutation populations undergo crossover operations.

(5) Based on the fitness function of the population (i.e., the revenue function of the energy station), the top ten populations with high fitness values are selected to generate a new generation of populations.

(6) The error between two iterations should be checked to see if it is less than 0.3% and if it has reached the maximum number of iterations. If the above conditions are met, the optimal selling prices for electricity and heat in the population, as well as the revenue of the energy station and users, are output. If not met, repeat Step (2).

5 Simulation

To validate the effectiveness of the energy station-user dual-layer optimization strategy proposed considering electric and thermal demand response, this paper takes a certain PIES as an example to simulate and analyze the multi energy trading strategy proposed. The thermal-electric co-optimization problem is transformed into a mixed integer bilevel nonlinear programming problem and solved in MATLAB using solvers CPLEX and YALMIP toolbox.

The relevant parameters for the energy station and users are presented in Tab.1 and Tab.2, and other parameter settings can be referred to in Jung et al. [22]. The typical electrical loads and heating loads of users, as well as the forecast curves for the PV and the WT, are shown in Fig.6. The parameters for the biogas production model are referenced from literature. The thermal capacity c_m and density ρ_m of biomass materials are 1.1×10^–3 and 1.1×10^–3 kWh/°C, respectively. The heating value of methane Q_{\mathrm{C}{\mathrm{H}}_4} is 11.5 kWh/m³, and the hydraulic retention time θ is set to 40 d.

This section conducts a feasibility analysis of the optimization strategy proposed. As shown in Fig.7, the profit curves for the energy station and users gradually converge after 38 iterations. In the initial stages, due to the collaborative interaction between the energy station and users, both parties experience significant profit fluctuations. With the increase in the number of iterations, the profits on both sides gradually reach equilibrium. Eventually, the profit of the energy station stabilizes at $13266.9, and the user’s profit reaches $19391.6. At this stage, neither party can independently improve its income by altering its strategy. This result indicates that the collaborative optimization strategy proposed achieves a win-win situation through close interaction and collaboration between both parties.

Fig.8 illustrates the comparison of user electrical load and heating load before and after demand response. From Fig.8(a), it can be observed that the electrical load has two peaks before demand response, occurring during the time intervals of 11:00–12:00 and 17:00–22:00. During these peak periods of electricity demand, the electricity prices are also relatively high, thereby incentivizing users to engage in demand response. Additionally, during the time span of 0:00–6:00, as the electricity demand decreases, the electricity prices also decline. Under the strategy proposed in this paper, the user’s peak electrical load is reduced, while the non-peak electrical load is increased. Users can be effectively guided in their transactions with the energy station, continuously optimizing and altering their electricity consumption behavior to facilitate the transfer of peak loads. Due to the influence of user comfort, the overall variation in user heating load is relatively small. The adjustment of heating load decreases during the peak heating period and increases during the non-peak heating period. The overall fluctuation in heating load demand is minimal, contributing to the stability of heat output.

Fig.9 illustrates the energy supply situation for various devices after demand response. As shown in Fig.10, the output of the PV and the WT is given priority from 6:00 to 18:00. When the user’s electricity demand exceeds the power provided by these sources, the GT or the BES is used to meet the user’s requirements. If none of the power supply devices within the energy station can meet the user’s electrical load, the energy station purchases electricity from the external grid. When the electricity demand is less than what is provided, the surplus electricity can be stored in the BES or traded with the external grid for economic benefit. Specifically, during the time intervals of 0:00–7:00 and 6:00–18:00, when the user’s electricity demand is relatively low, the energy station sells part of the surplus electricity generated by the power generation devices to the grid, storing the remaining portion in the energy storage devices. From 14:00 to 20:00, when the user’s electricity demand is at its peak, the gas turbine and wind power output may not be sufficient to meet the user’s requirements. The energy station adjusts the supply-demand balance by discharging the BES and purchasing some electricity from the grid.

This section also simulates the process of ADR methane production under the influence of temperature. As shown in Fig.10, with the increase in temperature, the hourly methane production in the ADR continuously rises. The methane production curve initially exhibits a significant variation, and later, after the anaerobic digestion temperature reaches 40 °C, it gradually tends to stabilize. This variation is attributed to the fact that the early increase in temperature facilitates a faster decomposition of biomass materials, leading to the generation of more methane. In the later stages, the reactions within the ADR reach an optimal state, resulting in a gradual levelling off of methane production.

To validate the rationality of the stepped carbon trading cost optimization strategy proposed in this paper, three cases are set as follows:

Case I: The stepped carbon trading cost in the optimization process (the strategy proposed in this paper) considered.

Case II: The traditional carbon trading cost in the optimization process is considered.

Case III: Carbon trading costs are not considered.

The DE-QP algorithm is employed to optimize the three cases mentioned above, and the optimization results for the energy station and the user, including profits, carbon trading costs, and carbon emissions, are presented in Tab.3.

From Tab.3, it can be observed that Case I achieves the lowest carbon emissions and carbon trading costs, and attains the highest user profit. Regarding the optimization results for the profit of the energy station, while Case I is not the best, its results are second only to Case II. Thus, after comparing the optimization results in various aspects, Case I is the optimal choice. In comparison with Case II, despite both considering carbon trading costs in the profit of the energy station, traditional carbon trading costs in Case II keep the carbon trading price per unit of carbon emissions constant, regardless of the actual carbon emissions of the system. In contrast, the stepwise carbon trading strategy in Case I adjusts the carbon trading price per unit of carbon emissions based on the actual carbon emissions of the system. This means that as the actual carbon emissions increase or decrease, the carbon trading price per unit of carbon emissions in the stepwise strategy will vary accordingly. In Case I, the strategy is more flexible and can better incentivize the energy station to reduce carbon emissions. The carbon trading costs and carbon emissions in Scenario 1 are reduced by 2.06% and 1.71%, respectively, compared to Case II. Case III, compared to Case I, does not consider carbon trading costs in the profit of the energy station. The carbon trading costs and carbon emissions in Case I are reduced by 5.23% and 2.54%, respectively, compared to Case III. In Case I, since the stepwise carbon trading strategy is not considered, the energy station tends to focus on maximizing its own profit. Consequently, during periods of low electricity prices, to reduce its operating costs, the energy station is more inclined to increase the amount of electricity purchased from the external grid, resulting in increased carbon emissions and carbon trading costs.

To examine the reasonableness of the ratio set between translatable electrical loads and reducible thermal loads in the user model, sensitivity tests were conducted by varying the parameters individually.

Fig.11 illustrates the variations in electrical load before and after demand response within a 10% fluctuation range of {\theta }_{\mathrm{t}\mathrm{r}\mathrm{s}}. Fig.12 provides a comparison of thermal load before and after demand response within a 10% fluctuation range of Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t. Specifically, Fig.10 depicts the electrical load before and after demand response obtained by varying the shifting ratio parameter in the electrical load model. It can be observed from Fig.10 that as the upper and lower limits of translatable rates fluctuate, there are noticeable changes in user electrical load errors, albeit with minimal magnitude. Thus, by preliminary assessment of the user electrical load model, the stability of the set translatable ratio can be validated.

Additionally, Fig.12 demonstrates the thermal load before and after demand response obtained by varying parameters in the user’s reducible thermal load model. In contrast to the comparison of translatable electrical loads before and after demand response, significant changes in user thermal load errors occur when the proportion of reducible thermal load, denoted as Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t, fluctuates. Furthermore, these changes exhibit a considerable magnitude. Within Fig.12, the thermal load, when Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t is set to 72 kW, is relatively consistent with the thermal load when Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t is set to 80 kW, while the thermal load shows a significant deviation when Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t is set to 88 kW, although the load remains relatively stable. Therefore, preliminary judgment suggests that the value of the reducible thermal load Q_{\mathrm{c}\mathrm{h}\mathrm{l},\mathrm{m}\mathrm{a}\mathrm{x}}^t should not be set excessively high.

To verify the effectiveness of the strategy proposed in this paper under the DE-QP algorithm, a comparison is made with the differential evolution and quadratic programming (GA-QP) algorithm. The comparison of the two in terms of iteration count and runtime is shown in Tab.4. When using the DE-QP algorithm, the revenues of the energy station and users have already converged by the 38th iteration. In contrast, it takes 65 iterations for the results to converge when using the GA-QP algorithm. Furthermore, the execution time using the DE-QP algorithm is reduced by 16.7% compared to the GA-QP algorithm.

To conduct a sensitivity analysis of the carbon trading mechanism, the reward factor and the penalty factor are varied to support different simulations. The simulation results for carbon emission quota, actual total carbon emissions, and carbon trading costs are shown in Tab.5. From Tab.5, it can be concluded that with the penalty factor held constant, as the reward factor increases, the actual total carbon emissions and carbon trading costs decrease. The reason for this is that, with greater rewards, there is a stronger incentive for users to participate in the carbon trading mechanism, leading to a greater reduction in carbon emissions. Similarly, with the reward factor held constant, as the penalty factor increases, the actual carbon emissions and carbon trading costs decreases as well.

6 Conclusions

This paper proposes a PIES low-carbon collaborative dual-layer optimization strategy considering electricity demand and heat demand response, transforming the traditional vertical integrated interaction mode into a thermal-electric collaborative interaction mode. Through interaction strategies, a multi-agent collaborative optimization operation is ultimately achieved. The introduction of user-side demand response strategies enables the system to adapt more flexibly to user energy demands while encouraging users to change their energy consumption behavior. A comparison of three different cases is made. The simulation results indicate that, compared to the traditional vertically integrated structure, the strategy proposed in this paper increases the revenue of energy stations and the consumer surplus of users by 5.09% and 2.46%, respectively. Additionally, the adoption of the stepped carbon trading strategy not only ensures the economic operation and energy utilization efficiency of the PIES, but also motivates the energy station and users to participate in carbon reduction efforts. The carbon trading costs and carbon emissions are reduced by 5.23% and 2.54% based on the strategy proposed in this paper.

However, this study only considered the participation of a single energy station in transactions within the park. In a given park, there may be multiple energy stations. To further enhance system operating efficiency, non-cooperative game issues among energy stations need further investigation. Besides, apart from energy transactions within each region, there can also be energy trading between different regions. Thus, non-cooperative game issues among multiple PIESs also need to be studied.

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