The PEB converts electricity into thermal energy for high-temperature heat supply. Operating under elevated pressure compared to atmospheric boilers, it achieves higher outlet temperatures and improved operational safety, offering significant technical advantages [21].

Eq. (8) states that the total energy output of the PEB is the sum of its heating power and thermal storage power. Eq. (9) limits the total energy output of the PEB to be less than its upper limit. Eq. (10) constrains the heating power to be non-negative. (8)QtEB=QtEB,H+QtEB,S (9)0≤QtEB≤XtEBQmaxEB (10)QtEB,H≥0where QtEB is the total energy output of the PEB at time t; QtEB,H and QtEB,S are the heating power and thermal storage power of the PEB at time t; QmaxEB is the upper limit of the TSEB’s energy output; XtEB is a binary variable indicating the operating state of the PEB.

The electricity consumption of the PEB is given by Eqs. (11)–(14). Eq. (11) describes the operating electric power of the PEB, taking into account the heat loss caused by the working fluid flow velocity [22]. Eq. (12) indicates that the working fluid velocity is a function of the water pump’s output [23]. (11)PtEB=QtEB+δwgtWP(TtEB,out−TtEB,in)ΔtηEB+XtEBPtEB,WP (12)gtWP=fEBXtEBPtEB,WP (13)TtEB,out=Ttg (14)TtEB,in=Tthwhere PtEB is the electricity consumption of the PEB at time t; ηEB is the energy conversion efficiency of the PEB; (TtEB,out−TtEB,in) is the temperature difference across the boiler; δw is a flow-dependent heat loss coefficient specific to the boiler’s design; gtwpis the instantaneous mass flow rate of the working fluid; PtEB,WP is the rated electricity consumption of the associated circulating water pump of the PEB; Ttg and Tth represent the supply temperature and return temperature of the hot pipeline.

The TSWT serves as the core component of the TSEB system. Its main function is to store and release thermal energy, thereby facilitating the temporal shift and utilization of heat. The device accumulates thermal energy produced by the PEB during off-peak electricity price periods and discharges it during times of high heating demand. However, extended storage durations lead to inevitable thermal energy losses due to heat dissipation. Prolonged thermal storage results in thermal energy loss.

The thermal energy stored in the TSWT is given by Eqs. (15) and (16). Eq. (17) constrain the heating power output of the TSWT. Eqs. (18) and (19) ensure the TSWT cannot store and release simultaneously. The binary variables are defined in Eq. (20). (15)WtWT=(1−εWT)Wt−1WT+QtEB,SΔt−QtWT,HΔt (16)WminWT≤WtWT≤Wmax,tWT (17)0≤QtWT,H≤XtWT,HQmax,tWT,H (18)0≤QtEB,S≤XtEB,SQmax,tEB,S (19)XtEB,S+XtWT,H≤1 (20)XtEB,S,XtWT,H∈{0,1}where WtWT is the thermal energy stored in the TSWT at time t; εWT is the thermal energy loss rate of the TSWT; Δt is the scheduling time interval; QtWT,H represents the heating power supplied by the TSWT at time t; WmaxWT and Wmax,tWT represent the upper and lower limits of the stored thermal energy in the TSWT; Qmax,tWT,H represent the upper limits of the TSWT’s heating power; XtEB,S and XtWT,H are binary variables. XtEB,S indicates the TSWT is charging, XtWT,H indicates the TSWT is discharging, and they cannot both be 1 simultaneously.

Considering the thermal stratification [24] and the load effect of the TSWT, its power upper limit should be subject to variation over time t, as shown by Eqs. (21)–(25). Eq. (21) illustrates that the maximum thermal storage capacity of the TSWT is influenced by its internal temperature distribution. Eq. (22) specifies that the maximum supply temperature of the TSWT should not exceed 98°C. Meanwhile, Eq. (23) indicates that the supply water temperature is a function of the total heating load. Under conditions of limited flow rate, the water temperature should increase accordingly to meet a higher heat demand. (21)Wmax,tWT=cwρVWT(TtWT,t−TtWT,b)3600 (22)TtWT,t=min{Ttg,98∘C} (23)TtWT,b=Tth (24)Ttg=δg,lLh (25)Qmax,tWT,H=Qmax,tEB,S=αWTWmax,tWTwhere cw is the specific heat capacity; ρ is the density of the working fluid; VWT is the TSWT volume; TtWT,t and TtWT,b is the top and bottom temperature of TSWT.

Accurate carbon accounting methods are fundamental to low-carbon emission reduction. Carbon emissions are categorized as direct carbon emissions and indirect carbon emissions, as shown in Fig. 2.

Direct carbon emissions are defined as CO₂ emitted directly by the sources. This includes emissions from on-site fossil fuel combustion or industrial processes. Indirect carbon emissions are those attributable to an entity’s activities but occurring from sources not directly under its control. These emissions typically arise from purchased electricity, heat, transportation services, or other activities associated with the entity’s consumption. While physically emitted by entities such as power generators or transportation providers, the responsibility for these indirect emissions is assigned to the consuming entity.

In this paper, the CO₂ emissions from the CHP unit burning natural gas are classified as direct carbon emissions. In contrast, the electricity consumption activities of the park and the energy losses of the EES devices [28] are regarded as indirect carbon emissions. Although these activities do not produce CO₂, they indirectly lead to CO₂ generation elsewhere. The classification of carbon emissions is shown in Table 1.

The carbon emissions generated by park operations are given by Eqs. (44)–(48). (44)Ecoal=φcoal∑tPcoal,tΔt (45)Egas=φgas∑tPgas,tΔt (46)Etotal=Ecoal+Egas (47)∑tPcoal,tΔt=PtGSHPΔt+PtEBΔt+ΔSOCtES+αTLPtTLΔt (48)∑tPgas,tΔt=PtCHPΔt+βTLPtTLΔtwhere Ecoal, Egas and Etotal represent the carbon emissions from coal consumption, natural gas consumption and the total indirect carbon emissions during park operations, respectively; φcoal and φgas are the carbon emission factors for coal and natural gas, respectively, representing the amount of CO₂ emitted per kWh of product generated due to coal or gas consumption during park operations; ∑tPcoal,t and ∑tPgas,t represent the total coal consumption and natural gas consumption during the optimization period; PtTL is the power purchased from the public grid; αTL and βTL are the proportional coefficients for the coal and gas components of public grid electricity, respectively. Their values are determined by the installed capacity of coal-fired and gas-fired power units in the power system [29].

Carbon emission trading (carbon market) treats carbon quotas as commodities. Its purpose is to reduce carbon emissions by economic instruments. The carbon market is divided into the primary carbon market and the secondary carbon market.

Government regulators set overall carbon quotas and allocate them to entities in the primary carbon market through methods such as free allocation and auctions. These entities can subsequently trade quotas in the secondary market. If an entity’s emissions exceed its allocated quota, it must purchase additional allowances, while those with surpluses may sell them to others. By the compliance deadline, entities are required to hold sufficient quotas; otherwise, they will incur penalties. The trading mechanism of the secondary carbon market is shown in Fig. 3.

This paper’s carbon quota allocation model addresses the primary carbon market, where government departments allocate quotas gratis to the park according to its carbon emission intensity. The carbon quota allocation model for park is given by Eqs. (49)–(51). (49)Ecoal,qu=φcoal,qu∑tPcoal,t (50)Egas,qu=φgas,qu∑tPgas,t (51)Etotal,qu=Ecoal,qu+Egas,quwhere Ecoal,qu, Egas,qu and Etotal,qu represent the allocated coal carbon quota, natural gas carbon quota, and total indirect carbon quota for the park, respectively; φcoal,qu and φgas,qu are the carbon quota allocation factors, representing the amount of carbon quota allocated per kWh of product generated due to coal or natural gas consumption during park operations.

The carbon price model reflects secondary market operations. The trading volume in the carbon market is defined as the difference between its carbon emissions. The park’s trading volume is defined in Eq. (52). A positive Etr indicates emissions exceeding the quota, necessitating additional quota purchases for compliance. A negative Etr denotes surplus quotas sellable in the market to generate revenue. (52)Etr=Eind−Eind,qu

The carbon market price function is given by Eq. (53). (53)CE={λEtr,Etr<lλ(1+ζ)(Etr−l)+λl,l≤Etr<2lλ(1+2ζ)(Etr−2l)+λ(2+ζ)l,2l≤Etr<3lλ(1+3ζ)(Etr−3l)+λ(3+3ζ)l,3l≤Etr<4lλ(1+4ζ)(Etr−4l)+λ(4+6ζ)l,Etr≥4lwhere, λ is the base carbon price, l is the interval length, and ζ is the price growth rate. The tiered carbon pricing mechanism charges more for higher carbon emissions. This structure imposes tighter constraints on high-emission activities, motivating parks to cut their emissions.

Fig. 4 shows the function of the tiered carbon price. The blue curve illustrates the park’s total carbon cost dependence on trading volume, which increases non-linearly. The red curve plots the unit price against trading volume, demonstrating a stepwise progression as volume rises.

The objective function of the park operations optimization model is to minimize the system operating costs over one scheduling cycle. Eq. (54) represents the daily operating cost (F1), comprising energy purchasing costs and SES service costs. Eq. (55) represents the carbon trading cost (F2). Eq. (56) defines the overall objective function (F) balancing F1 and F2. The weighting coefficients x1 and x2 can be adjusted to reflect the decision-maker’s preference. (54)F1=∑t=1NT[CtTLPtTL+CtNGPtCHP+CtSES(PtSES,out+PtSES,in)+ftstart] (55)F2=CEEtr (56)minF=x1F^1+x2F^2where CtTL is the electricity purchasing price at time t; CtNG is the unit price of natural gas at time t; CtSES is the unit service price for SES at time t; NT is the total number of time intervals in one scheduling cycle; x1, x2 are the weighting coefficients balancing operational cost and carbon trading cost minimization objectives, the impact of this weight allocation on the system’s performance is detailed in Appendix A and Fig. A1; ftstart is the start-up cost of the park equipment; F^1 and F^2 are the normalized sub-objective functions.

While SES incurs direct service fees for charge/discharge operations, EES operational costs are internalized through carbon emissions: Energy losses ΔSOCtES during EES operation increase indirect carbon emissions, thereby elevating carbon trading costs. This mechanism ensures both storage systems bear usage-related costs, preventing dispatch bias.

A balance must be maintained between the supply of energy carriers and the load demand at each time step t.

(1) Thermal Power Balance Constraint (57)QtGSHP+QtEB,H+QtWT,H+QtCHP,H=LtHwhere LtH is the total thermal load at time t.

(2) Electrical Power Balance Constraint

Eq. (58) represents the electrical power balance constraint, signifying that the total power supply in the park must equal the total power demand. (58){LtE+Ptdem=PtTL+PtsupPtdem=PtHP+PtEB+PtES,in+PtSES,inPtsup=PtCHP,E+PtPV+PtES,out+PtSES,outwhere Ptdem and Ptsup present the power demand and supplied of internal park equipment.

Frequent start-stop cycles adversely affect equipment lifespan and incur additional operational costs. Therefore, the CHP unit and PEB system within the TSEB are subject to the constraint that each device, once started, must operate continuously for at least 6 h. These constraints are formulated using binary state variables and expressed uniformly as: (59)St≥Ot−Ot−1 (60)St≤1−Ot−1 (61)St≤Ot (62)Ot+1,Ot+2,Ot+3,Ot+4,Ot+5≥St (63)Ot,St∈{0,1}where: St is the start-up flag at time t. St=1 indicates the device initiated a start-up at time t. Ot is the operation status flag at time t. Ot=1 indicates the device is in operation at time t.

The above equations define St as a transition detector: (64)St={10Ot−1=0andOt=1otherwisethis 0 to 1 state change marks the onset of an operational cycle.

When St=1, Eq. (62) enforces a minimum runtime: (65)Ot+k=1,k∈{1,2,3,4,5}ensuring 6 consecutive hours of operation after startup.

Based on the aforementioned general constraints, the operational start-stop constraint variables StCHP, StEB, OtCHP and OtEB for the CHP unit and the PEB can be established. The equipment start-up cost in the park at time t is derived from the start-up flag St, as shown in Eq. (66). (66)ftstart=∑iCtstartSti,i∈{CHP,PEB}

The economic performance across three operational scenarios is systematically compared in Table 2, highlighting the impact of carbon pricing mechanisms on cost structures.

In Scenario I, all emissions are indirect, originating from electricity consumption. The absence of carbon price signals impedes adoption of cleaner but potentially higher-cost options such as CHP.

Implementation of electricity-carbon synergy in Scenario II yielded simultaneous economic and environmental benefits, reducing carbon emissions by 42.64% while increasing total operational costs by 20.85% compared to Scenario I. This cost optimization primarily resulted from the internalization of carbon expenses through the tiered pricing mechanism, which effectively transformed emissions into controllable economic variables. The marginal carbon emission cost reaches 23.67 USD/ton, significantly exceeding the base carbon price of 4.4 USD/ton. This indicates that each ton of carbon emission reduced incurs an additional cost of 23.67 USD. The pursuit of environmental benefits thus necessitates a trade-off in economic performance, which aligns with theoretical expectations.

Scenario III demonstrated the economic viability of aggressive decarbonization strategies. Although natural gas consumption costs increased by 60.30% due to expanded CHP utilization, carbon trading expenses decreased by 25.10% relative to Scenario II. The marginal carbon emission cost in this scenario reaches 103.82 USD/ton, substantially higher than the 23.67 USD/ton observed in Scenario II. This indicates that increased preference for environmental benefits incurs progressively higher costs.

Due to the tiered carbon pricing mechanism, when parks proactively reduce emissions into lower price tiers, they not only reduce the volume of carbon traded but also avoid the high carbon prices in high-emission tiers. Conversely, to reduce emissions, the park purchases more expensive natural gas, driving up the energy purchase cost and ultimately increasing the total cost. This mechanism proved instrumental in maintaining cost controllability while achieving substantial emission reductions.

Fig. 6 shows the thermal power allocation in Scenario I, II and III.

Scenario I established a GSHP-dominated thermal supply structure supplemented by TSEB systems. During peak electricity price periods (t = 9, 19–21), GSHP output reduction was compensated by TSWT discharge, avoiding direct high-cost electric heating.

Scenario II fundamentally transformed the thermal supply paradigm with CHP becoming the dominant heat source. The TSEB system shifted to exclusive thermal storage mode during off-peak hours, while GSHP served as auxiliary supply during CHP inactivity.

Fig. 7 illustrates the electrical power balance in Scenario I, II and III.

Scenario I achieved real-time power balance through multi-energy sourcing. Purchased electricity and PV generation met demand, supported by EES and SES discharging during peak hours. Economic responsiveness was evidenced by energy storage during low-price periods (t = 0–7, 11–14, 17–18) and utilization during high-price intervals (t = 8–10, 19–23). PV output followed solar irradiance profiles.

Scenario II maintained similar storage strategies but reduced electricity purchases by 24.84% relative to Scenario I. This reduction was directly attributed to increased CHP utilization, which displaced grid electricity consumption.

Scenario III further decreased electricity purchases by 1.90% compared to Scenario II. Total electricity demand exhibited evident reduction as carbon pricing influenced consumption behavior.

Scenario III is selected for energy storage analysis due to its distinctive operational characteristics under carbon constraints. The storage systems exhibit evident differences compared to the scenario I.

Fig. 8 shows the operational status of the TSWT.

The TSWT demonstrates a high-frequency, low-volume charging pattern during off-peak periods (t = 3–7). This operational shift resulted in a 77% reduction in per-cycle storage volume compared to Scenario II, primarily constrained by the mandatory 6-h continuous operation requirement when the PEB activated. Discharge events occurred selectively during peak periods (t = 8, 9), with residual heat persisting at t = 23 due to operational constraints. The operation of TSWT reflects an adaptation to carbon pricing signals, minimizing direct electricity consumption for heating while maintaining thermal buffer capacity.

Fig. 9 shows the operational status of the EES device. Charging/discharging frequency of local EES reduced by 27.8% compared to Scenario II. This conservation strategy directly responded to carbon cost considerations, as energy losses during conversion processes contribute to indirect carbon emissions. State of charge maximum levels decreased by 18%, indicating deliberate capacity underutilization to minimize carbon cost. Charging occurred primarily during PV peak periods (t = 11–14) and low-price intervals (t = 0–7, 17–18), while discharging was limited to peak hours (t = 8–10, 19–22).

The calculated values for the round-trip efficiency [31] of the system across the Scenario I~III are presented in Table 3.

The TSWT demonstrates consistently high round-trip efficiency, with values ranging from 0.83 to 0.90. In contrast, the ES system exhibits a wider range of round-trip efficiency (0.55–0.72), indicating its higher sensitivity to operational parameters such as cycling rate and depth of discharge. The notably lower round-trip efficiency observed in Scenario III (0.55) suggests that excessive prioritization of environmental benefits may adversely affect equipment operating efficiency. Conversely, the highest efficiency occurs in Scenario II (0.72), indicating an operational profile that aligns more closely with the system’s optimal performance.

Fig. 10 shows the status of the SES system.

The SES system provided critical supplementation to local EES, exhibiting a 32% increase in charging activity during peak photovoltaic generation periods (t = 10–14) compared to Scenario II. In addition, SES operation generates zero indirect carbon emissions under the park’s accounting model, as its energy losses are not taken into account. This hierarchical storage strategy—prioritizing local EES with SES as backup—optimally leveraged external resources without incurring additional carbon cost for the park.

The impacts on key performance indicators, including economic costs, carbon emissions, and energy storage behavior, are systematically presented in Tables 4 and 5. This analysis aims to demonstrate the model’s ability to maintain stable and predictable operation under significant load fluctuations.

The total operational cost shows a positive correlation with variations in both electrical and thermal loads. A 20% increase in electrical load leads to an 8.62% rise in total cost, while a similar 20% increase in thermal load results in a more substantial 21.31% cost increase. This asymmetry reflects the different marginal costs associated with supplying electrical vs. thermal energy within the system. Notably, the carbon trading cost remains limited, contributing to less than 1.5% of the total cost across all scenarios, which underscores the effectiveness of the proposed carbon management mechanism in mitigating economic risks under uncertainty.

The system’s carbon emissions also exhibit a strong correlation with load variations, increasing as loads rise. For electrical load changes, emissions grow by 7.39% across the ±20% range, while thermal load variations produce a more pronounced 13.27% increase. The model’s response is consistent and without erratic nonlinearities, indicating a stable and environmentally predictable operation under load uncertainty.

Despite considerable perturbations in input loads, the model consistently generates stable, feasible, and economically viable scheduling solutions. Both cost and emission responses are smooth and predictable, without extreme nonlinearities or instability. Furthermore, the inclusion of electrical and thermal storage enhances the system’s resilience against uncertainty. These characteristics ensure the model’s practical applicability in real-world scenarios where load forecasting errors are inevitable.

The impacts on economic costs, carbon emissions, and average operating depth of the EES are summarized in the Table 6.

As irradiance decreases, both economic and environmental performance deteriorate predictably. A 50% reduction in PV output results in a 21.32% increase in total cost and a 10.74% rise in carbon emissions. Conversely, a 30% increase in PV power improves economic efficiency by reducing total cost by 10.05% and carbon emissions by 8.60%. The EES demonstrates adaptive responsiveness, with its average action depth rising from 354.17 to 746.02 as PV power rises, indicating greater utilization for energy shifting during periods of higher renewable generation.

The model exhibits stable and monotonic responses to PV variations without erratic fluctuations, highlighting strong robustness against renewable generation uncertainties. These results confirm the model’s practical applicability in real-world scenarios with intermittent renewable energy sources.

The peak-valley electricity price spread was adjusted by −20%, 0%, and +20% relative to the baseline scenario, with key results summarized in the Table 7.

The results demonstrate that a larger peak-valley price spread enhances the system’s economic viability. As the price spread widened by 20%, the total cost decreased by 15.5%, primarily driven by a reduction in operational cost. This trend confirms that the model effectively leverages arbitrage opportunities through optimal EES scheduling. Consequently, the EES’s average action depth increased from 674.64 to 732.41, indicating more active and rational utilization of storage under larger price differentials. Carbon emissions also decreased from 134.09 to 126.81, suggesting improved operational efficiency. The carbon trading cost remains stable across all scenarios, never exceeding 192.79, which underscores the resilience of the emission management strategy to electricity market volatility.

The model exhibits consistent and adaptive performance under varying price spreads, demonstrating strong robustness and the capability to achieve economical and low-carbon operation through rational storage dispatch.

This analysis aims to contrast the performance of the proposed tiered carbon pricing mechanism against a fixed carbon price model (4.4 USD/ton) and simulate scenarios with carbon price fluctuations of −50%, 0%, +50%, +100%, and +200%, with key results summarized in the Table 8.

Firstly, the model demonstrates strong economic robustness. Although the total cost is positively correlated with the carbon price, the rate of increase is significantly mitigated. For instance, when the carbon price rises by +200%, the total cost increases by approximately 12.4% (from 13,163.21 to 14,783.32 USD) relative to the benchmark scenario. This rate of increase is substantially lower than the magnitude of the carbon price shock itself, reflecting the model’s ability to adapt its scheduling strategy through flexible energy management to cushion the impact of extreme carbon market volatility on overall operational economics.

Secondly, the tiered carbon pricing mechanism proves effective in curbing carbon emissions compared to a fixed-price model. Under the fixed carbon price, the system’s carbon emissions reach 141.08 tons. In contrast, the proposed mechanism drives emissions down to 134.09 tons even at the benchmark price. Its environmental advantage becomes more pronounced as the carbon price rises: emissions monotonically decrease to 123.04 tons (+50%), 121.82 tons (+100%), and finally to 120.75 tons (+200%). This consistent downward trend generates that the tiered mechanism creates a sustained and powerful economic incentive for low-carbon transition. Even when the carbon price is halved (−50%), the emissions (144.48 tons) remain significantly lower than those under the fixed-price model, demonstrating the mechanism’s structural advantage in emission reduction.