Multi-Time-Scale Optimal Scheduling of Integrated Energy System Considering Transmission Delay and Heat Storage of Heating Network

: In the integrated energy system, signiﬁcant potential exists for the regulation of the heat storage capacity within the heating network. In relation to this attribute, the establishment of the quasi-dynamic model for the heating network is accomplished through the utilization of the ﬁctitious node method. Additionally, a method is introduced to quantify the heat storage within the heating network. Moreover, a multi-time-scale scheduling approach is proposed for the integrated energy system, with consideration given to the heat storage of the heating network. During the day-ahead scheduling phase, the active regulation of the heat storage within the heating network is carried out to enhance the economy of system operation. Transitioning to the intra-day upper scheduling phase, the heat storage capacity of the heating network is utilized to eliminate the transmission delay effect, thereby achieving the coordinated scheduling of both electricity and heat. Shifting to the intra-day lower scheduling phase, the heat storage capacity of the heating network is utilized to enhance the operational ﬂexibility of the power system. Simulation experiments demonstrate that the coordinated scheduling of electricity and heat in the integrated energy system can be effectively achieved through the utilization of the ﬁctitious node method. Furthermore, the proposed multi-time-scale scheduling method, making full use of the heat storage characteristics of the heating network, can effectively suppress ﬂuctuations in the new energy output and load demand while taking the economy into account. In this paper, it results in a 5.9% improvement in system operating economics and possesses the capacity to mitigate wind power ﬂuctuations with an error rate of approximately 20%. This capability signiﬁcantly enhances the integration of wind power as a sustainable energy source.


Introduction
Given the growing tension between energy supply and demand stemming from the depletion of fossil fuels, coupled with the escalating severity of environmental pollution, the imperative to establish a clean, low-carbon, safe, and efficient modern energy system has become increasingly evident.Such a system aims to elevate the energy utilization rate and mitigate environmental degradation [1][2][3].The integrated energy system (IES), which merges electricity, heat, and energy storage, emerges as a crucial conduit for advancing energy interconnection.This integration has the potential to significantly enhance both the economic viability and environmental stewardship of the system [1,4].Moreover, within the IES framework, the thermal component augments the operational flexibility of the power system.This instrumentally mitigates the challenges posed by the fluctuating nature of renewable energy on system operations, enhancing the capacity to integrate new energy sources into the system [1,2].
In IES, the thermal system exhibits a substantial transmission delay.This delay, spanning from the heat source to individual load points, typically varies among these load points and does not frequently align with the multiples of the IES scheduling interval.This nonconformity presents a challenge to the coordinated and optimized operation of the integrated electric and thermal energy system [5].Addressing this issue is crucial for achieving the accurate quasi-dynamic modeling of the thermal system.To address this challenge, ref. [6] proposed dividing heat users into distinct heat load regions and ensured that the transmission delay between each heat load region and the heat source aligns with multiples of the scheduling interval.Ref. [7] approximated the transmission delay for each load point by rounding it off to the nearest multiple of the scheduling interval, while ref.[8] rounded it down.The computational accuracy of these three methods depends on the size of the scheduling interval.In principle, the smaller the scheduling interval, the greater the calculation accuracy.Concerning the calculation of transmission delay, errors may be as high as half of the scheduling interval, and for scheduling intervals of 30 min or more, this discrepancy will introduce non-negligible scheduling errors.Ref. [9] employed the node method to precisely calculate the transmission delay within a heating network.Nevertheless, the application of this method to achieve optimal system scheduling introduces new optimization problems and necessitates multiple iterations, resulting in significant complexity in solving the model.To simplify the solution, ref. [10] enhanced the node method and introduced the water mass method.This method identifies the water mass delay using a binary variable, removing the necessity for variables in the optimal scheduling to be determined by the optimization problem.While achieving this reduction in computational complexity, it does result in a trade-off with some computational accuracy.Ref. [11] scheduled the electric and heat systems separately at each scheduling interval for the entire system.However, this approach would hinder the attainment of optimal scheduling for the system as a whole.Ref. [12] employed a decomposition strategy to partition the heat supply network into multiple parallel subsystems, each featuring a sole heat source and a single load.Similarly, in [13], from the load points towards the heat source, the heating network was gradually transformed into a new network where the load points were directly connected to the heat source.Ref. [14] formulated a heat flow model and utilized Ohm's law and Kirchhoff's law to derive the heat transfer matrix corresponding to the heating network, thereby drawing a parallel between the thermal system and an electric power system.All three of these approaches achieve the coordinated dispatch of heat and power within the system by simplifying the topology of the heating network.However, the change in topology leads to an increase in computational errors, with some periods experiencing deviations as high as 2 • C.
The implicit upwind model demonstrated its effectiveness in predicting the dynamic characteristics of pipelines while preserving temperature distribution information along the pipelines [15].In [16], the potential application of the implicit upwind method for system optimal scheduling was theoretically examined.However, it also delved into the idea that employing this method for optimal scheduling could result in significant computational complexity.Moreover, it is limited to theoretical analysis and lacks experimental validation.In [17], the fictitious node method was introduced.This method effectively balances computational accuracy and the computational burden by employing the calculation time.Furthermore, it holds the advantage of preserving temperature distribution information along the pipe network, similarly to the implicit upwind model.Nonetheless, it is important to emphasize that the aforementioned methods require further experimental verification and comparative analysis.
In order to enhance the operational flexibility of integrated energy systems, the integration of energy storage has become a widely adopted approach.Ref. [18] focused on enhancing system flexibility and stability by optimizing the capacity of thermal storage equipment.The concept of virtual energy storage was explored through the optimization of controllable loads within the system.These loads included centrally controlled air conditioners [19], electric vehicles engaged in orderly charging and discharging [20], customer demand response [21], thermal inertia of buildings [22], and user thermal comfort [23].This optimization effectively assisted in peak shaving and valley filling within the power system, all without the need for additional energy storage devices.Ref. [24] employed the node method to model the energy flow transmission characteristics within the heating network as a virtual thermal energy storage, resulting in a 1.9% reduction in operating costs.In [25], graph theory and Kirchhoff's law were employed to construct a dynamic model of a district heating network.This model characterized the thermal accumulation performance, resulting in a 2.41% improvement in economic performance and a 5.51% increase in wind energy accommodation.Ref. [26] delved into energy storage within district heating networks in a multi-district integrated energy system.The proposed virtual thermal storage and regulation model successfully achieves a trade-off between operating costs and heat losses.However, despite these efforts, the utilization of the thermal storage capacity within the heating network remains relatively homogeneous, primarily resembling that of thermal storage tanks.Additionally, the quantification of the thermal storage capacity of a heating network is still limited to port variables.
Multi-time-scale scheduling proves to be an effective approach for mitigating the influence of deviations in new energy output predictions and load predictions on the overall system.This strategy contributes to achieving a secure and stable operation of the system.Ref. [27] introduced a multi-time-scale scheduling strategy solely focused on the power system.In the intra-day scheduling, it leveraged more accurate source and load forecasts, employing rolling optimization and model prediction control.This approach effectively mitigated the impact of fluctuations in both the power source and the load on the operation of the system.Ref. [28] introduced a multi-time-scale scheduling framework for subsystem decomposition along with a corresponding economic model predictive control method.Furthermore, ref. [29] proposed a distributed model predictive control method aimed at decomposing the comprehensive optimization problem of the integrated energy system.Both of these approaches significantly contribute to enhancing the effectiveness of the model predictive control algorithm.Ref. [30] explored the role of a hybrid energy storage system in the multi-time scale scheduling of an integrated energy system.Various forms of energy storage were reasonably employed based on the distinct response times of the corresponding energy storage devices.This strategy enhances the advantages of the energy storage system in stabilizing power fluctuations while also extending the operational lifespan of the energy storage devices.Due to differences in transmission speeds among the energy subsystems, it is necessary to divide the entire system into distinct sub-layers for effective control.In the literature [31], the system is divided into two sub-layers with a focus on coordinating the scheduling of cooling and electrical energy on an intraday basis.Ref. [32] expands on this by incorporating the scheduling of thermal energy into the upper layer, while ref.[33] introduces an intermediate layer specifically aiming to optimize the scheduling of natural gas energy.Ref. [34] explored various modes of operation for demand response across different time scales, aiming to enhance the flexibility of system operation.Ref. [35] explored the multi-time-scale optimization of an integrated energy system while taking into consideration stepwise carbon trading and environmental requirements.
The aforementioned references analyzed various methods for calculating quasi-dynamic processes in heating systems, approaches for enhancing system flexibility through energy storage, with a particular focus on heat storage within heating networks, and various multi-time-scale scheduling schemes for these systems.The methods that involve rounding the transmission delay to a multiple of the scheduling interval exhibit significant errors.The calculation becomes intricate when employing the node method or its variations, and altering the heating network's topology can simplify the calculation but also introduces considerable errors.The fictitious node method effectively strikes a balance between the computation time and complexity, with the added advantage of preserving the temperature distribution along the pipeline network, a feature shared by the implicit upwind model.The implicit upwind model can ensure that the prediction error of the pipeline outlet temperature remains within ±0.5 • C [15], theoretically enabling its use for system optimization scheduling [16].However, the performance and adaptability of both the fictitious node method and the implicit upwind model require further experimental validation and comparative analysis.Currently, the utilization of energy storage to enhance system flexibility extends beyond conventional storage devices such as batteries and heat storage tanks, encompassing virtual energy storage mechanisms like demand response, building thermal inertia, and heat storage within heating networks.Nonetheless, research on and the utilization of the heat storage capacity within heating networks remain somewhat limited, primarily resembling traditional heat storage tanks.In the realm of multi-time-scale scheduling for integrated energy systems, the utilization of heat storage within heating networks remains underexplored.Furthermore, energy transmission delays are seldom considered in intra-day scheduling.
Therefore, the following yet unexplored work is performed in this paper.The fictitious node method is experimentally compared with the implicit upwind model to further verify the effectiveness of the former in achieving a balanced trade-off between computational accuracy and complexity.Additionally, it is used to demonstrate the impracticality of the implicit upwind model in optimal system scheduling.Leveraging the capability of the fictitious node method to retain temperature distribution information across the entire pipeline network, it is utilized to more precisely quantify heat storage within the network.A multi-time-scale scheduling approach is proposed for the integrated energy system, accounting for both heat transmission delays and heat storage in the heating network.In the day-ahead scheduling phase, the active regulation of the heating network's heat storage capacity is employed to enhance the overall system's operational efficiency.During the intra-day upper-level scheduling phase, the heating network's heat storage characteristics are harnessed to mitigate heat supply delays, simplifying the coordinated scheduling of power and heat systems.In the intra-day lower-level dispatch phase, variations in power system source and load are converted into heat system fluctuations, thus achieving smoothing effects.Given the substantial uncertainty associated with sustainable energy sources like wind power, this paper delves into the heat storage capacity of the heating network to mitigate the adverse impacts of wind power fluctuations, thereby reducing wind abandonment due to wind power uncertainty, improving the integration and consumption of wind power, and fostering the sustainability of energy sources.

Modeling of the Heating System
The quality regulation mode, which involves maintaining a constant mass flow while varying the temperature supply [7], is more commonly employed in practical engineering due to its stable hydraulic properties [3].Therefore, this study is based on the quality regulation mode.
This paper focuses on the thermal system of centralized heat supply, encompassing the heat source, primary heat exchange station, primary pipe network, secondary heat exchange stations, secondary pipe network, and heat users, as shown in Figure 1.Among these elements, the primary pipe network stands out as a transmission network characterized by noticeable transmission delay and prominent dynamic characteristics.Therefore, it holds significant research importance.Conversely, the secondary pipe network functions as a distribution network with shorter transmission delays and less pronounced dynamic characteristics, which is not considered in this paper.
This study of the heating network is based on the following assumptions [17]: (1) the water within the pipe is treated as an ideal fluid; (2) the focus is solely on the temperature distribution along the length of the pipe; (3) due to the significantly lower thermal conductivity along the length of the pipe compared to convective heat transfer, thermal conductivity in this direction is disregarded.This study of the heating network is based on the following assumptions [17]: (1) the water within the pipe is treated as an ideal fluid; (2) the focus is solely on the temperature distribution along the length of the pipe; (3) due to the significantly lower thermal conductivity along the length of the pipe compared to convective heat transfer, thermal conductivity in this direction is disregarded.

The Implicit Upwind Method
In accordance with the law of conservation of energy, the one-dimensional thermal dynamic model of the heating pipeline is represented by Equation (1).

(
) where ρ is the density of water; Ap represents the internal cross-sectional area of the pipe; cw is the specific heat capacity of water; m is the mass flow rate; T represents the temperature of the water in the pipe; t and x represent the temporal and spatial variables, respectively; λ represents the total heat transfer coefficient between the interior of the pipe and its surroundings; and T0 is the ambient temperature outside the pipe.As pipelines are typically buried underground, the ambient temperature surrounding the pipeline is the soil temperature, commonly treated as a constant [5].
The differentiation of temperature with respect to time is expressed using an implicit differential form, as demonstrated in Equation (2).The differentiation of temperature concerning space is denoted using the first-order upwind differential, as illustrated in Equation (3) [16].
where T i n denotes the temperature of the water in the pipe at time step i and space step n; δt1 represents the time step size of the implicit upwind method; δx denotes the space step size.
Substituting Equations ( 2) and (3) into Equation (1) and then simplifying these yields Equation (4): In a heating network employing the quality regulation mode, for a specific pipe, once δt1 and δx are chosen, all variables in Equation (4) become constants except for temperature.The temperature distribution along the specific water supply and return pipelines can be calculated using Equations ( 5) and ( 6):

The Implicit Upwind Method
In accordance with the law of conservation of energy, the one-dimensional thermal dynamic model of the heating pipeline is represented by Equation (1).
where ρ is the density of water; A p represents the internal cross-sectional area of the pipe; c w is the specific heat capacity of water; m is the mass flow rate; T represents the temperature of the water in the pipe; t and x represent the temporal and spatial variables, respectively; λ represents the total heat transfer coefficient between the interior of the pipe and its surroundings; and T 0 is the ambient temperature outside the pipe.As pipelines are typically buried underground, the ambient temperature surrounding the pipeline is the soil temperature, commonly treated as a constant [5].
The differentiation of temperature with respect to time is expressed using an implicit differential form, as demonstrated in Equation (2).The differentiation of temperature concerning space is denoted using the first-order upwind differential, as illustrated in Equation (3) [16].
∂T ∂t ∂T ∂x where T i n denotes the temperature of the water in the pipe at time step i and space step n; δt 1 represents the time step size of the implicit upwind method; δx denotes the space step size.
Substituting Equations ( 2) and (3) into Equation (1) and then simplifying these yields Equation ( 4): In a heating network employing the quality regulation mode, for a specific pipe, once δt 1 and δx are chosen, all variables in Equation (4) become constants except for temperature.The temperature distribution along the specific water supply and return pipelines can be calculated using Equations ( 5) and (6): where the coefficients B 1 , B 2 , and B 3 are constants.

The Fictitious Node Method
In the fictitious node method, the transmission delay of water in the pipeline is calculated using Equation (7), and the temperature loss of water along the pipeline is calculated using Equation (8).
where τ represents the transmission delay; L is length of pipe; T d and T in represent the water temperature in the pipeline at a distance d from the entry point of the pipe, and the initial temperature of the water at the inlet of the pipe, respectively.Fundamentally, Equations ( 7) and ( 8) offer approximate descriptions of the time delay and heat loss.Nevertheless, in cases where the heating network is of a small scale, the discrepancy between models (7), (8), and model ( 1) is exceedingly minor, allowing it to be disregarded [16].
The quasi-dynamic process calculation of the heating network using the fictitious node method comprises two distinct steps.In the first step, the calculation time is defined, and the length (or delay) of each pipe throughout the network is adjusted.This adjustment ensures that the transmission delay of each pipe is a multiple of the calculation time.The second step focuses on computing the heat loss between adjacent fictitious nodes.
For the calculation time, denoted by δt 2 , it is necessary for it to be a multiple of the system's scheduling interval, represented by ∆t.Opting for an appropriate calculation time can better strike a balance between the computational accuracy and computational burden.In [17], the selection of the calculation time is achieved through a simple experiment in advance.This involves assuming a consistent flow of water at a constant temperature into the pipe network from the starting point of the heating network.By calculating and comparing the errors in transmission delay and heat loss before and after adjustments to the length of pipes, the suitable calculation time is determined.
The process of changing the pipeline length in the heating network is illustrated in Figure 2. The pipeline length is gradually adjusted, starting from the heat source and extending towards each heat load point.In this figure, the pipelines that have not yet undergone length transformation are represented by black lines.The orange lines, on the other hand, depict the pipelines currently undergoing length adjustment, while the red lines represent the pipelines that had already undergone this transformation.Pipes linked to square nodes are currently experiencing length transformations.The solid orange line represents the pipe prior to a square node, where the length transformation is computed using Equation (9).On the other hand, the orange-dashed lines correspond to the pipes following the node, with the length transformation being calculated through Equation (10).In instances where there is no pipe following the node, it signifies that the node is connected to the secondary heat exchange station.In this scenario, only the length of the pipe preceding the node undergoes transformation.
where round [•] indicates rounding the value within the square brackets; τ p and τ p are the transmission delays before and after the length change, respectively, for the pipe prior to the square node; τ f and τ f are the transmission delays before and after the length change, respectively, for each pipe following the node.The number of fictitious nodes of pipe j can be calculated by (11).The distance lj between two neighboring fictitious nodes on pipe j can be calculated using (12):.
where Mj is the number of fictitious nodes on pipe j; τ ' j is the transmission delay of pipe j after the length of it is transformed; mj and Ap,j are the mass flow rate and the internal cross-sectional area of pipe j, respectively.
Up until now, fictitious nodes were placed at the beginning and end of each pipe, as illustrated in Figure 3.This makes it easy to compute transmission delays across the entire network, as well as the temperature distribution along each pipeline.The temperature distribution along pipe j for water supply and pipe j for water return can be calculated using Equations (13) and (14).( ) e x p ( ) where T i n,s,j and T i n,r,j represent the temperature of pipe j for water supply and return, respectively, at calculation time numbered i and fictitious node numbered n.

Other Constraints for Modeling the Thermal System
For a node connected by different pipes, the mass flow rate of water flowing into and out of the node is conserved, as is the energy flowing into and out of the node [19].The number of fictitious nodes of pipe j can be calculated by (11).The distance l j between two neighboring fictitious nodes on pipe j can be calculated using ( 12):.
where M j is the number of fictitious nodes on pipe j; τ j is the transmission delay of pipe j after the length of it is transformed; m j and A p,j are the mass flow rate and the internal cross-sectional area of pipe j, respectively.Up until now, fictitious nodes were placed at the beginning and end of each pipe, as illustrated in Figure 3.This makes it easy to compute transmission delays across the entire network, as well as the temperature distribution along each pipeline.The temperature distribution along pipe j for water supply and pipe j for water return can be calculated using Equations ( 13) and (14).
where T i n,s,j and T i n,r,j represent the temperature of pipe j for water supply and return, respectively, at calculation time numbered i and fictitious node numbered n.The number of fictitious nodes of pipe j can be calculated by (11).The distance lj between two neighboring fictitious nodes on pipe j can be calculated using ( 12):.
2 , where Mj is the number of fictitious nodes on pipe j; τ ' j is the transmission delay of pipe j after the length of it is transformed; mj and Ap,j are the mass flow rate and the internal cross-sectional area of pipe j, respectively.
Up until now, fictitious nodes were placed at the beginning and end of each pipe, as illustrated in Figure 3.This makes it easy to compute transmission delays across the entire network, as well as the temperature distribution along each pipeline.The temperature distribution along pipe j for water supply and pipe j for water return can be calculated using Equations ( 13) and ( 14). ( ) e x p ( ) ( ) e x p ( ) where T i n,s,j and T i n,r,j represent the temperature of pipe j for water supply and return, respectively, at calculation time numbered i and fictitious node numbered n.

Other Constraints for Modeling the Thermal System
For a node connected by different pipes, the mass flow rate of water flowing into and out of the node is conserved, as is the energy flowing into and out of the node [19].

Other Constraints for Modeling the Thermal System
For a node connected by different pipes, the mass flow rate of water flowing into and out of the node is conserved, as is the energy flowing into and out of the node [19].
In accordance with the energy conservation theorem, the heat exchange power of the primary heat exchange station and the secondary heat exchange station can be calculated using Equations ( 15) and ( 16): where H t p and H t d,q represent the heat transfer power of the primary heat exchange station and the qth secondary heat exchange station at time t, respectively; m p and m d,q represent the mass flow rate of the primary heat exchange station and the qth secondary heat exchange station, respectively; T i in,s and T i out,r , respectively, represent the temperatures at time i of the water supply pipe's inlet and the return pipe's outlet, both of which are connected to the primary heat exchange station; T i out,s,q and T i in,r,q , respectively, represent the temperatures at time i of the water supply pipe's outlet and the return pipe's inlet, respectively, both of which are connected to the qth secondary heat exchange station.
The heating network operation has upper and lower temperature constraints.The minimum and maximum allowable temperatures in the water supply and return network are expressed as T min s , T max s , T min r , and T max r , respectively.

Quantification of Heat Storage in Heating Network
The heating network contains a significant water volume that grants it a certain heat storage capacity.This capacity becomes evident when adjustments are made to the supply and return water temperatures.During a particular time span, if the heat generated by the heat source surpasses the corresponding heat load demand, the network's water temperature rises, signifying heat storage within the network.Conversely, a decrease in supply and return water temperatures indicates the release of stored heat from the network.Due to transmission delays and temperature fluctuations within the network, accurately determining the exact amount of heat stored at any given moment by observing temperatures at individual points is unfeasible.To provide a more precise quantification of heat storage, the concept of equivalent average temperatures is introduced through Equations ( 17) and ( 18): where T t s,eq and T t r,eq are, respectively, the equivalent average temperatures of the supply and return pipe network at time t; R is the total number of pipes; and M j is the number of fictitious nodes of the pipe j.
The amount of heat storage within the heating network can be calculated using Equation (19).
∆T t a,eq = (∆T t s,eq + ∆T t r,eq )/2 ∆T t s,eq = T t s,eq − T t,min s,eq ∆T t r,eq = T t r,eq − T t,min r,eq where S t h is the amount of heat storage within the heating network at time t; m Σ is the total water mass in the supply or return pipe network; ∆T t,min s,eq and ∆T t,min r,eq are the equivalent average temperatures of supply and return water without considering heat storage in the pipe network at time t, respectively; ∆T t a,eq , ∆T t s,eq , and ∆T t r,eq denote the changes in the equivalent average temperature of the whole network, the supply and return water at time t, in comparison to the value at the same time t when the network's heat storage is disregarded.

Structure of the System and Device Modeling
An energy system typically comprises four main components: the source, network, load, and energy storage.In an integrated energy system, the energy conversion equipment is also integrated to effectively use different energy sources in a complementary manner.The structure of the electro-thermal integrated energy system, as examined in this paper, is depicted in Figure 4.The structure is reused and modified from [36,37].This system encompasses the external power grid, wind turbine (WT), gas turbine (GT), gas boiler (GB), electric boiler (EB), and electric storage equipment (ES).Notably, separate heat storage tanks are not configured, with emphasis placed on heat storage in the heating network (HSHN).Given the brief transmission time of the electric power system, its impact on the coordinated scheduling of the electric-thermal integrated energy system is minimal.To maintain clarity of focus, this paper solely addresses the transmission process of the heat system.
Sustainability 2023, 15, x FOR PEER REVIEW 9 of 26 equivalent average temperature of the whole network, the supply and return water at time t, in comparison to the value at the same time t when the network's heat storage is disregarded.

Structure of the System and Device Modeling
An energy system typically comprises four main components: the source, network, load, and energy storage.In an integrated energy system, the energy conversion equipment is also integrated to effectively use different energy sources in a complementary manner.The structure of the electro-thermal integrated energy system, as examined in this paper, is depicted in Figure 4.The structure is reused and modified from [36,37].This system encompasses the external power grid, wind turbine (WT), gas turbine (GT), gas boiler (GB), electric boiler (EB), and electric storage equipment (ES).Notably, separate heat storage tanks are not configured, with emphasis placed on heat storage in the heating network (HSHN).Given the brief transmission time of the electric power system, its impact on the coordinated scheduling of the electric-thermal integrated energy system is minimal.To maintain clarity of focus, this paper solely addresses the transmission process of the heat system.GT generates both electrical and thermal energy through the combustion of natural gas.Its operational traits are represented by ( 20 where  GT generates both electrical and thermal energy through the combustion of natural gas.Its operational traits are represented by (20): where P t gt and H t gt are the electrical and thermal power output of GT at time t, respectively; η e gt and η h gt are the electrical and heat efficiency of GT, respectively; q gas is the calorific value of natural gas; V t gt is the volumetric flow rate of natural gas input to GT at time t; P min gt and P max gt are the minimum and maximum electrical power output of GT, respectively; R d gt and R u gt are, respectively, the downward and upward ramping power of GT.The operating characteristics of GB are represented by (21): where H t gb is the thermal power output of GB at time t; η gb is the thermal efficiency of GB; V t gb is the volumetric flow rate of the natural gas input to GB at time t; H min gb and H max gb are the minimum and maximum thermal power outputs of GB, respectively; R d gb and R u gb are, respectively, the downward and upward ramping power of GB.
The operating characteristics of EB are represented by ( 22): where H t eb is the thermal power output of EB at time t; P t eb is the electrical power input of EB at time t; η eb is the efficiency of EB; H min eb and H max eb are the minimum and maximum thermal power output of EB, respectively.
The operating characteristics of ES are represented by (23).The operational characteristics of HSHN are given by Equations ( 1)- (19).

Multi-Time-Scale Scheduling Modeling
Multi-time-scale scheduling stands out as a highly effective approach for mitigating the impact of deviations in predictions of new energy output and load on system operations.On one hand, the accuracy of the scheduling outcome increases with smaller scheduling time intervals.On the other hand, predictions become more accurate as the predicted time gets closer to the actual operational time.
The multi-time-scale scheduling model, as presented in this paper, is illustrated in Figure 5 and is categorized into three distinct timescales.
(1) In the day-ahead scheduling, the scheduling interval is 1 h.The new energy output and load demand are forecasted 24 h ahead, forming the basis for the scheduling plan.Notably, the day-ahead scheduling comprehensively accounts for the heat storage capacity of the heating pipe network.The coordinated scheduling of the electric-heat system is achieved, while the economy of the system operation is improved.(2) The intra-day upper-level scheduling employs a rolling optimization approach with a scheduling time window of 3 h and a control time domain of 1 h.This means that the scheduling interval is 1 h, predicting the new energy output and load demand for the upcoming 3 h and generating a corresponding scheduling plan.However, only the results of the first hour's scheduling are carried forward to the subsequent time scale.The rationale behind establishing the scheduling time window is to address constraints related to certain equipment's power ramping capabilities.By considering the power output of specific equipment for only a given hour, there is a risk of significantly impacting the economic performance of the system in the subsequent hour.In intra-day upper-level scheduling, the scheduling of thermal energy solely pertains to the portion of the heat load demand change.Due to the existence of heat storage in the heating network, the transmission delay effect of this portion of thermal energy is not taken into account.The fluctuation of the heat system source and load occurs simultaneously, aiming to achieve the coordinated scheduling of the electric-heat system.(3) The rolling optimization approach is also employed for the intra-day lower-level scheduling.Wind power output and load demand are forecasted 15 min in advance, and EB is utilized to smooth the source and load fluctuations of the electric power system.The EB scheduling interval is set at 15 min.Additionally, at the start of each full hour, the output of the GB is determined for the subsequent hour based on the output data of the EB from the previous hour.The GB scheduling interval spans 1 h.For instance, in the intra-day lower-level scheduling depicted in Figure 5, the output of the EB for the time period between 4:00 and 4:15 is established at 3:45.Concurrently, by leveraging the EB output between 3:00 and 4:00, the GB output for the timeframe of 4:00-5:00 is ascertained.Converting power system fluctuations into thermal system fluctuations can effectively attenuate the negative impact of wind power fluctuations on power system operation, helping to reduce wind abandonment due to wind power uncertainty and improve wind power integration.(1) In the day-ahead scheduling, the scheduling interval is 1 h.The new energy output and load demand are forecasted 24 h ahead, forming the basis for the scheduling plan.Notably, the day-ahead scheduling comprehensively accounts for the heat storage capacity of the heating pipe network.The coordinated scheduling of the electricheat system is achieved, while the economy of the system operation is improved.(2) The intra-day upper-level scheduling employs a rolling optimization approach with a scheduling time window of 3 h and a control time domain of 1 h.This means that the scheduling interval is 1 h, predicting the new energy output and load demand for the upcoming 3 h and generating a corresponding scheduling plan.However, only the results of the first hour's scheduling are carried forward to the subsequent time scale.The rationale behind establishing the scheduling time window is to address constraints related to certain equipment's power ramping capabilities.By considering the power output of specific equipment for only a given hour, there is a risk of significantly impacting the economic performance of the system in the subsequent hour.In intra-day upper-level scheduling, the scheduling of thermal energy solely pertains to the portion of the heat load demand change.Due to the existence of heat

Day-Ahead Scheduling Model
In the day-ahead scheduling phase, the minimum system operating cost is used as the objective function, which is expressed as Equation ( 24): where F da is the total cost of day-ahead scheduling for the system; β t grid is the unit price of electricity purchased from the external grid at time t; β gas is the unit price of natural gas purchased; P t,da grid is the electric power purchased from the grid in the day-ahead scheduling at time t; V t,da gt and V t,da gb are the volumetric flow rate of natural gas input to GT and GB at time t, respectively.
The electric power balance constraint is expressed as Equation ( 25): P t,da grid + P t,da wt + P t,da gt + P t,da esd = P t,da load + P t,da eb + P t,da esc (25) where P t,da wt and P t,da load are the forecasts of wind power output and load demand for the dayahead scheduling period at time t, respectively; P t,da gt is the electric power generated by GT during the day-ahead scheduling period at time t; P t,da eb is the electric power consumed by EB during the day-ahead scheduling period at time t; P t,da esd and P t,da esc are the ES discharging and charging power, respectively, for the day-ahead scheduling period at time t.
The thermal power balance constraint at the heat source is expressed in Equation ( 26): where H t,da gt , H t,da gb , and H t,da eb are the thermal powers emitted by GT, GB, and EB, respectively, in the day-ahead scheduling at time t; H t,da p is the heat transfer power of the primary heat exchange station in the day-ahead scheduling at time t.
The thermal power balance constraint for the heat load is expressed in Equation ( 27): where H t,da load is the predicted heat load for the day-ahead scheduling period at time t; H t,da d,q is the heat transfer power of the qth secondary heat exchange station for the day-ahead scheduling period at time t; and v is the number of secondary heat exchange stations.
To ensure sufficient regulation power for EB and GB in intra-day lower-level scheduling, add constraints as presented in Equations ( 28) and ( 29): where ε gb and ε eb are the power margin factors for GB and EB, respectively.The heating network constraints and equipment constraints are given in Equations ( 1)- (23).In order to increase the margin for the heat storage regulation of the heating network in intra-day scheduling, a margin of 1 • C is set for both the upper and lower limits of the pipeline temperature.

Intra-Day Upper-Level Scheduling Model
The minimization of the operating cost of the system for a scheduling time window is utilized as the objective function in the intra-day upper-level scheduling, expressed as Equation (30): where F iu is the total cost of a scheduling time window in intra-day upper-level scheduling for the system; the corner symbol iu indicates that the variable pertains to the intra-day upper-level scheduling, while the remaining letters retain the same meanings as in the day-ahead scheduling; ∆t 1 is the time interval for intra-day upper-level scheduling.This paper encompasses two scheduling time intervals, referred to as ∆t 1 and ∆t 2 .∆t 1 spans 1 h, serving as the scheduling time interval for the day-ahead scheduling, intra-day upper-level scheduling, and GB in intra-day lower-level scheduling.Meanwhile, ∆t 2 covers 15 min, serving as the scheduling time interval for power subsystems and EB in intra-day lower-level scheduling.
The electric power balance constraint is expressed as Equation ( 31): where the corner symbol iu indicates that the variable pertains to the intra-day upper-level scheduling, while the remaining letters retain the same meanings as in the day-ahead scheduling.
To mitigate the adverse effects of excessive frequent cycling, deep discharging, and overcharging on the service life of ES [36], and simultaneously, exclusively during dayahead scheduling, we can determine the electric storage for the entire day and identify its most cost-effective operational condition.Thus, this paper solely modifies the charging and discharging status of ES within the day-ahead scheduling.
In intra-day upper-level scheduling, the scheduling of thermal energy pertains solely to the portion of the heat load demand change.Due to the existence of heat storage in the heating network, the transmission delay effect of this portion of thermal energy is not considered.The fluctuation of the heat system source and load occurs simultaneously, as illustrated by Equation ( 32): The other constraints remain the same as those in day-ahead scheduling.

Intra-Day Lower-Level Scheduling Model
Heat demand usually fluctuates on an hourly scale, whereas renewable energy tends to vary on a minutely scale [31][32][33].Thus, this paper mitigates the fluctuations in the electric power system's source and load on a significantly smaller time scale, utilizing a scheduling interval of 15 min.The fluctuations in the power system's source and load are translated into fluctuations within the thermal system through EB, with this transformational relationship being illustrated by Equation (33): P t,il wt − P t,iu wt = P t,il load − P t,iu load + P t,il eb − P t,iu eb (33) where the corner symbol il indicates that the variable pertains to the intra-day lowerlevel scheduling, while the remaining letters retain the same meanings as in intra-day upper-level scheduling.Changes in the output of the EB result in fluctuations within the thermal system.This aspect of fluctuations requires mitigation.Additionally, given that the heat storage capacity within the heating network can attenuate a portion of these fluctuations, there is no immediate necessity to the real-time mitigation of the thermal system fluctuations arising from the EB.Because the GB is not an electro-thermal coupling device, this paper employs the GB to flatten the thermal system fluctuations with a scheduling interval of 1 h, with the correlation being represented by Equation (34): where I denotes a time period from one whole point hour to the next whole point hour, and (I−1) denotes the time period from the previous whole point hour to the current whole point hour.
The other constraints remain the same as those in intra-day upper-level scheduling.

Case Studies 4.1. The Setup of Simulation
In this paper, case studies are conducted using an integrated energy system as shown in Figure 4 and a 30-node heating network, as shown in Figure 6.The parameters of the heating network are presented in Table 1, and the operating parameters of the system are provided in Table 2.The multi-time-scale projections for electric load demand, thermal load demand, and wind power output are demonstrated in Figure 7, along with the unit price of electricity bought from the external grid.The flowchart of the model solving is shown in Figure 8.The model proposed in this paper is a linear model and is solved by invoking the optimization solver Cplex through Matlab.The solution process takes place on a laptop equipped with an Intel Core i7-12700H processor and 16 GB of RAM.
The other constraints remain the same as those in intra-day upper-leve

The Setup of Simulation
In this paper, case studies are conducted using an integrated energy sys in Figure 4 and a 30-node heating network, as shown in Figure 6.The para heating network are presented in Table 1, and the operating parameters of t provided in Table 2.The multi-time-scale projections for electric load dem load demand, and wind power output are demonstrated in Figure 7, along price of electricity bought from the external grid.The flowchart of the mo shown in Figure 8.The model proposed in this paper is a linear model an invoking the optimization solver Cplex through Matlab.The solution proce on a laptop equipped with an Intel Core i7-12700H processor and 16 GB of

Comparison of Fictitious Node Method and Implicit Upwind Method
The scheduling results, obtained through the fictitious node method and the implicit upwind method, are depicted in Figure 9 when the heat storage in the heating network is not taken into account.The time used for solving the scheduling model are listed in Table 3, with each time being calculated 20 times and then averaged.
Analyzing Table 3, it becomes evident that, in the case of the fictitious node method, a shorter calculation time corresponds to a quicker solution for the scheduling model.Similarly, in the context of the implicit upwind method, reducing the time step size and space step size leads to decreased time spent on solving the scheduling model.Upon comparing the two methods, it is notable that the fictitious node method exhibits significantly shorter solution times compared to the implicit upwind method.The fictitious node method is capable of solving the scheduling model in just a few tens of seconds, or even a few seconds, making it a suitable choice for optimal scheduling.In contrast, the implicit upwind method demands more time, ranging from a few minutes to even ten minutes.Furthermore, it is foreseeable that utilizing smaller time and space steps could lead to more accurate scheduling results; however, this could also result in the lengthening of the solution time to several tens of minutes.As a result, the implicit upwind method is not suitable for optimal scheduling.
When heat storage in the heating network is disregarded, the scheduling results aligning closer with the trend of the heat load demand curve indicate higher accuracy.Because the transmission delay of heat energy is not an integer multiple of the scheduling interval, theoretically, the ideal scheduling result is different from the shape of the heat load-demand curve, but the trend is the same.As evident from Figure 9a, accuracy in scheduling results is highly achieved at a calculation time of 30 s. Smaller calculation times lead to greater scheduling accuracy.Notably, minute disparities exist between scheduling results at calculation times of 60 s, 180 s, and 300 s.However, a substantial scheduling result error emerges at a 600 s calculation time.This observation underscores the nonlinear relationship between the computational error and calculation time.This is attributed to the fact that heat transmission delays typically fall within the range of tens of minutes to over an hour.When the calculation time does not exceed 300 s, the computation

Comparison of Fictitious Node Method and Implicit Upwind Method
The scheduling results, obtained through the fictitious node method and the implicit upwind method, are depicted in Figure 9 when the heat storage in the heating network is not taken into account.The time used for solving the scheduling model are listed in Table 3, with each time being calculated 20 times and then averaged.error related to transmission delay remains negligible, resulting in more accurate scheduling outcomes.However, when the calculation time extends to 600 s, the computation error associated with transmission delay becomes non-negligible.As evident in Figure 9b and Table 3, the accuracy of the scheduling results of the fictitious node method with δt2 of 600 s is larger than that of the implicit upwind model with δt1-δx of 150 m-25 s, but the computation time of the implicit upwind model with δt1-δx of 150 m-25 s is much larger than that of the fictitious node method with δt2 of 30 s.This indicates that the computational precision of the implicit upwind method remains inferior to that of the fictitious node method when a greater solution time is required.To attain more precise scheduling outcomes, the adoption of smaller time step and space step sizes becomes essential.However, this will unavoidably result in a further extended solution time for the scheduling model.The scheduling results presented in Figure 9b exhibit significant errors, primarily stemming from the use of a large time step.The implicit upwind method recommends a time step of 20 s and a spatial step of 30 m [15].However, it can be anticipated that selecting a time step and spatial step of 20 s and 30 m, respectively, will result in a scheduling solution time exceeding 15 min, rendering it impractical for the purposes of this paper.This underscores the fact that the implicit upwind method is inappropriate for the optimal scheduling of the electro-thermal integrated energy system.Considering the two scheduling intervals of 1 h and 15 min addressed in this paper, and the resemblance of computational errors at calculation times of 60 s, 180 s, and 300 s, subsequent case studies in this paper opt for a calculation time of 300 s.This decision factors in the time necessary for solving the model.Analyzing Table 3, it becomes evident that, in the case of the fictitious node method, a shorter calculation time corresponds to a quicker solution for the scheduling model.Similarly, in the context of the implicit upwind method, reducing the time step size and space step size leads to decreased time spent on solving the scheduling model.Upon comparing the two methods, it is notable that the fictitious node method exhibits significantly shorter solution times compared to the implicit upwind method.The fictitious node method is capable of solving the scheduling model in just a few tens of seconds, or even a few seconds, making it a suitable choice for optimal scheduling.In contrast, the implicit upwind method demands more time, ranging from a few minutes to even ten minutes.Furthermore, it is foreseeable that utilizing smaller time and space steps could lead to more accurate scheduling results; however, this could also result in the lengthening of the solution time to several tens of minutes.As a result, the implicit upwind method is not for optimal scheduling.When heat storage in the heating network is disregarded, the scheduling results aligning closer with the trend of the heat load demand curve indicate higher accuracy.Because the transmission delay of heat energy is not an integer multiple of the scheduling interval, theoretically, the ideal scheduling result is different from the shape of the heat load-demand curve, but the trend is the same.As evident from Figure 9a, accuracy in scheduling results is highly achieved at a calculation time of 30 s. Smaller calculation times lead to greater scheduling accuracy.Notably, minute disparities exist between scheduling results at calculation times of 60 s, 180 s, and 300 s.However, a substantial scheduling result error emerges at a 600 s calculation time.This observation underscores the non-linear relationship between the computational error and calculation time.This is attributed to the fact that heat transmission delays typically fall within the range of tens of minutes to over an hour.When the calculation time does not exceed 300 s, the computation error related to transmission delay remains negligible, resulting in more accurate scheduling outcomes.However, when the calculation time extends to 600 s, the computation error associated with transmission delay becomes non-negligible.As evident in Figure 9b and Table 3, the accuracy of the scheduling results of the fictitious node method with δt 2 of 600 s is larger than that of the implicit upwind model with δt 1 -δx of 150 m-25 s, but the computation time of the implicit upwind model with δt 1 -δx of 150 m-25 s is much larger than that of the fictitious node method with δt 2 of 30 s.This indicates that the computational precision of the implicit upwind method remains inferior to that of the fictitious node method when a greater solution time is required.To attain more precise scheduling outcomes, the adoption of smaller time step and space step sizes becomes essential.However, this will unavoidably result in a further extended solution time for the scheduling model.The scheduling results presented in Figure 9b exhibit significant errors, primarily stemming from the use of a large time step.The implicit upwind method recommends a time step of 20 s and a spatial step of 30 m [15].However, it can be anticipated that selecting a time step and spatial step of 20 s and 30 m, respectively, will result in a scheduling solution time exceeding 15 min, rendering it impractical for the purposes of this paper.This underscores the fact that the implicit upwind method is inappropriate for the optimal scheduling of the electro-thermal integrated energy system.Considering the two scheduling intervals of 1 h and 15 min addressed in this paper, and the resemblance of computational errors at calculation times of 60 s, 180 s, and 300 s, subsequent case studies in this paper opt for a calculation time of 300 s.This decision factors in the time necessary for solving the model.

Analysis of Scheduling Results for the Day-Ahead Phase
Figure 10 illustrates the electric and thermal power balances in the absence of considering heat storage in the heating network during day-ahead scheduling.The figure reveals that, during the valley time tariff hours of 0:00-5:00 and 22:00-24:00, the power system mainly relies on the external grid and stores electric power during these periods.During the 0:00-4:00 and 23:00-24:00 time periods, the GT operates at its lowest operational limit.However, there is a certain increase in the electric power output from the GT during the 4:00-5:00 and 22:00-23:00 time periods due to the limitations in GT climbing power.If the GT continues to operate at its lower limit during these two time periods, it will impact the economics of the system for the 5:00-6:00 and 22:00-23:00 time periods, thereby affecting the overall economic performance.Meanwhile, the heat system primarily depends on the EB during the time periods of 0:00-4:00 and 23:00-24:00.In contrast, during non-valley time tariff hours, the output of GT is elevated to optimize the system operation economy.However, given that electric load demand surpasses the thermal load demand during non-valley time tariff hours, the GT cannot further increase its output due to the thermal load demand constraints.Consequently, a substantial amount of electricity still needs to be procured from the external grid.As observed in Figure 10b, it becomes evident that the peak period of heat load demand lags behind the highest period of the heat source supply.This points to a significant energy transmission delay within the thermal system, and the transmission delay of the thermal system as a whole is approximately 1 h.Simultaneously, it is noteworthy that the power supplied by the heat source surpasses the power required by the heat load.This observation underlines a noteworthy energy loss occurring within the heat system during the process of heat energy transmission, amounting to approximately 9.3%.The output of the GB consistently operates at its lower operational limit.This is primarily because the EB is the more cost-effective option during valley time tariff hours, while the GT proves to be more economical during non-valley time tariff hours.

Analysis of Scheduling Results for the Day-Ahead Phase
Figure 10 illustrates the electric and thermal power balances in the absence of considering heat storage in the heating network during day-ahead scheduling.The figure reveals that, during the valley time tariff hours of 0:00-5:00 and 22:00-24:00, the power system mainly relies on the external grid and stores electric power during these periods.During the 0:00-4:00 and 23:00-24:00 time periods, the GT operates at its lowest operational limit.However, there is a certain increase in the electric power output from the GT during the 4:00-5:00 and 22:00-23:00 time periods due to the limitations in GT climbing power.If the GT continues to operate at its lower limit during these two time periods, it will impact the economics of the system for the 5:00-6:00 and 22:00-23:00 time periods, thereby affecting the overall economic performance.Meanwhile, the heat system primarily depends on the EB during the time periods of 0:00-4:00 and 23:00-24:00.In contrast, during non-valley time tariff hours, the output of GT is elevated to optimize the system operation economy.However, given that electric load demand surpasses the thermal load demand during non-valley time tariff hours, the GT cannot further increase its output due to the thermal load demand constraints.Consequently, a substantial amount of electricity still needs to be procured from the external grid.As observed in Figure 10b, it becomes evident that the peak period of heat load demand lags behind the highest period of the heat source supply.This points to a significant energy transmission delay within the thermal system, and the transmission delay of the thermal system as a whole is approximately 1 h.Simultaneously, it is noteworthy that the power supplied by the heat source surpasses the power required by the heat load.This observation underlines a noteworthy energy loss occurring within the heat system during the process of heat energy transmission, amounting to approximately 9.3%.The output of the GB consistently operates at its lower operational limit.This is primarily because the EB is the more cost-effective option during valley time tariff hours, while the GT proves to be more economical during non-valley time tariff hours.Figure 11 presents the outcomes of the system's optimal scheduling when accounting for heat storage within the heating network, while Figure 12 displays the corresponding heat storage in the network.From Figure 11, it becomes evident that during non-valley time tariff hours, the presence of heat storage within the heating network has a profound impact.The output of GT is no longer tightly constrained by the heat load demand.This shift allows the GT to operate in a decoupled electric-heat manner, resulting in a significant increase in the output of GT compared to scenarios where heat storage in the heating Figure 11 presents the outcomes of the system's optimal scheduling when accounting for heat storage within the heating network, while Figure 12 displays the corresponding heat storage in the network.From Figure 11, it becomes evident that during non-valley time tariff hours, the presence of heat storage within the heating network has a profound impact.The output of GT is no longer tightly constrained by the heat load demand.This shift allows the GT to operate in a decoupled electric-heat manner, resulting in a significant increase in the output of GT compared to scenarios where heat storage in the heating network is not taken into consideration.Notably, this consideration of heat storage leads to a substantial 5.9% enhancement in the system's overall efficiency.Nevertheless, during this time period, it is still necessary to procure a specific amount of power from the external grid.This requirement stems from the constraints imposed by the limited heat storage capacity within the heating network.From Figures 11b and 12, it is evident that the heating network releases heat during the valley time tariff period and stores heat during the non-valley time tariff period.The rationale behind the heating network's heat storage during the non-valley time tariff period is the significantly higher cost associated with purchasing electricity from the external grid at this time.This cost disparity outweighs the savings achieved by utilizing the EB for heat supply during the valley time tariff period.Correspondingly, the output of the EB decreases during the valley time tariff period.
nal grid.This requirement stems from the constraints imposed by the limited heat storage capacity within the heating network.From Figures 11b and 12, it is evident that the heating network releases heat during the valley time tariff period and stores heat during the non-valley time tariff period.The rationale behind the heating network's heat storage during the non-valley time tariff period is the significantly higher cost associated with purchasing electricity from the external grid at this time.This cost disparity outweighs the savings achieved by utilizing the EB for heat supply during the valley time tariff period.Correspondingly, the output of the EB decreases during the valley time tariff period.
As depicted in Figure 12, it is evident that only utilizing the water supply network to quantify heat storage within the heating network results in limitations.During the 19:00-20:00 period, the heating network begins to release heat.However, this time period also coincides with peak tariff hours, causing the GT output to be higher, and the heating network should be maintained in a state of heat storage.It is apparent that this method of calculating the heat storage quantity deviates significantly from the actual scenario.The same issue persists when quantifying heat storage within the heating network solely based on the return pipe network.During the 6:00-7:00 time period, which falls within the non-peak tariff hours, the heating network should refrain from continuing heat release, even when accounting for heat transmission delays.Furthermore, it is essential to acknowledge that both quantification methods have not adequately considered the impact of transmission delays within the heating system.Therefore, it is necessary to quantify the heat storage from the whole heating network.

Analysis of Scheduling Results for the Intra-Day Phase
Figure 13 presents a comparison of equipment outputs during different scheduling phases considering heat storage in the heating network.First of all, it should be pointed out that in intra-day upper-level scheduling, the scheduling of thermal energy solely pertains to the portion of the heat load demand change.Due to the existence of heat storage in the heating network, the transmission delay effect of this portion of thermal energy is not taken into account.The fluctuation of the heat system source and load occurs simultaneously, aiming for the simpler coordinated scheduling of the electric-heat system.The As depicted in Figure 12, it is evident that only utilizing the water supply network to quantify heat storage within the heating network results in limitations.During the 19:00-20:00 period, the heating network begins to release heat.However, this time period also coincides with peak tariff hours, causing the GT output to be higher, and the heating network should be maintained in a state of heat storage.It is apparent that this method of calculating the heat storage quantity deviates significantly from the actual scenario.The same issue persists when quantifying heat storage within the heating network solely based on the return pipe network.During the 6:00-7:00 time period, which falls within the nonpeak tariff hours, the heating network should refrain from continuing heat release, even when accounting for heat transmission delays.Furthermore, it is essential to acknowledge that both quantification methods have not adequately considered the impact of transmission delays within the heating system.Therefore, it is necessary to quantify the heat storage from the whole heating network.

Analysis of Scheduling Results for the Intra-Day Phase
Figure 13 presents a comparison of equipment outputs during different scheduling phases considering heat storage in the heating network.First of all, it should be pointed out that in intra-day upper-level scheduling, the scheduling of thermal energy solely pertains to the portion of the heat load demand change.Due to the existence of heat storage in the heating network, the transmission delay effect of this portion of thermal energy is not taken into account.The fluctuation of the heat system source and load occurs simultaneously, aiming for the simpler coordinated scheduling of the electric-heat system.The fluctuation of the heat system caused by transmission delay is mitigated by the heat storage in the heating network.For instance, if the heat load demand increases by 0.4 MW during the 0:00-1:00 time period, then the heat source output increases by the same 0.4 MW during this period, regardless of the transmission delay of this 0.4 MW of heat energy.If the heat storage role of the heating network is not taken into account, the transmission delay of heat energy will inevitably lead to difficulties in coordinating the dispatch of electricity and heat in intra-day scheduling.the GT output.Lastly, during the 16:00-18:00 period, when the net power of sources and loads in the power system decreases but the demand for heat loads increases, the most cost-effective approach is to increase the GT output and reduce the power purchased from the grid during the peak tariff period.In the case of intra-day lower-level dispatch, only the EB is employed to suppress the power system source and load fluctuations, as depicted in Figure 13c.Furthermore, for handling the thermal system fluctuations caused by the EB, the GB comes into play.When comparing Figure 13c,d, it becomes apparent that the fluctuation magnitude of GB is smaller than that of EB.This phenomenon arises from the heat storage capacity inherent in the heating network, enabling the mitigation of fluctuations in the thermal system generated by the EB.These fluctuations closely mirror the variations in electric power.Consequently, the GB can be delayed and employed to offset the thermal system's fluctuations with a smaller adjustment in power.For instance, during the 6:00-7:00 period, the GB is harnessed to compensate for the fluctuations generated by the EB during the 5:00-6:00 period, where the maximum power change of the EB is approximately 1 MW, while the GB experiences a reduction of around 0.3 MW.Figures 14 and 15 depict the heat storage within the heating network during intraday scheduling and its alterations relative to the preceding scheduling stage.In the context of intra-day upper-level scheduling, the dispatched thermal energy is only part of the heat load demand change, while disregarding the transmission delay associated with this thermal energy portion.When the heat system's load demand increases, and despite a Comparing the power output of the equipment between the day-ahead scheduling phase and the intra-day upper-level scheduling phase reveals specific trends.During the 0:00-1:00 period, there is an increase in heat load demand, while net fluctuations in power system sources and loads remain minimal.This leads to an augmentation of purchased power from the external grid and an increase in the output of the EB to satisfy the heat load demand, coinciding with the valley time tariff.This pattern also holds during the 3:00-5:00 period.During the 1:00-3:00 period, the reduction in heat load demand exceeds the net reduction in power system sources and loads, resulting in a decrease in EB output and a minor reduction in power purchased from the external grid.During the periods of 5:00-6:00, 11:00-14:00, and 21:00-24:00, the net power increase in power system sources and loads surpasses the increase in heat load demand.These periods fall outside the valley time tariff, and during such times, heat loads and a segment of electricity demand are primarily met by elevating the GT output.The remaining electricity demand is met by increasing the power purchased from the external grid.The 23:00-24:00 period falls under the valley time tariff, where the surge in electricity and heat demand is addressed by elevating power purchased from the external grid.However, during the 22:00-23:00 period, the increase in heat load demand is fulfilled by using GT instead of EB, primarily due to GT being restricted by ramping power limitations.Conversely, during the time periods of 6:00-8:00, 9:00-10:00, and 18:00-19:00, the reduction in net power of power system sources and loads surpasses the reduction in heat load demand.In these cases, the reduction in GT output is coupled with a decrease in power purchased from the external grid.In the time slots of 8:00-9:00, 10:00-11:00, and 14:00-16:00, fluctuations in electricity and heat demand are synchronized, allowing for the smoothing of fluctuations by merely adjusting the GT output.Lastly, during the 16:00-18:00 period, when the net power of sources and loads in the power system decreases but the demand for heat loads increases, the most cost-effective approach is to increase the GT output and reduce the power purchased from the grid during the peak tariff period.In the case of intra-day lower-level dispatch, only the EB is employed to suppress the power system source and load fluctuations, as depicted in Figure 13c.Furthermore, for handling the thermal system fluctuations caused by the EB, the GB comes into play.When comparing Figure 13c,d, it becomes apparent that the fluctuation magnitude of GB is smaller than that of EB.This phenomenon arises from the heat storage capacity inherent in the heating network, enabling the mitigation of fluctuations in the thermal system generated by the EB.These fluctuations closely mirror the variations in electric power.Consequently, the GB can be delayed and employed to offset the thermal system's fluctuations with a smaller adjustment in power.For instance, during the 6:00-7:00 period, the GB is harnessed to compensate for the fluctuations generated by the EB during the 5:00-6:00 period, where the maximum power change of the EB is approximately 1 MW, while the GB experiences a reduction of around 0.3 MW.
Figures 14 and 15 depict the heat storage within the heating network during intra-day scheduling and its alterations relative to the preceding scheduling stage.In the context of intra-day upper-level scheduling, the dispatched thermal energy is only part of the heat load demand change, while disregarding the transmission delay associated with this thermal energy portion.When the heat system's load demand increases, and despite a concurrent increase in heat source output, the presence of transmission delay hampers the timely delivery of this augmented heat output to heat users.Consequently, the increased heat demand is met by the heating network's storage, and the heightened heat source output is utilized to offset the decline in the network heat storage.Given the potential for a substantial or continuous increase in heat load demand during the intra-day compared to the day-ahead prediction, it becomes essential to incorporate a margin in the heat storage of the heating network during day-ahead scheduling.This precaution ensures that the quantity of stored heat during intra-day scheduling remains above zero, even as it undergoes reduction, as depicted in Figure 14b.Given that the overall increase in heat load demand surpasses the decrease, there is indeed a decrease in the heat storage within the heating network.Similarly, this principle applies to intra-day lower-level scheduling due to the delayed scheduling of the GB relative to the EB, as evident in Figure 15b.Because the increase in output by EB is generally smaller than the reduction, an overall reduction occurs in the heat storage within the heating network.This reduction primarily results from the delay in GB compensation.
a substantial or continuous increase in heat load demand during the intra-day compared to the day-ahead prediction, it becomes essential to incorporate a margin in the heat storage of the heating network during day-ahead scheduling.This precaution ensures that the quantity of stored heat during intra-day scheduling remains above zero, even as it undergoes reduction, as depicted in Figure 14b.Given that the overall increase in heat load demand surpasses the decrease, there is indeed a decrease in the heat storage within the heating network.Similarly, this principle applies to intra-day lower-level scheduling due to the delayed scheduling of the GB relative to the EB, as evident in Figure 15b.Because the increase in output by EB is generally smaller than the reduction, an overall reduction occurs in the heat storage within the heating network.This reduction primarily results from the delay in GB compensation.a substantial or continuous increase in heat load demand during the intra-day compared to the day-ahead prediction, it becomes essential to incorporate a margin in the heat storage of the heating network during day-ahead scheduling.This precaution ensures that the quantity of stored heat during intra-day scheduling remains above zero, even as it undergoes reduction, as depicted in Figure 14b.Given that the overall increase in heat load demand surpasses the decrease, there is indeed a decrease in the heat storage within the heating network.Similarly, this principle applies to intra-day lower-level scheduling due to the delayed scheduling of the GB relative to the EB, as evident in Figure 15b.Because the increase in output by EB is generally smaller than the reduction, an overall reduction occurs in the heat storage within the heating network.This reduction primarily results from the delay in GB compensation.

Further Discussion on Heat Storage in the Heating Network
The error in intra-day wind power prediction data typically exceeds 10% but only during some periods [38], with the maximum error reported at approximately 17% in reference [37].To delve further into the dampening effect of the heat storage capacity within the heating network on wind power fluctuations, we present the wind power output range obtained by adjusting the wind power output up or down by 20%, while accounting for equipment output constraints, as depicted in Figure 16a.This range is established using the intra-day upper-level wind power forecast data as a reference point.Notably, it becomes evident that the intra-day lower-level wind power forecast data fall within this established range.The upper and lower boundaries of this range are utilized as the intra-day lower-level wind power prediction data for calculating heat storage within the pipe network during the intra-day lower-level dispatch, as illustrated in Figure 16b.When the wind power data is the upper limit of the range, the dispatch process involves an initial increase in EB output to mitigate the surge in wind power output.Subsequently, the GB implements a delayed reduction in output to smooth out the thermal fluctuations induced by the EB.This approach leads to an increase in heat storage within the heating network.Conversely, when the wind power data align with the lower limit of the range, the heat storage of the heating network diminishes but remains above zero.This indicates that, in such scenarios, the heating network can effectively mitigate wind power fluctuations.
obtained by adjusting the wind power output up or down by 20%, while accounting for equipment output constraints, as depicted in Figure 16a.This range is established using the intra-day upper-level wind power forecast data as a reference point.Notably, it becomes evident that the intra-day lower-level wind power forecast data fall within this established range.The upper and lower boundaries of this range are utilized as the intraday lower-level wind power prediction data for calculating heat storage within the pipe network during the intra-day lower-level dispatch, as illustrated in Figure 16b.When the wind power data is the upper limit of the range, the dispatch process involves an initial increase in EB output to mitigate the surge in wind power output.Subsequently, the GB implements a delayed reduction in output to smooth out the thermal fluctuations induced by the EB.This approach leads to an increase in heat storage within the heating network.Conversely, when the wind power data align with the lower limit of the range, the heat storage of the heating network diminishes but remains above zero.This indicates that, in such scenarios, the heating network can effectively mitigate wind power fluctuations.The heat storage capability of the heating network, as explored in the aforementioned studies, aims to redistribute the heat energy emitted by the GT during peak-time tariff hours.This surplus heat energy is captured and utilized during other hours, effectively reducing the necessity to procure significant power from the external grid during peaktime tariff hours.This strategy contributes to enhancing the overall economic efficiency of the system's operation.The utilization of heat storage within the heating network can take various forms.For example, when the heat load demand surpasses electric load demand during peak tariff hours, limitations imposed by the electric power system constrain the GT's output.To address this, the EB output can be augmented during valley tariff hours, with excess heat energy stored within the network for subsequent release during peak tariff hours.Moreover, heat storage within the heating network has the potential to enhance the local consumption of renewable energy.Energy generated from renewable sources can be converted into heat energy and stored within the network during periods of high new energy output [17].Furthermore, heat storage within the network proves The heat storage capability of the heating network, as explored in the aforementioned studies, aims to redistribute the heat energy emitted by the GT during peak-time tariff hours.This surplus heat energy is captured and utilized during other hours, effectively reducing the necessity to procure significant power from the external grid during peak-time tariff hours.This strategy contributes to enhancing the overall economic efficiency of the system's operation.The utilization of heat storage within the heating network can take various forms.For example, when the heat load demand surpasses electric load demand during peak tariff hours, limitations imposed by the electric power system constrain the GT's output.To address this, the EB output can be augmented during valley tariff hours, with excess heat energy stored within the network for subsequent release during peak tariff hours.Moreover, heat storage within the heating network has the potential to enhance the local consumption of renewable energy.Energy generated from renewable sources can be converted into heat energy and stored within the network during periods of high new energy output [17].Furthermore, heat storage within the network proves valuable in decoupling thermo-electric coupling equipment, as exemplified by the GT studied in this research.In addition, the efficacy of the heat storage capacity within the heating network in alleviating the detrimental impacts of wind power volatility on system operations has been demonstrated.This effectiveness arises from the ability to transform wind power output fluctuations into thermal fluctuations through the use of faster-responding equipment, such as electric boilers or heat pumps.Thermal fluctuations, within a short timeframe, exert minimal influence on the heat system.Consequently, smoothing out the fluctuations in the heat system can be achieved using equipment with a longer response time.

Conclusions
In this paper, a multi-time-scale scheduling model for an integrated energy system is proposed, considering the heat storage in the heating network.The fictitious node method is used to calculate the quasi-dynamic process of the heat system and is compared with the implicit upwind method.Simulation experiments verify that the fictitious node method offers advantages in both computational accuracy and complexity.It is also confirmed that the implicit upwind method is not applicable to the optimal scheduling of the system.Considering the characteristics of the fictitious node method, the heat storage capacity of the heating network is quantified from the entire network.During the day-ahead scheduling phase, the heat storage in the heating network can be leveraged to improve the system's operational economy.In the intra-day scheduling phase, the heat storage in the heating network can be utilized to suppress the system's sources and load fluctuations, enhancing the operational stability and the system's ability to accommodate new energy sources.Due to wind power's fluctuating and intermittent characteristics, accurately predicting its output is challenging, both a day or even a few hours in advance.This unpredictability can diminish the system's ability to effectively harness wind power.Nevertheless, optimizing the utilization of heat storage capacity within the heating network can alleviate the wind abandonment phenomenon resulting from wind power's uncertainty, improving the integration and consumption of wind power, and contributing to the sustainability of energy development.
In future research, the intention is to delve into the role of heat storage within the pipeline network across various heating system operation modes.Additionally, there are plans to investigate the intricate interplay between heat storage in the pipeline network and other influential factors affecting the system's operation.The detailed modeling of equipment and systems is identified as a critical and intriguing area of study that warrants thorough exploration.Moreover, the role of heat storage within the heating network in the context of more intricately interconnected energy systems will also be included in the research agenda.

Figure 1 .
Figure 1.Structure of the heating network.

Figure 1 .
Figure 1.Structure of the heating network.

Sustainability 2023 ,
15, x FOR PEER REVIEW 7 of 26 the square node; τf and τ ' f are the transmission delays before and after the length change, respectively, for each pipe following the node.

Figure 2 .
Figure 2. First step of the fictitious node method: transformation of the length of pipes.

Figure 3 .
Figure 3. Second step of the fictitious node method: setup of fictitious nodes and calculation of heat loss.

Figure 2 .
Figure 2. First step of the fictitious node method: transformation of the length of pipes.

Sustainability 2023 ,
15, x FOR PEER REVIEW 7 of 26 the square node; τf and τ ' f are the transmission delays before and after the length change, respectively, for each pipe following the node.

Figure 2 .
Figure 2. First step of the fictitious node method: transformation of the length of pipes.

Figure 3 .
Figure 3. Second step of the fictitious node method: setup of fictitious nodes and calculation of heat loss.

Figure 3 .
Figure 3. Second step of the fictitious node method: setup of fictitious nodes and calculation of heat loss.

Figure 4 .
Figure 4. Structure of the system.

Figure 4 .
Figure 4. Structure of the system.

Figure 7 .
Figure 7. (a) The multi-time-scale projections of electric load demand; (b) The multi-time-scale projections of thermal load demand; (c) The multi-time-scale projections of wind power output; (d) The unit price of electricity bought from the external grid.

Figure 7 .
Figure 7. (a) The multi-time-scale projections of electric load demand; (b) The multi-time-scale projections of thermal load demand; (c) The multi-time-scale projections of wind power output; (d) The unit price of electricity bought from the external grid.

Figure 8 .
Figure 8. Flowchart of solving the model.

Figure 8 .
Figure 8. Flowchart of solving the model.

Figure 9 .
Figure 9. (a) Scheduling results of the fictitious node method (FNM); and (b) Scheduling results of the implicit upwind method.

Figure 9 .
Figure 9. (a) Scheduling results of the fictitious node method (FNM); and (b) Scheduling results of the implicit upwind method.

Figure 10 .
Figure 10.Day-ahead scheduling results when heat storage in the heating network is not considered.(a) Electric power balance; (b) Thermal power balance.

Figure 10 .
Figure 10.Day-ahead scheduling results when heat storage in the heating network is not considered.(a) Electric power balance; (b) Thermal power balance.

Figure 11 .
Figure 11.Day-ahead scheduling results considering heat storage in the heating network.(a) Electric power balance; (b) Thermal power balance.Figure 11.Day-ahead scheduling results considering heat storage in the heating network.(a) Electric power balance; (b) Thermal power balance.

Figure 11 . 26 Figure 12 .
Figure 11.Day-ahead scheduling results considering heat storage in the heating network.(a) Electric power balance; (b) Thermal power balance.Figure 11.Day-ahead scheduling results considering heat storage in the heating network.(a) Electric power balance; (b) Thermal power balance.Sustainability 2023, 15, x FOR PEER REVIEW 20 of 26

Figure 12 .
Figure 12.Heat storage in the heating network during the day-ahead scheduling phase.

Figure 14 .Figure 15 .
Figure 14.(a) Heat storage in the heating network during the intra-day upper-level scheduling phase; (b) Change in heat stored in the heating network during the intra-day upper-level scheduling phase relative to the day-ahead scheduling phase.

Figure 14 .
Figure 14.(a) Heat storage in the heating network during the intra-day upper-level scheduling phase; (b) Change in heat stored in the heating network during the intra-day upper-level scheduling phase relative to the day-ahead scheduling phase.

Figure 14 .Figure 15 .
Figure 14.(a) Heat storage in the heating network during the intra-day upper-level scheduling phase; (b) Change in heat stored in the heating network during the intra-day upper-level scheduling phase relative to the day-ahead scheduling phase.

Figure 15 .
Figure 15.(a) Heat storage in the heating network during the intra-day lower-level scheduling phase; (b) Change in heat stored in the heating network during the intra-day lower-level scheduling phase relative to the intra-day upper-level scheduling phase.

Figure 16 .
Figure 16.(a) Wind power output intervals obtained by adding or subtracting 20% to the intra-day upper-level wind power forecast data, taking into account equipment output constraints; (b) The heat storage in the heating network calculated using the black and red curves in (a) as the intraday lower-level wind power forecast data.

Figure 16 .
Figure 16.(a) Wind power output intervals obtained by adding or subtracting 20% to the intra-day upper-level wind power forecast data, taking into account equipment output constraints; (b) The heat storage in the heating network calculated using the black and red curves in (a) as the intraday lower-level wind power forecast data.

Funding:
This research was supported by Natural Science Foundation of Jiangsu Province (BK20210932) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX23_1202).Data Availability Statement: Not applicable. ): es is the self-discharge rate of ES; α t c and α t d are the charging and discharging states of ES at time t, respectively; η c and η d are, respectively, the charging and discharging efficiency of ES; S start es and S end es are the electricity storage capacity of ES at the beginning and end of the day's scheduling, respectively.
es is the electricity storage capacity of ES at time t; S max es is the maximum storage capacity of ES; P t esc and P t esd are the charging and discharging powers of ES at time t, respectively; P max esc and P max esd are the maximum charging and discharging power of ES at time t, respectively; σ

Table 1 .
Parameters of the heating network.

Table 1 .
Parameters of the heating network.

Table 2 .
Parameters of the system scheduling operation.

Table 2 .
Parameters of the system scheduling operation.

Table 3 .
The time used to solve the scheduling model using a fictitious node method and implicit upwind method.Fictitious Node MethodImplicit Upwind Method δt2(s) Time(s) δt1-δx(s-m) Time(s)

Table 3 .
The time used to solve the scheduling model using a fictitious node method and implicit upwind method.