1. Introduction
According to
GWEC’s Global Wind Report 2023, the global cumulative installed wind power capacity reached 906 gigawatts in 2022, with an additional 77.6 gigawatts of new capacity added worldwide [
1]. The rapid advancement of wind power and other emerging energy sources has generated a pressing need to accelerate the integration and consumption of these new energies. In the northern region of China, a substantial number of cogeneration facilities are in operation to fulfill the demand for winter heating. These cogeneration systems operate under the “Ordering Power by Heat” mode, which establishes a strong linkage between power generation and heating output. However, this approach results in a higher minimum power generation output to meet the system’s heating demands, leading to insufficient system peaking capacity. This scenario poses challenges for the effective consumption of new energy sources. Therefore, there is immense significance to be found in exploring the coordinated scheduling of heat and power, alongside the integration of renewable energy sources like wind power.
The electricity and heat integrated energy system is a complex network comprising two heterogeneous energy sources that are capable of achieving both economical and complementary energy utilization while efficiently reducing environmental pollution and carbon emissions [
2,
3,
4,
5,
6]. Collaborative planning and operation of these energy flows can enhance the safety and stability of the system [
7]. Meanwhile, the thermal energy system can provide significant flexibility to the operation of the power system and promote the consumption of renewable energy [
8,
9].
Liu et al. [
7] investigated the steady-state operation scenario of electricity and heat networks as an integrated whole. The study employed the hydraulic–thermal model of heat networks and the electrical power flow model, utilizing the Newton–Raphson method as the solution approach. However, it did not consider the disparity in transmission characteristics between electric and thermal energy. Specifically, the electric power system exhibits remarkable transmission speeds and swift response times, whereas the thermal energy system demonstrates considerable inertia and significant transmission delays. In [
6], a quasi-steady multi-energy flow model was proposed, considering the time-scale characteristics of the interactions between electricity systems and heating systems. Additionally, a heating network node-type transformation technique was developed to study the intermediate processes during the transition between steady states. Building upon this work, Pan et al. [
10] quantified the quasi-dynamic interactions between electricity systems and heating systems with a focus on the transmission delay. However, it should be noted that the methods proposed in both [
6,
10] are inapplicable within the context of optimal scheduling.
Li et al. [
11] incorporated the consideration of transmission delay into the optimal scheduling of the integrated system. In [
12], the transmission delay was added to the time when the source started changing the supply temperature. Xu et al. [
13] decomposed the district heating system into multiple equivalent subsystems with a single-producer–single-consumer structure for calculations. However, a significant limitation of these models is that they assume the transmission delay of the thermal system to be an integer multiple of the scheduling interval. Meeting this requirement is extremely difficult in practical systems [
14], making it challenging to achieve effective coordination in scheduling between the heat and power systems. To address this issue, Wu et al. [
15] proposed dividing heat users into multiple heat load zones, ensuring that the transmission delay of each zone aligns with an integer multiple of the scheduling time interval. Additionally, Gu et al. [
16,
17] approximated the transmission delay of the heat network by rounding it off to the nearest integer multiple of the scheduling time interval. Similarly, Wu et al. [
18] rounded down the transmission delay. It is worth noting that the accuracy of calculations using these methods and the choice of the scheduling time interval are closely interconnected, often resulting in significant errors in the resulting calculations. In order to enhance computational accuracy, researchers have employed various methods such as the characteristic line model [
19], the implicit upwind model [
20], and the orthogonal collocation and finite difference method [
21] to solve the partial differential equations describing the dynamic properties of pipes. However, Lu et al. [
22] noted that the accuracy of these methods relies on the time step and space step size. Achieving a balance between computational accuracy and computational burden has proven challenging, posing difficulties when applying these methods to optimal scheduling. To address this problem, Lu et al. [
22] proposed the utilization of the node method for optimal scheduling calculations, along with employing an accurate thermodynamic model to verify the obtained results.
The node method, as elaborated in [
23], offers precise calculations of heat network transmission delays. This method has been widely applied in the optimal scheduling of integrated energy systems [
24,
25,
26,
27]. However, when considering both water supply network and return network delays, the node method imposes a significant computational burden. As a solution, Chen et al. [
28] introduced a water mass method that employs 0–1 variables to identify the water mass delay. This approach reduces the computational complexity of the node method, albeit with a minor sacrifice in computational accuracy. Furthermore, in [
29], a hydraulic–thermal cooperative optimization model based on the water mass method was proposed that takes into account the influence of hydraulics on the operation of the system.
In addition, Lu et al. [
30] utilized an equivalent start network to simulate the radial district heating network. This approach gradually simplifies the network topology, starting from the load point and extending to the heat source. It results in a network that directly connects each secondary heat exchange station to the primary heat exchange station. However, the simplification of the internal states of the district heating network in the thermal inertia aggregation model introduces unavoidable errors. The experiments conducted in [
30] demonstrated that these errors can be significant during specific periods. Hao et al. [
31,
32,
33,
34] employed the heat current method to model the district heating network. They applied Ohm’s law and Kirchhoff’s law to deduce the corresponding heat transport matrix and proposed a basic thermal–electric analogy circuit for each fluid element. This method represents the district heating network as an electric power network for calculation. However, it is important to note that this method may result in the loss of some valuable information during the calculation process, such as the temperature distribution along the pipe.
The analysis of the references above underscores the significance of considering the transmission process of district heating networks in the context of optimal scheduling for integrated energy systems. It is essential to develop a method that is well-suited for optimal scheduling and takes into consideration both computational accuracy and computational burden while ensuring the retention of valuable information throughout the computational process.
Combined heat and power (CHP) units and gas turbine units are commonly employed as heat sources in district heating systems. During the heating season, they often operate in a mode known as “Ordering Power by Heat,” which leads to the generation of a certain amount of forced electrical power. This forced electrical power can impose limitations on the utilization of renewable energy sources, such as wind power. To enhance the utilization of new energy sources and improve the operational flexibility of the system, several studies have explored different approaches. Nuytten et al. [
35] focused on configuring heat storage tanks for CHP units, while Wu et al. [
15,
36] investigated the integration of heat storage tanks and electric boilers to improve the flexibility of CHP units. The impact of thermal inertia in buildings on system flexibility was analyzed in [
12,
16]. To further examine the effects on system flexibility, Liu et al. [
25,
37] simultaneously considered the use of heat storage tanks, electric boilers, thermal inertia in buildings, and the variability of heat demands. These studies investigated the individual and combined effects of these factors on system flexibility. Additionally, the influence of heat transfer constraints of heat storage equipment on system flexibility was studied in [
38,
39]. A district heating network reconfiguration was explored in [
40] as a means to manage congestion and enhance the consumption of new energy sources. Ma et al. [
41,
42] discussed the conversion of electrical power into hydrogen or natural gas, which can be stored more easily, with the aim of enhancing the capacity for new energy consumption. The analysis of the aforementioned references reveals that the inclusion of energy storage equipment or the conversion of power into heat or gas, which can be more readily stored, can enhance the capacity for new energy consumption and improve the operational flexibility of the system.
Based on the aforementioned literature analysis, it is evident that scholars have presented several solutions with varying degrees of effectiveness to address the challenge of achieving coordinated dispatch of electricity and heat. However, persistent deficiencies are observed in achieving the balance between computation time and computation accuracy, as well as in effectively preserving pertinent information during the computational process. Furthermore, ongoing studies aim at enhancing the flexibility of integrated energy systems that center around energy storage configuration, the incorporation of thermal inertia within buildings, and the conversion of electricity to gas. However, these studies often overlook the significant influence of heat storage within the heating network on overall system flexibility. To address these issues, the fictitious node method is proposed in this paper as a means to calculate the quasi-dynamic model of the heating network. Unlike traditional approaches, this method offers a specific calculation time. Notably, the computational accuracy of this method is not dependent on the scheduling interval but rather closely associated with the calculation time. By employing this method, a balance between computation accuracy and complexity is struck, while also retaining essential information, such as the temperature distribution along the pipeline. Additionally, based on the advantages of the fictitious node method, a methodology for quantifying the heat storage capacity of the heating network is proposed. Moreover, the heat storage capacity within the heating network was taken into full consideration in this study. A portion of the forced electrical power generated by CHP units and gas turbine units was converted into thermal energy through electric boilers. This thermal energy was then stored using the heat storage capacity of the heating network, thereby improving the flexibility of system operation and enhancing the local consumption capacity of wind power.
2. Quasi-Dynamic Mode of the Heating Network
The heating system usually operates in two regulation modes: quality regulation mode and quantity regulation mode [
43]. The quality regulation mode refers to maintaining a constant mass flow while varying the temperature supply [
17]. On the other hand, the quantity regulation mode refers to adjusting the mass flow while keeping the temperature supply constant [
43]. Due to its stable hydraulic properties, the quality regulation mode is more commonly used in practical engineering [
11,
16]. Therefore, this study was based on the quality regulation mode.
Several assumptions were made for the heating network [
44]:
- (1)
Water in the pipe network was considered an incompressible fluid.
- (2)
Friction heat was neglected.
- (3)
The thermal properties of the water were assumed to be constant.
- (4)
Turbulence effects were ignored, and the water flow in the pipeline was assumed to be stable.
- (5)
The longitudinal temperature distribution of the water flow was disregarded, and only the axial direction of the water flow temperature was considered.
The centralized heat supply system consists of the heat source, primary heat exchange station, primary network, secondary heat exchange stations, secondary network, and heat users. This study specifically focused on the primary network, which serves as a long-distance transport network characterized by significant transmission delays and dynamic properties. In contrast, the secondary network functions as a distribution network with shorter transmission delays and less pronounced dynamic characteristics [
10]. Consequently, the secondary network is not addressed in this paper. The thermal flow model of the primary network is depicted in
Figure 1. The return water passes through the primary heat exchange station, where it is heated by the heat source. Subsequently, the water, now at an elevated temperature, enters the water supply network and is distributed to each secondary heat exchange station. Finally, after undergoing heat exchange, the water flows back into the water return network. Throughout the transmission process, the water experiences transmission delays and incurs losses.
2.1. The Fictitious Node Method
The transmission delay of the pipe can be calculated by using (1):
where
τ is the transmission delay,
ρ is the density of water,
d is the inner diameter of the pipe,
L is the length of the pipe, and
m is the mass flow rate of the water.
The thermal loss, referred to as a temperature loss, experienced by water following its passage through a pipeline can be calculated using Equation (2) [
8,
14]:
where
T0,
Tin, and
Tx represent the ambient temperature surrounding the pipe, the initial temperature of the hot water at the entry point of the pipe, and the temperature of the hot water at a distance
x from the entry point of the pipe, respectively;
α is the total heat transfer coefficient between the interior of the pipe and its surroundings; and
cw is the specific heat capacity of water.
Given that pipelines are typically buried underground, the immediate surrounding medium is soil. The standardization of burial depth for directly buried heat pipes dictates a minimum depth of 0.7 m [
45]. Furthermore, at depths greater than 0.4 m, diurnal fluctuations in soil temperature display negligible variation [
46]. Thus, treating
T0 as a constant within this context is warranted [
14].
The fictitious node method is presented below as an example of a six-node heating network, as illustrated in
Figure 2. Point A is connected to the primary heat exchange station, while points D, E, and F are connected to their respective secondary heat exchange stations. This network exemplifies a typical branch heating network, featuring the pipe that connects to the primary heat exchange station, pipes that connect to the secondary heat exchange stations, and intermediate pipes.
To ensure accurate calculations, it is important to establish a suitable calculation time, denoted as δ
t, that is a factor of the scheduling interval Δ
t. Transform the transmission delay of each pipe in the entire heating network using (3).
where round[·] indicates the process of rounding the values within the square brackets,
τ represents the original transmission delay of a pipeline,
τ″ represents the intermediate calculation of the pipeline transmission delay,
τ′ represents the transformed transmission delay of a pipe, and the subscript denotes the respective pipe number.
The transformation method for the transmission delay of pipe
j can be summarized using Equation (4):
where
τj and
τj,p are the original transmission delays of pipe
j and the pipe before pipe
j, respectively;
and
are the transformed transmission delays of pipe
j and the pipe before pipe
j, respectively; and
is an intermediate variable of pipe
j in the computation of
. It is important to note that when pipe j is connected to a primary heat exchange station, such as pipe 1 in
Figure 2, it implies that there is no previous pipe for this pipe and that
= τj.
The transformed length of each pipe can be calculated by using (5):
where the subscript
j denotes the number of the respective pipe and
is the transformed length of the pipe numbered
j.
Up to this point, it has been ensured that the transmission delay of each pipe is a multiple of the calculation time δ
t. Within the entire heating network, the primary and secondary heat exchange stations, as well as the pipeline connections, are positioned at a hypothetical node. Consequently, the calculation of the transmission delay for the entire network and the temperature distribution along the pipeline becomes straightforward.
Figure 2 illustrates this concept, using pipeline 1 as an example. Assuming that the transformed length of the pipe results in a transmission delay equivalent to M-1 times the calculation time, M fictitious nodes are established along pipe 1, and these fictitious nodes encompass both ends of the pipe.
The distance
lj between two neighboring fictitious nodes on pipe
j can be calculated by using (6):
The temperature distribution along the water supply and return pipes denoted by subscript
j can be calculated by using (7) and (8):
where
is the temperature of the water supply pipe numbered
j at calculation time numbered
i and fictitious node numbered
n and
is the temperature of return pipe
j at calculation time
i and fictitious node
n + 1.
2.2. Heat Flow Model
For a node connected by different pipes, the mass flow rate of water entering the node is equal to the mass flow rate of water exiting the node. This relationship can be mathematically expressed as (9):
where
In(k) and
Out(k) represent the sets of pipes connected to node k that carry flow into and out of the node, respectively.
For a node connected by different pipes, water of varying temperatures enters the node for temperature mixing and subsequently exits the node at the same temperature. In this process, the heat energy entering the node equals the heat energy leaving the node. The calculation of temperature mixing is determined by (10):
where
is the temperature at the outlet of pipe
j at time t;
is the temperature at node
k after temperature mixing at time
t and also denotes the temperature at the inlet of the pipes flowing out of node
k at time
t.
The primary heat exchange station is responsible for heating the returning water from the pipe network and supplying it to the water supply network. In accordance with the law of conservation of energy, the heat exchange power of the primary heat exchange station can be calculated by using (11):
where
represents the heat transfer power of the primary heat exchange station at time
t,
mp is the mass flow rate of the primary heat exchange station,
is the temperature at the inlet of the water supply pipe connected to the primary heat exchange station at time
i, and
is the temperature at the outlet of the return pipe connected to the primary heat exchange station at time
i.
A secondary heat exchange station receives thermal energy transferred from the primary network and further distributes it to the secondary network for delivery to the heat users. The heat exchange power of the secondary heat exchange stations can be calculated using (12):
where
is the total heat exchange power of v secondary heat exchange stations at time
t,
mq is the mass flow rate of the qth secondary heat exchange station,
is the temperature at the outlet of the water supply pipe connected to the qth secondary heat exchange station at time
i, and
is the temperature at the inlet of the return pipe connected to the qth secondary heat exchange station at time i.
The temperature limits for the heating network are represented by (13) and (14):
where
is the temperature of water supply pipe j at fictitious node
n and time
t; and
are the maximum and minimum allowable temperatures in the water supply network, respectively; and the subscript
r denotes the return pipe.
To ensure the safe and stable operation of the heating network, it is necessary to impose constraints on the rate of temperature change in the pipe network. This can be achieved by controlling the rate of temperature change at the inlet of the supply and return network, as expressed in (15) and (16):
where
is the maximum allowable temperature change at the inlet of the water supply pipe connecting the primary heat exchange station between time
t and
t + 1;
is the maximum allowable temperature change at the inlet of the return pipe connecting each secondary heat exchange station between time
t and
t + 1.
The quasi-dynamic model of the heating network can be established through Equations (1)–(16). The determination of the scheduling interval for the system typically relies on the system’s operational requirements and load forecast data. For the thermal system, the day-ahead scheduling interval commonly assumes values of 15 min, 30 min, or 1 h. Within each scheduling interval, an assumption is made regarding the constancy of the heat load demand. This assumption implies that the heat exchange power of the secondary heat exchange station remains constant, as indicated in Equation (11). Simultaneously, the heat source output is determined through optimal scheduling, while the heat exchange power of the primary heat exchange station also remains constant within a given scheduling interval, as expressed in Equation (12). Regarding the source and load powers, the entire system operates under the framework of steady-state scheduling. However, the presence of transmission delays within the heating network introduces dynamic characteristics to the hot water transmission process. To address this dynamic behavior, a calculation time was introduced, which is consistently set to the factor of the scheduling interval. This selected calculation time enables a more precise analysis of the transmission characteristics of the heating network at a finer time scale. The alteration in temperature of the return water is depicted by employing the calculation time as the temporal scale. The return water sequentially passes through the primary heat exchange station, having undergone an identical temperature increase within a given scheduling interval, and subsequently enters the water supply pipeline. This principle equally applies to the secondary heat exchange station. This method seeks to achieve a more accurate calculation of the heat transmission process through quasi-dynamic modeling, aiming to enhance the coordinated scheduling of heat and power.
2.3. Quantification of Heat Storage in the Heating Network
The transmission delay effect in the thermal system allows the heating network to possess a certain heat storage potential. This potential capacity can be utilized when the supply and return water temperatures can be adjusted. The heat storage capacity of the heating system is indicated by changes in the return water temperature. During a specific time period, if the heat output from the heat source exceeds the corresponding heat load demand, heat is transmitted through the pipe network and exchanged at secondary heat transfer stations, resulting in an increase in the temperature of the return water network, indicating heat storage within the pipe network. Conversely, a decrease in the return water temperature indicates heat release from the pipe network. Heat storage in a heating network resembles a heat storage tank, as both contribute to enhanced system operational economics. However, the distinct advantage of employing heat storage in the heating network lies in its avoidance of the need for additional equipment configuration, resulting in a cost reduction. Moreover, while employing heat storage tanks, it is crucial to account for the impact of heat transmission delays. In comparison, the utilization of heating network heat storage offers superior flexibility. Due to temperature fluctuations, the heat storage capacity of the network at any given moment cannot be accurately determined by observing the temperature at a single location. To better quantify the heat storage, the return water equivalent average temperature is defined by (17):
where
is the equivalent average temperature of the return pipe network at time
t, R is the total number of pipes, and M
j is the number of fictitious nodes of pipe
j.
The magnitude of heat storage in the heating network can be calculated by using (18):
where
is the amount of heat stored in the heating network at time t,
mΣ is the total water mass in the return pipe network, and
is the minimum equivalent average temperature of the return network.