## Abstract

This study investigates the urban heating system (UHS) by taking a look at the heat transportation flexibility of the system. We propose the heating system flexibility (HSF) concept to represent UHS’s capability of meeting the heating demand under different operation conditions during the heating season and give out the corresponding evaluation method. Based on the evaluation method, we investigate the impact of heating network enhancement measures upon HSF by taking a real UHS in Beijing as a demo site. We pick network-wise topological change (extra pipe) and booster pump installation as two representative renovation measures. When an extra pipe close to end-user is introduced to the network, the average flexibility increases but the median flexibility drops. The results show that the introduction of the extra pipe does not reduce the hydraulic imbalance among different substations. A booster pump is more suitable for improving local substation HSF, although such a measure is only effective to a portion of the substations. Overall, the concept of HSF has the potential of being used as an important criterion in the design, operation, and control of UHS and other energy systems.

## 1 Introduction

### 1.1 Background.

China is undergoing a rapid process of urbanization. The urbanization rate increases from 26.4% in 1990 to 56.1% in 2015 [1]. By the end of 2016, the urban resident population had already reached 790 million, making up 57.35% of the total population [2]. One consequence of urbanization is the increase in primary energy consumption. For example, according to Zhao [3], the primary energy consumption has increased by 336% from 28.9 billion GJ in 1990 to 126 billion GJ in 2015.

Meanwhile, one achievement of China’s urbanization is that northern cities have built centralized urban heating system (UHS), which consumes a significant proportion of the primary energy and provides heat to both urban and rural areas. UHS uses water or steam as a working medium. Compared with steam, water is more used in the urban heating system. In urban UHS, the system first drives water to absorb the heat from the heating sources (power plants, boilers, waste heat recovery devices). The heated water is not sent directly to the end-users. Instead, the heated water is sent to a series of heating substations distributed in the covered heating area. The typical structure of UHS is shown in Fig. 1. In Fig. 1, the heating substations then further deliver heat to the end-users through plate heat exchangers and the corresponding secondary loop. As shown in Fig. 1, UHS has a complicated and distributed topological structure, which includes a variety of heat sources, substations, and valves. Meanwhile, along with the introduction of renewable energy-based heating sources, some heating sources are fluctuating sources when compared with conventional stable boilers. Such a system has high thermal inertia and shows strong nonlinearity and time-dependency. For example, in megacities such as Beijing, the centralized heating system covers a total area of more than 600 million m2 [4]. A single action from the heating source could take the farthest heating substations a couple of hours to respond.

To manage UHS, the existing approach is to generate a series of dispatch scenarios with each scenario reflecting a state of thermal and hydraulic balance under a presumed working condition (such as warm-up operation, emergent operation, etc.). For each scenario, the operators assume its return water temperature and water mass flowrates based on previous operation experience. Before the heating season, the operators need to generate hundreds of scenarios. During the heating season, operators adjust heating substations (the pumps and valves installed within the substation) primarily based on temperature control. In the meanwhile, the overall mass flowrate distribution is kept untouched to avoid disturbance to the established hydraulic balance [5]. Overall, such an operation approach tends to cause overheating. Once the system is affected by severe conditions (such as extremely low ambient temperature and equipment failure) and the working condition is not included in the preset scenarios, it is challenging for operators to setup a new operation mode of the system. One aspect of reducing such risk is to carry out comprehensive energy planning of UHS. For example, Alibabaei et al. [6] discussed the differences between system energy requirements and operating costs by proposing three different models for the same system during the heating and cooling seasons to find the proper energy planning model. Another drawback of such an approach is the lack of optimization capability since the scenarios are usually setup before the heating season. Moreover, such an approach is also incapable of supporting flexible management and operation control when dealing with heating sources with operational uncertainty (such as heating sources based on renewable energy). For example, International Energy Agency (IEA) [7] mentioned that higher shares of variable renewables raise flexibility needs and call for reforms to deliver investment in power plants, grids, energy storage, and demand-side response.

In addition to that, such an approach is also incapable of supporting the optimization of dispatching management and operation control with the heating sources with operational uncertainty. According to Hua’s research [8], the trend of China’s UHS is the introduction of renewable energy-based heating sources such as industrial waste heat, wind power, solar energy in addition to the existing clean coal-based combined heating and power (CHP) plants. The introduction of these heating sources brings uncertainty to the heating sources per the fluctuating nature. For instance, Li et al. [9] considered the uncertainty of wind power and study the integrated electrothermal system to enhance the robustness of the system model. Moreover, in the heat demand side, the renovation of heating metering devices, and the development of delicate public building heating adds up to the uncertainty in the end-user side. These factors altogether constitute a new challenge of heating system operation in UHS, that is, to find a new way of coordinating and dispatch all the elements involved in UHS.

The key is to understand the transport capability of the heating system. The development of the heating network mainly marks the development of UHS in China. For instance, to achieve a better balance between the heating sources and heating demand, the researchers introduced a looping structure into the once linear and radial heating network. To absorb the full potential of renewable energy-based heating sources, the heating network needs to carry out both temperature-based and mass-based operations in coordination with large-scale thermal storage devices. In order to find the heating network layout that maximizes robustness, Pizzolato et al. [10] studied the use of topology optimization under an investment constraint. Cammarata et al. [11] established a general mathematical model that considered the irreversibility due to temperature and pressure losses and evaluated the exergy cost of each branch to find the best solution. In practice, the heating network management system needs to conduct a network-wise energy balance check every 10 min to offset the fluctuation from the renewable energy-based heating resources.

### 1.2 Urban Heating System Flexibility.

To solve these challenges, researchers have investigated a series of new methods or system configurations that could enhance the flexibility of UHS, although the term flexibility might not clearly be defined as the core of studies. For example, Chen et al. [12] proposed new measures, including renewable energy, such as photovoltaic and wind energy, heating, ventilation, and air conditioning systems, energy storage, building thermal quality, etc. Cai et al. [13] put forward a realistic demand-side management (DSM) mechanism to improve system efficiency and manage congestion issues from the view of building modeling. From the source side, researches focus more on flexible operation. For example, Liu et al. [14] employed a genetic algorithm to search for the lowest exergy loss of heat pumps by optimizing the hourly load distribution rate. Usually, the non-intelligent operation of multiple units will lead to unbalanced and result in the low energy efficiency of the unit. In this case, Jin et al. [15] changed the operating load rate of each unit to determine the optimal energy efficiency ratio and to achieve more flexible dispatch. To increase the flexibility of combined heat and power plants, Mollenhauer et al. [16] presented the optimal operation of a thermal energy storage unit and a heat pump in combination with combined heat and power plants as well as synergies or competitions between them. Ji et al. [17] proposed a heat source and heating load energy-saving operation optimization control system by studying the relationship of flow and heating load in the quality—flow adjusting process. Based on both supply and demand, Lund et al. [18] offered a method to evaluate related flexibility measures of various energy systems and introduced renewable energy integrated with a large number of variable power. Compared with UHS, in the area of energy system studies, flexibility related studies mainly focus on energy storage. For example, to describe how the target building responds to the grid’s need for flexibility in energy system control, Junker et al. [19] defined energy system flexibility as a dynamic function suitable. A key part of this study is that the building envelop should be part of the flexibility calculation. Similar energy system dispatch, response, and transport capability studies could also be found in IEA’s “Energy in Buildings and Communities Programme Annex 67” [20]. As for the enhancement of energy system flexibility, Finck et al. [21] analyzed the demand flexibility of thermal energy storage tanks integrated with building heating system and found power-to-heat devices, water tanks, phase change material tanks, and thermochemical material tanks can be designed to provide the flexibility of short duration. Yilmaz et al. [22] found that the flexibility of the power-to-heat system could be achieved by adding thermal energy storage. Their work found that flexibility analysis of the energy system is critical in the application of novel technologies such as thermal storage. However, when compared with demand or source side, the issue of heating system flexibility (HSF) is rarely discussed in existing studies.

### 1.3 Motivation.

Due to the intrinsic uncertainty of renewable energy resources, the conventional operation approach could not support the ever-growing heating networks marked by renewable energy-based heating sources and a variety of heating end-users. It builds up the core motivation of this study. This paper will discuss how HSF emerges from different sectors and proposes its evaluation methodology. Based on the methodology, we will use a UHS in Beijing metropolitan area as a demo site to carry out HSF analysis. This case study will demonstrate the effectiveness of HSF and how it is reflected in the heating network enhancement measures. The paper will be concluded with our findings in HSF research and future work.

## 2 Heating System Flexibility

### 2.1 Flexibility Definition.

The essence of UHS is a pump-driven fluid network that transports the heat from the supply side to the demand side. In this sense, heating flexibility is the measurement of how UHS could achieve a precise match between supply and demand sides in different operation conditions. Figure 2 shows all the possible elements in modern UHS. It includes different types of end-users, heating sources, distributed energy supplies, and heating storage devices. Flexibility is reflected by the coordination capability of all elements of “source-network-load-storage”. During the heating season, the goal of the operation is to find a combined device dispatch mode that could minimize the loss in the network and meet the various demands in substations, which could be represented in Fig. 3.

Figure 3 illustrates the change of the operation mode while realizing dynamic load/supply balance during the heating season operation of UHS. For UHS, as shown in Fig. 2, the heating sources include both conventional power plants and boilers, as well as distributed energy supply, solar heating, and industrial waste heat recovery. The conventional power plants might be affected by emission control regulations. Distributed energy supply, if driven by biomass or other renewable energy sources, might be affected by pricing issues. With more renewable energy-based heating sources entering the heating network, the supply side is less steady, as shown in Fig. 3. In the meanwhile, the demand side is also changing due to the weather conditions and heating DSM operation. To realize the dynamic match between demand and supply, the operation modes need to be changed. A centerpiece in operation mode change is its operation state change of devices because it is difficult to change the topological structure (renovation or extension) once entering the heating season. In such a case, to calculate flexibility is to find out the network’s potential of maintaining a balance between supply and demand under all the possible device operation combinations, which differ from reliability analysis of network [23]. Once the boundary (such as the hardware limits of the pumps) and initial conditions (such as the last steady-state operation conditions) are given, the researchers could estimate the flexibility of the heating system. This aspect of analysis could also help researchers find out the optimal transport scheme for dispatching by making the most of its flexibility.

To simplify the formulation of HSF concept, the following assumptions are introduced into this study:

• We assume that the heating system is under a quasi-steady state during the operation.

• We assume that the flexibility of the end-users is reflected by their capability of regulating heating demand intensity and their surplus heat (if any) is not sent into the heating network.

• We assume that energy storage could be considered as an operating device and should be classified as a source of flexibility.

In real operation, due to the heat loss and the limited transport capacity, the heating supply and demand balance on any working condition are as follows:
$∑Qs=∑Qd+∑RQd+∑Qstorage$
(1)
where QS describes the amount of supply from heating sources, including different types of heating sources; Qd describes the load demand of heating substations; R is the transport loss matrix of heating system pipe network describing the loss in the pipes; and QStorage is the amount of thermal storage.

The flexibility sources are as follows.

• HSF from the heating sources

HSF from the heating sources is reflected as the capability of coordination operation of CHP plants and renewable energy-based heating sources. Such capability is the prime drive of HSF analysis of the entire UHS. During the operation, QS fluctuates are as follows:
$Qs→=Qs→(P→c,Xs→,δs→)$
(2)
where Pc is the capacity of the heating source; Xs(t) is the dispatch scheme of heating source; δs(t) is the uncontrollable factors on the source side, such as accidents.
• HSF from the load side

HSF from the load side could be reflected as the capability of realizing the heating DSM. The conventional UHS control in China does not consider the load side. Instead, the load side is merely represented by the change of operation state in heating substations. Once the HSF in the load side is introduced, the heating DSM comes into the scope of UHS control, that is, to incorporate the shaving of the heating load as part of HSF enhancing measures. Similarly, Qd is affected by the following factors:
$Qd→=Qd→(P→d,Wd(t)→,λd(t)→)$
(3)
where Pd describes the expected load demand in the heating substations, Wd(t) describes the change of the load due to weather conditions, λd(t) describes the disturbance in the heating substations.
• HSF from the heating network

HSF from the heating network is reflected as the number of possible paths for the heating network to deliver the required heat to the heating substations (including both regular and emergent operation). It is affected by the elasticity of the topological structure of the heating network. An essential issue in heating network HSF is the selection of redundancy of the structure. With an excessively high level of structural redundancy, the HSF may be reduced due to the increased thermal and hydraulic resistance. In the pipes, the loss R is affected by the following factors:
$R=R(S,E(t),X(t),ST)$
(4)
where S is topological structure; E(t) describes the transport parameter matrix in the heating network, that is, the thermal and hydraulic performance of the pipes; X(t) is the operating matrix, such as the operation combination of devices; ST is the hardware constraints in the network.
• HSF from the devices (thermal, hydraulic)

HSF from the devices such as thermal storage devices and hydraulic components (valves and pumps) are reflected as a flexible operation range based on the characteristics of the devices. In conventional UHS control, the dynamics of these components is rarely considered. Take storage devices, for example, QStorage is affected by the following elements:
$Qstorage=Qstorage(PS,CS,QSh,Xst,TS)$
(5)
where Ps describes the system configuration of thermal storage equipment, Cs is the designed capacity of equipment, Qsh is the instant discharging power of thermal storage equipment, Xst describes charge and discharge switch conditions, and Ts is thermal storage charge and discharge duration.

All these sources of flexibility contribute to the system’s heat transport capability under various working conditions. The analysis and evaluation of HSF is thus a reflection of the hardware and operation limits. It could be used to estimate the effectiveness of design and renovation measures such as adding new pipes and introducing new heating sources.

### 2.2 Evaluation Methodology.

To calculate thermodynamic states in the heating network and carry out HSF analysis, we divided the heating network into several loops containing “nodes” (valves, pumps, plants, etc.) and “section” (pipelines) through graph theory. Therefore, the topology of the heating network can be represented as a directed graph combined of the directed node Vx and the pipe section Ex, as shown in Fig. 4.

Based on the graph, we use Kirchhoff's law to establish the relationship between the flowrate and head loss of each loop. One benefit of this approach is that we could treat different system structures and working media (especially the steam-based system). During the modeling process, the data acquisition system of the heating source plant and the heating network provide the boundary conditions (overall heating capacity and source-side outlet mass flowrate). We use this information, together with the aforementioned network formulation, to solve the thermohydraulic equation sets, whose solution describes the states (water temperature, flowrate, and pressure) of the heating network in both the supply and return loop. The key to this process is first calculating the status (mass flow rate, temperature, and pressure) in each “node.” With the status of nodes in hand, we could further calculate mass/heat transport in each line and use this information to update the network until the preset convergence criterion is achieved. The detailed modeling approach could be found in our previous work [24]. In our earlier work, we took a steam network as a demo site and carried out the validation of such a modeling approach. Once the state of the system is achieved, we could evaluate the impact of diverse splitting strategies of pumps or valves and calculate HSF.

In this study, we propose the followed practical evaluation methodology for HSF in any heating substation in UHS. We set the demand for heating substation as Q1, …, QM, and the weight of substation in user side is α1, …, αM. We use H1, …, HN to describe the heat every substation can gain. Therefore, we could define of HSF of every substation.

Based on the above assumptions, we could calculate the satisfaction degree δi of each substation via Eq. (6), which describes the ratio of heat supply to user’s demand
$δi=min(1,HiQi)$
(6)
We could calculate the satisfaction γ of the overall network in a certain working condition via Eq. (7)
$γ(x)=maxy∑i=1Nαiδi∑i=1Nαi$
(7)
where x describes the supply and demand condition (uncontrollable part) and y describes adjustable parts such as heating source, valves, pumps, etc. For each heating substation, its flexibility is essentially its capability of meeting the demand, i.e., satisfying the demand. However, for the network, its flexibility could be marked by a series of weights assigned to different substations. Some substations (such as hospitals and schools) might be critical in terms of its flexibility evaluation.
Similarly, instant HSF at a specific time t is
$HSFt=∫Ωγ(x)dx∫Ω1dx$
(8)
where Ω is the collection of uncontrollable events combinations. It should be noted that this could also be used to calculate substation-wise flexibility.
Finally, the average HSF during a period T is defined as
$HSFT=∫0THSFtT$
(9)
The HSF calculation flow chart mentioned above is shown in Fig. 5. It is possible to carry out a substation-wide HSF analysis based on operation scheme and network structure with this methodology. It should be noted that although the substation concept is frequently used in this context, its meaning could be more general than typical substations. For example, in substantial metropolitan heating networks, it is possible to carry out partitioning of the network. In such conditions, each partition could be taken as a single substation from the view of the entire network.

## 3 Case Study

To further explain the HSF concept, this study took a UHS located in Beijing as an example and carried out HSF analysis. The heating system model is shown in Fig. 6. At present, there are three sets of central heat sources in the heating system. Among them, heating sources #1 and #2 are first-stage heat sources, which are designed to provide heat load of 500 MW with coal-fired CHP units; #3 and #4 are second-stage heat sources, which are designed to provide heat load of 620 MW with the same coal-fired CHP units. Another set of the heat source is made up of four gas-fired hot water boilers with a capacity of 116 MW, which are connected to the main heating network in the heating season.

There are 230 heating substations in the system. The overall heating area is divided into two main lines, line A (thin line of the picture) and line B (bold line of the picture). The heating area of line A is 8.5 million m2, which is mainly supplied by #1 and #2 power plants. The heating area of line B is 7 million m2, which is mainly supplied by #3 and #4 power plants. The used inputs are shown in Table 1.

This study calculated the HSF of these heating substations of line A based on the modeling and evaluation approach mentioned above. To simplify calculations, 20 critical heating substations were selected for HSF evaluation. We assumed that the system was in steady-state operation, and the water supply temperature was kept at 60 °C. We also assumed that there was no priority in terms of substations, which means that the heating network operation did not consider the differences in the substations and the weighting factors could be ignored.

## 4 Results and Discussions

Based on the modeling of UHS and the HSF evaluation mentioned above, we evaluated substation HSF of the target UHS and then analyzed the effect of the change of topological structure, and the introduction of booster pumps on the substation flexibility quantitatively.

### 4.1 Heating System Flexibility Analysis.

HSF of the 20 key substations in the target UHS under the given condition is shown in Fig. 7. The distribution parameters of heating substation HSF are listed in Table 2.

From the calculation of HSF above, substations #1, #2, #4, and #5 have insufficient flexibility (less than 1), which means these stations could not meet the heating demand from the side of the building with all possible operating conditions considered. Therefore, it is necessary to optimize the heating system to improve flexibility. In this study, to facilitate the discussion, substations #1, #2 #4, #5, #9, #10, and #12 were selected as exemplary substations when discussing the impact of topological structure changes and the introduction of the booster pump.

### 4.2 Impact of Topological Structure.

There are a variety of ways of changing the topological structure of UHS. In real UHS systems, one of the most used measures aiming at improving its transport capability and flexibility is to add new pipes. Therefore, we choose this as an example to show its impact on the flexibility of the substations.

At end substations, we study the impact of topological changes on HSF by considering adding additional pipelines, as shown in Fig. 8, where the substations are shown as nodes in the topological structure. When compared with Fig. 6, Fig. 8 has an extra pipeline close to end-users, that is, to increase the heating transport capability of remote end-users. The HSF changes are shown in Fig. 9. The heating substations HSF distribution parameters are shown in Table 3.

When the topological changes happen close to end-users, the average flexibility increases from 0.805 to 0.833, but the median flexibility decreases from 0.913 to 0.814. The reason is that in the network-wise pipe resistance increases because of the introduction of the new pipe structure. In addition to the increase in overall transport capacity, it also leads to a rebalancing of flow distribution in the network.

In this case, we found that HSF of substations #1, #9, #10, and #12 could rise from baseline’s 0.269, 0.230, 0.997, and 0.7124 to 0.617, 0.794, 1.000, and 0.731 while the average HSF decrease. It suggests a higher transport capacity in this case benefit only in these substations, and the rest of the substations are barely affected. Although the weight of each substation is set to equal in this study to facilitate the discussion, the real UHS has priority over the substations. For example, substations could not benefit from the overall increase of flexibility due to the design of the heating network could be, in fact, critical to the safety and security of the heating system. These substations could belong to the areas with low-income communities, areas with school and office buildings, or areas with health care. In such cases, the increase of piping does not solve the issue of low regional flexibility. More local measures (such as local booster pumps) should be considered. The analysis based on HSF then serves as a useful indicator of choosing the best locations for these local measures.

### 4.3 Impact of Booster Pump.

The impact of a booster pump on HSF change was also investigated in this study. Compared with topological structure changes, the booster pump option is intuitively more of a local HSF enhancement measure. Modern UHS tend to have underground pipes to reduce the space occupied by the heating network, especially in the downtown area of UHS. In practice, the usage of booster pumps is more found in remote substations where more space could be provided to the installation of pumps. With these factors considered, this study investigated adding a booster pump to the substation at the end of the remote branches. In the demo site, a booster pump with a capacity of 800 t/h and maximal hydraulic head of 26 m under rated conditions were selected. The installation location is as shown below in Fig. 10.

The corresponding HSF changes are shown in Fig. 11. The HSF distribution parameters of heating substations are shown in Table 4.

From Fig. 11 and Table 4, the additional booster pump could increase the average flexibility from 0.805 to 0.832 and the median flexibility from 0.913 to 0.998. This is because the pump is newly added on the branch near the end, and the transport capacity of the heating substation at the end of the branch is obviously improved, which can overcome the problems such as the long distance between the heating source and the stations as well as the resistance of the circulating water in the pipe network. Therefore, it helps to increase the transportation efficiency, and the energy consumption of the pipe network was reduced as well as improving hydraulic balance degrees and balancing heating effect. For example, in substations #2, #4, and #5, HSF could be increased from 0.904, 0.808, and 0.592 to 1.000, respectively, with the booster pump in the branches. However, the problem of local measures is that installing booster pumps needs to include HSF analysis as part of the decision-making process. For instance, adding a booster pump could not effectively enhance its flexibility when in a high-HSF area. Instead, the introduction of a booster pump in such an area can cause local hydraulic imbalances that can damage the HSF in nearby areas, as found in the case of substations #12 and #13. Therefore, it is necessary to carry out HSF analysis as the prerequisite of introducing local enhancement measures.

## 5 Conclusions

We implement our HSF analysis methodology in the UHS of Beijing and achieve the following conclusions through HSF evaluation:

1. Topological structure enhancement can improve the overall HSF.

2. When applying topological structure enhancement, the results show that additional pipe does not guarantee an increase in HSF for all substations. This is because an increase in pipe resistance results in an imbalance in mass flow distribution, especially in some marginal stations.

3. Booster pumps are local enhancement measures and have a higher impact than topological structure change such as an extra pipe. However, the success of local measures does not apply to all substations. It is necessary to carry out preliminary HSF analysis before implementing these measures. Otherwise, the introduction of local measures in the high-HSF area could impair the hydraulic balance in nearby areas.

## 6 Future Work

One of the limits of this study is that we have not taken the HSF analysis to the industrial steam systems and see how it differs from urban heating systems. Since the centralized steam network becomes more and more popular in modern industrial parks, it is necessary to take a look at the steam supply system and investigate the impact of topological structure change on its HSF.

As for future work, a better understanding of flexibility could simplify the operation optimization of UHS. We believe the following topics would be of interest to researchers in this field. The first is to explore the impact of other improvement measures on HSF and quantify their specific impact degree. The second is to make intelligent fault diagnosis based on flexibility assessment.

## Acknowledgment

This work is supported by the National Key R & D Program of China (Grant No. 2017YFA0700302). This work is in part supported by the National Natural Science Foundation of China (Grant No. 51806190). This work is also in part supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F030007) and key project of Beijing Municipal Science and Technology Commission “Blue Sky Project” (Grant No. D171100001217001).

The authors also would like to thank the financial support from Zhejiang University & Guolian Huaguang Smart Energy System Joint Research Center.

## Nomenclature

### Symbols

• i =

substation ID

•
• x =

uncontrollable working condition (supply and demand)

•
• y =

adjustable parts such as heating sources and pumps

•
• R =

transport loss matrix of the heating system pipe network

•
• S =

topological structure

•
• Cs =

capacity of thermal storage equipment

•
• E(t) =

transport matrix of the heating network

•
• Ex =

directed pipe section of the heating network

•
• Hi =

heating achieved by substation

•
• Pc =

capacity of the heating source

•
• Pd =

expected load demand in heating substations

•
• Ps =

system configuration of thermal storage equipment

•
• Qd =

heat user/heating substation load demand

•
• Qi =

heating demand of substation

•
• Qsh =

heating power of thermal storage equipment

•
• QS =

heat supply of heating source, which includes different types of heating source

•
• QStorage =

thermal storage

•
• Ts =

thermal storage charge/discharge duration

•
• Vx =

directed node of the heating network

•
• Wd(t) =

change of demand of user side along with weather

•
• Xs =

dispatch scheme of heating source

•
• Xst =

charge and discharge switch conditions

•
• X(t) =

operating matrix of the heating network

•
• ST =

constraints in the pipe network

•
• αi =

weight of substation

•
• γ =

satisfaction of the heating network under a certain working condition

•
• δi =

satisfaction degree of a single heating substation

•
• δs =

uncontrollable factors on the source side

•
• λd(t) =

disturbance in the heating substations

•
• Ω =

collection of possible combinations of all uncontrollable factors

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