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A multienergy inertiabased power support strategy with gas network constraints
Protection and Control of Modern Power Systems volumeÂ 8, ArticleÂ number:Â 18 (2023)
Abstract
An integrated energy system with multiple types of energy can support power shortages caused by the uncertainty of renewable energy. With full consideration of gas network constraints, this paper proposes a multienergy inertiabased power support strategy. The definition and modelling of gas inertia are given first to demonstrate its ability to mitigate power fluctuations. Since partial utilization of gas inertia can influence overall gas network parameters, the gas network is modelled with an analysis of network dynamic changes. A multienergy inertiabased power support model and strategy are then proposed for fully using gasthermal inertia resources in integrated energy systems. The influence of gas network constraints on strategy, economy and power outputs is analyzed. Special circumstances where the gas network can be simplified are introduced. This improves the response speed and application value. The feasibility and effectiveness of the proposed strategy are assessed using a real scenario.
1 Introduction
With increasing access of renewable energy to the power system, the uncertainty of the output power from renewable energy sources may reduce the reliability of power grid operation [1, 2]. Currently, there is exploration of the potential for ensuring continuous and reliable power supply using energy reserves such as wind power or solar energy [3, 4]. Given the flexibility to realize the conversion of various energy sources [5, 6], an integrated energy system (IES) can also be adopted to provide power support after disturbances [7]. The electricity power can be converted from other energy forms with configurations such as combined heat and power (CHP) units to support power shortages [8, 9], or converted into heat or gas for storage to absorb excess electricity in the power system [10].
In addition to direct energy conversion, inertial resources contained in slowdynamic thermal and gas systems in IESs can also resist power fluctuations and support power shortages. For thermal systems, there is a lot of research on the storage potential of district heating networks [9]. As thermal power can be extracted from thermal sources without significantly influencing heat consumers [11], this has led to many studies focusing on the combined use of thermal inertia and demand response [12]. Current modelling methods of thermal inertia are relatively mature, and so will be used directly in our work.
However, for a gas system, the modelling methods of gas inertia still lack sound solutions. Most research focuses on gas storage characteristics, referring to the natural gas remaining in pipelines because of pressure difference between pipe heads and ends [13]. It has been shown that gas storage can be released in a short time for shortterm power support [14], or integrated with the electrical power system to deal with uncertain power supply in IESs [15]. Given the similar resistance characteristics of a gas system to power fluctuations as thermal inertia, a similar concept of gas inertia is proposed. As well as gas storage characteristics, reference [16] further enriches the definition of gas inertia by including a gas systemâ€™s longtime response to the demand changes in the gas inertia characteristics. Nevertheless, despite the expansion of the gas inertia definition, the corresponding mathematical forms have not been given to verify the newly proposed characteristics. Therefore, this paper presents a novel gas inertia model with an intuitive mathematical form to represent both the gas storageâ€™s capability to mitigate the power impact, and gas its longtime response capability to delay the impact arrival time.
As for the application scenarios of natural gas system resistance characteristics, gas storage is often used in emergency power support and frequency stability maintenance. Gas storage can support emergent power shortage in seconds along with thermal inertia [17], or maintain frequency stability in minutes, serving as a backup energy for frequency second dip suppression [18]. To simplify the network modelling and improve the computing speed of optimization strategies, gas storage is only considered as an output energy form with the network influence and interactions between pipelines being overlooked [17,18,19]. To consider the gas networkâ€™s influence on system operation, the mutual constraint relationship between nodal pressure and gas flow in the gas network is included in [20], and the flow difference between pipe heads and ends brought by consideration of gas storage is included in [21]. Improvements are further made in [22] with concrete analysis of natural gas usage effects on power/gas loads as well as the storage in pipelines.
It can be seen that research on natural gas systems tends to develop towards the refinement of modelling of internal gas networks. Thorough consideration of gas network constraints will inevitably increase the complexity of system operational modelling and reduce computing speed. These are especially influential in emerging power support scenarios. Therefore it is necessary to clarify the influence of partial gas inertia use on overall gas network power support capacity, so as to study gas network simplification conditions. Refinement of modelling of internal gas networks and consideration of scenario applicability of the power support strategies are first examined in this paper. Different from the aforementioned research, the power support strategy proposed in this paper not only considers gas network constraints in detail, but also gives conditions where gas network constraints can be simplified to ensure the practicability of the strategy in real application.
Therefore, based on the intuitive mathematical modelling of gas inertia resistance characteristics, this paper aims at fully using multienergy inertia resources in power support optimization, and, in particular, refers to thermal energy inertia and natural gas energy inertia in IESs. For thermal inertia, the supply of thermal power at the load side can be reduced to support the electric power shortages. Thus, the influence of power fluctuations on the power balance of electric power system can be reduced by using the resistance ability of the thermal system to power fluctuations. For gas inertia, the prereserved gas storage in gas pipelines can be released quickly to support electric power shortages by energy conversion. The influence of power fluctuations on the power balance of electric power system can be reduced by the inertia characteristics of natural gas systems. Then, the strategy of economy and power output are compared in scenarios of considering/not considering gas network constraints. The different power support results confirm the necessity of gas network constraints consideration, while the size of the difference varies with different IES parameters and power shortage conditions. Through the analysis of possible influence factors, conditions whereby gas network constraints can be simplified are given. These involve the power shortage spots in the network and the upper limits of thermal inertia.
The major contributions of this paper are as follows:

1.
The capability to accommodate power fluctuations using gas inertia is presented by solving the gas inertia model. To fully use the inertial resources contained in IESs, gas inertia is used along with thermal inertia to support power shortages with improved efficiency.

2.
A novel multienergy inertiabased power support strategy considering gas network constraints is proposed. The strategy of economy and power output are precisely evaluated by analysis of gas network dynamic changes.

3.
Since detailed consideration of gas network constraints can increase the computational burden of a power support strategy, certain circumstances where the network can be simplified are introduced. The simplification can reduce modelling complexity and increase computing speed to improve the speed of power support response.
The rest of the paper is organized as follows. SectionÂ 2 presents the definition of gas inertia in contrast to thermal inertia, and the modelling of gas inertia to reveal its resistance characteristics. SectionÂ 3 establishes the power support model using gasthermal inertia of IESs with detailed consideration of gas network dynamic changes, and a power support strategy is then proposed with precise measurement of gas storage consumption throughout networks. The feasibility, effectiveness and application value of the proposed strategy are tested in Sect.Â 4, and Sect.Â 5 draws the conclusions.
2 Use of gas inertia considering network influence
Given that the two approaches have similar resistance characteristics to power fluctuations, the definition of gas inertia in contrast to thermal inertia is presented in Sect.Â 2.1. A novel gas inertia model is then established representing both its capability to mitigate the power impact and delay the impact arrival time in an intuitive mathematical form in Sect.Â 2.2. With further consideration of the internal gas network constraints, a refined modelling of a gas network is presented in Sect.Â 2.3.
2.1 Definition of gas inertia
For a natural gas system shown in Fig.Â 1, gas inertia deals with power fluctuations by restoring/releasing gas storage in pipelines depending on the input and output flow difference. With load fluctuations, gas inertia mainly manifests as the response delay of the arrival of the minimum gas pressure and the available gas storage.
The definition of gas inertia is shown in Fig.Â 1b, while thermal inertia is shown in Fig.Â 1a. When the input power fluctuates on the thermal sources, thermal inertia mainly manifests as: (1) the response delay of the arrival of the minimum indoor temperature; and (2) the comfort range within which temperature is allowed to change. Therefore, the load fluctuation in a gas system is equivalent to the input power fluctuation in the thermal system, and the slowspeed response of gas pressure at pipe ends in a gas system is equivalent to the slowspeed response of indoor temperature in a thermal system.
2.2 Modelling of gas inertia
From the response characteristics providing buffer space for power fluctuations, the power support model of gas inertia based on the dynamic model of a gas system is established.
The relationship between the natural gas pressure, density and pipeline flow can be expressed as [23]:
where \({P}^{G}{/q}^{G}/{T}^{G}\) are the pressure/flow rate/temperature of natural gas. \({R}_{M}\) is the quotient of gas constant to molar mass, \(\rho\) is the density of natural gas, \({v}^{G}\) is the velocity of natural gas, and A is the crosssectional area of the gas pipe.
The transient transmission process of a gas pipeline can be expressed as [24]:
where \(x\) is the space variable and \(t\) is the time variable. \(\theta\) is the inclination of the pipe to the horizontal plane, D is the inner diameter of pipe, and g is the acceleration due to gravity.
To simplify the dynamic model of a natural gas pipe after the substitution of (1) into (2), the following assumptions are made: a) the differential term \({v}^{2}\) is ignored given its small influence on the gas pressure; and b) \(\theta\) is considered to be 0.
The gas pressure response of pipe ends is the direct reflection of gas inertia dealing with load fluctuations. This can be expressed through the simplification of a finite element approximation, as:
where \({L}^{G}\) is the length of the natural gas pipeline, \({q}_{out}^{G}(t)/ {P}_{out}^{G}(t)\) are the flow rate\(/\)pressure of pipeline outlet at time \(t\), and \({P}_{in}^{G}\) is the pressure of pipeline inlet and assumed to be constant.
Assuming the load demand increases instantaneously at \({t}_{1}\) and the gas flow at pipe ends drops from the normal value \({q}_{1}^{G}\) to \({q}_{2}^{G}\), the process can be expressed as:
where \({q}_{1}^{G} \mathrm{and} {q}_{2}^{G}\) are the initial and dropped values of gas flow rates, and \({q}_{out}^{G}(t{t}_{1})\) is the flow rate of the gas pipe outlet at time \(t{t}_{1}\).
Substituting (5) into (4) through calculation methods of Laplace\(/\)inverse Laplace transforms, the power support model of gas inertia based on the gas pressure response of pipe ends can be obtained as:

1.
\({a}_{1}=A{L}^{G}/({R}_{M}{T}^{G})\);

2.
\({a}_{2}=A\lambda {v}^{G}{L}^{G}/(2D{R}_{M}{T}^{G})\);

3.
\({a}_{3}=A/{L}^{G}\);

4.
\({b}_{1}=\lambda {v}^{G}/(2D)\);

5.
\({b}_{2}=A{P}_{in}^{G}/{L}^{G}\);

6.
\({P}_{out}^{G}\left({0}^{}\right)={\dot{P}}_{out}^{G}\left({0}^{}\right)=0\);

7.
\({x}_{1}\) and \({x}_{2}\) are the two roots of the equation \({a}_{1}{s}^{2}+{a}_{2}s+{a}_{3}=0\).
$$\begin{aligned} P_{out}^{G} \left( t \right) &= {{\left( {b_{2}  b_{1} q_{2}^{G} + b_{1} q_{1}^{G} } \right)} \mathord{\left/ {\vphantom {{\left( {b_{2}  b_{1} q_{2}^{G} + b_{1} q_{1}^{G} } \right)} {\left( {a_{1} x_{1} x_{2} } \right)}}} \right. \kern0pt} {\left( {a_{1} x_{1} x_{2} } \right)}} \hfill \\ &\quad {{\left\{ \begin{gathered} {{\left[ {\left( {x_{1}  b_{1} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right)  b_{2} } \right]e^{{  x_{1} t}} } \mathord{\left/ {\vphantom {{\left[ {\left( {x_{1}  b_{1} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right)  b_{2} } \right]e^{{  x_{1} t}} } {x_{1} }}} \right. \kern0pt} {x_{1} }} \hfill \\  \left. {{{\left[ {\left( {b_{1}  x_{2} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right) + b_{2} } \right]e^{{  x_{2} t}} } \mathord{\left/ {\vphantom {{\left[ {\left( {b_{1}  x_{2} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right) + b_{2} } \right]e^{{  x_{2} t}} } {x_{2} }}} \right. \kern0pt} {x_{2} }}} \right] \hfill \\ \end{gathered} \right\}} \mathord{\left/ {\vphantom {{\left\{ \begin{gathered} {{\left[ {\left( {x_{1}  b_{1} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right)  b_{2} } \right]e^{{  x_{1} t}} } \mathord{\left/ {\vphantom {{\left[ {\left( {x_{1}  b_{1} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right)  b_{2} } \right]e^{{  x_{1} t}} } {x_{1} }}} \right. \kern0pt} {x_{1} }} \hfill \\  \left. {{{\left[ {\left( {b_{1}  x_{2} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right) + b_{2} } \right]e^{{  x_{2} t}} } \mathord{\left/ {\vphantom {{\left[ {\left( {b_{1}  x_{2} } \right)\left( {2q_{1}^{G}  q_{2}^{G} } \right) + b_{2} } \right]e^{{  x_{2} t}} } {x_{2} }}} \right. \kern0pt} {x_{2} }}} \right] \hfill \\ \end{gathered} \right\}} {\left[ {a_{1} \left( {x_{2}  x_{1} } \right)} \right]}}} \right. \kern0pt} {\left[ {a_{1} \left( {x_{2}  x_{1} } \right)} \right]}} \hfill \\ \end{aligned}$$(6)where \({x}_{1}\ge 0\) and \({x}_{2}\ge 0\), and are given by:
$$\begin{gathered} x_{1} = {{\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}  \sqrt {\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} \right)^{2}  4{{a_{3} } \mathord{\left/ {\vphantom {{a_{3} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} } \right)} \mathord{\left/ {\vphantom {{\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}  \sqrt {\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} \right)^{2}  4{{a_{3} } \mathord{\left/ {\vphantom {{a_{3} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} } \right)} 2}} \right. \kern0pt} 2} \ge 0 \hfill \\ x_{2} = {{\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }} + \sqrt {\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} \right)^{2}  4{{a_{3} } \mathord{\left/ {\vphantom {{a_{3} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} } \right)} \mathord{\left/ {\vphantom {{\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }} + \sqrt {\left( {{{a_{2} } \mathord{\left/ {\vphantom {{a_{2} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} \right)^{2}  4{{a_{3} } \mathord{\left/ {\vphantom {{a_{3} } {a_{1} }}} \right. \kern0pt} {a_{1} }}} } \right)} 2}} \right. \kern0pt} 2} \ge 0 \hfill \\ \end{gathered}$$(7)
The above gas inertia model reveals its support capability in response to abrupt power shortages. The negative exponential function form of gas inertia response mitigates the step function form of power fluctuations and delays the arrival of a power fluctuation shock to the system. Enough buffer space is provided to deal with power shortages. This establishes the foundation for gas inertia utilization in the power support strategy.
2.3 Modelling of natural gas network
The utilization of gas inertia mainly manifests as the use of gas storage, which involves the change of pressure, flow rate and gas storage of multiple gas pipelines in the network. Therefore, gas inertia providing support to power shortages at one load might influence the use of gas inertia at other loads, leading to their consequent decrease in power support capability.
This paper adopts a typical 8node gas network structure shown in Fig.Â 2 to analyze the network influence on gas inertia utilization. The gas network is supplied by two gas sources \({GS}_{1}\) and \({GS}_{2}\). Three load groups Load 1 \({L}_{1}\), Load 2 \({L}_{2}\) and Load 3 \({L}_{3}\) are connected respectively at the end of the gas pipes \(12\), \(34\), and \(67\). The corresponding gas flows at the load groups are \({q}_{L1}^{G}\), \({q}_{L2}^{G}\) and \({q}_{L3}^{G}\), respectively. The gas flows at the head\(/\)end of the gas pipe \(mn\) are \({q}_{mn,in}^{G}/{q}_{mn,out}^{G}\), the pressures of natural gas at nodes \(m/n\) are \({P}_{m}^{G}/{P}_{n}^{G}\), and the gas storage in the gas pipe \(mn\) at time \(t\) is \({V}_{mn}^{G}(t)\).
The gas storage is defined as a certain amount of gas stored in the pipeline because the quantities of incoming and outgoing flows may differ [22]. The gas storage within a gas pipe from node \(m\) and node \(n\) at time \(t\) can be expressed as:
where \({\rho }_{0}\) is the density of natural gas under standard conditions (0.7174 (kg/m^{3})). \({L}_{mn}^{G}/{d}_{mn}^{G}\) are the length/diameter of the gas pipe \(mn\), and \({\overline{P} }_{mn}^{G}(t)=({P}_{m}^{G}+{P}_{n}^{G})/2\) represents the average gas pressure in the pipeline.
The change of the gas storage amount from time \(t1\) to time \(t\) can be expressed as:
where \({V}_{mn}^{G}(t1)\) is the gas storage in the gas pipe \(mn\) at time \(t1\), and \({q}_{mn,in}^{G}(t)/{q}_{mn,out}^{G}(t)\) are the gas flows at the head/end of the gas pipe \(mn\) at time \(t\).
Usually, the relationship between the gas pressure and flow in the static natural gas system between node \(m\) and node \(n\) without compressors is given as (10):
where \({P}_{0}^{G}\) and \({T}_{0}^{G}\) are the pressure (101.325Â kPa) and temperature (278Â K) of natural gas under standard conditions, respectively. \({Z}_{0}^{G}\) is the compression factor of natural gas under standard conditions, and Z is the compression factor of natural gas. \({q}_{mn}^{G}\) is the flow rate of natural gas in pipe \(mn\).
When considering gas storage and accompanying gas flow difference between pipe heads and pipe ends, Eq.Â (10) should be replaced by:
where \({\overline{q} }_{mn}^{G}=({q}_{mn,in}^{G}+{q}_{mn,out}^{G})/2\) represents the average gas flow in the pipeline.
Based on the above gas network modelling, the condition of gas storage consumption is presented in detail through the flow rate changes of pipelines in Fig.Â 3. The power shortages occur at \({L}_{1}\), \({L}_{2}\) and \({L}_{3}\). Gas storage is increased when the flow rate at pipe heads is greater than the flow rate at pipe ends and is consumed when the flow rate at pipe heads is less than that at pipe ends. The flow rate of natural gas at the loads usually remains consistent with the change in load, and only changes when the load demand changes.
Since the three fault cases in Fig.Â 3 follow the same natural gas dynamic change mechanism, the case of power shortages at \({L}_{1}\) shown in Fig.Â 3a is chosen as an example to be explained in detail. To support power shortages at \({L}_{1}\), the electricity demand will increase suddenly, resulting in a sudden increase in the flow rate at \({L}_{1}\) and Node 2. During this process, the flow rate at Node 1 is less than that of Node 2, suggesting that the gas storage in pipe12 is consumed. Following this principle, the gas storage is also consumed in pipe23, pipe34, pipe36 and pipe67 during power shortages at \({L}_{1}\). When the power shortages disappear, the flow rate at \({L}_{1}\) returns to normal and the flow rate at Node 1 is greater than that of Node 2, suggesting that the gas storage in pipe12 is increased. Following this principle, the gas storage in pipe23, pipe34, pipe36 and pipe67 is also increased after the recovery of power shortages. In summary, the following three phenomena are illustrated by Fig.Â 3:

1.
Most pipelines will contribute a portion of gas storage at the beginning of power shortages and then slowly store the natural gas at the end of power shortages.

2.
Power shortages at one load not only consume the gas storage in the nearby pipelines, but also influence the amount of gas storage in the whole gas network. This can further affect the usage of natural gas by other loads as well as reduce their power support potential if not regulated.

3.
Power shortages at different load spots have different gas storage consumption results. This suggests the need to study different power shortage spots in the network.
Therefore, it is important to consider the network influence of a natural gas system in the use of gas inertia.
3 Power support model of multienergy inertia considering gas network constraints
During power grid operation, power fluctuations often occur because of the imbalance between power supply and demand. Current methods of supporting power shortages usually focus on the exclusive use of inertial resources in IESs, whereas the similar characteristics to resist power fluctuations between different inertial resources can be further explored and applied in multienergy inertiabased power support to improve the efficiency of IES operation. Despite the joint use of gas inertia with other energy forms in some research, the gas network is often considered as an entirety with the internal dynamic changes and consequent influences being overlooked. Therefore, this section introduces a new power support strategy considering multienergy inertial resources in gas and thermal systems. The gas network is modelled in detail with load groups connected to different locations. Use of gas inertia is likely to affect the normal operation of load groups without power shortages. Therefore, the output of different energy forms and power support strategy economy will be different from when the gas network is considered as an entirety.
As for the network influence of a thermal system, since the support time scale of thermal inertia is up to hours compared with the minutelevel of gas inertia, the dynamic changes of a natural gas network is much larger than that of thermal network over the same power support time. Therefore, the network influence of thermal system will not be considered in detail in this paper.
3.1 Internal structure of the integrated energy system
The internal structure of IESs studied in the paper is shown in Fig.Â 4, where two gas sources and N groups of loads (\(N=1, 2, 3\)) are presented with the 8node gas network structure of Fig.Â 2. Three types of energy are involved in this multienergy system, i.e., thermal, natural gas and electric. The electric load is supplied by the external power grid and CHP units of the respective load groups, while the thermal load is supplied by the electric boiler and CHP units of the respective load groups. The input and output model of the energy hub is given as:
where \({L}_{n}^{E}/{L}_{n}^{H}\) is the electric load/thermal load of \({L}_{n}\) (nâ€‰=â€‰1, 2, 3). \({\lambda }_{dis}^{E}\) is the electric energy distribution coefficient. \({\eta }_{T}/{\eta }_{EB}\) is the transformer/electric boiler conversion efficiency, and \({\eta }_{{L}_{n}}^{CHPE}/{\eta }_{{L}_{n}}^{CHPH}\) is the gaselectricity/gasheat conversion efficiency of the CHP units of \({L}_{n}\) (nâ€‰=â€‰1, 2, 3). \({P}_{1}^{GS}/{P}_{2}^{GS}\) are the external gas sources 1 and 2, while \({P}^{ES}\) is the external power supply.
3.2 Gasthermal inertiabased power support strategies considering gas network constraints
Supposing the power shortages occur at one load group in the IES, gas and thermal inertia mainly respond in two ways involving the transmission or conversion of the three energy forms of electric, thermal and gas, as:

1.
To make up for the power shortages at one load, electricity from the power grid can be directly rearranged to supply the electric load from the original route of the electric boiler. This method requires no conversion of any energy but only the control centre to rearrange the power transfer route. This method will lead to the decrease of thermal power on the load side, but is still feasible because of the existence of thermal inertia. Therefore, although this method is essentially the transmission of electrical energy, it is called thermal inertial output.

2.
To obtain more electricity within a short time, natural gas stored in gas pipelines can be converted to electricity through CHP units. This method takes advantage of the fast response speed of CHP units and prestored gas storage at the expense of increased thermal power from the output of CHP units.
During the power support process, thermal power on the load side may decrease because of the first method but increase because of the second method. The total thermal deviation caused by the two methods can affect the user comfort on the load side, but with a time delay, while it mitigates power fluctuations and temperaturetolerance from the users given the existence of thermal inertia. Gas inertia mainly contributes to the prestored gas storage and the delayedresponse of gas pressure, which delays the time of the pipe pressure reaching the lower limit and allows more time for the power support provided from gas inertia.
The influence of the gas network is studied based on the 8node gas network structure shown in Fig.Â 2 with three load groups connected to different spots. Assuming that the power shortages occur at \({L}_{1}\), three output forms are considered in the whole power support process as shown in Fig.Â 5:

1.
Gas inertia output uses gas storage from different pipelines in the gas network to support the electricity shortages at \({L}_{1}\) through the single CHP unit at \({L}_{1}\).

2.
The electricity transferred from the electric boiler to the transformer may come from three different load groups, called the thermal inertia output given that the feasibility of this method is to make use of: (a) the tolerance of load side to temperature change; (b) the time delay; and (c) the reduced fluctuation influence of the thermal source on the load side.

3.
Given thermal fluctuations resulting from gasthermal inertia output on the load side, the demand side output is also considered based on usersâ€™ willingness to tolerate total thermal deviation on the load side brought by various power support methods.
The above three output forms act simultaneously in no particular order while there are differences in the unit output costs between the same output form. These are marked with different colours in Fig.Â 5. The main response force among the three load groups should be \({L}_{1}\) and the pipelines nearby when the power shortages are at \({L}_{1}\). Therefore, when the outputs of other load groups are mobilized because of the failure of \({L}_{1}\), it may affect their normal operation and even the ability to deal with possible power failure in the future, so the unit cost of output would increase.
3.3 The objective function
This paper deals with power shortages using power support from gasthermal inertia. To ensure the system reliability level, the objective function is to minimize the total cost of the power support during the power failure period, given as:
where \(Cost, {C}_{RG},\mathrm{ and }{C}_{CHP}\) are the total cost, and costs of gas inertia output, thermal inertia output, demand side output, and heatelectricity ratio adjustment, respectively.
The calculation methods for the different costs are shown as:
where T is the duration of the power shortage, and \({t}_{0}\) is the initial time of the power shortage occurrence. \({c}_{RG,{L}_{n}}^{a} {c}_{RG,{L}_{n}}^{b}\) are the respective unit costs of gas storage in pipe \(a(a+1)\) and pipe \((b+1)(b+2)\) with power shortages at \({L}_{n}\) (\(a\)=1, 2, 3; \(b\)=4, 5). \({c}_{RG,{L}_{n}}^{6}\) is the unit cost of gas storage in pipe 36 with power shortages at \({L}_{n}\) (nâ€‰=â€‰1, 2, 3), and \({c}_{RH,{L}_{n}}^{c}\) is the unit cost of thermal inertia at \({L}_{c}\) with power shortages at \({L}_{n}\) (c, nâ€‰=â€‰1, 2, 3). \({c}_{COM,{L}_{n}}^{c,m}\) is the unit cost of demand side output of \({L}_{c}\) in level \(m\) with power shortages at \({L}_{n}\) (c, nâ€‰=â€‰1, 2, 3), while \({c}_{CHP}^{w}\) is the unit cost of heatelectricity ratio adjustment in level \(w\). \({P}_{RH}^{c}\)(\(t\)) is the power output of thermal inertia of \({L}_{c}\) at time \(t\) (câ€‰=â€‰1, 2, 3), \({P}_{RG}^{n,w}\)(\(t\)) is the power output of gas inertia in level \(w\) at time \(t\), and \({L}_{n}{e}_{heat}^{c,m}\)(\(t\)) is the thermal deviation of \({L}_{c}\) in level \(m\) at time \(t\) (câ€‰=â€‰1, 2, 3)., M is the total level of demand side step price, W is the total level of heatelectricity ratio adjustment step price, and C is the total load group (Câ€‰=â€‰3).
Among these costs, the unit costs of gas and thermal inertia are fixed prices, which are set according to the spot of power shortages and load groups. The unit cost of thermal inertia output from other load groups will be higher than that of the power shortage location. The unit cost of gas inertia output from the pipelines far from the power shortage spot will be higher than that in the nearby pipelines of the power shortage spot, and will be even higher if the end of the pipeline is directly connected to other load groups given that the possibility of affecting the normal gas supply of other loads will increase.
The demand side output adopts the step price and is divided into M response levels, where M is the total step number of demand side step price. When one load group is operating normally and needs to provide demand side output for other loads with power shortages, the demand side step price is higher than that in the power shortage spot.
CHP units usually have an optimal heatelectricity ratio, under which the energy utilization rate is the highest, and the generated thermal power and increased fuel consumption are optimally balanced. The optimal heatelectricity ratio is 1.0 in this paper, and the corresponding heatelectricity ratio adjustment cost is the lowest given its highest energy utilization rate. When the heatelectricity ratio is increased for more thermal power, the heatelectricity ratio adjustment cost at the same gas consumption will also increase. Therefore, consumption of gas under different heatelectricity ratios corresponds to different prices. The heatelectricity ratio adjustment cost adopts the step price and is divided into W response levels, where W is the total step number of heatelectricity ratio adjustment cost.
3.4 The constraint conditions

1.
Electrical power balance constraint
$$P_{RG}^{n} \left( t \right)\eta_{{L_{n} }}^{CHPE} + \left[ {P_{RH}^{1} \left( t \right) + P_{RH}^{2} \left( t \right) + P_{RH}^{3} \left( t \right)} \right]\eta_{T} = e_{ele}^{{}} \left( t \right)$$(18)where \({P}_{RG}^{n}\)(\(t\)) is the power output of gas inertia at \({L}_{n}\) at time \(t\), and \({e}_{ele}(t)\) is the electric power shortage at time \(t\), where \(t\) is between \({t}_{0}\) and \(T\). The electric power shortages are supported by the electricity output of CHP units and the transfer of electricity from the three load groups.

2.
Thermal power balance constraints
Based on the principle of minimizing the influence of power support on the normal operation of other load groups, only the CHP unit in the power shortage spot consumes the gas storage in response to the power shortages. Therefore, different power shortage spots correspond to different responding CHP units. Supposing the power shortages are at \({L}_{G}\):
Thermal deviation at \({L}_{G}\) is caused by the increased heat output of CHP units at \({L}_{G}\) and the reduced electricity of electric boilers at \({L}_{G}\), whereas thermal deviations at \({L}_{K}\) are caused by the reduced electricity of electric boilers at \({L}_{K}\).

3.
Total gas inertia output constraint
$${\text{q}}_{{L_{{\text{n}}} }}^{G,m} \left( {\text{t}} \right)  {\text{q}}_{{L_{{\text{n}}} }}^{G,m} \left( {{\text{t}}_{0} } \right){ = }\frac{{\sigma_{{{\text{conv}}}} P_{RG}^{n} \left( t \right)}}{{1000H_{{{\text{gas}}}} }} \quad n = 1,2,3$$(20)where \({q}_{{L}_{n}}^{G,m}\)(\(t\)) and \({q}_{{L}_{n}}^{G,m}\)(\({t}_{0}\)) are the gas flow of \({L}_{m}\) at time \(t\mathrm{ and }{t}_{0}\) with power shortages at \({L}_{n}\) (m, nâ€‰=â€‰1, 2, 3), respectively. \({\sigma }_{conv}\) is the conversion coefficient from â€˜kWhâ€™ to â€˜Jâ€™, and \({H}_{gas}\) is the calorific value of gas. Different measuring units of \({P}_{RG}^{n}\)(\(t\)) and \({q}_{{L}_{n}}^{G,m}\)(\(t\)) are unified using (22) shown below.

4.
Gas flow, pressure and storage related constraints
The gas flow, pressure and storage related constraints are the extensions of the abovementioned (8), (9), and (11), given as:
$$\left\{ {\begin{array}{ll} {V_{mn}^{G} \left( {\text{t}} \right) = \frac{\pi }{4}\frac{{L_{mn}^{G} d_{mn}^{G} }}{{R_{M} T^{G} Z\rho_{0} }}\frac{{P_{{\text{m}}}^{G} \left( {\text{t}} \right) + P_{n}^{G} \left( {\text{t}} \right)}}{2}} \hfill \\ \left[ {P_{m}^{G} \left( t \right)} \right]^{2}  \left[ {P_{n}^{G} \left( t \right)} \right]^{2} \hfill \\ \,\,\,\,\,\, = 1.62\lambda \rho_{0} P_{0}^{G} \frac{{T^{G} }}{{T_{0}^{G} }}\frac{Z}{{Z_{0}^{G} }}\frac{{L_{mn}^{G} }}{{\left( {d_{mn}^{G} } \right)^{5} }}\left[ {\frac{{q_{mn,in}^{G} \left( t \right) + q_{mn,out}^{G} \left( t \right)}}{2}} \right]^{2} \hfill \\ {V_{mn}^{G} \left( {t_{0} } \right) = V_{mn}^{G} \left( {t_{0}  T} \right) + T\left[ {q_{mn,in}^{G} \left( {t_{0} } \right)  q_{mn,out}^{G} \left( {t_{0} } \right)} \right]} \hfill \\ \end{array} } \right.$$(21)
5.
Nodal flow of natural gas constraints

5.
The nodal flow of natural gas is similar to the nodal flow of electricity and follows the principle that the total incoming flow at the node equals the total outcoming flow. The flow rate of natural gas at the loads remains consistent with the change in load, and changes only when the load demand changes.
When the power shortages are at \({L}_{1}\), \({q}_{{L}_{1}}^{G,2}\)(\(t\)) and \({q}_{{L}_{1}}^{G,3}\)(\(t\)) remain the same:
When the power shortages are at \({L}_{2}\), \({q}_{34,out}^{G}\)(\(t\)) changes with \({q}_{{L}_{2}}^{G,2}\)(\(t\)) while \({q}_{{L}_{2}}^{G,1}\)(\(t\)) and \({q}_{{L}_{2}}^{G,3}\)(\(t\)) remain the same:
For power shortages at \({L}_{3}\), \({q}_{67,out}^{G}\)(\(t\)) changes with \({q}_{{L}_{3}}^{G,3}\)(\(t\)) while \({q}_{{L}_{3}}^{G,1}\)(\(t\)) and \({q}_{{L}_{3}}^{G,2}\)(\(t\)) remain the same, as:

6.
Power output related upper and lower limits constraints
$$\left\{ {\begin{array}{*{20}c} {0 \le P_{RH}^{n} \left( t \right) \le P_{RH,\max }^{n} } \\ {0 \le P_{RG}^{n} \left( t \right) \le P_{RG,\max }^{n} } \\ \end{array} } \right.$$(25)where \({{P}_{RH,max}^{n},\mathrm{ and }P}_{RG,max}^{n}\) are the upper limits of thermal and gas inertia outputs at \({L}_{n}\) (nâ€‰=â€‰1, 2, 3), respectively. Both outputs of thermal and gas inertia have upper limits to ensure the safe operation of IESs.

7.
Gas related upper and lower limits constraints
$$\left\{ {\begin{array}{*{20}l} {V_{mn,\min }^{G} \le V_{mn}^{G} \left( {\text{t}} \right) \le V_{mn}^{G} \left( {{\text{t}}_{0} } \right)} \hfill \\ {P_{a,\min }^{G} \le P_{a}^{G} \left( t \right) \le P_{a,\max }^{G} \quad 1 \le a \le 7} \hfill \\ {P_{c + 1}^{G} \left( t \right) \le P_{c}^{G} \left( t \right) \quad 1 \le c \le 6,c \ne 4} \hfill \\ \end{array} } \right.$$(26)
\({V}_{mn,min}^{G}\) is the lower limit of gas storage in pipe \(mn\), while \({{P}_{a,min}^{G}\mathrm{ and }P}_{a,max}^{G}\) are the lower and upper limits of gas pressure at node a (\(1\le a\le 7\)), respectively. Excessive consumption of gas storage will lead to abnormal low nodal pressure, thus affecting the downstream load gas supply.

8.
Heatelectricity ratio constraint
$$\eta_{{L_{n} }}^{CHPE} \le \eta_{{L_{n} }}^{CHPH} \le 0.95$$(27)
The adjustment range set in this paper is 1.0â€“1.9 so that the power support model can adjust the heatelectricity ratio of CHP units by itself according to the requirements of the objective function.
4 Case study
This case is simplified from a northern electricity, gas, and heat coupling IES. The typical summer day of the electric power load and thermal load data is shown in Fig.Â 6, which illustrates the background load condition of the studied IES. Based on this background load condition, the topology structure of the gas network in this case adopts the 8node gas network structure diagram shown in Fig.Â 2, and is commonly used in related natural gas network research. The internal structure containing the gas network topology of the studied IES is shown in Fig.Â 4.
All relevant constant parameters are shown in Table 1, and parameters of the gas network are shown in Table 2. The unit costs of gas storage \({c}_{RG,{L}_{N}}^{n}\) and thermal inertia \({c}_{RH,{L}_{N}}^{n}\) are shown in Tables 3 and 4, respectively. The demand side output and the heatelectricity ratio adjustment adopt step prices shown in Tables 5 and 6, respectively. The power support optimization is carried out in MATLAB 2019 with the toolbox of YALMIP. The concepts of setting up the next three case studies are as follows:

1.
SectionÂ 4.1 provides an analysis of the gas network conditions including the changes of gas pressure, gas flow and gas storage under different power shortage spots. Various changes in the gas network verify the value of considering gas network constraints in the power support model, and establish the foundation for the following study.

2.
To further prove the significance of considering gas network constraints, the proposed strategy in this paper is compared with similar power support strategies considering the gas network as an entirety [17] in Sect.Â 4.2.

3.
Based on the comparisons in Sect.Â 4.2, it is found that power shortage spots in the gas network may affect power support results, which are analysed in Sect.Â 4.3.

4.
To improve the application value of the proposed strategy, circumstances are given in Sect.Â 4.4 where the gas network can be simplified to reduce modelling complexity and increase computing speed.
4.1 Analysis of the network condition with different power shortage spots
Based on the 8node gas network structure shown in Fig.Â 2, the changes of gas pressure, gas flow rate and gas storage in the pipelines of the whole gas network are studied assuming power shortages of 500 kWh/s occurring respectively at \({L}_{13}\) for 3600Â s. The network conditions with the power shortages at \({L}_{13}\) are shown in Figs. 7, 8 and 9, respectively. FiguresÂ 7a and 8a demonstrate the consumptions of gas storage in each pipe, while Figs. 7b and 8b demonstrate the gas flows in the steady state before power shortages and at pipe heads and ends after power shortages. FiguresÂ 7c and 8c demonstrate the changes of gas pressure at different nodes and the corresponding change trends of pressure before and after power shortages. The results of power support regarding costs and outputs of power forms are listed in Table 7aâ€“c.
Instead of considering the gas network as an entirety and focusing only on input and output flows, the parameters of each pipeline can vary in a more complex way when taking into account the gas network. Therefore, the power support model considering unified gasthermal inertia can be established more comprehensively when the variations are considered in detail in the optimization model.
From Figs. 7a, 8a and 9a, the increase of load demand at one load group will consume gas storage in almost all the pipelines in the network, especially in the pipelines closest to the power shortage spot.
In Figs. 7b, 8b and 9b, the gas flows at pipe heads and ends remain the same in the steady state but deviate after power shortages. Gas inertia output is increased to support power shortages at one load, and this leads to the sudden rise in the gas flow, especially at the end of the pipe where the load group is connected. Therefore, the respective gas flows at the end of the pipes \({PL}_{12}\), \({PL}_{34}\), and \({PL}_{67}\) increase more significantly than other flow variations in pipelines when power shortages are at \({L}_{1}\), \({L}_{2}\), and \({L}_{3}\), respectively. The change of flow rate can also reflect the consumption of gas storage by the area between the head and end flows. Gas storage is consumed/stored when the flow rate at pipe heads is smaller/larger than the flow rate at pipe ends. Therefore, the variation trends of the area between the head and end flows in Figs. 7b, 8b and 9b are consistent with the variation trends of the gas storage in Figs. 7a, 8a and 9a.
In Figs. 7c, 8c and 9c, the change of pressure is caused by the variation of gas flow. Since the natural gas flowing out of the pipeline is too fast to be replenished in a short time, the rise in the gas flow will lead to a delayed fall in pressure. The pressure drop at the power shortage spot is particularly large given the largest gas storage supply, so the lowest point of the pressure change trend before and after power shortages is always at the power shortage spot.
In Table 7aâ€“c, both power forms of thermal inertia and demand side have three outputs representing the outputs from the three load groups. Table 7 shows that power shortages at different spots can result in different total costs and power outputs. Although the difference is not so large with the same failure size and duration, the results still prove the value of considering gas network constraints in the power support model. Next, the proposed strategy is compared with existing power support strategies without full consideration of the gas network constraints [17]. The comparison will further prove the significance of considering gas network constraints.
4.2 Analysis of the strategy comparisons between considering and not considering gas network constraints
Without considering the network influence, the gas network is considered in its entirety and the power support model no longer distinguishes from which pipeline the gas inertia output comes. The unit cost of gas inertia is uniformly set as 2.5065 Â¥/(NÂ m^{3}) while other forms of power support remain the same. The results of the power support strategies considering and not considering gas network constraints are compared in Figs. 10, 11 and 12 with the power shortage occurring at \({L}_{13}\). The comparison of the total costs between considering and not considering gas network constraints is specifically shown in Table 8.
Table 8 demonstrates the differences of total costs between considering and not considering the gas network constraints. The total costs considering gas network constraints under different sizes of power shortages are all higher than those not considering gas network constraints. The reason is that the unit cost of gas inertia is uniformly set as 2.5065 Â¥/(N m^{3}) in the case of not considering the gas network. This is the lowest in the case of considering gas network. Considering the impact of power shortages at one location on the whole gas network, the unit cost of gas inertia will be increased. Therefore, the total cost considering gas network constraints is bound to rise. However, the changing trend of the total cost difference between considering and not considering gas network constraints CostCostN in Table 8 shows that CostCostN is generally lower when the power shortages are small, and begins to increase when the power shortage size reaches a certain degree. This trend implies that under certain circumstances, the size of the total cost can be estimated considering the simplification of gas network constraints. Specific simplified scenarios will be described in the following paragraphs.
FiguresÂ 10a and 11a evaluate the power support strategy economy by presenting the total costs of the strategy, with and without considering gas network constraints, under different power shortages. The difference of total costs between considering and not considering the network is also plotted for clearer observation. In general, the total costs considering the network under different power shortages are higher than those without considering the network. The main reason is that the gas inertia unit cost without considering networks is the lowest among all gas inertia unit costs considering networks, while unit costs of the other outputs are the same. However, there is a turning point on the cost difference curve. The cost difference remains unchanged regardless of how much the power shortages are before the turning point, but increases with the increase of power shortages after the turning point.
FiguresÂ 10b and 11b and Figs. 10c and 11c present the heatelectricity ratio and gas inertia output considering and not considering the gas network, respectively. It can be seen that there are also turning points on both the heatelectricity ratio difference curve and the gas inertia output difference curve. There are fluctuating differences between considering and not considering the gas network in the heatelectricity ratio and gas inertia output before the turning point. These drop to zero after the turning point regardless of the size of the power shortages. FiguresÂ 10d and 11d show the thermal inertia output differences from three load groups. Except for the thermal inertia output from the power shortage spot, which does not differ between considering and not considering the gas network, other thermal inertia output differences all drop to zero after the turning point.
It is worth noting that all the turning points are at the same power shortage size of 370 (kWh/s), which is not a coincidence. By observing each power output size and total cost before and after the turning point, the phenomenon can be explained as follows:

1.
The unit cost of thermal inertia is lower than that of gas inertia. Assuming that user comfort is not significantly affected, the optimization model of power support will always prefer the use of thermal inertia output until it reaches the upper limit.

2.
Before the turning point, the power shortage size is not very large, and the thermal inertia can respond to power shortages with gas inertia output without reaching its upper limit. In this way, the optimization model considering the gas network is not constrained by the upper limit of thermal inertia, and can always flexibly use more thermal inertia output to avoid the higher cost of gas inertia, since the biggest difference between considering and not considering the gas network is the higher unit cost of gas inertia. Therefore, outputs when considering and not considering the gas network are different under different power shortages but the extra total costs remain the same.

3.
After the turning point, both the outputs of thermal inertia considering and not considering the gas network reach the upper limits and the optimization flexibility is constrained. Therefore, with the increase of power shortages, the output of gas inertia will definitely increase and the difference of the total cost when considering and not considering the gas network will also increase, given the higher unit cost of gas inertia considering the gas network under the same gas inertia output.
Considering the gas network can improve the power support model. The differences in the total cost, heatelectricity ratio, and gasthermal inertia output in the comparison results also verify the research significance. However, based on the abovementioned same turning point at different power shortage spots, it can also be concluded that in some conditions the network can be simplified if only a rough estimate is to be made in practical engineering application. In the case studied in Sect.Â 4.2, the turning point is the power shortage size of 370 (kWh/s). Solving the optimization model without considering the gas network can directly obtain the heatelectricity ratio and gasthermal inertia output when the actual power shortages are larger than 370 (kWh/s). In contrast, the total cost without considering the gas network has the smallest deviation from the total cost considering the gas network when the actual power shortages are smaller than 370 (kWh/s). The advantages of the estimate without considering the network influence include easier modelling and faster computing speed, which are especially crucial to the strategy response speed. The computing speeds of the models with and without considering the gas network are 0.49Â s and 0.07Â s, respectively. However, the scales of actual engineering projects are much larger than the 8node gas network used in the paper, so the differences in computing speed and modelling complexity will be further widened, and thus the importance of simplifying the gas network is further increased.
As for the power shortages at \({L}_{3}\) in Fig.Â 12, the turning point of the total cost difference is still 370 (kWh/s), but the turning points of the other three outputs are smaller while the overall differences in the three outputs have all decreased. This phenomenon does not affect the above conclusion obtained by setting the turning point as 370 (kWh/s), but it does indicate that the impact of the network is reduced when the power shortages are at \({L}_{3}\). Therefore, it is worth studying whether power shortage spots in the gas network can affect the power support results.
4.3 Analysis of the influence of power shortage spots
The respective differences in total cost, heatelectricity ratio and gasthermal inertia output with and without considering the network influence during the power shortages at \({L}_{13}\) are shown in Fig.Â 13.
Generally, the four differences with power shortages at \({L}_{12}\) are close in size while the four differences with power shortages at \({L}_{3}\) are especially small. To analyse this phenomenon, all pipelines can be divided into 2 groups, with those in group 1 directly connected to the load and group 2 the remaining pipes in the gas network. Therefore, \({PL}_{12}\) and \({PL}_{23}\) are in group 1 when the power shortages are at \({L}_{1}\), \({PL}_{34}\) is in group 1 when the power shortages are at \({L}_{2}\), while \({PL}_{67}\) is in group 1 when the power shortages are at \({L}_{3}\). According to Figs. 7, 8 and 9, pipelines in group 1, given their direct connection to the load when power shortages occur, contribute more gas storage than those in group 2. However, there is only a small difference in the contribution of gas storage between the pipelines in group 1 and 2 when the power shortages are at \({L}_{12}\), but a great difference when the power shortages are at \({L}_{3}\). In other words, the contribution of gas storage is largely evenlydistributed to all the pipelines when the power shortages are at \({L}_{12}\), but mostly distributed to pipelines in group 1 when the power shortages are at \({L}_{3}\). The unit cost of gas storage in pipelines of group 1 is the same as that of gas storage without considering the gas network, which explains the reason why the four differences with the power shortages at \({L}_{3}\) are not so large in comparison with \({L}_{12}\).
As for the reason of the low gas network impact on \({L}_{3}\), it may be due to the fact that the load is at the downstream of the entire natural gas network and will not further affect the downstream gas storage because of the decrease in pressure caused by power shortage support. Although \({L}_{2}\) is also connected to the end of the pipe \({PL}_{34}\), it is not really the downstream load given that the gas flow in \({PL}_{36}\) is from node 3 to node 6, which to some degree constrains the gas storage consumption in \({PL}_{34}\). Therefore, the power shortage spots can also be used as the basis to determine whether the network influence can be ignored. However, the 8node gas network studied in this paper only consists of three load groups and the analysis of power shortage spots influence is inexhaustive. At the next stage, the power support model will be further refined based on the gas network with more load groups so that more data can be obtained for analysis. However, an improved wellgrounded basis, which considers the power shortage spots influence to decide when the gas network influence can be ignored, has yet to be discovered.
4.4 Analysis of the influence factor of the power shortage turning point
The power shortage turning point of 370 (kWh/s) mentioned in Sect.Â 4.2 has been proved to provide the basis for the rough estimate of the system operational condition without considering network influence. The rough estimate is practical in the prediction of the total cost/power output and the improvement of the strategy response speed in engineering application. Typically, this specific turning point cannot be directly obtained from the basic parameters of IESs, which are necessary for the indepth study of the influence factor of the power shortage turning point.
Based on the analysis in Sect.Â 4.2, it is found that the influence factor of the power shortage turning point is mainly the upper limit of the thermal inertia output, which can be directly obtained as the parameters of the IES itself. Therefore, the power shortage turning points \({S}_{P}\) under different upper limits \({U}_{L}\) of thermal inertia output are shown in Fig.Â 14 with a fitted curve of the relationship between \({S}_{P}\) and \({U}_{L}\). The curve appears as the form of a linear function, which provides an approach to obtain \({U}_{L}\) when \({S}_{P}\) of the IES is known.
In order to illustrate the linear relationship between \({S}_{P}\) and \({U}_{L}\), the specific relationships of thermal inertia output and power shortage size under five different upper limits of thermal inertia output are given in Fig.Â 14. Since the thermal inertia output comes from three different load groups, this section only chooses thermal inertia output from one load with higher unit output cost to study while the same is true for the other two loads. Before reaching the upper limit, the thermal inertia output increases proportionally with the increase of power shortages and remains unchanged after reaching the upper limit, which is exactly the time of the flexibility of thermal inertia output being constrained and the appearance of the power shortage turning point. With the regular increase of the upper limit of thermal inertia output, the corresponding curve will also shift to the upright proportionally. Therefore, the five curves can be expressed before reaching the upper limit of thermal inertia output, as:
The five turning points \(\left({S}_{1}^{m},{P}_{RH1}^{m}\right)\), \(\left({S}_{2}^{m},{P}_{RH2}^{m}\right)\), \(\left({S}_{3}^{m},{P}_{RH3}^{m}\right)\), \(\left({S}_{4}^{m},{P}_{RH4}^{m}\right)\), and \(\left({S}_{5}^{m},{P}_{RH5}^{m}\right)\) fit in each of these five formulas:
The slopes of the lines formed between these five points are found to be the same, which can prove that these five points are on the same line and the slope is given in (30). These 5 points are just 5 special cases on the curve of the relationship between \({S}_{P}\) and \({U}_{L}\), but the illustration of these 5 points is enough to explain the forming reason of the linear \({S}_{P}\)\({U}_{L}\) curve.
The fitted curve in Fig.Â 14 has application value in real projects. When the upper limit of the thermal inertia output in an IES is known, the actual size of power shortages can be compared with the power shortage turning point to decide whether the gas network constraints can be simplified and which parameters can be roughly estimated.

1.
If the actual power shortages are larger than the power shortage turning point, the heatelectricity ratio and gasthermal inertia output can be directly estimated without considering the gas network.

2.
If the actual power shortages are smaller than the power shortage turning point, the total cost without considering the gas network has the smallest deviation from that considering the gas network.
Therefore, the curve of S_{P}U_{L} can guide IES power output schemes in engineering practice. Since the upper limit of thermal inertia output is usually the known parameter of IESs, the power output scheme and the strategy economy can be estimated and evaluated before actual power support takes place.
5 Conclusion
In this paper, the definition and modelling of gas inertia in integrated energy systems are described. A multienergy inertiabased power support strategy considering the gas network constraints is proposed with its effectiveness verified in an actual scenario. The following conclusions can be drawn:

1.
Gas inertia can provide buffer space to accommodate power fluctuations. This provides the foundation for gasthermal inertia utilization in the power support strategy.

2.
The significance of considering gas network constraints is verified through the comparisons of results of not considering gas network constraints. By consideration of the gas network, the optimization model accuracy for total cost has been improved by about 1.8â€“3.0%.

3.
Despite the gas network influence on the power support results, it can still be simplified under certain circumstances to obtain a rough estimate of the power support condition. The prediction error of the total cost after simplification is about 2.8%, and the prediction error of heatelectricity ratio, gasthermal inertia outputs after simplification is about 0.1%. Gas network simplification can reduce modelling complexity and increase the computing speed by about 34% for the improvement of the response speed of the power support strategy.
In future research, the multienergy inertiabased power support model will be further refined based on a more specific natural gas network model considering the gas compressor modelling or more connected load groups. The current relatively conservative integrated energy system power support scheme can be improved for expanded application in a more practical complex system.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
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Funding
This work was supported by National Key R&D Program of China (No. 2019YFE0118000).
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CM: Methodology, Data curation, Investigation, WritingOriginal draft preparation. QW:Â Conceptualization, Methodology, Supervision, Reviewing. YT: Supervision.Â All authors read and approved the final manuscript.
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Miao, C., Wang, Q. & Tang, Y. A multienergy inertiabased power support strategy with gas network constraints. Prot Control Mod Power Syst 8, 18 (2023). https://doi.org/10.1186/s4160102300292z
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DOI: https://doi.org/10.1186/s4160102300292z