Abstract
The essential task of integrated electricity-heat systems (IEHSs) is to provide customers with reliable electric and heating services. From the perspective of customers, it is reasonable to analyze the reliabilities of IEHSs based on the ability to provide energy services with a reasonable assurance of continuity and quality, which are termed as service-based reliabilities. Due to the thermal inertia existing in IEHSs, the heating service performances can present slow dynamic characteristics, which has a great impact on the service satisfaction of customers. The neglect of such thermal dynamics will bring about inaccurate service-based reliability measurement, which can lead to the inefficient dispatch decisions of system operators. Therefore, it is necessary to provide a tool which can analyze the service-based reliabilities of IEHSs considering the impacts of thermal dynamics. This paper firstly models the energy service performance of IEHSs in contingency states. Specifically, the nodal energy supplies are obtained from the optimal power and heat flow model under both variable hydraulic and thermal conditions, in which the transmission-side thermal dynamics are formulated. On this basis, the energy service performances for customers are further determined with the formulation of demand-side thermal dynamics. Moreover, a service-based reliability analysis framework for the IEHSs is proposed utilizing the time-sequential Monte Carlo simulation (TSMCS) technique with the embedded decomposition algorithm. Furthermore, the indices for quantifying service-based reliabilities are defined based on the traditional reliability indices, where dynamic service performances and service satisfactions of customers are both considered. Numerical simulations are carried out with a test system to validate the effectiveness of the proposed framework.
WITH the growing need for improving energy utilization efficiency and reducing environmental pollution, integrated energy systems (IESs) have gained rapid development during the past few decades [
With the increasing interdependence between EPS and DHS, the customers may suffer energy-related service risks due to the random failures in IEHSs. For example, the malfunction of coupled components such as combined heat and power (CHP) units may result in the degradation or interruption of both the electric and the heat power supply for customers, which further affects the corresponding energy services of customers [
The reliabilities of the individual energy subsystems, e.g., EPSs and DHSs, have been well studied in the previous research works [
As the two forms of energy service for customers, the electric and heating services can show different dynamic characteristics in contingency states. In specific, the electric services would degrade instantaneously when the failure occurs, while the heating services would degrade gradually and the heating requirements of customers can be maintained for a period of time [
There have been some previous research works [
In order to quantify the service-based reliabilities of IEHSs, the pertinent reliability indices are needed. The existing reliability indices for EPSs are usually steady-state which are utilized to reflect the system reliability levels over a long period [
In order to bridge the research gaps, this paper contributes in the following aspects:
1) The service-based reliability of IEHS is firstly defined and analyzed in this paper. The system reliability is evaluated based on the energy service conditions of customers rather than the energy supply conditions in the traditional reliability research works.
2) The energy service performance of IEHS for customers in contingency states is modelled considering thermal dynamics. In specific, the nodal energy supplies are obtained from the optimal power and heat flow (OPHF) model under both variable hydraulic and thermal conditions, in which the transmission-side thermal dynamics are formulated. Then, the energy service performances for customers are further determined with the formulation of demand-side thermal dynamics.
3) A service-based reliability analysis framework for the IEHSs is proposed based on the time-sequential Monte Carlo simulation (TSMCS) technique. During the inner loop of the TSMCS, a hydraulic-thermal decomposition algorithm is embedded to realize the tractable calculation of OPHF model, which characterizes the chronological impacts of thermal dynamics on service-based reliabilities.
4) Conventional steady-state reliability indices are re-defined to quantify the time-varying service-based reliabilities of IEHSs. The dynamic service performances and service satisfactions of customers are both considered in the definition of the indices to ensure the accuracy and effectiveness of reliability evaluation.
In this section, the impacts of thermal dynamics on service-based reliabilities of IEHSs are firstly described with the schematic diagram shown in

Fig. 1 Thermal dynamics in IEHSs and their impacts on service-based reliabilities. (a) Transmission-side and demand-side thermal dynamics in IEHSs. (b) Energy service performances affected by thermal dynamics.
In a certain contingency state, the energy service performances would be affected by thermal dynamics in IEHSs, as shown in the shaded area of
As analyzed above, the performances of energy services present dynamic variation processes in contingency states. With various kinds and degrees of failures, the dynamic performances would take different forms such as the slight degradation for minor disturbances and severe degradation for large failures in IEHSs. Considering the possible scenarios with different performances, the overall service-based reliability could be determined, which denotes the ability of IEHSs to provide adequate energy services for customers. Moreover, the reliability levels of IEHSs in this manner would be affected by thermal dynamics, which should be consequently considered in the service-based reliability analysis.
To analyze the service-based reliabilities of IEHSs, the energy service performances in contingency states are modelled and calculated considering thermal dynamics in this section. Firstly, the reliability models of components in IEHSs are proposed to characterize the state transition processes, where both the electric and heat performances in each state are considered for coupled components especially. Then, the transmission-side and the demand-side thermal dynamics are considered, respectively, in the two stages for modelling the energy service performances in contingency states. Specifically, the contingency electric and heat power supply for customers at each node in IEHSs is determined considering the thermal storage of pipelines. According to the nodal contingency energy supply, the energy service performances of customers are measured considering thermal inertia of buildings.
Random failures of coupled components (e.g., generator units, electric lines, pipelines) could make the IEHS enter the contingency state. To analyze the contingency performance of the IEHS, the reliability modelling of coupled components is conducted firstly. In practice, the components usually present more than two exclusive states (e.g., states) from perfectly functioning s0 to complete failure sM. The intermediate states of components () are characterized by the partially functioning performances considering performance degradation [
(1) |
where and are the failure rate and repair rate from states i to j, respectively.
Different from the other components related to one type of energy, the coupling components in IEHSs, e.g., CHP units and electric boilers (EBs), have both the electric and the heat performances. To comprehensively measure the performances of the coupling component in a certain state, a two-dimensional parameter represented as can be used, where and denote the electricity and heat generation capacities in state s, respectively. Especially for electric heating devices such as EBs, is negative to describe the electric power consumption behavior. Therefore, the two-performance multi-state reliability model for coupling components can be developed, as shown in

Fig. 2 Two-performance multi-state reliability model for coupling components.
B. Contingency Energy Supply Determination for Customers Considering Transmission-side Thermal Dynamics
In the contingency state, the IEHS will be re-dispatched and suffer the risk of energy load curtailment. Under this circumstance, the contingency energy supply for customers can be determined by calculating the energy load curtailment based on the thermal dynamic model of pipelines and the OPHF technique [
As modelled in the previous subsection, failures of coupled components may lead to the reduction of their heat generation capacities. After that, heating services could be maintained for a while considering the propagation time from the failure location to geographically distributed customers, which has been theoretically analyzed in Section II. Such dynamic process is related to the storage capability of insulated pipelines in IEHSs, which could be modelled based on the node method [
The basic idea of this method is to represent the outlet temperatures of pipelines using historic inlet temperatures [
Firstly, the outlet temperature is represented as the mean temperature of the outflowing mass flow blocks at the tail of the pipeline:
(2) |
(3) |
where and are the temperatures of the two outflowing mass flow blocks in pipeline p; is the mass flow rate in pipeline p; and are the proportion coefficients corresponding to the two outflowing mass flow blocks; and are the two masses for determining and , respectively; and , , and are the pipeline parameters denoting the mass density, cross-section area, and total length, respectively.
Then, the temperatures of outflowing mass blocks can be calculated by historic inlet temperatures at different time steps considering time delays in pipelines:
(4) |
(5) |
where and are the current outlet temperatures and historic inlet temperatures of the two out-flowing mass flow blocks without heat losses, respectively; and and are the integers for time intervals calculated by rounding the precise transfer delays.
Besides time delays, heat losses during heat transmission along the pipeline should also be characterized. Hence, inlet temperatures considering heat losses can be modelled as:
(6) |
(7) |
where and are inlet temperatures of pipeline p at times and , respectively; and are the heat loss factors; is the environment temperature of the pipeline; is the heat transfer coefficient; and are the equivalent lengths; is the specific heat capacity of water; and is a symbol for variables without considering heat losses.
Combining (2), (4), and (6), the relationship between the outlet and inlet temperatures considering both time delays and heat losses is formulated as:
(8) |
When the mass flow rate is fixed, , , , and will become constant parameters and the relationship formulated in (8) will be linear. This feature will be used by the decomposition algorithm to solve the following optimal load curtailment model which integrates the formulated thermal dynamic model of pipelines. The detailed algorithm will be given in Section IV-B.
Based on the OPHF technique, the optimal load curtailment model of IEHSs is formulated which can determine the nodal contingency energy supply. The objective of the model is to minimize the total system operation cost in the contingency state s during the studied period:
(9) |
denotes the cost related to nodal electric power supply, including the generation cost of CHP units and non-CHP thermal units along with the electric load curtailment cost. denotes the cost related to nodal heat power supply considering both heat generation of CHP units and heat load curtailments. Moreover, the electric loads are divided into heating and non-heating loads considering the electric heating devices at the demand side, and the curtailment of the former can affect the heating service performances.
The objective function is subject to the constraints of the IEHS, including the coupled component constraints, DHS constraints, and EPS constraints. It should be noted that the superscripts s for variables in (9) are omitted for simplicity.
Coupled components in IEHSs mainly include the CHP units and the electric heating devices such as EBs. Each coupled component generates the electric or heat power within or for a certain state as defined in Section III-A:
(10) |
(11) |
where and are the generated electric power and heat power of the CHP unit c, respectively; and are the consumed electric power and generated heat power of the EB k at time t, respectively; and is the operator for calculating the absolute value.
Besides, the electric and heat energy behaviors for each state of these components are strongly coupled. Regarding the electric heating devices, the generated heat power is proportional to the consumed electric power, as denoted in (12). Regarding the CHP units, the coupling relation between electric and heat generation can be described by the polyhedron feasible operating region [
(12) |
(13) |
where is the conversion efficiency of EB k; is the number of extreme points in polyhedron operating region of CHP unit c; and are the electric power and heat power corresponding to the
Moreover, the generated heat power of the components can heat the mass flow at the nodes where they locate in IEHSs:
(14) |
where is the mass flow rate at node i and time t; and are the supply and return mass flow temperatures at node i, respectively.
Similar to (14), in the heat load nodes, the heat load power of customers is satisfied by the heated mass flow of the connected pipelines in IEHSs:
(15) |
where and are the original heat load and the heat load curtailment at node i and time t, respectively.
In order to guarantee the heating service quality and prevent steam forming, temperatures of both supply water and return water are bounded as:
(16) |
where , and , are the upper and lower boundaries of and , respectively.
The temperature of the confluence node is the weighted average value of outlet temperatures in all pipelines ending at that node [
(17) |
where is the set of pipelines ending at node i; is the outlet temperature of pipeline p at time t; and is the mixed temperature at node i and time t.
According to the Darcy-Weisbach equation [
(18) |
where is the pressure loss along pipeline p at time t; is the coefficient of pressure loss in pipeline p; is the set of pipelines forming a closed loop; and denotes that the directions of mass flow in pipeline p and the loop are consistent, while denotes that the two directions are opposite.
The EPS operation in the contingency state mainly subjects to the following constraints.
(19) |
(20) |
(21) |
(22) |
(23) |
where and are the upper and lower boundaries for power output of non-CHP generation unit g at node m, respectively; and are the upward and downward ramping capabilities of non-CHP generation unit g at node m, respectively; , and , are the original non-heating and heating electric loads and the corresponding load curtailments at node m, respectively; , , and are the shift distribution factors from unit g, unit c, and node m to line l, respectively; and is the transmission capacity determined by dynamic thermal rating (DTR) [
Since the conventional static thermal rating (STR) generally sets the fixed low ratings of power lines with the conservative assumption of weather conditions, it could lead to the underutilization of line capacities [
(24) |
where , , , and are the line conductor temperature, environment temperature, wind speed, and incident wind angle, respectively; is the line current; is the line conductor resistance changing with ; is the convection heat loss as a function of , , , and ; is the radiated heat loss as a function of and ; and and are the heat gains from solar radiation and line conductivity, respectively. In DTR model, the line rating denotes the current that yields the maximum allowable conductor temperature with the given weather parameters, which further determines the line capacity in (23).
After obtaining the nodal load curtailment in the formulated optimization problem (9)-(23), the contingency energy supply for customers can be calculated using (25) and (26). Here, the contingency electric power supply is divided into the heating and non-heating parts, as represented in (26).
(25) |
(26) |
where and are the heating and non-heating parts of the contingency electric power supply at node m, respectively; and is the contingency heat power supply at node i.
According to the contingency energy supply conditions in the previous stage, the energy service performances of IEHSs for customers can be further calculated. Since the electric service interruptions are mostly static, the corresponding contingency performances can be measured by the supplied load power directly. In contrast, the heating service performances are greatly affected by the demand-side thermal dynamics, which are measured by physical variables related to time-varying temperatures in this paper.
In the demand side of IEHSs, the heating services are mainly provided by the heat power from DHSs. Besides, for customers equipped with electric heating devices, the partial heat power demands can also be satisfied by the electric power supply. Therefore, the total contingency heat power supply for customers is expressed as:
(27) |
where is the total contingency heat power supply for customers at heat node i; is the heat power at node i converted from the electric power at node m; and is the conversion efficiency.
When the heat power supply is insufficient, the heat loss of the building is a slow and transient process due to the insulation structures, which can be demonstrated by the dynamic indoor temperatures. Since the heating service satisfaction of customers is usually determined by the indoor temperatures, they are utilized as one of the physical variables to measure the heating service performances in contingency states of IEHSs, as shown in

Fig. 3 Thermal dynamic model of buildings in the demand side.
The indoor temperature variation is related to both the contingency heat power supply from the IEHSs and the heat exchange with the outdoor environment, which is formulated utilizing the first-order equivalent thermal parameter (ETP) model [
(28) |
where Tib is the indoor temperature of the equivalent building b at node i; and and are the heat capacity and thermal resistance [
Furthermore, the variation of the indoor temperature at each time slot is obtained by discretizing the differential equations as modelled in (28).
Obviously, the indoor temperature variation is limited by the time interval as illustrated in (29), and the temperature would reduce over time before the heat equilibrium is achieved. In other words, the heating services will not be lost immediately from the perspective of customers.
(29) |
To reflect dynamic energy losses during the temperature degradation period, the heating service performances also need to be measured by power. Hence, the equivalent heat load power modelled by temperatures is used as another physical variable for heating service performances, which is defined as follows:
(30) |
In the above subsections, thermal dynamics have been characterized in the two stages for determining energy service performances in contingency states. These dynamic processes are equivalent to buffers for the energy supply shortage in the contingency states, which are beneficial for the IEHS to provide reliable energy services. Therefore, it is valuable to analyze the factors that affect the dynamic processes for investigating the potential improvement of the service-based reliabilities.
According to the specific thermal dynamic models, there are several factors that could affect the transmission-side and demand-side thermal dynamics. Regarding the transmission-side thermal dynamics as formulated in (2)-(8), the factors include the pipeline parameters such as the length L and the cross-section area A. These factors for dynamic processes are related to the failure propagation time in the transmission side of IEHS in contingency states. In addition, based on the model (28) and (29), the factors affecting the demand-side dynamic processes mainly include the heat capacity C and thermal resistance R of the equivalent buildings. These factors determine the thermal inertia of customers, which are related to the dynamic service losses when the energy supply is insufficient. In a word, all of the above factors would have impacts on the service-based reliability levels of IEHS through different dynamic processes in contingency states.
The conventional indices such as the loss of load probability and the expected energy not supplied have been widely used to evaluate the reliability of the EPSs [
Firstly, the equivalent heat load power and dynamic temperatures are used to define the reliability indices for heating services, as shown in
(31) |

Fig. 4 Equivalent heat load power and dynamic temperatures used for defining service-based reliability indices. (a) Heat generation capacity of heat source. (b) Heat load curtailment considering thermal dynamics. (c) Dynamic temperatures of buildings.
Different from the conventional static load curtailment, the equivalent heat load curtailment can reflect the demand-side dynamic heat loss processes, as shown in
(32) |
where is the equivalent heat load curtailment at node and time at the
Moreover, the failure probability regarding heating services with dynamic performances is not suitable to be calculated by simply judging the occurrence of load curtailment. Instead, dynamic temperatures are used to achieve the accurate judgement which is commonly related to service satisfaction of customers, as explained in
(33) |
where is defined as a sign function, and when , and when ; and is the indoor temperature of the equivalent building b at node i and time t at the iteration of TSMCS.
Since the electric service performances are measured by the supplied load power in contingency states, the corresponding reliability indices are calculated according to the time-varying nodal load curtailment. Specifically, the loss of electric service probability and the expected electric service not supplied of the EPS can be expressed as:
(34) |
(35) |
where is the number of nodes in the IEHS.
Since the hydraulic conditions, e.g., mass flow rate, and thermal conditions, e.g., supply temperature, can all be variable, there are several non-linear terms in DHS constraints (14)-(17). Besides, the pipeline dynamic model formulated in (2)-(8) is also non-convex. As a result, it would bring about the computational burden when solving the optimal load curtailment model of IEHSs formulated in Section III-B. Note that both these DHS constraints and the pipeline dynamic model will be linear and convex when the mass flow rates are fixed. On this basis, a hydraulic-thermal decomposition algorithm [
Algorithm 1 : hydraulic-thermal decomposition algorithm |
---|
1: Set the index , and set the temperature variables to their lower boundaries 2: while TRUE do 3: Determine the heat losses at the transmission side and obtain the optimal output of heat sources satisfying the summation of heat loads and losses 4: Determine the mass flow rate variables using (14), (15) and (17), (18) 5: Taking the mass low rates as the fixed values , solve the optimal load curtailment model (2)-(26) 6: Update the temperature variables 7: if then: 8: Stop while loop, and output the solutions for index 9: else 10: Set , , and return to 2 11: end if 12: end while |
The TSMCS technique is used to evaluate service-based reliabilities through numerous iterative simulations for the IEHS. At each simulation, the system state sequence is created during the study period ST based on the state sampling of components, and then the energy service performances of IEHS are further measured. At the last simulation, the proposed indices could be obtained as the final service-based reliability evaluation results for IEHS.
According to the focused time scale, the reliability could be evaluated for either the long term or the short term. Compared with long-term reliability, the short-term reliability could accurately incorporate the time-varying system operating conditions, which is consistent with the concept of operational reliability [
Based on the strong law of large numbers and the central limit theorem, the TSMCS will converge after a certain number of iterative simulations and the solution could satisfy the confidence level [
The flowchart for evaluating service-based reliabilities is as follows.
Step 1: input initial parameters of components in IEHS. Build the multi-state reliability models as expressed in (1) of the components, where or is used to represent the component performance in each state.
Step 2: suppose all the components perform perfectly in the initial time and determine the initial operation condition of IEHS. Set the iteration index .
Step 3: conduct the TSMCS sampling for components and create their state sequences during the whole simulation period. Determine the contingency state of IEHS according to the state sequences of components.
Step 4: obtain the nodal electric and heat power supply in the contingency state of IEHS by solving the optimal energy load curtailment model represented as (2)-(26) and utilizing the hydraulic-thermal decomposition algorithm in Section IV-B.
Step 5: measure the energy service performances for customers using (27)-(30) according to the contingency nodal power supply obtained in Step 4.
Step 6: achieve the service-based reliability indices defined as (31)-(35) according to the energy service performances measured in Step 5.
Step 7: calculate the variance coefficients and check the stopping criterion presented in (36). If (36) is satisfied for the whole period, go to Step 8; otherwise, set and go back to Step 3 for the next iteration.
(36) |
where is the variance coefficient at time t; is the corresponding criterion value for stopping TSMCS; and and are the variances of and , respectively.
Step 8: output the service-based reliability indices of the IEHS obtained in the final iteration.
The proposed method is tested on an IEHS which contains a 30-bus EPS modified from [

Fig. 5 Topology of test IEHS.
The reliability parameters of the components in the test system are presented in Tables
State | s0 | s1 | s2 | s3 |
---|---|---|---|---|
s0 | 0.0022 | 0.0022 | 0.0011 | |
s1 | 0.020 | 0.0010 | 0.0021 | |
s2 | 0.020 | 0.0010 | 0.0021 | |
s3 | 0.020 | 0.0100 | 0.0100 |
Component | (1/hour) | (1/hour) | (MW) | (MW) |
---|---|---|---|---|
G1 | 0.0021 | 0.050 | 10 | 10 |
G2 | 0.0011 | 0.020 | 16 | 16 |
G3 | 0.0010 | 0.025 | 16 | 16 |
G4 | 0.0022 | 0.020 | 20 | 20 |
EB | 0.0021 | 0.050 | 8 | 8 |
The original electric and heat loads in the IEHS are set to their peak values. The equivalent heat capacities of buildings are 0.1 MWh/℃. The initial indoor temperature, environment temperature, and the minimum acceptable temperature of customers are 24 ℃, -4 ℃, and 14 ℃, respectively. The modeling and simulations are conducted by MATLAB R2018a, which is performed on a computer with an 1.80 GHz Inte
This case study is performed in a specific contingency state to demonstrate the energy service performances of the test system utilizing the proposed method. The study period and the time interval are 72 hours and 15 min, respectively. The contingency state of the test system is caused by the complete failure of CHP1 from hours to hours, when and are decreased to zero. The computation time of this case is 6.5 s, which is equivalent to the required time for a single simulation of TSMCS. The simulation results in this contingency state are presented as follows.
Firstly, the nodal electric and nodal equivalent heat load curtailments (PC and EHC) at some typical nodes are shown in

Fig. 6 Nodal electric and nodal equivalent heat load curtailments. (a) Nodal electric load curtailments. (b) Nodal equivalent heat load curtailments.
Regarding the heating service performances, the nodal equivalent heat load curtailments have transient forms with the contingency heat power supply. On the one hand, the dynamic variations can be observed for these nodes during the failure period, which originate from the thermal inertia of buildings. On the other hand, the delay times of load curtailments can be observed from the enlarged view for the initial failure period, which result from the transfer delays of pipelines. These time delays are related to the distances from the failure location to different heat nodes. For heat node 30 (H30) near CHP1, the delay time is only 34 min. However, it reaches 2.81 hours for heat node 16 (H16) with a long distance from CHP1.
Regarding the electric service performances, the load curtailments at different electric nodes present distinct patterns. On the one hand, most of the loads at electric nodes are curtailed immediately at the beginning of the failure period with the contingency electric power supply. On the other hand, different from the stable values of load curtailments at most electric nodes, load curtailments at some electric nodes such as E8 and E28 have transient processes and are delayed for about 34 min. These load curtailments are caused by the decrease of the electric power output of CHP2. When CHP1 fails, CHP2 is re-dispatched and increases its heat power output to compensate for the heat power supply shortage because of the higher priority of heat load. At the same time, the electric power output of CHP2 has to be decreased due to the electricity-heat coupling feature of the extraction-condensing CHP unit, and some electric nodal loads are consequently curtailed. Since the electric power output of CHP2 is related to the heat loads, these electric load curtailments are indirectly affected by thermal dynamics and thus present transient patterns.
As another measure for heating service performances, the indoor temperatures of buildings at the above-mentioned heat nodes are shown in

Fig. 7 Indoor temperatures of buildings.
In the second case, the service-based reliability indices of the test system are calculated in two scenarios based on the proposed framework. Scenario A neglects the transmission- and demand-side thermal dynamics, which are modelled in Scenario B. The study period and the time interval are one week and 15 min, respectively. The tolerance level of variance coefficient is .
The computation time of TSMCS is presented in
Scenario | Number of iterations | Total time (hour) |
---|---|---|
A | 4135 | 6.09 |
B | 4322 | 8.16 |
The LOESP and LOHSP and EESNS and EHSNS in two scenarios are presented in Figs.

Fig. 8 LOESP and LOHSP in two scenarios. (a) LOESP. (b) LOHSP.

Fig. 9 EESNS and EHSNS in two scenarios. (a) EESNS. (b) EHSNS.
Regarding the evaluation results for the whole study period, it can be observed from
As analyzed in Section III-D, thermal dynamics could be affected by several factors such as the pipeline parameters at the transmission side and the equivalent building parameters at the demand side. In this case, the impacts of the typical factors on the reliability of IEHS are investigated through different scenarios. Except for the studied factors, other settings in this case are the same as those in Scenario B of Case 2.
Firstly, three scenarios are considered and distinguished by different pipeline lengths in IEHS, which belong to the factors for the transmission-side thermal dynamics. In these scenarios, the pipeline lengths are set to be 0.5, 1.0, 2.0 times of original values, respectively. The LOESP and LOHSP for different pipeline lengths are shown in

Fig. 10 LOESP and LOHSP for different pipeline lengths. (a) LOESP. (b) LOHSP.
In addition, the impacts of the heat capacity of the equivalent buildings, as the factor for the demand-side thermal dynamics, are also evaluated in this case. There are also three scenarios where the heat capacity levels of the equivalent buildings in IEHS are set to be 0.5, 1.0, and 2.0 times the original values respectively. The indices EESNS and EHSNS in these scenarios are used to illustrate the reliability results, as shown in

Fig. 11 EESNS and EHSNS for different heat capacity levels of equivalent buildings. (a) EESNS. (b) EHSNS.
In summary, the factors for both the transmission-side and demand-side thermal dynamics could affect the reliability of IEHS, since they determine the potential energy storage capabilities of the system. Either the longer pipelines or the larger building heat capacities correspond to the greater energy storage capabilities with more significant dynamic processes. Under these circumstances, the energy service provisions for customers are better ensured in contingency states and thus the reliability levels of IEHS could be improved.
Apart from thermal dynamic factors, the DTR could also have impacts on the reliability of IEHS, which is investigated in Case 4. Here, two scenarios are considered where the STR and DTR are applied in IEHS, respectively. The maximum allowable conductor temperature of power lines is set to be 100 ℃. The conductor coefficients and weather data for DTR are referred to [
The reliability indices EESNS and EHSNS and their differences in STR and DTR are shown in

Fig. 12 Reliability indices EESNS and EHSNS and their differences in STR and DTR. (a) EESNS and EHSNS in STR and DTR scenarios. (b) Differences of reliability indices between STR and DTR scenarios.
Although the service-based reliability concerned in this paper is more related to the short-term timeframe, the long-term reliability evaluation is also worth studying. In this case, the one-year simulation is conducted using the proposed method, where the study period and time interval are 8760 hours and 1 hour, respectively. The scenario settings are consistent with the previous cases including Scenarios A and B. The tolerance level of variance coefficient is .
The computation time of TSMCS for two scenarios is presented in
Scenario | Number of iterations | Total time (hour) |
---|---|---|
A | 3691 | 8.20 |
B | 3974 | 9.41 |
Scenario | LOESP | LOHSP | EESNS (MWh) | EHSNS (MWh) |
---|---|---|---|---|
A | 0.092 | 0.077 | 1857.8 | 2532.2 |
B | 0.088 | 0.072 | 2004.5 | 2830.3 |
The service-based reliabilities of IEHSs from the perspective of customers are defined and analyzed in this paper, where thermal dynamics are considered to ensure the accuracy of the reliability analysis. The energy service performances of IEHSs in contingency states are firstly modelled. On this basis, the framework for service-based reliability analysis is further proposed. Besides, the pertinent reliability indices are proposed considering both the dynamic service performances and service satisfactions of customers.
The simulation results demonstrate that the proposed framework can effectively quantify the reliability levels of IEHSs based on energy services for customers. The reduced EESNS and EHSNS indicate that thermal dynamics can lessen the damage of IEHSs in contingency states. And the LOHSP can specifically represent the probability of the heating service loss by judging the acceptable or unacceptable states utilizing temperature-measured service performances. Moreover, both the thermal dynamic factors and the DTR could affect the reliability of IEHS, since they determine the potential energy storage capabilities of the system. During the long-term period, the service-based reliability of IEHS could benefit more from thermal dynamics due to the temporally cumulative effects. The reliability analysis results obtained from the proposed framework can provide the decision-making guidance for operators to ensure the reliable energy services for customers in IEHS.
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