An Explicit Representation Method for Operational Reliability Constraints in Multi-Energy Coupled Low-Carbon Distribution Network

Abstract

Multi-energy coupled low-carbon distribution networks (MEC-LCDNs) face growing risks from extreme weather and high-order contingencies. Traditional deterministic criteria (e.g., N-1) often overlook these low-probability, high-impact events, while existing simulation-based probabilistic methods suffer from excessive computational burdens and a lack of intuitive visualization. To address these challenges, this paper proposes an explicit representation method for MEC-LCDN operational reliability constraints based on the probabilistic reliability region (PRR). This approach transforms the abstract probabilistic reliability criterion—loss of load probability (LOLP)—into a visualizable geometric space. Specifically, a fast contingency screening technique (FCST) is developed to identify a minimal set of boundary scenarios that anchor the target reliability threshold. Subsequently, complex probabilistic constraints are decoupled into deterministic N-k security constraints under these boundary scenarios, enabling the analytical construction of the PRR boundary. A case study demonstrates that the proposed method reduces the number of required contingency scenarios by over 90% and slashes computation time from 78.8 s to 3.1 s compared to traditional N-k truncation methods. Furthermore, the method accurately quantifies the system’s total supply capability (TSC) at 44.501 MW while providing intuitive visualizations of reliability boundaries that satisfy stringent LOLP criterion.

1. Introduction

1.1. Background and Motivation

Driven by the global energy transition and “dual-carbon” goals [1], the multi-energy coupled low-carbon distribution network (MEC-LCDN) breaks the barriers of traditional single-energy systems through the synergistic complementation of heterogeneous energy systems such as electricity, gas, and heat [2,3,4]. It has become a critical carrier for improving comprehensive energy efficiency and accommodating high penetrations of distributed renewable energy sources [5]. MEC-LCDN not only realizes the cascade utilization of energy flows but also significantly reduces the overall system carbon emissions through multi-energy complementarity [6].

In particular, with the rise of carbon trading markets [7], the operational paradigm of distribution networks is undergoing profound changes. System operation no longer merely responds to physical load demands but must also deeply respond to price signals and emission constraints from carbon markets [8]. As pointed out in the recent literature [9], the deep coupling and synergistic integration of flexible resources, such as electric vehicles, within electricity and carbon markets provide a new dimension for the low-carbon operation of power grids; however, they also significantly increase the complexity of system operation. Furthermore, recent research [10] highlights that demand response strategies utilizing flexible loads in modern active distribution systems can significantly enhance operational efficiency by smoothing load curves and mitigating power quality issues such as voltage violations.

However, despite the significant advantages of MEC-LCDN in enhancing energy efficiency and low-carbon environmental protection, its increasingly complex physical architecture has also introduced new security risks. Recently, the frequent occurrence of extreme weather events (e.g., typhoons, cold waves, rainstorms) due to global climate change has posed severe challenges to the reliable energy supply of energy systems [5]. Unlike traditional single power grids, the tightly coupled components within MEC-LCDN, while facilitating energy flow, also constitute “bridges” for cross-system fault propagation. Minor disturbances or faults in local components can easily be amplified and diffused to the entire system through coupling links, triggering cross-system cascading failures or even large-scale blackouts [11,12,13]. Relevant studies [11,14,15] indicate that the vulnerability of multi-energy coupled systems under extreme weather cannot be ignored, which urgently requires us to conduct more rigorous reliability analysis and assessment for MEC-LCDN.

1.2. Literature Review

To address the aforementioned challenges, scholars worldwide have conducted extensive and fruitful research on reliability assessment, resilience enhancement, probabilistic security assessment, and optimal dispatch of integrated energy systems (IES).

• In terms of reliability assessment: The first category relies on simulation-based approaches. Extensive research has been conducted to enhance the depth and breadth of these simulations. For instance, ref. [16] incorporated hydrogen integration into the evaluation model to address new coupling forms. To unify the assessment of heterogeneous energy qualities, ref. [17] introduced exergy-based reliability indices. Addressing the computational inefficiency for rare events, ref. [18] proposed an accelerated cross-entropy method based on subset simulation. Moreover, recent studies have extended these methods to capture dynamic behaviors: Ref. [19] utilized dynamic optimal energy flow to account for thermal inertia, while [12] analyzed the bidirectional propagation of cascading failures in interdependent systems.
• In terms of resilience enhancement: Ref. [20] first statistically verified through a comprehensive review that sector coupling—particularly with two energy types—yields significant resilience improvements compared to single-energy baselines. Building on this, research on defense and restoration against extreme disasters has been extensively conducted. For pre-disaster defense, ref. [21] proposed a three-layer defense–attack–defense robust model for integrated gas–electricity systems. Regarding disaster impact assessment, ref. [22] focused on extreme windstorms, establishing a framework to quantify cascading failure propagation. For post-disaster restoration, ref. [23] developed a coordinated strategy combining network reconfiguration with the dynamic dispatch of mobile power sources, while [24] introduced a decentralized restoration paradigm using multi-energy microgrids based on multi-task reinforcement learning.
• In terms of probabilistic security assessment: The necessity of shifting from deterministic to probabilistic frameworks was first systematically clarified by the authors of [25], who highlighted that traditional deterministic criteria (e.g., N-1) become inadequate under high renewable penetration. To address these uncertainties, ref. [26] developed a probabilistic scheme explicitly modeling the correlation among nodal power injections and frequency regulation. From a mathematical perspective, ref. [27] introduced a Lebesgue integration formulation based on high-order moments, transforming risk evaluation into a semi-definite programming problem. Expanding the assessment scope, ref. [28] proposed a unified framework combining resource adequacy with dynamic security assessment using sequential Monte Carlo simulation. Furthermore, ref. [29] extended the risk definition to integrated power systems to assess vulnerabilities against cyber–physical threats.
• In terms of optimization considering reliability constraints: Various strategies have been developed to balance economic efficiency and system reliability. Ref. [30] proposed a linearized model for energy hubs explicitly introducing maximum-allowable loss of load probability (LOLP) constraints. Expanding on this, ref. [5] addressed hydrogen integration in regional multi-energy systems, formulating a mixed-integer linear programming (MILP) model that enforces reliability constraints on critical devices. Moving towards probabilistic operation, ref. [31] developed a day-ahead security-constrained unit commitment model incorporating expected unserved energy cost via a solvable mixed-integer second-order cone programming (MISOCP). Addressing long-term planning, ref. [32] utilized fuzzy set theory to cluster contingency states in reserve expansion, while [33] introduced a reliability-constrained two-stage stochastic model for power-to-gas placement, embedding sequential Monte Carlo within the optimization framework.

However, despite the theoretical support provided by the aforementioned studies for the reliable operation of MEC-LCDN, most existing approaches rely predominantly on either “point-wise verification” or optimization dispatch subject to reliability constraints. These methods suffer from several significant limitations:

• Computational inefficiency: Point-wise methods typically generate fault scenarios via state enumeration or Monte Carlo simulation (MCS) and perform load-shedding calculations for each individual scenario to evaluate reliability indices. This exhaustive, repetitive process leads to prohibitive computational complexity and excessive calculation times, rendering them unsuitable for online applications.
• Lack of situational awareness (black-box nature): Point-wise verification can only assess reliability for discrete, finite operating points. It fails to characterize the system’s global reliability region, making it impossible to explicitly obtain critical operational information such as reliability margins or adjustment directions for current operating points.
• Vulnerability to boundary risks: Relying solely on reliability constraints in optimization models often drives the optimal solution to reside near the reliability boundary. Without explicit knowledge of the complete boundary geometry, even minor local disturbances can push the system into an unreliable state.

The “region-based” methods offer a promising pathway to overcome the efficiency bottleneck. The concept of security region (SR), originally developed for power systems, has been successfully extended to IES. Ref. [34] defined the SR for integrated electricity–gas–heat systems, utilizing piecewise hyperplane approximations to describe boundaries considering multi-energy constraints. Focusing on subsystem characteristics, ref. [35] specifically established the pressure and transmission security regions for natural gas networks. To enhance situational awareness, ref. [36] proposed a regional IES security region (RIESSR) model based on energy hubs and N-1 criteria, achieving 3D visualization of security boundaries. Addressing the challenges of uncertainty and dynamic coupling, researchers have further refined these models. Ref. [37] constructed a robust security region (RSR) using a convex hull approach to guarantee secure operation under wind power fluctuations, while [38] introduced an interval security region model to effectively quantify the impact of renewable uncertainty and optimize observation variables. From a dynamic perspective, ref. [39] leveraged the two-time-scale feature of IES to decompose fast and slow subsystems, deriving an exact steady-state security region with significantly reduced computational burden. Ref. [40] applied the load feasible region (LFR) concept to accelerate operational reliability evaluation. By calculating the minimum Manhattan distance to the LFR boundary, it enabled rapid estimation of optimal load shedding.

However, despite the efficiency of region-based methods, existing works (e.g., refs. [34,35,36,37,38,39,40]) are fundamentally rooted in deterministic or robust criteria (satisfying constraints strictly or under worst-case scenarios). They define a ‘Hard Boundary’ (e.g., strictly N-1 secure). In contrast, the operational reliability of MEC-LCDN is governed by probabilistic indices (e.g., $LOLP\le 0.01$), which implies a ‘probabilistic soft boundary’. Currently, there is no explicit method to model such a probabilistic reliability region (PRR) where every point within the boundary satisfies a specified reliability probability threshold, rather than a deterministic contingency set.

1.3. Contributions and Organization

To address these challenges, this paper proposes an explicit representation method for operational reliability constraints of MEC-LCDN based on PRR model. Unlike traditional black-box assessments, the PRR maps the abstract probabilistic reliability requirements into a visualizable geometric space. It defines a feasible polyhedral region for pipeline loads such that, under random N-k contingencies, any operating point located within this region strictly guarantees that the system’s LOLP remains below a prescribed threshold. This explicit geometric characterization enables operators to intuitively perceive reliability margins and perform rapid online security checks without repetitive simulations. A comprehensive comparison between the proposed method and existing methodologies is summarized in

Table 1. Comparison between the proposed method and the existing literature.

References	Core Methodology	Reliability Criterion	Model Representation	Computational Characteristics
[12,16,17,18,19]	MCS, state enumeration	LOLP	Implicit	Low Efficiency
[20,21,22,23,24]	Dynamic cascading analysis, restoration optimization	Resilience metrics	Implicit/Optimization	Low Efficiency
[25,26,27,28,29]	Analytical methods (e.g., moments), SMCS	Risk, probability	Implicit/Semi-explicit	Medium/Low Efficiency
[5,30,31,32,33]	Stochastic programming, MILP/MISOCP	Probabilistic constraints embedded in optimization	Implicit/Integrated	Low Efficiency
[34,35,36,37,38,39]	Geometric approximation (hyperplanes, convex hulls)	Deterministic (N-1) or robust worst-case	Explicit	High Efficiency
Proposed method	Boundary scenario inversion, FCST	$LOLP\le$ 0.01 Covers N-k risks	Explicit	High Efficiency

.

The main contributions are as follows:

• A new reliability-constraint construction paradigm based on a boundary scenario set (BSS) inversion is proposed, converting LOLP constraint into no-load-shedding conditions under boundary scenarios.
• An efficient BSS identification strategy based on a fast contingency screening technique (FCST) is developed to accurately extract the BSS anchoring a given reliability threshold from massive hypothetical contingency sets.
• Leveraging the identified BSS, complex probabilistic reliability constraints are decoupled into a set of deterministic N-k security inequality constraints, yielding an explicit analytical model for PRR-based operational reliability constraints.

The rest of this paper is organized as follows. Section 2 introduces the concept of the PRR for MEC-LCDN and presents an efficient boundary scenario identification strategy based on FCST. Section 3 formulates the explicit PRR model by transforming probabilistic reliability requirements into deterministic N-k security constraints under boundary scenarios. Section 4 develops the TSC model and the simulation-based fitting method for PRR boundary construction. Section 5 provides comprehensive case studies to validate the effectiveness and efficiency of the proposed method. Finally, Section 6 concludes the paper and discusses potential directions for future research.

2. Boundary Scenario Identification for MEC-LCDN Oriented to Reliability Criterion

2.1. Structure of the MEC-LCDN

A typical configuration of an MEC-LCDN is illustrated in Figure 1. The energy hub (EH) converts the input electricity and natural gas into electricity, heat, and gas, which are delivered to end users through a radial medium-voltage distribution network, a ring-shaped heating network, and a gas network. The internal structure of an EH is divided into two parts: a substation and a regional energy station (REST). The key equipment in the substation consists of transformers, and the corresponding critical pipelines are power feeders. In the REST, the key equipment includes a compressor, a gas boiler (GB), combined heat and power (CHP) units, and a circulation pump (CP), while the critical pipelines comprise electricity, heat, and gas pipelines. This paper focuses on high-reliability MEC-LCDN composed of multiple interconnected EHs [41]. Accordingly, following N−k contingencies, the system can perform load transfer by closing tie lines and operating valves, thereby achieving loss-of-load-free operation under fault scenarios.

2.2. Definition of the PRR of MEC-LCDN

Drawing on the concept of the security region [36], PRR is defined as the set of all operating points that satisfy the system reliability constraint $LOLP\le \overline{LOLP}$, while explicitly accounting for all contingency scenarios ${\Omega }_s$ and their associated probabilities.

If we let the loads of the output-side pipeline segments associated with energy coupling equipment at EHs (e.g., gas boilers and combined heat and power units), together with related components such as transformers and compressors, form the operating point vector L = [L₁, L₂, …, L_m, L_M], then the PRR can be expressed as:

$${\Omega }_{\mathrm{PRR}}=\left\{\mathit{L}\in \mathcal{L}\mid LOLP(\mathit{L})\le \overline{LOLP}\right\}$$

(1)

where $\mathcal{L}$ denotes the set of feasible operating points that satisfy operational and security constraints. The system loss of load probability at operating point L is given by [42]

$$LOLP(\mathit{L})={\displaystyle \sum _{s\in {\Omega }_s}P}(s)·I(s,\mathit{L})$$

(2)

The state indicator function $I(s,\mathit{L})$ depends on the minimum load curtailment $I(s,\mathit{L})$ under scenario s. The value of $C_{curtail}^{\min }(s,\mathit{L})$ is obtained by solving an optimal load shedding model [19].

A schematic illustration of the PRR is shown in Figure 2, where any operating point within the region satisfies the prescribed reliability criterion.

2.3. Mathematical Definition of Boundary Scenarios and the Analytical Equivalence Theorem

The direct construction of the PRR is confronted with two major challenges: the curse of dimensionality and implicit constraints. The massive size of the contingency set ${\Omega }_s$ leads to exponential growth in computational complexity, while the presence of the logical variable $I(s,\mathit{L})$ renders the reliability constraint highly nonlinear and nonconvex, making analytical boundary characterization intractable. To overcome these difficulties, this paper introduces the concept of boundary scenarios, with the objective of identifying a minimal BSS denoted by ${\mathcal{S}}^{*}$, through which probabilistic reliability constraints can be equivalently transformed into deterministic no–load-shedding constraints.

A partial order among contingency scenarios is first defined. Based on the monotonicity of security performance, if for any operating point L the minimum load curtailment under scenario $s_i$ is always no greater than that under scenario $s_j$, i.e.,

$$C_{curtail}^{\min }(s_i,\mathit{L})\le C_{curtail}^{\min }(s_j,\mathit{L})$$

(3)

then scenario $s_i$ is said to dominate scenario $s_j$, denoted as $s_i\succcurlyeq s_j$. Based on this ordering relation, the following theorem is established to ensure the sufficiency of the equivalence transformation.

Theorem 1.

Given a reliability threshold $\overline{LOLP}$, if there exists a scenario subset ${\mathcal{S}}^{*}\subset {\Omega }_s$ such that the cumulative probability condition ${\displaystyle \sum _{s\in {\Omega }_{safe}({\mathcal{S}}^{*})\cup {\mathcal{S}}^{*}}P(s)}\ge 1-\overline{LOLP}$ is satisfied, where ${\Omega }_{safe}({\mathcal{S}}^{*})=\{s\in {\Omega }_s\mid \exists s^{*}\in {\mathcal{S}}^{*},s\succcurlyeq s^{*}\}$ denotes the set of covered safe scenarios, then the absence of load shedding under all boundary scenarios, i.e., $C_{curtail}^{\min }(s^{*},\mathit{L})=0,\forall s^{*}\in {\mathcal{S}}^{*}$ , guarantees that the system reliability index satisfies $LOLP(\mathit{L})\le \overline{LOLP}$.

Proof of Theorem 1.

Under the above condition, security monotonicity implies that for any $s\in {\Omega }_{safe}({\mathcal{S}}^{*})$, there exists an $s^{*}$ such that $C_{curtail}^{\min }(s,\mathit{L})\le C_{curtail}^{\min }(s^{*},\mathit{L})=0$. Therefore, no load shedding occurs under all scenarios in ${\Omega }_{safe}({\mathcal{S}}^{*})$. The system loss-of-load probability is thus upper-bounded by the total probability of uncovered scenarios, yielding $LOLP(\mathit{L})\le 1-{\displaystyle \sum _{s\in {\Omega }_{safe}}P}(s)\le \overline{LOLP}$. □

Theorem 1 reveals that boundary scenarios constitute the probabilistic dividing surface between reliable and unreliable operating regimes. Through this transformation, complex probabilistic constraints are decoupled into a finite number of deterministic N–k security constraints, allowing the PRR to be analytically expressed as the intersection of linear inequality constraints:

$${\Omega }_{\mathrm{PRR}}={\displaystyle \underset{s\in {\mathcal{S}}^{*}}{\cap }\left\{\mathit{L}\mid C_{curtail}^{\min }(s,\mathit{L})=0\right\}}$$

(4)

This transformation converts black-box probabilistic assessment into a transparent geometric boundary description, thereby providing a rigorous theoretical foundation for the efficient identification of BSS.

2.4. Efficient BSS Identification Strategy Based on FCST

The detailed implementation procedure of the proposed FCST strategy is formalized in Algorithm 1.

If we consider a system comprising $N$ components, with the component set given by $\mathcal{C}=\{c_1,c_2,\dots ,c_N\}$, and let $q_i$ denote the unavailability of component $c_i$, then a system state $s$ is represented by a binary vector $x_s={[x_1,x_2,\dots ,x_N]}^T$, where $x_i=0$ indicates normal operation and $x_i=1$ indicates component failure.

Assuming statistically independent component failures, the occurrence probability of state $s$ is given by [42]

$$P(s)={\displaystyle \prod _{i=1}^N\left[{(1-q_i)}^{1-x_i}·q_i^{x_i}\right]}$$

(5)

To reduce computational complexity and avoid repeated multiplications, an auxiliary variable ${\lambda }_i={\displaystyle \frac{q_i}{1-q_i}}$ is introduced, and the probability of the all-normal (N–0) state $s_0$ is denoted by

$$P_{s_0}={\displaystyle \prod _{i=1}^N(1-q_i)}$$

(6)

Accordingly, the probability of any state s can be equivalently expressed as

$$P(s)=P_{s_0}·{\displaystyle \prod _{i\in {\mathcal{F}}_s}{\lambda }_i}$$

(7)

where ${\mathcal{F}}_s=\{i\mid x_i=1\}$ denotes the index set of failed components in state $s$. This transformation indicates that the relative magnitude of state probabilities is monotonically determined by the product of the ${\lambda }_i$ values associated with failed components, thereby providing a theoretical foundation for ranking-based state screening.

If we let $\Omega$ denote the set of hypothetical contingency scenarios, then cumulative probability is defined as

$$P_{cum}(\Omega )={\displaystyle \sum _{s\in \Omega }P}(s)$$

(8)

Given a maximum allowable loss-of-load probability threshold $\overline{LOLP}$ (e.g., 10⁻³),the corresponding target cumulative probability threshold is $P_{th}=1-\overline{LOLP}$. The objective of boundary scenario identification is to identify, within the vast system state space, a scenario set ${\mathcal{S}}^{*}$ with the minimum cardinality such that its cumulative probability satisfies the required reliability coverage.

Algorithm 1: Efficient BSS identification strategy based on FCST

Input:$\mathrm{Component} \mathrm{set} \mathcal{C}=\{c_1,c_2,\dots ,c_N\}$$; \mathrm{Unavailability} q_i$$; \mathrm{Target} \mathrm{cumulative} \mathrm{probability} \mathrm{threshold} P_{th}=1-\overline{LOLP}$

Output:$\mathrm{BSS} {\mathcal{S}}^{*}$

1: Step 1: Preprocessing

2: Calculate odds ratio ${\lambda }_i=q_i/(1-q_i)$$\mathrm{for} \mathrm{all} c_i$.

3: Sort components such that ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_N$.

4: Step 2: Initialization

5: Define root state $s_0$$(N\text{-}0 \mathrm{state}); \mathrm{Calculate} P(s_0)$ by (6)

6: Initialize Max-Heap $\mathcal{H}\leftarrow \left\{s_0\right\}$.

7: $P_{cum}\leftarrow 0$$;{\Omega }_B\leftarrow Ø$.

8: Step 3: Screening Loop

9: While $P_{cum}<P_{th}$ and $\mathcal{H}\ne Ø$ do

10:   $s_k\leftarrow \mathrm{Pop}\text{-}\mathrm{Max}(\mathcal{H})$;

11:   ${\Omega }_B\leftarrow {\Omega }_B\cup \left\{s_k\right\}$;

12:   $P_{cum}\leftarrow P_{cum}+P(s_k)$;

13:   Let idx be the index of the last added component in $s_k$.

14:    //Operation A: Vertical expansion

15:   if idx < N then

16:     Generate $s_{\mathrm{new}}=s_k\cup \left\{c_{idx+1}\right\}$;

17:     $P(s_{\mathrm{new}})=P(s_k)·{\lambda }_{idx+1}$;

18:     $\mathrm{Push}(\mathcal{H},s_{\mathrm{new}})$;

19:   end if

20:    //Operation B: Horizontal replacement

21:   if$s_k\ne s_0$ and idx < N then

22:     Generate $s_{\mathrm{new}}=(s_k\setminus \{c_{idx}\})\cup \left\{c_{idx+1}\right\}$;

23:     $P(s_{\mathrm{new}})=P(s_k)·({\lambda }_{idx+1}/{\lambda }_{idx})$;

24:     $\mathrm{Push}(\mathcal{H},s_{\mathrm{new}})$;

25:   end if

26: end while

27: return ${\mathcal{S}}^{*}$

According to the greedy principle, if all 2^N system states can be strictly ordered in descending order of probability (i.e., $P(s_1)\ge P(s_2)\ge \dots$), then the subset formed by the first k states necessarily yields the minimum-cardinality solution that satisfies the cumulative probability constraint.

All components are first sorted in descending order of the failure odds ratio ${\lambda }_i$ and reindexed such that ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_N$. Since the state probability P(s) is proportional to the product of $\lambda$ values corresponding to the failed components, this ordering ensures that, for the same contingency order, states involving components with larger ${\lambda }_i$ values exhibit higher occurrence probabilities.

Based on the reordered component sequence, a state search tree is constructed with the all-normal (N–0) state as the root. To guarantee monotonic descent in state probability, two child-generation operators are defined.

(1) Horizontal replacement: The failed component with the largest index i is replaced by component j (j > i), yielding $P(s_{\mathrm{new}})=P(s_{\mathrm{old}}){\displaystyle \frac{{\lambda }_j}{{\lambda }_i}}$. Since ${\lambda }_j\le {\lambda }_i$, it follows that $P(s_{new})\le P(s_{old})$.

(2) Vertical expansion: A new failed component j (j > i) is appended to the current state, yielding $P(s_{\mathrm{new}})=P(s_{\mathrm{old}}){\lambda }_j$. Given that component failures are rare events, ${\lambda }_j<1$, the resulting state probability is significantly reduced.

A max-heap data structure is employed to maintain the candidate state set. At each iteration, only the most probable state is extracted, and its child states are generated and inserted into the heap. In this manner, system states are extracted in strictly descending order of probability $P_1,P_2,\dots ,P_k$ without exhaustive enumeration. The algorithm terminates when ${\displaystyle \sum _{i=1}^k{P}_i}\ge P_{th}$, and the resulting subset constitutes the globally optimal minimum-cardinality scenario set ${\mathcal{S}}^{*}$.

To clearly illustrate the proposed method, the overall framework of the FCST strategy is depicted in Figure 3. The process comprises three sequential stages: (1) component sorting based on the failure odds ratio to establish a priority sequence; (2) state tree generation with implicit enumeration to explore high-probability states; and (3) greedy selection via a max-heap structure to efficiently identify the BSS.

3. Modeling of PRR Using BSS

3.1. PRR Model of MEC-LCDN

After identifying the BSS ${\mathcal{S}}^{*}$ that anchors the system reliability threshold $\overline{LOLP}$ using the proposed strategies, the construction of the PRR can be reformulated as follows: determine the largest steady-state operating region such that, for any operating point L within this region, the system not only satisfies all physical operating constraints under the base-case condition, but also remains in steady-state equilibrium without load shedding under any contingency scenario in ${\mathcal{S}}^{*}$.

Based on the previously derived boundary scenarios and inspired by the classical N–1 security region formulation [36], the constraint model of the probabilistic reliability region is established. It consists of two categories of constraints: (1) multi-energy flow balance constraints of MEC-LCDN, and (2) boundary scenario security constraints. The latter includes security constraints on critical EH equipment and security constraints on critical pipeline outlets of EHs under boundary scenarios, collectively ensuring that no load shedding occurs for any scenario in ${\mathcal{S}}^{*}$.

Accordingly, the PRR can be analytically expressed as

$${\Omega }_{\mathrm{PRR}}=\left\{\mathit{L}\mid \mathit{h}(\mathit{L})=\mathbf{0}, {\mathit{g}}_s(\mathit{L})\le \mathbf{0}, \forall s\in {\mathcal{S}}^{*}\right\}$$

(9)

where ${\mathcal{S}}^{*}$ denotes BSS; $\mathit{h}(\mathit{L})=0$ represents the multi-energy flow balance equations; and ${\mathit{g}}_s(\mathit{L})$ denotes the system security inequality constraints under scenario s.

Energy flow analysis forms the foundation of PRR construction. Electric power systems, natural gas systems, and district heating systems are typical subsystems of MEC-LCDN. Power flows in electrical networks can be solved using the forward–backward sweep method, while energy flows in gas and heating networks can be solved using the Newton–Raphson method. As multi-energy flow solution techniques are well established in the literature [43], they are not detailed here. Instead, emphasis is placed on boundary scenario security constraints.

To visually illustrate this explicit modeling framework, the overall process of transforming the identified boundary scenario set into the final PRR mathematical model is depicted in Figure 4. As shown, the PRR is rigorously defined by the intersection of the feasible regions corresponding to all boundary scenarios.

3.2. Boundary Scenario Security Constraints of MEC-LCDN

According to Theorem 1, constructing the PRR of MEC-LCDN is equivalent to identifying a set of operating points that satisfy all physical and operational constraints under every boundary scenario $s\in {\mathcal{S}}^{*}$. In practical engineering applications, reliability requirements are often stringent (e.g., LOLP less than or equal to 0.01), and boundary scenarios therefore commonly involve N–2, N–3, or even higher-order contingencies. Consequently, it is necessary to generalize conventional N–1 security constraints [36] to N-k security constraints. This section formulates the boundary scenario security constraint set from two perspectives: critical equipment within EH and critical pipeline outlets of EH.

3.2.1. Constraints of Key EH Equipment Under Normal Operation

• Pipeline Load Equality Constraints of Multiple Energy Types

Electric feeders typically have multi-section structures due to sectional switches (normally closed), while natural gas and thermal pipelines generally lack such segmentation. Therefore, the outlet load of a pipeline under different energy types can be expressed as follows [36]:

$$L_k^{\mathrm{en}}=\left\{\begin{array}{ll}{\displaystyle \sum _{a\in {\mathcal{S}}_k}{L}_{k_a}^{\mathrm{shift},e}},\hfill & \mathrm{en}=e\hfill \\ L_k^{\mathrm{shift},v},\hfill & \mathrm{en}=v\hfill \end{array}\right.$$

(10)

where $L_k^{\mathrm{en}}$ is the outlet load of pipeline k and ${\mathcal{S}}_k$ denotes the set of feeder sections for electric feeder k. For electric loads (en = e), $L_k^{\mathrm{en}}$ is the sum of section loads $L_{k_a}^{\mathrm{shift},e}$, accounting for load transferred to other feeders if a section fails. For multi-energy loads (en = v, e.g., thermal or gas), as these pipelines are not segmented, $L_k^{\mathrm{en}}$ equals the load of the pipeline segment. Here, $L_{k_a}^{\mathrm{shift},e}$ represents the portion of load shifted to other feeders when feeder section ka fails, while $L_k^{\mathrm{shift},v}$ represents the multi-energy load transferred to other pipelines when pipeline k is out of service.

• Load Equality Constraints between Key Equipment and Pipelines

The load served by the key equipment and the corresponding pipeline outlet loads are related as follows [36]:

$$E_j={\displaystyle \sum _{k\in {\mathbb{L}}_j}{L}_k^{\mathrm{en}}},\hspace{1em}\forall j$$

(11)

where E_j is the total load supplied by equipment j, and ${\mathbb{L}}_j$ denotes the set of pipeline outlets connected to equipment j.

3.2.2. Security Constraints of Key EH Equipment Under Boundary Scenarios

If we consider a boundary scenario s (assumed to be an N-k contingency), which comprises a set of failed equipment ${\mathbb{Z}}_s$ and failed pipelines ${\mathbb{Y}}_s$, i.e., $s={\mathbb{Z}}_s\cup {\mathbb{Y}}_s$, then when the equipment in ${\mathbb{Z}}_s$ fails, the system performs topological reconfiguration via tie switches or valves, and the load in the affected area must be transferred by the remaining interconnected equipment set ${\mathbb{W}}_s^{\mathrm{shift}}$. The system must satisfy the following three levels of constraints:

• Pipeline Load Transfer Equality after Equipment Failure

Under boundary scenario s, for each failed key equipment $i\in {\mathbb{Z}}_s$, the load in the affected area is supplied via interconnected equipment $j\in {\mathbb{W}}_s^{\mathrm{shift}}$. The transferred load is given by:

$$L_{ij}^{\mathrm{shift},\mathrm{en}}(s)={\displaystyle \sum _{m\in {\mathbb{L}}_i}{\displaystyle \sum _{n\in {\mathbb{L}}_j}{L}_{mn}^{\mathrm{shift},\mathrm{en}}}}(s),\hspace{1em}\forall i\in {\mathbb{Z}}_s,j\in {\mathbb{W}}_s^{\mathrm{shift}}$$

(12)

where $L_{ij}^{\mathrm{shift},\mathrm{en}}(s)$ denotes the load transferred from failed equipment i to equipment j, and $m\in {\mathbb{L}}_j$ and $n\in {\mathbb{L}}_j$ indicate the pipelines associated with equipment i and j, respectively. The term $L_{mn}^{\mathrm{shift},\mathrm{en}}(s)$ represents the load shifted from pipeline m to pipeline n under scenario s:

$$L_{mn}^{\mathrm{shift},\mathrm{en}}(s)=\left\{\begin{array}{ll}{\displaystyle \sum _{a\in {\mathcal{S}}_m}{\displaystyle \sum _{b\in {\mathcal{S}}_n}{L}_{m_a{n}_b}^{\mathrm{shift},e}}}(s),\hfill & \mathrm{en}=e \hfill \\ L_{mn}^{\mathrm{shift},v}(s),\hfill & \mathrm{en}=v \hfill \end{array}\right.$$

(13)

where, $L_{m_a{n}_b}^{\mathrm{shift},e}$ denotes the load transferred from feeder segment ma to mb for electric loads, and $L_{mn}^{\mathrm{shift},v}$ represents the load transferred from pipeline m to n for multi-energy loads.

• Load Transfer Balance Constraint

The total load originally served by each failed equipment $i\in {\mathbb{Z}}_s$ must be fully accommodated by interconnected equipment:

$${\displaystyle \sum _{j\in {\mathbb{W}}_s^{\mathrm{shift}}}{L}_{ij}^{\mathrm{shift},\mathrm{en}}}(s)=E_i,\hspace{1em}\forall i\in {\mathbb{Z}}_s$$

(14)

• Capacity Margin Constraint of Interconnected Equipment

After receiving transferred loads, the total load on each interconnected equipment j (original load E_j plus transferred load) must not exceed its permissible short-term overload capacity:

$$0\le E_j+{\displaystyle \sum _{i\in {\mathbb{Z}}_s}{L}_{ij}^{\mathrm{shift},\mathrm{en}}}(s)\le k_j·C_j,\hspace{1em}\forall j\in {\mathbb{W}}_s^{\mathrm{shift}}$$

(15)

where C_j is the rated capacity of equipment j, and k_j is the allowable short-term overload coefficient following the contingency.

3.2.3. Security Constraints of Key EH Pipeline Exits Under Boundary Scenarios

Similarly, when a boundary scenario s involves pipeline failures (failure set ${\mathbb{Y}}_s$), the transmission capacity of the remaining pipeline network must be verified. Specifically, for electric feeders, the complex load transfer logic following sectional switch actions must be considered. For gas and thermal pipelines, which do not have sectional structures, the load is typically redistributed through looped backup paths.

• Load Transfer Balance Constraint for Pipelines

If we let $L_{mn}^{\mathrm{shift},\mathrm{en}}(s)$ denotes the load transferred from a failed pipeline $\forall m\in {\mathbb{Y}}_s$ to a receiving pipeline $\forall n\in {ℝ}_s^{\mathrm{shift}}$ under boundary scenario s, then the original load on pipeline m, $L_m$, must be fully transferred without load shedding:

$${\displaystyle \sum _{n\in {ℝ}_s^{\mathrm{shift}}}{L}_{mn}^{\mathrm{shift},\mathrm{en}}}(s)=L_m^{\mathrm{en}},\hspace{1em}\forall m\in {\mathbb{Y}}_s$$

(16)

• Capacity Limit Constraint for Receiving Pipelines

The total flow on each receiving pipeline n must not exceed its rated capacity C_n:

$$0\le L_n^{\mathrm{en}}+{\displaystyle \sum _{m\in {\mathbb{Y}}_s}{L}_{mn}^{\mathrm{shift},\mathrm{en}}}(s)\le C_n,\hspace{1em}\forall n\in {ℝ}_s^{\mathrm{shift}}$$

(17)

4. PRR Boundary Modeling of MEC-LCDN and Solution

4.1. Total Supply Capability Model of MEC-LCDN and Solution

The supply capability of MEC-LCDN is quantified by three key indices: System Supply Capability (SSC), Minimum Supply Capability (MSC), and Total Supply Capability (TSC) [36].

SSC is defined as the total load supplied by the system at a given operating point, expressed as follows [36]:

$$\mathrm{SSC}={\displaystyle \sum _{i\in {\Omega }_L}{L}_i}$$

(18)

where $L_i$ represents the load of the $i$-th variable node, and ${\Omega }_L$ denotes the set of load nodes.

Correspondingly, MSC represents the lower bound of the feasible supply range, which corresponds to the system’s idle or no-load state:

$$\mathrm{MSC}=\min {\displaystyle \sum _{i\in {\Omega }_L}{L}_i}=0$$

(19)

Based on these definitions, TSC of MEC-LCDN measures the system’s maximum supply limit under specific security criteria. In traditional deterministic frameworks, TSC is usually constrained by the N-1 security criterion. Within the probabilistic reliability assessment framework proposed in this paper, the TSC based on the PRR is defined as the maximum load that the system can supply within a given operational region while satisfying the probabilistic reliability threshold (e.g., $LOLP\le 0.01$) and considering multi-energy flow balance and actual operating constraints. The TSC corresponds to the point on the PRR boundary delivering the maximum supply with the highest efficiency. It not only reflects the system-wide maximum load but also the corresponding load distribution at that operating point.

According to this definition, the optimization model incorporating multi-energy flow balance and probabilistic reliability constraints can be formulated as:

$$\mathrm{TSC}=\min (-SSC)=\min (-\mathit{L})=\min \left(-{\displaystyle \sum _{m=1}^M{\mathit{L}}_m}\right)$$

(20)

$$\mathrm{s}.\mathrm{t}. \left\{\begin{array}{c}\mathit{h}(\mathit{L})=0\hfill \\ {\mathit{g}}_{\min }\le \mathit{g}(\mathit{L})\le {\mathit{g}}_{\max }\hfill \end{array}\right.$$

(21)

where, −SSC is the optimization objective; h(L) represents the set of energy balance equality constraints [43]; g(L) represents the boundary scenario security constraints of MEC-LCDN given by (10)–(17); and L denotes the set of key pipeline outlet loads at the EH.

This paper adopts the primal-dual interior-point method to solve the maximum supply capability of MEC-LCDN. Numerous studies [44,45] have demonstrated that this method features fast convergence, strong robustness, and polynomial-time complexity, making it one of the most widely used and efficient algorithms. The TSC solution process is illustrated in Figure 5.

4.2. Simulation-Based Fitting and Solution of PRR Boundary

The PRR is composed of the set of operating points that satisfy the probabilistic reliability constraints under all boundary scenarios ${\mathcal{S}}^{*}$. The PRR boundary, serving as the critical demarcation between reliable and unreliable operating points, is formed by all critical operating points that exactly meet the probabilistic reliability constraints. Therefore, by simulating and validating each operating point individually, all critical points can be identified. These points are then fitted to construct a curve that represents the PRR boundary.

Since the system’s TSC is constrained by the PRR model, the TSC operating point necessarily lies on the PRR boundary. To enable visual observation of the system’s PRR, the probabilistic reliability boundary is solved around the TSC operating point using a simulation-based fitting method. Here, the system’s steady-state energy flows are computed via a decoupled solution approach [43]. This section uses a three-dimensional PRR as an example to detail the procedure for obtaining PRR boundaries in three-dimensional space, as shown in Figure 6. The method is equally applicable to other low-dimensional PRR boundaries.

Stage I: Solving the TSC and Load Distribution. Initialize system parameters, including topology, equipment characteristics, and algorithmic settings. Using the TSC of MEC-LCDN as the optimization objective and PRR model as constraints, apply the original primal–dual interior point method following the workflow shown in Figure 4. This yields the TSC operating point and the corresponding full-network load distribution.

Stage II: Constructing the Critical Point Array. Select any combination of multi-energy pipeline outlet loads $L_b=(L_m,L_n,L_o)$ as independent variables, while fixing the remaining M-3 variables at their TSC values. Incrementally increase $L_m$ and $L_n$ toward their upper limits $(L_m^u,L_n^u)$ using a step size $\Delta L$. At each step, solve for Lo under the multi-energy flow energy balance constraints, ensuring the PRR model is critically satisfied. Record each operating point $L_b=(L_m,L_n,L_o)$ into the critical point array $\mathbf{X}$.

Stage III: Fitting the Critical Point Array. Apply the least squares method to fit the critical point array $\mathbf{X}$, thereby obtaining a three-dimensional visualized PRR boundary.

5. Case Study

5.1. Parameter Setting

This paper constructs an enhanced MEC-LCDN test case based on the topology from [36]. The selection and modification of this test system were driven by the necessity to verify high-reliability performance. Since the proposed PRR targets a stringent reliability threshold (e.g., $\overline{LOLP}=0.01$), traditional radial network topologies designed solely for the deterministic N-1 criterion lack sufficient redundancy to cope with the high-order contingencies (N-2, N-3) considered in this study. Consequently, we reinforced the electrical, gas, and heating subsystems by adding critical equipment and corresponding pipelines, and explicitly introducing multiple tie-lines to form a “hand-in-hand” loop structure. This capacity expansion and topological enhancement ensure that the system possesses adequate load transfer capability to avoid load shedding via topological reconfiguration under extreme high-order fault scenarios. This configuration serves as a representative benchmark for high-reliability urban energy systems [46,47,48]. The topology is illustrated in Figure 7.

The system comprises two core EHs, and the parameter settings of each subsystem are specified as follows. The electric power subsystem is based on a modified IEEE-RBTS-BUS4 test system [46], with the total substation capacity set to 42 MVA. The heating subsystem adopts a modified 16-node ring-type district heating network model [47], including both supply and return pipelines, with a maximum design flow velocity of 3.0 m/s. The natural gas subsystem is modeled using a modified 14-node gas network [48], in which the maximum allowable pipeline flow velocity is set to 10 m/s.

In this case study, the overload factor of key equipment within each EH is set to 1.0, and the overall load power factor of the system is assumed to be 0.85. The detailed parameters of pipeline parameters are listed in

Table 2. Key parameters of EH pipelines.

Pipeline	Type of Trunk Pipeline	Rated Capacity/Flow of Trunk Pipeline
Electric feeder L4–L9	JKLYJ-120 (type I)	10 MVA
Electric feeder L10–L12	JKLYJ-150 (type II)	12 MVA
Thermal pipeline	DN400	376.99 kg/s
Natural gas pipeline	DN120	0.5 MMCFD

, the key parameters of equipment in the EHs are provided in

Table 3. Key parameters of EH equipment.

EH Type	Object	Equipment	Equipment Parameters	Equipment Capacity
EH #1	RES #1	GB	Efficiency: 0.9	10 MW
		CHP1, CHP2	cm: 0.8, cv: 0.1	Electricity: 12 MW, Heat: 12 MW
		CP1, CP2, CP3	Efficiency: 0.6	0.3 MW
	Substation #1	T1	Voltage ratio: 35 kV/10 kV	15 MVA
		T2	Voltage ratio: 35 kV/10 kV	15 MVA
		T3	Voltage ratio: 35 kV/10 kV	15 MVA
EH #2	RES #2	C1	Compressor ratio: 1.15	11 MW
		C2	Compressor ratio: 1.20	10 MW
		C3	Compressor ratio: 1.20	10 MW
	Substation #2	T4	Voltage ratio: 35 kV/10 kV	12 MVA
		T5	Voltage ratio: 35 kV/10 kV	12 MVA
		T6	Voltage ratio: 35 kV/10 kV	12 MVA

, and the reliability parameters of system components are listed in

Table 4. Reliability parameters of system components.

Component Type	Component ID	Unavailability
Transformer	T1–T3	0.020
	T4–T6	0.015
Circulation pump	CP1–CP3	0.030
Thermal pipeline	L1–L3	0.004
Natural gas pipeline	L13–L15	0.004
Electric feeder	L4–L9	0.005
	L10–L12	0.006
Combined heat and power	CHP1, CHP2	0.015
Gas boiler	GB	0.013
Compressor	C1–C3	0.030
	C4–C6	0.020

.

5.2. Efficiency Analysis of Boundary Scenario Identification

To validate the effectiveness of the proposed boundary scenario identification strategy and to examine the characteristics of system boundaries under different reliability targets, this subsection investigates the aforementioned practical test system. The screening efficiency of the traditional N-k truncation method is compared with that of the proposed approach, and the construction of BSS is analyzed in conjunction with the LOLP threshold $\overline{LOLP}$. In this comparative analysis, the ‘Traditional N-k Truncation Method’ is defined as a scenario generation strategy that exhaustively enumerates all contingency combinations up to a specific order (e.g., N-1, N-2, …, N-k) until the cumulative probability of the scenario set satisfies the system reliability requirement ($1-\overline{LOLP}$). Unlike the proposed FCST strategy, which screens scenarios based on their individual probability magnitude, this traditional approach performs a coarse-grained truncation based solely on fault order. Consequently, to satisfy a stringent reliability target (e.g., $\overline{LOLP}=0.01$), it is strictly compelled to include the entire set of high-order contingencies (e.g., all N-3 scenarios) regardless of their actual impact, which inherently leads to significant computational redundancy.

Figure 8 compares the screening efficiency of the two methods under different target cumulative probabilities. The proposed method consistently demonstrates significant dimensionality-reduction advantages across all probability levels. At the commonly adopted engineering probability level of 0.99, the traditional method requires evaluating 6018 scenarios, whereas the proposed method needs to screen only 589 key scenarios, achieving a scenario reduction rate exceeding 90%. Under more stringent reliability requirements (a target cumulative probability of 0.999), the traditional method must traverse nearly 47,000 scenarios, including all N–4 contingencies, while the proposed method requires only 5655 scenarios to achieve equivalent probability coverage.

Taking $\overline{LOLP}$ = 0.01 as an illustrative example, Figure 9 highlights the differences in the detailed composition of the scenario sets produced by the two methods. For comparable total numbers of scenarios, the traditional N-k truncation method necessarily includes a large proportion of N–3 long-tail scenarios with negligible impact on system reliability (up to 90.7%) in order to satisfy the probability requirement. In contrast, the scenario set obtained using the proposed method exhibits a more balanced structure, consisting of 5.6%N–1 scenarios, 87.2%N–2 scenarios, and 7.1% high-impact N–3 scenarios. This cross-order combination ensures that, under limited computational resources, risk events with substantive influence on system reliability are captured to the greatest possible extent. Meanwhile, Figure 9 profoundly reveals the inadequacy of the traditional N-1 criterion under stringent reliability requirements. Statistically, the cumulative probability of all N-1 contingency scenarios in this system sums to only approximately 0.9233. This implies that even if the system survives all N-1 checks, the resulting LOLP would still be as high as 0.0767, failing to meet the target of LOLP ≤ 0.01 (i.e., cumulative probability ≥ 0.99). Consequently, to bridge this probability gap, the BSS is compelled to incorporate a substantial number of N-2 and high-impact N-3 scenarios, resulting in the dominance of high-order contingencies within the boundary set.

5.3. Comparison with Two Existing Mainstream Methods

To further demonstrate the superiority of the proposed method, this section conducts a comprehensive comparison with two existing mainstream methods: the traditional N-k truncation method (based on point-wise simulation) and the deterministic N-1 SR method [36]. All simulations were performed on a computing platform equipped with an Intel^® Core™ i5-9300H CPU @ 3.20 GHz and 8.00 GB of RAM. The proposed model was implemented in MATLAB (version R2023a, MathWorks, Natick, MA, USA), and the optimization problems were solved using the Gurobi Optimizer (version 12.0.2, Gurobi Optimization, LLC, Beaverton, OR, USA). The comparison focuses on three dimensions: computational efficiency, reliability coverage, and visualization capability. $\overline{LOLP}$= 0.01.

5.3.1. Comparison with Traditional N-k Truncation Method

The traditional N-k truncation method evaluates reliability by exhaustively simulating all contingency combinations up to a certain order (e.g., N-3) for a specific operating point. Specifically, based on a given load state, this ‘point-wise’ simulation approach sequentially simulates equipment and pipeline failures. It determines whether the system incurs load shedding by verifying if post-contingency static security constraints are satisfied, thereby aggregating these results to calculate the LOLP index. As shown in

Table 5. Comparison of computational performance and features among different methods.

	Traditional N-k Truncation	N − 1 SR [36]	Proposed Method
Assessment scope	Point-wise (specific point)	Region-wise (whole space)	Region-wise (whole space)
Scenario basis	Exhaustive (N − 1 to N − 3)	Only N − 1	Boundary scenarios (N-k)
Number of scenarios	6017	33	588
Computation time (s)	78.8358 s (per point)	0.0032 s	3.0645 s
Reliability criterion	Quantified (LOLP)	Unquantified (Deterministic)	Quantified (LOLP ≤ 0.01)
High-order risk	Covered	Ignored	Covered
Visualization	No	Yes	Yes

, this “point-wise” assessment suffers from significant computational redundancy.

• Computational efficiency: For a single reliability assessment, the traditional method requires traversing 6017 scenarios, consuming approximately 78.8358 s. In contrast, the proposed FCST-based method screens out 90.2% of redundant scenarios, reducing the scenario set to 588 and shortening the computation time to 3.0645 s.
• Online application: The traditional method is a “black-box” assessment. If the system operating point changes, the entire simulation process must be repeated, making it unsuitable for real-time applications. The proposed method constructs an explicit “region-wise” boundary offline. For online applications, operators only need to determine the geometric relationship between the operating point and the PRR boundary (a logical judgment), which has negligible computational cost (millisecond level).

5.3.2. Comparison with N-1 SR

N-1 SR [36] is a widely used geometric method that defines the safe boundary considering only single-component failures (N-1).

• Reliability coverage: Although the SR is computationally efficient (checking only 38 N-1 scenarios in this case), it fails to capture high-order risks. Under the strict reliability threshold ($\overline{LOLP}$ = 0.01) required for MEC-LCDN, the reliability boundary is largely determined by N-2 and high-impact N-3 events. Consequently, the operating region defined by the SR is overly optimistic and larger than the actual PRR, leaving potential security blind spots for high-order contingencies.
• Methodological advantage: The proposed PRR method integrates the efficiency of the “region-based” approach with the accuracy of “probabilistic” assessment. It effectively identifies the “shrinkage” of the feasible region caused by high-order risks, ensuring that the system meets the quantified reliability target.

5.4. TSC Calculation Result of the Case

The TSC of the MEC-LCDN is calculated to be 44.501 MW. The allowable loads of key pipeline segments when the system operates at the TSC are reported in

Table 6. Allowable load of TSC.

Pipeline ID	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Load Lm(MW)	2.237	4.332	2.431	2.752	2.672	3.237	3.076	2.761	2.679	3.000	2.583	2.740	2.719	2.568	4.714

.

5.5. Visualization of the PRR in Two-Dimensional Space

Heating pipelines L₁ and L₂ are selected as free variables, forming the pair (L₁, L₂), to visualize the PRR in two-dimensional heating space. The terminal heating loads of these two pipelines vary synchronously, while the loads of all remaining pipelines are fixed at the TSC operating point. The reliability boundary in the two-dimensional heating space is obtained using the simulation-based fitting approach, and the results are shown in Figure 8. As illustrated in Figure 10, all reliability boundaries identified through critical-point fitting are linear, and the closed region enclosed by multiple boundaries forms a rectangular shape. For operating points located along the diagonal within the reliability region, the total system load reaches its maximum limit, corresponding to the calculated TSC of 44.500 MW (i.e., 44,500 kW) by (20) and (21), whereas all other operating points exhibit load levels below the TSC. Meanwhile, the electrical and gas network load distributions remain at the TSC.

The electrical feeder pairs (L₆, L₈) are selected as free variables, with the terminal feeder loads varying synchronously. Using a step size of 0.1 MW, the reliability boundaries in the two-dimensional electrical space are obtained, and the results are shown in Figure 11. As illustrated in Figure 9, all reliability boundaries identified through critical-point fitting are linear. For operating points located along the diagonal within the reliability region, the total system load reaches its maximum limit, corresponding to the calculated TSC of 44.500 MW (i.e., 44,500 kW) by (20) and (21), whereas all other operating points exhibit load levels below the TSC. Meanwhile, the load distributions of the heating and gas networks remain at the TSC.

Similarly, the gas network compressor outlet loads L₁₃ and L₁₄ are selected as free variables, forming the combination (L₁₃, L₁₄), with the terminal natural gas loads varying synchronously. Using a step size of 0.1 MW, the security boundary in the two-dimensional gas space is obtained, as shown in Figure 12. As illustrated in Figure 12, all reliability boundaries identified through critical-point fitting are linear, and the enclosed region exhibits a triangular shape. For operating points located along the diagonal within the reliability region, the total system load reaches its maximum limit, corresponding to the calculated TSC of 44.500 MW (i.e., 44,500 kW) by (20) and (21), whereas all other operating points exhibit load levels lower than the TSC. Meanwhile, the load distributions on the electrical and heating network sides remain at the TSC.

The above results indicate that two-dimensional reliability regions enable visualization only for single-energy subsystems, i.e., purely electrical, purely heating, or purely gas networks, and are applicable only when the load distributions of the non-observed energy networks remain at the TSC. However, in practical operation, scenarios in which all multi-energy subsystems simultaneously operate at the TSC are relatively uncommon. For example, in the heating network reliability region shown in Figure 10, observing only the two-dimensional heating domain does not allow the reliability states of non-TSC operating points in the electrical and gas networks to be identified, leading to incomplete reliability information. By contrast, visualizing reliability regions that simultaneously consider electricity–heat or electricity–gas subsystems enables reliability assessment of non-TSC operating points in multiple energy networks, which is more consistent with practical operational requirements. Consequently, it is necessary to investigate three-dimensional reliability regions based on the outlet loads of electricity–heat or electricity–gas pipelines.

5.6. Visualization of the PRR in Three-Dimensional Space

To visualize the reliability of pipelines subject to supply fluctuations, the multi-energy pipeline combination (L₁, L₂, L₈) is selected as the set of free variables. Consistent with the two-dimensional analysis, a step size of 0.1 MW is adopted for the variation of free variables to derive the three-dimensional reliability region, as shown in Figure 13.

The TSC operating point corresponds to point H (2.85 MW, 3.56 MW, 3.34 MW), representing the upper bound of the supply capability. The MSC operating point corresponds to point C (0, 0, 0), which aligns with the lower bound condition (MSC = 0) defined in Equation (19). All other operating points within the triangular prism exhibit load levels within the feasible range (MSC < SSC < TSC). According to the definition in Equation (18), the variation in SSC is analyzed along the reliability boundaries. Specifically, along the boundaries AH, BG, DE, CF, DA, EH, FG, and CB, the SSC increases monotonically, whereas along boundaries AB and DC, the SSC decreases monotonically. Along boundaries HG and EF, the SSC remains unchanged. As shown in Figure 13, point I represents an unreliable operating point located outside the reliability region. At this operating point, neither the electrical network nor the heating network operates at the TSC, and its reliability state cannot be identified using the two-dimensional reliability regions in Figure 10 and Figure 11. This observation highlights the necessity of three-dimensional reliability region visualization.

Similarly, the multi-energy pipeline combination (L₁₁, L₁₃, L₁₅) is selected as the set of free variables to derive the three-dimensional electricity–gas reliability region of MEC-LCDN, as shown in Figure 14.

In Figure 14, the closed region formed by the critical-point-fitted reliability boundaries constitutes a triangular prism. The TSC operating point corresponds to point A (3.25 MW, 0, 5.25 MW), and the MSC operating point corresponds to point C (0, 0, 0). All operating points along boundary AE operate at the TSC, whereas all remaining points within the triangular prism exhibit load levels greater than the MSC and lower than the TSC. Along reliability boundaries DA, DE, CB, and CF, the SSC increases monotonically; along boundaries AB, DC, and EF, the SSC decreases monotonically; and along boundaries AE and BF, the SSC remains unchanged.

In summary, three-dimensional visualization allows simultaneous observation of supply capability along three different directions and across two energy types (or even a single energy type). For MEC-LCDN, such visualized three-dimensional reliability regions provide a more comprehensive and multi-energy perspective for assessing system reliability.

6. Conclusions

This paper presents an explicit representation framework for operational reliability constraints in MEC-LCDN based on the PRR. By introducing the concept of boundary scenarios and employing FCST, the proposed method transforms implicit probabilistic reliability requirements into a finite set of deterministic N-k security constraints. This transformation effectively mitigates the curse of dimensionality and enables an analytical characterization of the PRR boundary through simulation-based fitting. The resulting PRR provides an intuitive and interpretable description of the safe operating region under a prescribed loss-of-load probability threshold, facilitating efficient reliability assessment and supporting reliability-aware operational decision-making in strongly coupled multi-energy systems.

Despite its effectiveness, several limitations of the proposed approach merit further investigation. First, the current framework is established under steady-state post-contingency assumptions and does not explicitly capture dynamic behaviors such as transient electrical responses, gas pressure dynamics, or thermal inertia effects. Second, component failures are assumed to be statistically independent, whereas correlated failures induced by extreme weather events may significantly impact system reliability in practice. In addition, as the number of critical pipelines increases, the dimensionality of the PRR may pose challenges for visualization and boundary fitting. Future research will focus on extending the PRR framework to dynamic and correlated failure models, developing scalable dimensionality-reduction techniques, and embedding explicit PRR constraints into multi-period operational and planning optimization of MEC-LCDN.