Delivery Reliability Assessment for a Multistate Smart-Grid Network with Transmission-Loss Effect

Abstract

Assessing the performance of the smart-grid system (SGS) under uncertainty is essential for ensuring a reliable energy supply from the perspective of the grid operator. In this study, a multistate smart-grid network (MSGN) is developed to evaluate the delivery capability of the SGS. An MSGN consists of multiple interconnected facilities, where nodes represent energy sources or converters and arcs denote feeders. The output of each facility in the MSGN is modeled as multistate, as maintenance activities and partial failures can result in multiple possible output levels. During power delivery, transmission losses may arise due to heat dissipation and feeder aging, potentially resulting in insufficient power supply at the demand side. From a smart-grid management perspective, delivery reliability, defined as the probability that the MSGN can successfully deliver sufficient power from energy sources to the destination under transmission loss, is adopted as a performance index for evaluating SGS capability. To compute delivery reliability, a minimal-path-based algorithm is developed. A practical SGS is presented to demonstrate the applicability of the proposed model and to provide managerial insights into smart-grid performance and operational decision-making.

1. Introduction

The smart-grid system (SGS) represents the evolutionary advancement of the conventional power system into an intelligent and adaptive cyber–physical energy system. By integrating energy resources, converters, and feeders within a digitalized monitoring and control framework, the SGS enables bidirectional flows of both power and information [1]. The overall performance of an SGS depends on its capability to maintain a reliable energy supply in the presence of various operational uncertainties, including component aging, inverter failures, fluctuations in renewable energy generation, and transmission losses. Moreover, external stressors such as extreme weather events (e.g., typhoons, earthquakes, and floods) further exacerbate system vulnerability and may lead to islanding events, load shedding, or partial system outages, ultimately impairing grid stability and efficiency. Under such conditions, ensuring high performance and operational stability of the SGS is critical for the grid operator to sustain local economic activities and energy security [2]. Therefore, developing a quantitative framework to measure SGS performance under uncertainty has become a key issue in modern power management.

Performance measurements in SGS can be approached from several perspectives. From an economic perspective, performance is often assessed in terms of cost-effectiveness, operational efficiency, and long-term investment sustainability [3,4,5]. For instance, Mehrpooya et al. [4] developed an integrated renewable-energy-driven seawater reverse osmosis desalination system and conducted a dynamic assessment of its economic and environmental impacts using the transient system simulation tool. Similarly, Wang et al. [5] designed a molten salt-coupled steam accumulator system to achieve heat–power decoupling in a combined heat and power unit. The thermodynamic performance, operational flexibility, and economic feasibility of this system are evaluated and compared with those of a conventional molten-salt storage system. On the other hand, reliability provides the most direct index for evaluating SGS performance. Conventional reliability indices, such as the system average interruption duration index (SAIDI), loss of load probability (LOLP), and expected energy not supplied (EENS), focus primarily on customer-oriented or energy-oriented indices, which consider the service quality of the user and interruption of energy supply [6,7,8,9]. Jain and Verma [8] constructed a reliability-based stochastic unit commitment model that incorporates K-means clustering and Markov chain transition probabilities to model wind power volatility, and evaluated its reliability using LOLP and EENS. dos Reis and Raizer [9] introduced a multi-objective optimization model to improve lightning protection in overhead distribution systems by balancing reliability and investment costs. SAIDI is utilized to measure the number of sustained interruptions in overhead distribution systems. While these indices are valuable for evaluating SGS interruptions, the viewpoint of the facility in these studies is deterministic, which is insufficient to represent the uncertainty of the facility during the power delivery process.

To address this limitation, network analysis is a practical approach to evaluate system performance by modeling the system as a multistate flow network [10,11,12,13,14,15,16,17]. Network analysis has been successfully applied in diverse real-world systems, including supply chain [10,11,12,13], manufacturing system [14,15], and computing system [16,17]. Similarly, the SGS is represented by a network topology, where each node represents an energy source or a converter, and each arc denotes a feeder. In the SGS, each facility consists of multiple devices: energy sources comprise generators, converters comprise transformers, and feeders comprise conductors. The output of each facility should be multistate because partial outputs may degrade or require maintenance [18,19]. Consequently, the SGS can be modeled as a classical multistate flow network in which each facility exhibits multiple output levels, leading to the formulation of a multistate smart-grid network (MSGN). Previous studies related to the MSGN have considered the allocation of feeders [18] and correlated facility failure [19]. Subsequent work has incorporated vulnerability analysis [20,21] to investigate the invulnerability of the MSGN. Another concern is whether the amount of power supplied to the user can meet the demand. To be worthy of attention, the amount of power transmitted to the destination in these studies is always undamaged. However, the electricity would decrease during the power delivery process due to heat dissipation and feeder aging [22,23,24]. That is, the generated power cannot satisfy the user demand, and such a situation is referred to as transmission loss. Hence, the amount of power delivered to the destination must be greater than the demand. Yeh et al. [23] evaluated delivery reliability in a large-scale power system, considering transmission losses. However, they do not fully capture the operational characteristics of the SGS, which involve renewable energy sources and multiple power delivery paths through various converters and feeders. Niu et al. [24] incorporated transmission losses between facility pairs in the power system. However, in practical applications, the transmission loss rate is typically determined by measuring the difference between the total supplied power and the total received power, as recorded by power meters [25].

Therefore, developing a quantitative framework to measure SGS performance under uncertainty has become increasingly important for grid operators. In this study, the multistate output of the facility and transmission loss are considered when evaluating the power delivery capability of SGS. The SGS is constructed as an MSGN. A performance index, delivery reliability, which is the probability that the MSGN can fulfill the power supply of the user under transmission loss, is proposed to evaluate the power delivery performance of the MSGN and for facility improvement. To compute the delivery reliability of the MSGN, an algorithm is designed. The remainder of this article is arranged as follows. The representation of the proposed MSGN model, which considers multistate output and transmission loss, is introduced in Section 2. The relationship between electricity flow and output is described in Section 3, and the delivery reliability is further defined. The computation algorithm is also presented in this section. In Section 4, an illustrative example is provided to demonstrate the practicality of the MSGN model. The concluding remarks and future research direction are given in Section 5.

2. Representation of the Multistate Smart-Grid Network

The MSGN consists of multiple multistate facilities, including energy sources, converters, and feeders, forming a complex power generation and delivery structure. The MSGN model is formulated as follows. Let N = {b_i|i = 1, 2, …, u + v} refer to a set of nodes, where the subset {b_i | i = 1, 2, …, u} denotes a set of u energy sources (source nodes), and the subset {b_i | i = u + 1, u + 2, …, u + v} denotes a set of v converters (intermediate nodes). The symbol T = {b_i | i = u + v + 1, u + v + 2, …, u + v + w} represents a set of w feeders (arcs). Thus, the MSGN can be represented as (N, T). In the MSGN, each facility b_i is composed of m_i devices, which means that b_i can operate totally m_i output levels. The corresponding available output levels are denoted as ε_i,1, ε_i,2, …, ε_i,mi, where 0 = ε_i,1 ≤ ε_i,2 ≤ … ≤ ε_i,mi, i = 1, 2, …, u + v + w. A minimal path (MP) is defined as a set of arcs that connects a source node to the destination node (sink) such that the removal of any arc in the path results in the disconnection between the source and the destination [23]. In addition, for a pair of energy source b_i and the destination, there exist θ_i MPs connecting them, where i = 1, 2, …, u. Let P_i,j denote the jth MP linking the ith source node to the destination node, where i = 1, 2, …, u, j = 1, 2, …, θ_i. To make the relationship between MSGN facilities easy to understand, a simple MSGN including two energy sources (b₁, b₂), one converter (b₃), three feeders (b₄, b₅, b₆), and one destination t is shown in Figure 1, which has two MPs: P_1,1 = {b₁, b₃, b₄, b₆} and P_2,1 = {b₂, b₃, b₅, b₆}.

3. Multistate Smart-Grid Network Model

In this section, the MSGN model will be constructed first. The MSGN model can be characterized by the output pattern X ≡ (x₁, x₂, …, x_u+v+w), the electricity flow pattern F ≡ (f_1,1, f_1,2, …, f_{1, θ1}, f_2,1, …, f_u,θu), and the transmission loss pattern Z ≡ (z_1,1, z_1,2, …, z_{1, θ1}, z_2,1, …, z_u,θu). The notation x_i represents the current output of facility b_i, f_i,j represents the current electricity flow through P_i,j, and z_i,j represents the transmission loss rate of P_i,j. The following assumptions should be considered to develop the MSGN model:

• The outputs of different facilities are statistically independent of each other.
• The electricity flow in (N, T) required to follow the flow-conservation law [26].

Note that flow conservation applies specifically to the undamaged electricity flow, which refers to the amount of electricity evaluated before considering transmission loss.

3.1. Relationship Between Electricity Flow and Output

To ensure the demand d is supplied, any electricity flow pattern F should meet the following equation:

$${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j=1}^{{\theta }_i}{f}_{i,j}} = d,}$$

(1)

where ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j=1}^{{\theta }_i}{f}_{i,j}}}$ is the total undamaged electricity flow delivered through the network (N, T). Such an F without considering loss in the power delivery process is referred to as an undamaged electricity flow pattern. However, the power received at the destination is reduced by the transmission loss along each MP. Based on long-term operational data, the transmission loss rate z_i,j can be obtained by averaging the proportion of power lost along the path P_i,j. Apparently, each undamaged electricity flow passing through P_i,j can thus be calculated as f_i,j× (1 − z_i,j), for j = 1, 2, …, θ_i. To ensure that the amount of power delivered to the destination is sufficient to fulfill the demand, let h_i,j represent the required flow delivered to the destination via the P_i,j, and H = (h_1,1, h_1,2, …, h_{1, θ1}, h_2,1, …, h_u,θu) is represented as a required flow pattern. Based on the feasible electricity flow f_i,j and the corresponding transmission loss rate z_i,j, the required flow h_i,j can be determined by:

$$H_{i,j} = {\displaystyle \frac{f_{i,j}}{1-z_{i,j}}} \mathrm{for} i = 1, 2, \dots , u, j = 1, 2, \dots , {\theta }_i.$$

(2)

Clearly, the required flow h_i,j must be greater than the undamaged electricity flow f_i,j, except in the special case where z_i,j = 0. Let L ≡ (ε_1,m1, ε_2,m2, …,ε_u+v+w,mu+v+w) represent the maximal output pattern. Any required flow pattern H is said to be feasible under L if it satisfies the following constraint:

$${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}} \le {\epsilon }_{i,m_i} \mathrm{for} i = 1, 2, \dots , u + v + w,}$$

(3)

where ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$ is the total required flow transmitted through the facility b_i. Constraint (3) indicates that the required flow passing facility b_i cannot be greater than its maximal output, i = 1, 2, …, u + v + w. Similarly, any H satisfying the following constraint is said to be feasible under the output pattern X = (x₁, x₂, …, x_u+v+w):

$${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}} \le X_i \mathrm{for} i = 1, 2, \dots , u + v + w.}$$

(4)

For simplicity, let H_X denote the set of all Hs under X.

3.2. Delivery Reliability Evaluation for an MSGN

In the MSGN, delivery reliability R_d is defined as the probability that the MSGN can meet the demand d within the limitations of transmission loss. Let Ψ ≡ {X | X satisfies demand d} for convenience, and R_d can be represented as ${\displaystyle {\sum }_{X\in \Psi }\mathrm{Pr}\left\{X\right\}}$. By assumption 2, Pr{X} is computed as the product of Pr{x₁}, Pr{x₂}, …, and Pr{x_u+v+w}. Due to the substantial size of the network, it becomes unrealistic to compute all feasible X and combine their probabilities to obtain R_d. To efficiently assess R_d, the concept of lower output patterns (LOPs) is adopted to reduce the computational burden associated with delivery reliability evaluation. Before defining the LOP, the comparison rules between two output patterns are introduced as follows:

Rule 1:

Q ≤ S: (q₁, q₂, …, q_p) ≤ (s₁, s₂, …, s_p) if and only if q_α ≤ s_α for each α.

Rule 2:

Q < S: (q₁, q₂, …, q_p) < (s₁, s₂, …, s_p) if and only if X ≤ Y and x_α < y_α for at least one α.

Under these rules, the LOP is defined as follows:

Definition 1:

Any LCV contained in the set Ψ is referred to as anLOP. Equivalently, if X is anLOP, then there exists no outputpattern Ysuch that Y < Xwith Y  ∈  Ψ.

Assume that there are δ LOPs in total, denoted by X₁, X₂, …, X_δ. Then, the delivery reliability of the MSGN can be expressed as:

$$R_d = {\displaystyle {\sum }_{X\in \Psi }\mathrm{Pr}\left\{X\right\}}= \mathrm{Pr}\left({\displaystyle {\cup }_{i=1,2,\dots ,\delta }\left\{X|X\ge X_i\right\}}\right).$$

(5)

For Equation (5), various techniques can be utilized to calculate this probability, including the inclusion–exclusion principle [27,28], the state-space decomposition [29], and the sum of disjoint products. Among these approaches, the recursive sum of disjoint products (RSDP) has been demonstrated to offer better computational efficiency than the comparable approaches [30]. Thus, the RSDP method is adopted to calculate R_d in this study.

3.3. Theory of Generating All LOPs

To generate all LOPs, the relationship between an LOP X and an H ∈  H_X is illustrated as follows:

Theorem 1.

If X is anLOP, then there exists at least one H∈ H_X satisfying the demand d, such that

$$x_i \ge {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}} \mathrm{for} i = 1, 2, \dots , u + v + w.}$$

(6)

Proof of Theorem 1.

If X is an LOP, then there exists at least one H ∈ H_X satisfying d. That is, we have x_i ≥${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$ for i = 1, 2, …, u + v + w. Assume that there exists a facility b_λ such that x_λ > η_λ,k ≥ ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$> η_λ,k−1 for a k and x_i  ≥  ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$ for all i ≠ λ. Let T = (t₁, t₂, …, t_{u + v + w}) with t_λ = η_λ,k and t_i = x_i for all i ≠ λ. Then, T  <  X and t_i ≥ ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$ ∀ i. This implies that T ∈ Ψ and contradicts the fact that X is a LOP. The proof is thus completed. □

The value ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$ is the loading on facility b_i, and x_i is the lower output necessary to satisfy the loading. According to Theorem 1, each feasible H derived under constraints (1–3) can be transformed into a corresponding output pattern X via Equation (6), and ensures that X can satisfy the demand d. However, not all Xs necessarily qualify as accurate LOPs; they are therefore initially treated as LOP candidates. Let π denote the set of LOP candidates, and π_min represent the subset containing only the lower patterns in π. The following theorem confirms that π_min precisely constitutes all LOPs.

Theorem 2.

π_min is equivalent to the set of LOPs.

Proof of Theorem 2.

Suppose that X ∈ π_min but is not an LOP. It means that X ∈ π, and there exists an LOP Y with Y < X. It also means Y ∈ π, which contradicts X ∈ π_min. Thus, any X ∈ π_min is an LOP. Conversely, suppose that LOP X does not belong to π_min. X ∈ π depends on Theorem 1. Therefore, there exists at least one Y ∈ π_min such that Y < X. We have previously proved that Y must be an LOP. It contradicts the fact that X is an LOP, and we thus conclude that π_min is the set of LOPs. □

According to Theorem 2, each LOP candidate X must be examined to verify whether it qualifies as an LOP or not. Assume that there are m LOP candidates in π, delete non-lower patterns from LOP candidates to obtain π_min, and the process is presented in

Table 1. The process of obtaining the π_min.

Line: 1	Initialize πmin ← ∅ // πmin stores the identified minimal LOPs
Line: 2	Initialize A ← ∅. // A stores the indices of non-minimal LOP candidates
Line: 3	FOR i = 1 TO m − 1 DO
Line: 4	IF i ∈ A THEN CONTINUE
Line: 5	FOR j = i + 1 TO m DO
Line: 6	IF j ∈ A THEN CONTINUE
Line: 7	IF Xi > Xj THEN
Line: 8	A ← A ∪ {i}
Line: 9	BREAK // Xi is not minimal
Line: 10	ELSE IF Xi ≤ Xj THEN
Line: 11	A ← A ∪ {j}
Line: 12	END IF
Line: 13	END FOR
Line: 14	IF i ∉ A THEN
Line: 15	πmin ← πmin ∪ Xi
Line: 16	END IF
Line: 17	END FOR
Line: 18	RETURN πmin

.

3.4. The Proposed Algorithm to Evaluate the Delivery Reliability

Based on the above formulation, an algorithm is developed based on the MSGN model for calculating R_d and presented as follows:

• Input: An MSGN (N, T), a demand d, a transmission loss pattern Z = (z_1,1, z_1,2, …, z_{1, θ1}, z_2,1, …, z_u,θu), a set of MPs {P_1,1, P_1,2, …, P_u,θu}.
• Output: Delivery reliability R_d.
• Step 1. Identify all electricity flow patterns F that satisfy the following equation:

  $${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j=1}^{{\theta }_i}{f}_{i,j}} =d. }$$

  (7)
• Step 2. Convert each feasible F into the required flow pattern H = (h_1,1, h_1,2, …, h_{1, θ1}, h_2,1, …, h_u,θu) via

  $$h_i{,}_j=⌈{\displaystyle \frac{f_{i,j}}{{1-z}_{i,j}}}⌉ \mathrm{for} i = 1, 2, \dots , u, j = 1, 2, \dots , {\theta }_i.$$

  (8)
• Step 3. Check whether the H derived from Step 2 satisfies the maximal output pattern L or not by following constraint:

  $${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}} \le {\epsilon }_{i,m_i} \mathrm{for} i = 1, 2, \dots , u + v + w.}$$

  (9)
• Step 4. For each feasible H, find the minimal x_i such that

  $$x_i \ge {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}} \mathrm{for} i = 1, 2, \dots , u + v + w.$$

  (10)
• The obtained X = (x₁, x₂, …, x_{u + v + w}) is an LOP candidate.
• Step 5. Remove those non-lower ones in the set of LOP candidates to obtain all LOPs.
• Step 6. Suppose there are q LOPs: X₁, X₂, …, X_q obtained from Step 5. Apply the RSDP method to calculate the delivery reliability R_d:

  $$R_d = \mathrm{Pr}\left({\displaystyle {\cup }_{\mathrm{i}=1,2,\dots ,\delta }\left\{X|X\ge X_i\right\}}\right).$$

  (11)

The overall process for delivery reliability evaluation is illustrated in Figure 2, providing an overview of the proposed algorithm.

3.5. Time Complexity Analysis of the Proposed Algorithm

In the proposed algorithm, let the total number of feasible electricity flow patterns satisfying the demand constraint (e.g., solutions of Equation (7)) be denoted as τ = $\left(\begin{array}{c}{\displaystyle {\sum }_{i=1}^u{\theta }_i}+d-1\\ d\end{array}\right)$ in the worst case, where ${\displaystyle {\sum }_{i=1}^u{\theta }_i}$ represents the number of MPs connected to the electricity region. For convenience, define ρ = ${\displaystyle {\sum }_{i=1}^u{\theta }_i}$ as the total number of MPs in the MSGN.

1.

Time complexity of Steps 1 and 2: Each electricity flow pattern is generated by solving Equation (7) and then transformed into the corresponding required flow pattern using Equation (8), which takes at most O(ρ) time. Thus, generating all required flow patterns in Steps 1 and 2 takes O(ρ‧τ) time in the worst case.

2.

Time complexity of Step 3: Each solution from Equation (8) requires O(ρ) time to test whether it fulfills ${\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_i\in P_{i,j}}{h}_{i,j}}}$≤ m_i for each i, and O(w‧ρ) time for all i. Thus, Step 3 requires O(w‧ρ‧τ) time in the worst case.

3.

Time complexity of Step 4: In Step 4, each feasible required flow pattern takes O(w) time to be transformed into the corresponding output pattern X. Since there are τ LOPs in the worst case, Step 4 requires O(w‧τ) time to generate all LOP candidates.

4.

Time complexity of Step 5: Step 5 involves removing non-minimal LOP candidates. Testing whether each LOP candidate is an LOP takes O(w‧τ) time, resulting in a total complexity of O(w‧τ²) for all candidates in the worst case.

5.

Time complexity of Step 6: The number of LOPs is |Ω| = τ in the worst case, and applying the RSDP method requires at most O(${\displaystyle \sum _{\kappa =1}^{\tau }\kappa }$) time.

In summary, given that τ = $\left(\begin{array}{c}\rho +d-1\\ d\end{array}\right)$ = ${\displaystyle \frac{(\rho +d-1)\times (\rho +d-2)\times \dots \times \rho }{d\times \left(d-1\right)\times \dots \times 2\times 1}}$ which implies τ  > > ρ, and the number of MPs ρ is at most the number of arcs w. Moreover, because the number of nodes in a network generally does not exceed the number of arcs, the numbers of energy sources and converters are both less than ρ. Therefore, the computational complexity of the proposed algorithm in the worst case is dominated by Step 5 and can be expressed as O(w‧τ²) = O(ρ‧τ) + O(w‧ρ‧τ) + O(w‧τ) + O(w‧τ²) + O(${\displaystyle \sum _{\kappa =1}^{\tau }\kappa }$).

4. An Illustrative Example of the MSGN

An MSGN, as illustrated in Figure 3, is constructed to demonstrate the proposed delivery reliability evaluation. The MSGN consists of three energy sources: a photovoltaic cell (b₁), a wind turbine farm (b₂), and an energy storage station (b₃), two converters (b₄, b₅), and eight feeders (b₆, b₇, b_8, b₉, b₁₀, b_11, b₁₂, b₁₃). In total, six MPs are available for delivering power from source nodes to the destination node: P_1,1  =  {b₁, b₄, b₆, b₁₂}, P_1,2 = {b₁, b₅, b₇, b₁₃}, P_2,1 = {b₂, b₄, b₈, b₁₂}, P_2,2 =  {b₂, b₅, b₉, b₁₃}, and P_3,1 = {b₃, b₄, b₁₀, b₁₂}, and P_3,2 = { b₃, b₅, b₁₁, b₁₃}. The transmission loss rates associated with these MPs are summarized in

Table 2. Transmission loss rates of Figure 3.

Minimal Path (Pi,j)	Transmission Loss Rate (zi,j)
P1,1	0.04
P1,2	0.07
P2,1	0.05
P2,2	0.06
P3,1	0.03
P3,2	0.08

, which is determined based on a historical database provided by the power company.

In the MSGN, each facility consists of multiple devices, and the output of these devices follows a probability distribution. These devices often have distinct specifications due to differences in design, retrofitting, or operational constraints. Unlike models that assume homogeneous devices, the proposed MSGN model is designed to accommodate facilities consisting of heterogeneous devices with varying outputs and reliabilities. As a result, the output of each facility is multistate, and the probability of each output state is determined by a multivariate Bernoulli distribution [31]. Each component (e.g., generator, transformer, conductor) is treated as an independent Bernoulli trial with its own success probability (reliability). All combinations of device states yield different available output levels, and the corresponding probabilities are calculated by enumerating all possible combinations of device states. For example, consider a thermal energy source equipped with four generating units rated at 400 kW, 350 kW, 250 kW, and 200 kW, with corresponding reliability values of 0.9521, 0.9174, 0.8856, and 0.9347, respectively. Depending on the operational status of these generators, the possible output states of the thermal energy source include 0 kW, 200 kW, 250 kW, 350 kW, 400 kW, 550 kW, 600 kW, 750 kW, 950 kW, 1000 kW, 1150 kW, and 1200 kW, reflecting all possible combinations of functioning generators. To illustrate the probability calculation, consider the output state of 750 MW, which occurs when the 400 kW and 350 kW generators are operational, while the other two units fail. The probability of this state is computed as 0.9521 × 0.9174 × (1 − 0.8856) × (1 − 0.9347) = 0.00568. By applying the component reliability data presented in

Table 3. Output capacity (kW) and reliability of devices in energy sources, converters, and feeders.

Facility (bi)	Total Output of bi	Device	Output of the Component	Reliability	Facility (bi)	Total Output of bi	Device	Output of the Component	Reliability
b1	500 kW	generator 1	100 kW	0.9797	b8	250 kW	conductor 1	50 kW	0.9542
		generator 2	100 kW	0.9296			conductor 2	50 kW	0.9542
		generator 3	150 kW	0.9504			conductor 3	50 kW	0.9542
		generator 4	150 kW	0.9433			conductor 4	50 kW	0.9542
							conductor 5	50 kW	0.9542
		generator 2	90 kW	0.9485	b9	250 kW	conductor 1	50 kW	0.8977
		generator 3	90 kW	0.9476			conductor 2	50 kW	0.8977
		generator 4	140 kW	0.9764			conductor 3	50 kW	0.8977
		generator 5	140 kW	0.9753			conductor 4	50 kW	0.8977
							conductor 5	50 kW	0.8977
		generator 2	100 kW	0.9941	b10	250 kW	conductor 1	50 kW	0.9655
		generator 3	100 kW	0.9369			conductor 2	50 kW	0.9655
		generator 4	100 kW	0.9173			conductor 3	50 kW	0.9655
		generator 5	100 kW	0.9255			conductor 4	50 kW	0.9655
		generator 6	100 kW	0.9644			conductor 5	50 kW	0.9655
b4	350 kW	transformer 1	70 kW	0.8901	b11	250 kW	conductor 1	50 kW	0.9122
		transformer 2	70 kW	0.9433			conductor 2	50 kW	0.9122
		transformer 3	70 kW	0.8972			conductor 3	50 kW	0.9122
		transformer 4	70 kW	0.9064			conductor 4	50 kW	0.9122
		transformer 5	70 kW	0.9653			conductor 5	50 kW	0.9122
b5	350 kW	transformer 1	70 kW	0.9312	b12	200 kW	conductor 1	40 kW	0.9832
		transformer 2	70 kW	0.9564			conductor 2	40 kW	0.9832
		transformer 3	70 kW	0.9564			conductor 3	40 kW	0.9832
		transformer 4	70 kW	0.9145			conductor 4	40 kW	0.9832
		transformer 5	70 kW	0.9357			conductor 5	40 kW	0.9832
b6	250 kW	conductor 1	50 kW	0.8553	b13	200 kW	conductor 1	40 kW	0.9591
		conductor 2	50 kW	0.8553			conductor 2	40 kW	0.9591
		conductor 3	50 kW	0.8553			conductor 3	40 kW	0.9591
		conductor 4	50 kW	0.8553			conductor 4	40 kW	0.9591
		conductor 5	50 kW	0.8553			conductor 5	40 kW	0.9591
b7	250 kW	conductor 1	50 kW	0.9604
		conductor 2	50 kW	0.9604
		conductor 3	50 kW	0.9604
		conductor 4	50 kW	0.9604
		conductor 5	50 kW	0.9604

, the output-state distribution of each facility b_i can be derived based on the multivariate Bernoulli distribution.

4.1. Delivery Reliability Evaluation of the MSGN

Given the MSGN demand is d  =  360 kW, the following steps outline the proposed solution procedure to obtain all the LOPs for calculating the delivery reliability R_d.

• Step 1. Identify all the electricity flow patterns F = (f_1,1, f_1,2, f_2,1, f_2,2, f_3,1, f_3,2) that satisfy the following equation:

  f_1,1 + f_1,2 + f_2,1 + f_2,2 + f_3,1 + f_3,2 = 360.

  (12)

In this step, a total of 53,130 electricity flow patterns are generated, which are listed in the first column of

Table 4. Calculation results of steps 1–5 of the proposed algorithm.

Step 1	Step 2	Step 3	Step 4	Step 5
F = (f1,1, f1,2, f2,1, f2,2, f3,1, f3,2).	H = (h1,1, h1,2, h2,1, h2,2, h3,1, h3,2).	Is H feasible for L or not?	Convert H into X = (x1, x2, …, x13).	Find all the LOP.
F1 = (0, 0, 0, 0, 180, 180)	H1 = (0, 0, 0, 0, 185.5670, 195.6522)	Feasible for L	X1 = (0, 0, 400, 210, 210, 0, 0, 0, 0, 20, 20, 20, 20)	X1 = (0, 90, 300, 210, 210, 0, 0, 0, 100, 200, 100, 200, 20)
F2 = (0, 0, 0, 0, 190, 170)	H2 = (0, 0, 0, 0, 195.8762, 184.7826)	Feasible for L	X2 = (0, 90, 300, 210, 210, 0, 0, 0, 100, 200, 100, 200, 200)	X2 = (100, 90, 200, 210, 210, 0, 100, 0, 100, 200, 0, 200, 20)
F3 = (0, 0, 0, 10, 170, 180)	H3 = (0, 0, 0, 10.6383, 175.2577, 195.6522)	Feasible for L	X3 = (0, 90, 300, 210, 210, 0, 0, 0, 100, 200, 150, 200, 200)	X3 = (150, 140, 100, 210, 210, 150, 0, 50, 100, 0, 100, 200, 20)
F4 = (0, 0, 0, 10, 180, 170)	H4 = (0, 0, 0, 10.6383, 185.5670, 184.7826)	Feasible for L	X4= (0, 90, 300, 210, 210, 0, 0, 50, 50, 150, 200, 200, 200)	X4= (200, 90, 100, 210, 210, 200, 0, 0, 100, 0, 100, 200, 20)
⁝	⁝	⁝	⁝	⁝
F53,130 = (200, 160, 0, 0, 0, 0, 0, 0, 0, 0)	H53,130 = (208.3333, 172.0430, 0, 0, 0, 0)	Unfeasible for L	X3978 = (400, 90, 100, 210, 210, 200, 200, 50, 0, 50, 0, 200, 200)	X428 = (250, 90, 200, 210, 210, 100, 150, 50, 0, 100, 50, 200, 20)

.

• Step 2. Convert each F obtained from Step 1 into the required flow pattern H = (h_1,1, h_1,2, h_2,1, h_2,2, h_3,1, h_3,2). For example, F₁  =  (0, 0, 0, 0, 180, 180), the equations shown below are used to obtain the corresponding required flow pattern H₁  =  (0, 0, 0, 0, 185.5670, 195.6522).

  $$\begin{array}{c}{h}_{1,1} ={\displaystyle \frac{f_{1,1}}{1-z_{1,1}}} = {\displaystyle \frac{0}{1-0.04}} = 0,\\ h_{1,2} ={\displaystyle \frac{f_{1,2}}{1-z_{1,2}}} = {\displaystyle \frac{0}{1-0.07}} = 0,\\ h_{2,1} ={\displaystyle \frac{f_{2,1}}{1-z_{2,1}}} = {\displaystyle \frac{0}{1-0.05}} = 0,\\ h_{2,2} ={\displaystyle \frac{f_{2,2}}{1-z_{2,2}}} = {\displaystyle \frac{0}{1-0.06}} = 0,\\ h_{3,1} ={\displaystyle \frac{f_{3,1}}{1-z_{3,1}}} = {\displaystyle \frac{18}{1-0.03}} = 185.5670,\\ h_{3,2} ={\displaystyle \frac{f_{3,2}}{1-z_{3,2}}} = {\displaystyle \frac{18}{1-0.08}} = 195.6522.\end{array}$$

  (13)

All required flow patterns are listed in the second column of Table 4.

• Step 3. Based on the maximal output pattern L = (500, 550, 600, 350, 350, 250, 250, 250, 250, 250, 250, 200, 200), any electricity flow pattern that fails to satisfy the following constraints is excluded:

  h_1,1 + h_1,2 ≤ 500,
  h_2,1 + h_2,2 ≤ 550,
  h_3,1 + h_3,2 ≤ 600,
  h_1,1 + h_2,1 + h_3,1 ≤ 350,
  h_1,2 + h_2,2 + h_3,2 ≤ 350,
  h_1,1 ≤ 250,
  h_1,2 ≤ 250,
  h_2,1 ≤ 250,
  h_2,2 ≤ 250,
  h_3,1 ≤ 250,
  h_3,2 ≤ 250,
  h_1,1 + h_2,1 + h_3,1 ≤ 200,
  h_1,2 + h_2,2 + h_3,2 ≤ 200.

  (14)

This filtering step reduces the number of feasible electricity flow patterns from 53,130 to 3978, which are listed in the third column of Table 4.

• Step 4. Convert each H into a corresponding LOP candidate X = (x₁, x₂, …, x₁₃). As an example, for H₁  =  (0, 0, 0, 0, 185.5670, 195.6522), the loading of each facility is computed as follows:

  $$\begin{array}{c}(\mathrm{for} b_1) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_1\in P_{i,j}}{f}_{i,j}}} = f_{1,1} + f_{1,2} = 0,\\ (\mathrm{for} b_2) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_2\in P_{i,j}}{f}_{i,j}}} = f_{2,1} + f_{2,2} = 0,\\ (\mathrm{for} b_3) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_3\in P_{i,j}}{f}_{i,j}}} = f_{3,1} + f_{3,2} = 381.2192,\\ (\mathrm{for} b_4) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_4\in P_{i,j}}{f}_{i,j}}} = f_{1,1} + f_{2,1} + f_{3,1} = 185.5670,\\ (\mathrm{for} b_5) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_5\in P_{i,j}}{f}_{i,j}}} = f_{1,2} + f_{2,2} + f_{3,2} = 195.6522,\\ (\mathrm{for} b_6) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_6\in P_{i,j}}{f}_{i,j}}} = f_{1,1} = 0,\\ (\mathrm{for} b_7) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_7\in P_{i,j}}{f}_{i,j}}} = f_{1,2} = 0,\\ (\mathrm{for} b_8) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_8\in P_{i,j}}{f}_{i,j}}} = f_{2,1} = 0,\\ (\mathrm{for} b_9) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_9\in P_{i,j}}{f}_{i,j}}} = f_{2,2} = 0,\\ (\mathrm{for} b_{10}) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_{10}\in P_{i,j}}{f}_{i,j}}} = f_{3,1} = 185.5670,\\ (\mathrm{for} b_{11}) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_{11}\in P_{i,j}}{f}_{i,j}}} = f_{3,2} = 195.6522,\\ (\mathrm{for} b_{12}) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_{12}\in P_{i,j}}{f}_{i,j}}} = f_{1,1} + f_{2,1} + f_{3,1} = 185.5670,\\ (\mathrm{for} b_{13}) {\displaystyle {\sum }_{i=1}^u{\displaystyle {\sum }_{j: b_{13}\in P_{i,j}}{f}_{i,j}}} = f_{1,2} + f_{2,2} + f_{3,2} = 195.6522.\end{array}$$

  (15)

The output pattern that satisfies the load requirements is therefore X₁ = (0, 0, 400, 210, 210, 0, 0, 0, 0, 20, 20, 20, 20). Consequently, a total of 3978 output patterns, denoted as X₁, X₂, …, X₃₉₇₈, are generated and summarized in the fourth column of Table 4, which are considered as the LOP candidates.

• Step 5. Compare the 3978 output patterns and remove the non-lower X patterns to determine whether each pattern qualifies as an LOP. Finally, a total of 428 LOPs are identified and summarized in the fifth column of Table 4.
• Step 6. The RSDP method is employed to calculate delivery reliability, yielding an R_d value of 0.74114328. This result indicates that the MSGN has approximately a 74% possibility of supplying sufficient power to meet the demand. To demonstrate computational efficiency, the proposed algorithm is compared with the enumeration method, which computes delivery reliability by exhaustively identifying all LOPs and aggregating their corresponding probabilities. The results indicate that the proposed algorithm completes the evaluation in 10.3272 s, whereas the enumeration method requires 403.6667 s, highlighting the significant reduction in computational time achieved by the proposed approach.

4.2. Discuss the Performance of the MSGN

To measure the performance of the MSGN, an experiment is conducted under different demand levels, with an interval of 20 kW between each neighboring setting. The MSGN delivery reliability is shown in Figure 4. Based on Figure 4, delivery reliability decreases as demand increases, reflecting the tightening of feasibility constraints under transmission loss. If the grid operator intends to maintain a total demand of 340 kW, the possibility of the MSGN meeting the demand is approximately 0.9.

Furthermore, to identify the most sensitive MP affecting power supply quality, consider the example where the demand is set to d = 360 kW and the transmission loss rate z_i,j is initially set to zero for all i = 1, 2, …, u, j = 1, 2, …, θ_i. The delivery reliability, excluding the transmission loss rate on each MP, is 0.96249020. To examine the sensitivity of each MP, the transmission loss rate of one MP is increased from 0 to 0.9, while the transmission loss rates of all remaining MPs are maintained at 0. For example, when z_1,1 is set to 0.9, and all other rates remain unchanged, the delivery reliability decreases to 0.96249018. By repeating this process for each MP in turn, the experiment results are obtained and presented in

Table 5. Sensitivity analysis for the transmission loss rate on each MP.

Change the Transmission Loss Rate on Pi,j	Delivery Reliability	The Difference in Delivery Reliability Without Damage	Rank
P1,1	0.96249018	−0.00000002	6
P1,2	0.96248402	−0.00000618	1
P2,1	0.96248862	−0.00000158	3
P2,2	0.96248959	−0.00000061	5
P3,1	0.96248682	−0.00000338	2
P3,2	0.96248921	−0.00000099	4

. Among all MPs, the transmission loss rate associated with P_1,2 exhibits the most significant influence on the overall delivery reliability, demonstrating that P_1,2 plays a critical role because it combines a higher transmission loss rate with a key topological position connecting a major energy source to the destination. Therefore, enhancing the power delivery capability of this MP should be prioritized.

5. Conclusions

In this study, a multistate smart-grid network (MSGN) model is developed to evaluate the capability of the smart-grid system (SGS). The delivery reliability, denoted as R_d, is assessed through an algorithm constructed based on the concept of minimal path. The proposed algorithm consists of two phases. In the first phase, all lower output patterns, denoted as LOP, are identified. In the second phase, the LOPs are used to compute the delivery reliability of the MSGN. The results of delivery reliability provide practical insights for the grid operator, supporting maintenance prioritization, facility planning, and strategic system reinforcement. Once the proposed delivery reliability evaluation framework is established, it is inherently generalizable and can be applied to larger and more complex SGS. In real-world SGS operation, however, transmission losses depend on loading conditions and ambient temperature and exhibit nonlinear characteristics. Moreover, correlations among facilities and devices frequently arise due to shared environmental conditions, standard device aging, and regional weather disruptions. Under such circumstances, the current MSGN model may not represent the characteristics of real-world SGS. Accordingly, future research should focus on developing a correlation-aware MSGN model with nonlinear loss representation, as well as defining corresponding reliability indices that accurately reflect interdependent system dynamics.