Improved Imprecise Dirichlet Model–Improved Transitional Markov Chain Monte Carlo for Power System Reliability Assessment

Abstract

Component outage records in power systems are often limited, which makes it difficult to represent failure probabilities with deterministic point estimates. To address this issue, this paper proposes a reliability assessment framework that combines an improved Imprecise Dirichlet Model (IDM) with improved Transitional Markov Chain Monte Carlo (iTMCMC). The improved IDM introduces a sample-size-dependent hyperparameter to construct adaptive outage-probability intervals for different equipment categories. These interval probabilities are then propagated through iTMCMC to obtain interval-valued system reliability indices. In the sampling process, a reliability-oriented likelihood function is used to guide system-state exploration, and self-normalized weights are applied to maintain estimator consistency. A case study is conducted on a standard IEEE reliability test system. The results show that the improved IDM can provide adaptive component outage-probability intervals, while iTMCMC achieves more stable LOLP and EENS estimates than MC and MCMC. The interval propagation results further demonstrate that the proposed framework can transfer component-level probability uncertainty into system-level reliability-index intervals. The proposed method provides a practical tool for reliability assessment when component failure records are incomplete or insufficient.

1. Introduction

Reliability assessment is a fundamental task in power system planning and operation. It evaluates the ability of a power system to maintain electricity supply under component outages, load variations, and extreme operating conditions. The resulting reliability indices provide technical support for generation-resource planning, resilience analysis, resource adequacy evaluation, and electricity-market design [1,2,3].

Power system reliability assessment methods are commonly divided into analytical methods and simulation-based methods. Analytical methods include fault-tree analysis, Markov models, and network-reliability analysis, whereas simulation-based methods mainly include Monte Carlo simulation and its variants. Analytical methods are computationally efficient when the system-state model is explicit, but their scalability is limited as the number of components and operating states increases. Simulation-based methods avoid exhaustive state enumeration by sampling component operating and outage states, making them suitable for large-scale reliability assessment [4,5,6].

Monte Carlo simulation and related sampling techniques have been widely used in probabilistic power system analysis and reliability assessment [7,8,9]. Conventional MC, however, often requires a large number of independent samples to obtain stable estimates, especially in high-dimensional state spaces or rare-event reliability evaluation. Markov Chain Monte Carlo (MCMC) introduces a Markov transition mechanism into the sampling process and generates dependent samples whose stationary distribution approximates the target distribution [10].

Traditional MCMC methods usually rely on predefined transition kernels and proposal distributions. In complex or high-dimensional state spaces, the resulting chains may mix slowly or remain in local regions for many iterations, reducing sampling efficiency [11]. Transitional Markov Chain Monte Carlo (TMCMC) improves this process by introducing a sequence of intermediate distributions between the prior and target distributions. Recent refined TMCMC strategies further incorporate adaptive mechanisms to improve sampling efficiency in complex probability spaces [12].

Although Monte Carlo simulation and Markov Chain Monte Carlo methods have been widely used in power system reliability assessment, their computational efficiency may decrease when the system-state space is large and loss-of-load events are rare. To address this issue, several rare-event simulation and variance-reduction techniques have been developed. Cross-Entropy-based Monte Carlo methods construct an auxiliary sampling distribution by modifying the reliability parameters of system components, so that important failure states can be sampled more frequently [13]. This idea has been further extended to composite generation–transmission systems, where the unavailabilities of both generation and transmission equipment are distorted to improve the convergence of reliability-index estimation [14]. Subset Simulation provides another efficient strategy by expressing a small failure probability as a product of larger conditional probabilities and generating conditional samples from intermediate failure domains [15]. Importance Sampling methods reduce estimator variance by changing the sampling distribution of generation, transmission, and load states and correcting the estimates with likelihood ratios [16]. More recent studies have also improved sequential Cross-Entropy simulation and developed likelihood-ratio corrections for composite system reliability indices [17]. In addition, Cross-Entropy, Subset Simulation, and adaptive computational-burden criteria have been combined to improve parameter updating and stopping-rule selection in composite system reliability evaluation [18]. These studies provide important foundations for improving the efficiency of rare-event reliability assessment.

Table 1. Comparison of reliability assessment methods.

Reference	Main Method	Main Focus
[7]	Monte Carlo simulation	Basic probabilistic sampling
[10]	Markov Chain Monte Carlo	Large-scale state-space sampling
[12]	Refined Transitional Markov Chain Monte Carlo	Adaptive transitional sampling
[13]	Cross-Entropy-based Monte Carlo simulation	Generating-capacity reliability assessment
[15]	Subset Simulation	Rare failure event extraction
[16]	Improved Importance Sampling	Variance reduction for composite systems
[18]	Cross-Entropy with Subset Simulation	Parameter updating and computational-burden reduction

summarizes representative published studies related to sampling-based power system reliability assessment. Existing studies mainly focus on reducing the required sample size, improving the exploration of rare failure states, or decreasing the computational burden of state evaluation under given component reliability parameters.

Cross-Entropy and Importance Sampling improve rare-state exploration through biased sampling distributions, while Subset Simulation relies on nested intermediate failure events. These methods mainly focus on sampling efficiency under prescribed component outage probabilities, with uncertainty in failure-probability estimation generally treated separately from system-level reliability evaluation. In contrast, this study incorporates interval-valued component failure probabilities into the system-state prior distribution and couples them with iTMCMC-based transitional sampling. The transitional distributions and Gibbs updates retain the topology-dependent coupling of outage states without requiring a single predefined biasing distribution or a common threshold sequence for both LOLP and EENS. The resulting framework integrates component-level uncertainty, reliability-oriented state exploration, and load-curtailment-based index evaluation within a unified process. The main contributions are summarized as follows:

• An uncertainty-aware component failure-probability modeling mechanism is developed based on an improved IDM. Instead of using fixed outage probabilities, interval-valued failure probabilities are constructed through sample-size-dependent hyperparameter adjustment and used to form the prior distribution of system states.
• A reliability-oriented iTMCMC sampling mechanism is established for large-scale power system states. Transitional distributions, importance-weight correction, and resampling are used to improve the exploration of reliability-relevant system states in high-dimensional state spaces while ensuring consistency with the original system-state probability model.
• A coupled reliability assessment framework is constructed by linking component-level failure uncertainty, iTMCMC-based state sampling, and a DC-power-flow-based minimum-load-shedding model, enabling interval-valued LOLP and EENS to be obtained.

2. Basic Principles

In power system reliability assessment, component-level failure parameters provide the statistical basis for system-state sampling and reliability-index calculation. However, failure records for newly upgraded or rarely failed equipment are often sparse, especially when the system contains many heterogeneous components. In such cases, single-valued probability estimates may be sensitive to limited observations and may not adequately represent epistemic uncertainty. Imprecise probability models provide an alternative by representing component failure probabilities as intervals rather than fixed values.

2.1. Precise and Imprecise Probabilities

Traditional precise point probability requires sufficient historical failure data to derive a specific probability value for an event. In contrast, imprecise probability expresses probability as an interval, accounting not only for the possibility of the event occurring but also for the uncertainty in the sample data. This approach effectively addresses situations where sample data is insufficient.

2.2. Fundamental Principles of the Dirichlet Model

The Dirichlet model is a Bayesian probabilistic model commonly used to describe uncertainty in multinomial state probabilities. For a component with several mutually exclusive states, the state-probability vector can be assigned a Dirichlet prior, whose hyperparameters reflect prior information and regulate the influence of observations on posterior estimates. The probability density function of the Dirichlet distribution can be expressed as:

$$f(x;\alpha )={\displaystyle \frac{\mathsf{\Gamma }({\displaystyle \sum _{i=1}^k}{\alpha }_i+M)}{{\displaystyle \prod _{i=1}^k}\mathsf{\Gamma }({\alpha }_i+m_i)}}{\displaystyle \prod _{i=1}^k}{x}_i^{{\alpha }_i+m_i-1}$$

(1)

In the equation, α = (α1, α2, …, αk) is the hyperparameter introduced in the Dirichlet model, representing the influence of the assumed random variable on probability estimation; x = (x1, x2, …, xk) denotes the probability vector for each precise state of a specific assumed random variable; M = (m1, m2, …, mk) represents the actual sample data.

When the sample size is small, however, fixed hyperparameters may exert a disproportionate influence on the posterior probability estimates, leading to biased or overly narrow results.

2.3. Imprecise Dirichlet Model

To improve the robustness of component failure-probability estimation under limited observations, the Imprecise Dirichlet Model (IDM) is introduced. The Imprecise Dirichlet Model (IDM) dynamically adjusts hyperparameters αi based on sample size by modifying the values of prior parameters within these hyperparameters. This transforms the state probability vector into an interval form that contains precise probabilities. Consequently, it enhances the credibility of probability estimates while ensuring computational accuracy. Under limited samples and incomplete data, the probability interval provides a more cautious representation of epistemic uncertainty than a single point estimate.

Specifically, when the actual sample size m is small, the hyperparameter α is set to a larger value to ensure that the probability estimate does not suffer from significant bias. When the actual sample size m is large, the hyperparameter α is set to a smaller value to ensure that the probability estimate interval is narrowed [19,20].

2.4. Power System Reliability Analysis Based on an Improved Imprecise Dirichlet Model

The hyperparameter $\alpha$ controls the strength of the imprecise prior in the IDM probability interval. A larger $\alpha$ produces a wider probability interval, indicating stronger epistemic uncertainty, whereas a smaller $\alpha$ makes the interval estimate more dependent on the observed failure records. Therefore, $\alpha$ should be related to the amount of available statistical evidence. When the effective sample size m is small, the observed failure records are insufficient to support a confident point-valued estimate, and a relatively larger $\alpha$ is needed to retain interval uncertainty. As m increases, the influence of the imprecise prior should gradually decrease so that the probability interval is increasingly dominated by the observed data. Considering the dependence of the IDM strength on the available sample size, the hyperparameter is specified as follows:

$$\alpha (m)=\left\{\begin{array}{ll} \theta \hfill & 0<m\le 2\hfill \\ {\displaystyle \frac{\theta }{\ln (m)}}\hfill & m>2\hfill \end{array}\right.$$

(2)

When 0 < m ≤ 2, the available failure records are extremely limited, so $\alpha$ is kept at $\theta$ to retain sufficient interval uncertainty and avoid unstable estimates. For m > 2, the logarithmic form is used because ln(m) increases slowly with m, allowing $\alpha$ to decrease gradually rather than shrink too rapidly under limited records. The threshold m = 2 is used to avoid numerical instability when ln(m) is close to zero or not suitable for controlling the hyperparameter. This specification allows the IDM strength to decrease gradually with increasing sample size and is used as a practical modeling choice in this study, rather than being regarded as theoretically optimal.

In power system reliability assessment, the improved IDM enables interval estimation of equipment failure probabilities under limited historical data. Moreover, the size of the probability estimation interval can be dynamically adjusted based on changes in sample size, thereby enhancing the reliability of the probability estimation results.

3. Equipment and System Risk Assessment

3.1. Fundamental Principles of the iTMCMC Method

The improved Transitional Markov Chain Monte Carlo (iTMCMC) algorithm is a progressive sampling method developed from conventional MCMC. Unlike standard MCMC, which uses a single Markov chain to approach the target distribution, iTMCMC introduces a sequence of transitional distributions between the prior distribution and the target distribution. At each transitional stage, sample weighting, resampling, and local moves are performed. This strategy reduces sample degeneracy, improves the exploration of high-dimensional state spaces, and alleviates the dependence of conventional MCMC on initial samples and fixed proposal settings [21,22].

In power system reliability assessment, the sampled object is the system operating-state vector. For a system with N components, the state of component i is represented by a binary variable x_i:

$$x_i\in \{0,1\}, i=1,2,\dots ,N$$

(3)

where x_i = 0 denotes the normal operating state and xi = 1 denotes the outage state. The system-state vector is defined as:

$$x=(x_1,x_2,\dots ,x_N)$$

(4)

The prior distribution of the system state is constructed from the component failure probabilities estimated by the improved IDM. To avoid confusion with the IDM hyperparameter, the outage probability of component i is denoted by $p_i^f$. Assuming independent component outages at the prior-sampling stage, the prior probability of a system state can be expressed as:

$$p(x)={\displaystyle \mathbf{\prod }_{i=1}^N{\left(p_i^f\right)}^{x_i}}{\left(1-p_i^f\right)}^{1-x_i}$$

(5)

$p_i^f$ is the prior outage probability of component i. In this article, $p_i^f$ is obtained from the failure-probability interval estimated by the improved IDM. The component outage-probability intervals obtained from the improved IDM are incorporated into the prior distribution to represent component-level uncertainty in system-state sampling. The target distribution for system-state sampling is written in a posterior-type form:

$$\pi (x)={\displaystyle \frac{p(x)L(x)}{Z}}$$

(6)

p(x) is the prior distribution of the system state, L(x) is defined as a reliability-oriented function, and Z is a normalization constant. For any indicator z, min–max normalization is defined as:

$$\mathcal{N}(z_k)={\displaystyle \frac{z_k-{\min }_r{z}_r}{{\max }_r{z}_r-{\min }_r{z}_r}}$$

(7)

where z_k denotes the k-th value of the indicator being normalized, and the minimum and maximum are calculated over all values of the same indicator. If the denominator is zero, all normalized values are set to zero.

The risk score of branch b is calculated as:

$$r_b=\mathcal{N}\left[0.65\mathcal{N}\left({\displaystyle \frac{\left|P_b^0\right|}{{\overline{P}}_b}}\right)+0.20\mathcal{N}\left(\left|X_b\right|\right)+0.15T_b\right]$$

(8)

where $P_b^0$ is the active power flow of branch b obtained from the intact base-case DC power flow, ${\overline{P}}_b$ is the branch-flow limit used in the reliability calculation. X_b and T_b denote the branch reactance and the transformer indicator, respectively. Here, T_b = 1 for a transformer branch and T_b = 0 otherwise.

The individual risk weight of equipment i is:

$$h_i=\mathcal{N}\left({\gamma }_i{r}_{b(i)}\right)$$

(9)

where b(i) denotes the branch associated with equipment i, and γi is set to 1.15, 1.05, and 0.95 for transformers, transmission lines, and circuit breakers.

The coupling weight is defined as:

$${\mathrm{c}}_{\mathrm{ij}}=0{.35\mathrm{a}}_{\mathrm{ij}}$$

(10)

where a_ij = 1 if equipment i and j are associated with the same branch or with branches sharing an endpoint bus; otherwise, a_ij = 0, with a_ii = 0.

The state-risk score is then expressed as:

$$H(x)={\displaystyle {\displaystyle \sum _{i=1}^n}{h}_i}{x}_i+{\displaystyle \frac{1}{2}}{\displaystyle {\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n{c}_{ij}}}{x}_i{x}_j$$

(11)

Here, H(x) is defined as the state-risk score of outage state x. A larger H(x) indicates that the outage state is more relevant to system reliability deterioration or load curtailment.

And the reliability-oriented function L(x) is:

$$L(x)=\exp [\eta H(x)]$$

(12)

where x_i = 1 indicates the failure outage of component i, and x_i = 0 indicates the normal operating state. The coefficients in the risk model are parameter values adopted in this study to guide the exploration of reliability-relevant states and are not interpreted as physical probabilities.

Direct sampling from π(x) can be inefficient for large-scale power systems, especially when the state space is high-dimensional and reliability indices are affected by low-probability outage states. To improve sampling efficiency, iTMCMC constructs a sequence of transitional distributions:

$${\pi }_j(x)={\displaystyle \frac{p(x)L{(x)}^{{\beta }_j}}{Z_j}}$$

(13)

where j denotes the transitional stage, Z_j is the normalization constant at stage j, and β_j is the transitional parameter satisfying the following:

$$0={\beta }_0<{\beta }_1<\cdots <{\beta }_j<\cdots <{\beta }_m=1$$

(14)

When β₀ = 0, the transitional distribution reduces to the prior distribution:

$${\pi }_0(x)\propto p(x)$$

(15)

When β_m = 1, the transitional distribution becomes the target distribution:

$${\pi }_m(x)\propto p(x)L(x)$$

(16)

The gradual increase of β_j controls the difference between adjacent transitional distributions and prevents severe sample-weight degeneration. The equivalent likelihood function L(x) is gradually introduced through the transitional parameter βj. This avoids an abrupt transition from the prior distribution to the reliability-oriented target distribution and helps reduce sample-weight degeneration.

At the j-th transitional stage, the incremental weight of sample $x_i^{(j)}$ is calculated according to the likelihood increment between two adjacent transitional distributions:

$$\begin{array}{l}{\omega }_i^{(j)}={\left[L\left(x_i^{(j)}\right)\right]}^{\Delta {\beta }_j}\hspace{1em}\\ \Delta {\beta }_j={\beta }_{j+1}-{\beta }_j\end{array}$$

(17)

The normalized sample weight is then obtained as:

$${\tilde{\omega }}_i^{(j)}={\displaystyle \frac{{\omega }_i^{(j)}}{{\displaystyle \sum _{k=1}^{N_s}{\omega }_k^{(j)}}}},\hspace{1em}{\displaystyle \sum _{i=1}^{N_s}{\tilde{\omega }}_i^{(j)}=1}$$

(18)

where N_s is the number of samples at each transitional stage. Samples with larger normalized weights are more likely to be retained during resampling, whereas low-weight samples are gradually eliminated. This process guides the sample population toward the next transitional distribution and reduces sample degeneracy.

The effective sample size (ESS) is used to measure the degeneracy of the weighted sample set and to control the selection of the transitional parameter. For the normalized weights at stage j, ESS is calculated as:

$$ES{S}_j={\displaystyle \frac{1}{{\displaystyle \sum _{i=1}^{N_s}{\left({\tilde{\omega }}_i^{(j)}\right)}^2}}}$$

(19)

If the convergence criterion is expressed as an effective sample proportion, it can be defined as:

$$ESS\%={\displaystyle \frac{ES{S}_j}{N_s}}\times 100\%$$

(20)

The target effective sample proportion is set to 20%. At each transitional stage, ESS% is used as a stage-wise criterion for selecting the next transitional parameter ${\beta }_{j+1}$. Since the incremental weights depend on the candidate value of ${\beta }_{j+1}$, ESS% is recalculated for each candidate transitional parameter. If the terminal value ${\beta }_{j+1}$ = 1 still satisfies the target effective sample proportion, the final transitional stage is reached directly:

$$ESS\%(1)\ge 20\%, {\beta }_{j+1}=1$$

(21)

Otherwise, ${\beta }_{j+1}$ is adaptively determined by solving:

$$ESS\%({\beta }_{j+1})=20\%, {\beta }_j<{\beta }_{j+1}<1$$

(22)

After ${\beta }_{j+1}$ is determined, the incremental weights are normalized, and resampling is performed to generate an equally weighted sample population for the next transitional stage. For binary component-state sampling, the local update step is implemented using the Gibbs state sampler described in Section 3.2.

3.2. Gibbs State Sampler in Power System Reliability Assessment

In Gibbs sampling, the system operating state is updated component by component according to the conditional state probability of each component [23]. This study considers the random outages of the main reliability-related components, including transformers, circuit breakers, and transmission lines. Each sampled component is represented by a two-state Bernoulli model. The outage probability $p_i^f$ represents the probability that component i is in the outage state, while 1 − $p_i^f$ represents the probability of normal operation. Here, $p_i^f$ is obtained from the component failure probability estimated by the improved IDM. The load is treated as deterministic in this sampling stage, and load uncertainty is not included in the present model. The sampled component states are then used to generate system operating states for subsequent reliability assessment.

X represents the state vector of the sampled components, consisting of m binary state variables. Here, m denotes the number of components included in state sampling, including transformers, circuit breakers, and transmission lines. The k-th Gibbs sample is written as $x^{(k)}=\left(x_1^{(k)},x_2^{(k)},\dots ,x_m^{(k)}\right)$, where $x_{\mathrm{i}}^{(k)}$ denotes the state of component i at the k-th sampling step. The value of $x_{\mathrm{i}}^{(k)}$ is either 0 or 1, with 0 denoting the normal operating state and 1 denoting the outage state.

The Gibbs sampler is initialized with all sampled components in the normal state. At each Gibbs iteration, the state of component i is updated according to its full conditional probability. Given the current states of all other components, the outage probability of component i is calculated from the conditional distribution as:

$${\eta }_i^{(k+1)}=P\left(x_i=1\mid x_{-i}^{(k+1/k)}\right)$$

(23)

where $x_{-i}^{(k+1/k)}$ denotes the state vector excluding component i, with the first i − 1 components updated in the current iteration and the remaining components retained from the previous iteration.

The conditional probability is evaluated using the most recently available states of the remaining components. The conditional probability is expressed by comparing the target probabilities of the two possible component states. At the j-th transitional stage, the conditional outage probability of component i is calculated as:

$${\eta }_i^{(k+1)}={\displaystyle \frac{{\pi }_j\left(x_i=1,x_{-i}^{(k+1/k)}\right)}{{\pi }_j\left(x_i=1,x_{-i}^{(k+1/k)}\right)+{\pi }_j\left(x_i=0,x_{-i}^{(k+1/k)}\right)}}$$

(24)

This ensures that the update of each component state is consistent with the current transitional distribution used in the iTMCMC procedure.

Generate a random number u following a uniform distribution U(0, 1). The state of component i is determined by comparing u with ${\eta }_i^{(k+1)}$:

$$x_i^{(k+1)}=\left\{\begin{array}{l}1, u\le {\eta }_i^{(k+1)}\hfill \\ 0, u>{\eta }_i^{(k+1)}\hfill \end{array}\right.$$

(25)

The updated value of xi is then used immediately when updating the next component in the same Gibbs iteration.

By repeatedly updating all component states, the Gibbs sampler generates a Markov chain whose stationary distribution is consistent with the target system-state distribution. After convergence, the sampled system states are used as inputs for power-flow checking and load-curtailment assessment. The resulting statistical outputs are then used to calculate reliability indices.

3.3. System-State Assessment Model and Reliability Metrics

The power system network model employs a DC power flow model, while the generation rescheduling model utilizes a minimum load shedding model [24]. When component outages lead to line overloads or power-balance violations, generator outputs are first rescheduled within their operating limits. If rescheduling alone cannot restore a feasible operating state, load curtailment is introduced as a corrective action. The objective is to minimize the total curtailed load while satisfying power-balance, generation-capacity, and branch-flow constraints. The resulting minimum-load-shedding model is formulated as follows:

$$\left\{\begin{array}{ll}\underset{P_g,r,\theta }{\min }\hfill & {\displaystyle \sum _{i\in {\mathcal{N}}_L}{r}_i}\hfill \\ \hspace{1em} \mathrm{s}.\mathrm{t}.\hfill & B\theta =P_g-P_d+r\hfill \\ \hfill & P_{g,i}^{\min }\le P_{g,i}\le P_{g,i}^{\max }, i\in N_G\hfill \\ \hfill & 0\le r_i\le P_{d,i}, i\in N_L\hfill \\ \hfill & -P_l^{\max }\le P_l\le P_l^{\max }, l\in L\hfill \\ \hfill & P_l=b_l^{\mathrm{T}}\theta , l\in L\hfill \end{array}\right.$$

(26)

where NL, NG, and L denote the sets of load buses, generators, and branches, respectively; ri is the curtailed load at load bus i, Pd,i is the load demand, and Pg,i is the generator output. $P_{g,i}^{\min }$ and $P_{g,i}^{\max }$ are the lower and upper limits of generator output, θ is the vector of bus voltage angles, B is the bus susceptance matrix, Pl is the active power flow of branch l; Plmax is the branch flow limit; and bl is the branch-flow coefficient vector under the DC power flow approximation.

For each sampled outage state x_k, the DC-power-flow-based minimum-load-shedding model is solved to obtain the corresponding load curtailment C(x_k). The obtained load-curtailment values are then used to calculate LOLP and EENS. Since iTMCMC-Gibbs samples are generated from a reliability-oriented transitional distribution rather than directly from the original system-state probability model, self-normalized importance weights are introduced in the reliability-index estimation.

(1)

Loss of Load Probability (LOLP)

LOLP is defined as the probability that the system cannot fully supply the load under sampled outage states. A sampled state is counted as a loss-of-load state when its load curtailment is greater than zero. LOLP is calculated as follows:

$$LOLP={\displaystyle \sum _{x\in \mathsf{\Omega }}{p}_0}(x)I(C(x)>0)$$

(27)

For a sampled outage state x_k, the state is counted as a loss-of-load state when the minimum load curtailment C(xk) is greater than zero. The loss-of-load indicator is defined as

$${\mathrm{I}}_{\mathrm{k}}{=\mathrm{I}\left(\mathrm{C}\right(\mathrm{x}}_{\mathrm{k}})>0)$$

(28)

For MC and MCMC-Gibbs, the sampled states are generated from the original system-state probability model, and therefore wk = 1. For iTMCMC-Gibbs, the sampled states are generated from the reliability-oriented distribution constructed by L(x). To estimate reliability indices with respect to the original system-state probability model, the self-normalized importance weight is defined as:

$$w_k={\displaystyle \frac{1}{L(x_k)}}=\exp [-\eta H(x_k)]$$

(29)

Therefore, LOLP is estimated as:

$$\widehat{LOLP}={\displaystyle \frac{{\displaystyle \sum _{k=1}^N{w}_k}I(C(x_k)>0)}{{\displaystyle \sum _{k=1}^N{w}_k}}}$$

(30)

For MC and MCMC-Gibbs, wk = 1, and the above equation reduces to the conventional sample-average estimator.

(2)

Expected Energy Not Supplied (EENS)

EENS represents the expected amount of unsupplied energy over the study period. It combines the probability of sampled outage states with the corresponding load curtailment obtained from the system-state assessment model. For a study period T, EENS is calculated as follows:

$$EENS=T{\displaystyle \sum _{x\in \mathsf{\Omega }}{p}_0}(x)C(x)$$

(31)

where T is the study duration and C_s denotes the load curtailment under system state s. For annual reliability assessment, T = 8760 h. Ω denotes the system outage-state space, p₀(x) is the original probability of outage state x, and C(x) is the minimum load curtailment under state x.

For sampled outage states, the sampled estimator of EENS is:

$$\widehat{EENS}=8760{\displaystyle \frac{{\displaystyle \sum _{k=1}^N{w}_k}C(x_k)}{{\displaystyle \sum _{k=1}^N{w}_k}}}$$

(32)

In addition to the stage-wise ESS used to determine the transitional parameter, the ESS of the final importance weights is calculated to evaluate weight degeneration in the self-normalized estimator. The normalized final weight is defined as:

$${\tilde{w}}_k={\displaystyle \frac{w_k}{{\displaystyle \sum _{r=1}^N{w}_r}}}$$

(33)

The corresponding effective sample size and effective sample proportion are calculated as:

$${\mathrm{ESS}}_w={\displaystyle \frac{1}{{\displaystyle \sum _{k=1}^N{\tilde{w}}_k^2}}}={\displaystyle \frac{{\left({\displaystyle \sum _{k=1}^N{w}_k}\right)}^2}{{\displaystyle \sum _{k=1}^N{w}_k^2}}}$$

(34)

$${\mathrm{ESS}}_w\%={\displaystyle \frac{{\mathrm{ESS}}_w}{N}}\times 100\%$$

(35)

The stage-wise ESS is used to determine the increment of β_j, whereas ESS_w is used to evaluate the efficiency of the final self-normalized reliability-index estimator.

4. Case Study Analysis

4.1. Test System and Experimental Settings

The case study in this paper involves the IEEE RTS-96 reliability test system, which consists of 73 buses, 96 generating units, and 120 branches [25]. According to the branch data used in this study, the 120 branches are divided into 104 transmission lines and 16 transformers. This system is used as the network topology and operating condition for the DC-power-flow-based reliability assessment.

The public outage-rate and outage-duration data of the IEEE RTS-96 system are used to parameterize the prior outage probabilities of the corresponding equipment types. Three equipment categories are considered in this study: transmission lines, transformers, and circuit breakers. Since circuit breakers are not represented as independent branches in the original RTS-96 network topology, one equivalent circuit breaker is assigned to each AC branch, resulting in 120 equivalent circuit breakers. The equipment counts and the corresponding IDM observation sample sizes are listed in

Table 2. Equipment counts and IDM observation sample sizes.

Equipment Type	Number of Equipment Units	Observation Period/Year	IDM Sample Size m
Transformer	16	3	48
Transmission line	104	3	312
Circuit breaker	120	3	360

. In the IDM estimation, the effective sample size m is represented by the component-year sample size, which is calculated from the number of equipment units and the equivalent observation period.

Generator outages are not considered in the sampled outage states; the generator data are only used to define the operating condition and available generation capacity in the load-shedding calculation. The obtained equipment outage probabilities are then used for system-state sampling and DC-power-flow-based minimum-load-shedding calculation, from which LOLP and EENS are evaluated.

The main numerical parameters used in the case study are summarized in

Table 3. Main numerical settings of the case study.

Parameter	Value
Test system	IEEE RTS-96
IDM parameter θ	2.0
Risk-guidance coefficient η	3.0
Target ESS proportion	20%
Evaluated samples	50,000
High-precision MC reference	500,000
Repeated runs	30
Maximum outage order	3

. These parameters specify the baseline settings for IDM interval estimation, reliability-oriented iTMCMC sampling, repeated-run comparison, and high-precision MC reference calculation.

4.2. IDM Interval Construction and Parameter Analysis

The component outage-probability intervals based on the improved IDM are first constructed and analyzed. Based on the IDM formulations introduced in Section 2, this subsection examines the variation of the sample-size-dependent hyperparameter α, compares the interval-width behavior of the traditional IDM and the improved IDM, and analyzes the sensitivity of θ. Finally, the lower and upper outage-probability bounds of different equipment categories are obtained and used as the component-level uncertainty inputs for the subsequent IDM-iTMCMC reliability assessment.

The variation of α with the effective sample size is shown in Figure 1. As the sample size increases from 1 to 500, α decreases rapidly at the beginning and then gradually enters a slow-decreasing stage. This trend indicates that the strength of the imprecise prior is reduced as more statistical evidence becomes available. Therefore, the improved IDM does not use a fixed hyperparameter for all sample sizes but adjusts the interval strength according to the amount of available data.

To further compare the interval-width control behavior, the traditional IDM with s = 1, the traditional IDM with s = 2, and the improved IDM with θ = 2.0 are compared under different effective sample sizes. The corresponding interval-width curves are shown in Figure 2.

As shown in Figure 2, the interval widths of the traditional IDM and the improved IDM both decrease as the effective sample size increases. For the traditional IDM, the strength parameter s is fixed, so the interval-width reduction is only driven by the increase in sample size. For the improved IDM, the hyperparameter α decreases with the sample size, leading to a faster reduction in interval width. This result shows that the improved IDM can adaptively shrink the uncertainty interval when more sample information is available, while still preserving interval-based uncertainty representation. Unlike the fixed strength parameter s, α(m) remains relatively large for extremely small samples and decreases as the sample size increases. Therefore, the improved IDM preserves greater uncertainty when statistical evidence is scarce and reduces the interval width as more evidence becomes available.

After the interval-width behavior with respect to the sample size is analyzed, the sensitivity of θ is further examined. The value of θ controls the baseline strength of the improved IDM. A larger θ leads to a larger α and therefore produces wider outage-probability intervals. In this study, several values of θ are tested to evaluate the influence of this hyperparameter on the IDM interval estimation.

As shown in

Table 4. Sensitivity of IDM probability intervals to θ.

θ	Transformer Width	Transmission Line Width	Circuit Breaker Width
0.5	2.3527 × 10−4	3.6200 × 10−7	2.9084 × 10−6
1.0	4.6929 × 10−4	7.2379 × 10−7	5.8155 × 10−6
1.5	7.0205 × 10−4	1.0854 × 10−6	8.7211 × 10−6
2.0	9.3358 × 10−4	1.4468 × 10−6	1.1625 × 10−5
3.0	1.3930 × 10−3	2.1689 × 10−6	1.7430 × 10−5

, the interval widths of all three equipment categories increase with θ. This result is consistent with the role of θ in the improved IDM: a larger θ increases the prior strength and leads to a more conservative probability interval. Among the three equipment categories, the transformer interval is more sensitive to the variation of θ, while the transmission-line interval remains relatively narrow. This is mainly related to the different equipment sample sizes and outage characteristics. In the following analysis, θ = 2.0 is adopted as the baseline setting for constructing the equipment outage-probability intervals.

Under the baseline setting θ = 2.0, the final IDM-based outage-probability intervals of the three equipment categories are obtained, as shown in

Table 5. IDM-based outage-probability intervals of equipment.

Equipment Type	Lower Bound	Upper Bound	Interval Width
Transformer	1.7348 × 10−3	2.6683 × 10−3	9.3358 × 10−4
Transmission line	5.0950 × 10−4	5.1094 × 10−4	1.4468 × 10−6
Circuit breaker	8.7452 × 10−5	9.9077 × 10−5	1.1625 × 10−5

.

The values reported in this table are dimensionless steady-state unavailability probabilities used for component-state sampling, rather than annual failure rates.

4.3. System Reliability Estimation

Before comparing the three sampling methods, the influence of the risk-guidance coefficient was examined for η = 1, 2, 3, 4, 5. For each value, five independent calculations were performed with 50,000 evaluated samples, while the remaining system and sampling settings were kept unchanged. The mean and standard deviation of LOLP and EENS, together with the effective sample proportion calculated from the final importance weights, are summarized in

Table 6. Sensitivity of the reliability estimates and final weight ESS to η.

η	Mean LOLP	LOLP σ	Mean EENS	EENS σ	Final ESS Proportion
1	0.060287	0.000762	41,010.48	777.85	94.53%
2	0.059852	0.000539	40,829.11	653.93	76.24%
3	0.059767	0.000171	40,795.76	426.92	43.56%
4	0.059895	0.001206	40,679.74	499.58	11.78%
5	0.060312	0.0018	41,397.91	1255.31	1.17%

.

As shown in Table 6, increasing η strengthens the guidance toward reliability-relevant states but gradually reduces the effective sample size of the final importance weights. When η increases from 1 to 3, the standard deviations of both LOLP and EENS decrease. However, when η is further increased to 4 and 5, the final ESS proportions decrease to 11.78% and 1.17%, respectively, and the repeated-run variability increases. Among the tested values, η = 3 gives the smallest standard deviations for both LOLP and EENS while retaining a final ESS proportion of 43.56%. Therefore, η = 3 is used in the following comparison as a balance between reliability-oriented state exploration and final-weight stability.

Based on the above sensitivity analysis, η = 3 is adopted for the following iTMCMC calculations. MC, MCMC-Gibbs, and iTMCMC-Gibbs are then used to estimate LOLP and EENS. The sampling traces of the three methods are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. To reduce the influence of random fluctuations, each method is independently executed 30 times. If all repeated traces are plotted directly, the curves become too dense and difficult to distinguish. Therefore, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 are presented in the form of mean ±1σ band plots, where the solid line represents the mean value of 30 independent runs and the shaded region represents one standard deviation around the mean.

In addition, a 500,000-sample MC simulation is conducted as the high-precision numerical reference. The reference LOLP is 6.0096 × 10⁻², and the reference EENS is 4.0275 × 10⁴ MWh/year. Based on 30 independent repeated runs, the mean values and standard deviations of LOLP and EENS obtained by the three methods are summarized in

Table 7. Statistical comparison of reliability estimates.

Method	Mean LOLP	LOLP σ	Mean EENS	EENS σ
MC	5.9617 × 10−2	1.0752 × 10−3	4.0512 × 104	8.6998 × 102
MCMC	5.9689 × 10−2	1.0767 × 10−3	4.0399 × 104	1.0491 × 103
iTMCMC	5.9783 × 10−2	6.7872 × 10−4	4.0376 × 104	6.0731 × 102
High-precision MC	6.0096 × 10−2	—	4.0275 × 104	— —

and

Table 8. Variance comparison of reliability estimates.

Method	LOLP Variance	EENS Variance
MC	1.1560 × 10−6	7.5687 × 105
MCMC	1.1593 × 10−6	1.1005 × 106
iTMCMC	4.6066 × 10−7	3.6882 × 105

.

The corresponding variances of the six statistical quantities in Table 6 are further listed in Table 8. The calculation method of variance is shown in Equation (36):

$$s_Y^2={\displaystyle \frac{1}{R-1}}{\displaystyle \sum _{r=1}^R{\left(Y_r-\overline{Y}\right)}^2}$$

(36)

As shown in Table 7 and Table 8, the three methods provide LOLP and EENS estimates close to the 500,000-sample MC reference. Among them, iTMCMC gives the closest mean estimates to the high-precision MC reference for both LOLP and EENS. In terms of repeated-run stability, iTMCMC also achieves the smallest standard deviation and variance. Its LOLP variance is reduced by approximately 60.15% compared with MC and 60.26% compared with MCMC. Its EENS variance is reduced by approximately 51.27% compared with MC and 66.49% compared with MCMC. These results indicate that iTMCMC provides better estimation consistency and repeated-run stability.

Under the adopted setting of η = 3, the final importance-weight ESS is 21,780.76 for 50,000 evaluated samples, corresponding to an effective sample proportion of 43.56%. The coefficient of variation of the final weights is 1.138, and the weights span approximately 5.05 orders of magnitude. Although the final weights cover a relatively wide range, the effective sample size does not collapse, indicating that the self-normalized estimator retains an adequate effective sample population.

4.4. Interval-Valued LOLP and EENS Propagated by IDM-iTMCMC

After the sampling performance of iTMCMC is verified, the lower and upper outage-probability bounds obtained by the improved IDM are further propagated through iTMCMC. In this way, the uncertainty in component outage probabilities is transferred to the system-level reliability indices. The resulting interval-valued LOLP and EENS are listed in

Table 9. Interval-valued LOLP and EENS obtained by IDM-iTMCMC.

Probability Input	LOLP	EENS
IDM lower probability	5.9131 × 10−2	3.9190 × 104
IDM upper probability	7.2419 × 10−2	4.8574 × 104

.

As shown in Table 9, the LOLP interval obtained by IDM-iTMCMC is [5.9131 × 10⁻², 7.2419 × 10⁻²], and the EENS interval is [3.9190 × 10⁴, 4.8574 × 10⁴] MWh/year. The upper probability input leads to larger LOLP and EENS values than the lower probability input, which is consistent with the effect of component outage probabilities on system reliability risk.

The 500,000-sample MC reference is also located within the obtained interval. These values are closer to the lower-bound result because the IDM probability intervals are not symmetric around the nominal point. For equipment categories with limited equivalent failure counts, the lower IDM bound is close to the nominal probability, while the upper bound expands more evidently. Therefore, the resulting LOLP and EENS intervals are also asymmetric.

These results show that the proposed IDM-iTMCMC framework can propagate component-level outage-probability uncertainty into system-level reliability-index intervals. Compared with point-valued reliability estimates, the interval results provide a more informative representation of reliability risk when component reliability records are limited.

5. Conclusions

This paper proposes a power system reliability assessment framework based on an improved IDM and iTMCMC. The improved IDM is used to construct outage-probability intervals for different equipment categories, while iTMCMC is used to estimate LOLP and EENS and to propagate component-level probability uncertainty to system-level reliability indices. Compared with the traditional IDM using fixed strength parameters, the improved IDM introduces a sample-size-dependent hyperparameter, which allows the interval width to vary with the effective sample size. This provides a more flexible way to represent component outage-probability uncertainty when reliability records are limited.

The case study is conducted on the IEEE RTS-96 reliability test system. The results show that MC, MCMC, and iTMCMC provide comparable mean estimates of LOLP and EENS. Among the three methods, iTMCMC achieves the smallest standard deviation and variance for both indices. Its LOLP variance is reduced by approximately 60% relative to MC and MCMC, while its EENS variance is reduced by approximately 51% and 66%, respectively. These results indicate moderate but consistent improvements in repeated-run stability.

The interval propagation results show that the proposed IDM-iTMCMC framework can transfer component-level outage-probability uncertainty into system-level LOLP and EENS intervals. Compared with point-valued reliability assessment, the interval-valued results provide additional information on the possible variation range of reliability risk. This is useful when component outage probabilities cannot be represented adequately by a single deterministic value.

The limitations of this study are as follows:

(1)

The reliability assessment is based on a DC-power-flow model. This treatment improves computational efficiency in repeated sampling, but it does not fully represent voltage constraints, reactive power limits, network losses, and other AC-power-flow-related operating constraints. Therefore, the influence of voltage-security constraints on load shedding is not included in the present model.

(2)

The load level is treated as deterministic in the case study. In practical power systems, load demand may vary with season, weather, user behavior, and operating conditions. Renewable generation uncertainty is also not included in the present model. Therefore, the proposed framework mainly focuses on component outage-probability uncertainty, while operating-condition uncertainty is not fully considered.

(3)

Component outages are modeled as independent events. This assumption simplifies the system-state sampling process, but it does not describe correlated failures caused by common weather conditions, protection-device interactions, maintenance dependence, or common-cause disturbances. Such correlated outage mechanisms may have a significant impact on reliability risk in practical systems.

(4)

Cascading outage processes are not modeled. The present study evaluates the load-shedding consequence of sampled outage states, but it does not simulate dynamic cascading propagation, protection actions, frequency dynamics, or post-contingency corrective control. Therefore, the results mainly reflect static reliability risk rather than dynamic cascading-failure risk.

Future work will focus on extending the proposed framework to AC-power-flow-based reliability assessment, stochastic load and renewable generation modeling, correlated component outage modeling, and cascading-failure simulation. These extensions can further improve the physical accuracy and engineering applicability of IDM-iTMCMC-based reliability assessment.