Reliability Assessment Methods for Power Supply Systems Considering the Technical Condition of Electrical Equipment: A Critical Review

Abstract

The reliability assessment of power supply systems is complicated by the branched network topology, redundancy, switching operations, post-failure restoration, operating constraints, and uneven equipment degradation. Additional difficulties arise because calculations based on averaged normative parameters often fail to reflect the actual condition of specific components. This review critically compares reliability assessment methods for power supply systems, with particular attention to the technical condition of electrical equipment. Fault tree analysis and the logic–probabilistic method are discussed as structural approaches. Markov models are examined as tools for describing transitions between equipment states. Monte Carlo-based simulation is considered for calculating adequacy deficits and the operational consequences of supply interruptions. Reliability parameterization based on health indices, correction factors, and consumed life is also analyzed. The analysis indicates that no single methodology is equally effective at all problem levels. The main contribution is to separate technical condition parameterization, structural operability modeling, and adequacy assessment into interacting layers. This structure links condition indicators, such as health indices or consumed life, with system-level reliability and deficit indices through element-state probabilities. The review therefore supports a multi-level framework in which technical condition is considered at the element level, structural reliability is evaluated by the logic–probabilistic method, and failure consequences are assessed by simulation.

1. Introduction

The reliability of power supply systems is a key characteristic of energy infrastructure. It determines whether a system can supply consumers with electrical energy of the required quality and capacity during component failures, post-failure restoration, and external disturbances [1,2]. For industrial enterprises, distribution networks, and autonomous electrical complexes, supply interruptions lead not only to process losses but also to increased operational risks, economic damage, and the reduced stability of technological processes [3,4].

The assessment task has become more complex as modern electrical networks increasingly include redundancy, distributed generation, energy storage systems, automated control systems, and varying operating modes. These features increase the number of admissible and inadmissible system states, which cannot be adequately described only by averaged indices of individual components [5,6]. For power supply systems of fuel-and-energy enterprises and autonomous electrical complexes, this problem is further intensified by the combination of on-site power sources, power electronics, and increased requirements for power quality [7,8].

Operational data, diagnostic information, and risk-oriented maintenance are therefore becoming increasingly important for reliability studies [9,10]. Studies on asset management and predictive maintenance show that components with identical normative characteristics may differ substantially in their actual failure probabilities because of differences in electrical loading, thermal stresses, maintenance history, and accumulated wear [9,11]. This explains the growing interest in models that treat reliability parameters as time-varying rather than constant over the entire operating interval [12].

Despite this progress, converting diagnostic results and technical condition information into quantitative reliability parameters remains a significant methodological problem. This difficulty is reflected in studies on electrical equipment condition assessment, the probabilistic assessment of network assets, transformer health indices, and risk-oriented maintenance [11,13]. In the existing literature, individual classes of methods usually solve only part of the overall problem. Structural methods describe the network topology and the operability criterion. Markov models account for state transitions and restoration. Simulation methods calculate failure consequences, while condition-based approaches provide element-level parameterization [3,14]. As a result, no single methodology is equally effective across all levels of the problem [3].

The purpose of this review is to critically compare reliability assessment methods for power supply systems with particular attention to four issues. These are network structure representation, restoration modeling, the calculation of supply interruption consequences, and the incorporation of the actual technical condition of electrical equipment. Unlike works limited to listing methods, this article identifies the assessment level at which each approach is most appropriate and explains how these approaches can be combined within a multi-level reliability assessment framework.

2. Review Scope, Source Selection Strategy, and Analytical Framework

This review focuses on reliability assessment methods for power supply systems, especially distribution networks, industrial power supply systems, autonomous electrical complexes, and related facilities. For these objects, the complex supply topology, redundancy, post-failure restoration, and equipment condition directly influence reliability indices [15,16]. The selected publications consider reliability not only at the component level but also at the levels of system topology, operating mode, and supply interruption consequences.

The review follows a critical narrative design. Its purpose is not to provide an exhaustive formal record of all publications within a rigid protocol. Instead, it compares methodological approaches that occupy different positions in the overall reliability assessment problem. Priority was given to studies that compare methods with respect to four issues central to the article. These issues are network structure representation, state transitions and restoration, adequacy-related and operational consequences, and the incorporation of the actual technical condition of electrical equipment.

The source selection strategy was organized around four interrelated directions. The first direction included works on the methodological foundations of reliability analysis for electric power systems. A key source in this group is the paper by D.S. Krupenev on the system reliability analysis of electric power systems [15]. The second direction focused on technical condition accounting. It included the monograph by A.N. Nazarychev and D.A. Andreev on the integrated technical condition assessment of electrical equipment [16] and a review of transformer health indices that discusses the strengths and limitations of this approach [13].

The third direction included publications that link reliability with operating practice, maintenance, and asset management. Important sources in this group include the review by T.D. Gladkikh on electric network maintenance [17] and the paper by W. Naim, P. Hilber, and E. Shayesteh on data challenges in distribution system asset management [10]. These sources show that the quality of reliability assessment depends on the completeness and comparability of operational information. They also connect reliability assessment with practical tasks of monitoring, diagnostics, and technical decision-making.

The fourth direction included studies that connect calculated reliability indices with failure consequences and the practical use of the results. One representative example is a paper on cable reliability assessment based on sequential Monte Carlo simulation [18]. A related applied formulation is presented by D.A. Boyarkov, B.S. Kompaneets, and V.I. Sinitsin, who use risk assessment results to prioritize the renewal of electric grid equipment [19]. Together, these publications help to compare methods not only by their mathematical apparatus but also by the practical usefulness of their results for operational measures.

The analytical workflow comprised four stages. First, the object of the review and the target class of reliability problems were delimited. Second, publications were selected and grouped according to their methodological roles. Third, the selected approaches were compared as structural methods, state-based models, simulation-based methods, and technical condition approaches. Fourth, the comparison results were synthesized into a multi-level reliability assessment framework and a set of unresolved methodological problems. Figure 1 presents this workflow.

The review also includes selected publications on industrial electrical complexes, where reliability is considered in relation to actual operating conditions, operating modes, and power quality [7,8]. These sources are relevant because industrial and autonomous power supply systems cannot be analyzed adequately without considering the specifics of their real operation.

The review scope is therefore limited to works that allow reliability assessment methods to be compared according to four criteria. These criteria are system structure representation, accounting for state transitions and restoration, suitability for failure consequence assessment, and parameterization based on actual equipment conditions [13,15]. This scope supports the purpose of the article, which is to move beyond the simple listing of methods toward their critical comparison for the analysis and operation of power supply systems.

Table 1. Literature categories, search focus, and analytical dimensions of the review.

Literature Category and Search Focus	Representative Literature Focus	Reason for Inclusion and Analytical Dimension	Corresponding Section
General reliability methodology and reliability indices	Studies on reliability assessment of power supply and electric power systems, including structural reliability, adequacy reliability, power deficit indicators, and interruption consequences	To define the main groups of reliability assessment problems and the indices used to describe them	Section 2 and Section 3
Structural methods	Studies on fault tree analysis, logic–probabilistic modeling, Boolean operability functions, decision diagrams, minimal cut sets, topology, redundancy, and switching logic	To analyze methods that formalize system structure and determine combinations of element states corresponding to operability	Section 4
State-based models	Studies on Markov models, state transitions, degradation levels, restoration, repairable systems, and dynamic reliability assessment	To examine approaches that describe time-dependent changes in the states of elements and subsystems	Section 5
Simulation-based methods	Studies on Monte Carlo simulation, sequential simulation, stochastic operating scenarios, adequacy assessment, power deficit, expected energy not supplied, and restoration sequences	To analyze methods used to calculate deficit-related and consequence-related reliability indices under uncertainty	Section 6
Technical condition approaches	Studies on technical condition assessment, health indices, condition-based maintenance, consumed-life models, and asset diagnostics	To determine how diagnostic and operating information can be converted into element reliability parameters	Section 7
Asset management and decision support	Studies linking reliability indices with renewal prioritization, maintenance planning, monitoring, and risk-oriented asset management	To connect calculated reliability indices with practical decisions in operating power supply systems	Section 8, Section 9 and Section 10

summarizes the correspondence between the literature categories, search focus, reasons for inclusion, analytical dimensions, and subsequent manuscript sections.

Table 1 groups the reviewed literature not only by method name but also by function within the reliability assessment problem. This organization explains why structural methods are discussed separately from Markov models, Monte Carlo simulation, and technical condition approaches. Structural methods define topology and operability criteria. State-based models describe degradation and restoration. Simulation-based methods assess adequacy and failure consequences. Technical condition approaches provide element-level parameterization for system-level assessment and decision support. The following sections therefore move from reliability problems and indices to method-specific analysis, comparative synthesis, and the proposed multi-level reliability assessment framework.

3. Problem Formulation and Reliability Metrics

When reliability assessment methods are compared, three groups of problems should be distinguished. The first group concerns structural operability, namely whether the system can perform its power supply function for given element states and redundancy logic. The second group concerns state dynamics over time, including the probabilities of operable, partially degraded, failed, and restored states. The third group concerns disturbance consequences for load supply and power quality, including power deficit probabilities, expected energy not supplied, and adequacy indices [20,21].

Studies on the system reliability of electric power systems distinguish structural reliability from adequacy reliability [15,21]. A system may remain structurally connected and formally operable while still experiencing a power deficit because of failures, generation constraints, emergency repairs, or unfavorable load conditions [22,23]. Therefore, modern methodologies use not only the probability of failure-free operation and availability factors but also adequacy and deficit indices. These indices reflect the ability of the system to satisfy the demand under stochastic conditions [24,25].

V.P. Oboskalov’s works discuss algorithmic aspects of calculating probabilistic power deficit indices and show that these calculations must account for the generation status, network constraints, and the structure of the interconnected system [20,24]. This line of research was further developed in studies on the power deficit probability, rare event modeling, and the clustering of electric power systems into reliability zones [23,26]. Together, these studies show that system-level reliability should be interpreted not only as a property of components and topologies but also as a characteristic of load adequacy under uncertainty [25,27].

For distribution networks and consumer power supply systems, target indices increasingly include the interruption frequency and duration, expected outage consequences, risk indices for individual consumer groups, specific damage, and energy-not-supplied indices [28,29]. Studies devoted to these indicators emphasize that reliability assessment has practical value only when the calculated indices can be linked to measures or technical actions for a specific power supply object [19,30].

This leads to an important methodological conclusion. Model selection should depend not only on the available mathematical apparatus but also on the required output index. If the objective is to determine the set of operable network states, structural methods are required. If time-dependent state probabilities and restoration must be represented, Markov models are more appropriate. If adequacy indices must be assessed under complex stochastic influences, simulation-based approaches become necessary [31,32].

4. Structural Reliability Assessment Methods for Power Supply Systems

4.1. Fault Tree Analysis

Fault tree analysis is among the most interpretable structural methods because it starts from a top event, such as system failure or loss of a specified function, and traces it to combinations of basic events that can cause this outcome [33,34]. In engineering practice, this logic is useful because it links system failures to specific disturbance mechanisms. It also supports the identification of minimal cut sets and the ranking of critical components or event combinations [35,36].

In recent years, fault tree analysis has developed in both classical and modified forms. Reviews emphasize its relevance because the method combines cause-and-effect transparency with quantitative processing [33,37]. Its application has also expanded from energy and electrical systems to autonomous ships, wind power, hardware–software protection systems, and photovoltaic plants [38,39]. This broad use is relevant for power supply system reliability because it confirms the applicability of top-event logic and minimal cut sets to complex technical objects [36,40].

For power supply systems, fault tree analysis is especially useful when a specific function must be analyzed. Examples include loss of supply to a critical consumer, the failure of a reserve channel, relay protection malfunction, or the loss of readiness of a particular network node [41,42]. The cause-and-effect structure of the method supports not only the calculation of the top-event probability but also the interpretation of the most dangerous combinations of basic failures. This feature is important in operating practice, where decisions on modernization, redundancy, or repair depend not only on final index values but also on their interpretation [19,34].

An important advantage of fault tree analysis is the use of minimal cut sets and element importance analysis. In studies on industrial power supply systems, fault tree models are used to assess component importance and identify the elements that have the strongest effects on overall system reliability [34,41]. For enterprise power supply systems, this link is important because it connects calculation results with the prioritization of technical actions [19,41].

As the system topology becomes more complex, these advantages are accompanied by significant limitations. First, the number of minimal cut sets and logical failure combinations grows rapidly, which makes tree construction and maintenance labor-intensive [37,39]. Second, fault trees are better suited to analyzing a specific top event than to describing the full set of admissible and inadmissible states in a complex network. Third, frequent changes in network structure, switching logic, or reserve facilities require model updates. In real operation, this updating process may become a separate organizational problem [33,41].

Another limitation is the static nature of the classical fault tree formulation. Although the probabilities of basic events may be corrected using diagnostic data, the method itself does not provide a direct apparatus for describing restoration, multi-level degradation, or non-stationary parameter changes over time [36,40]. Extended and hybrid schemes, including quantum fault trees and fuzzy or ontology-based modifications, have been proposed to address this drawback. However, for power supply systems with complex real topologies, these approaches still play an auxiliary rather than a core role [35,38].

Fault tree analysis is therefore a useful tool for cause-and-effect structural analysis and for identifying critical elements and failure scenarios. Its main value lies in relating a top event to underlying causes, evaluating minimal cut sets, and producing an interpretable picture of system risk [33,34]. As the sole core method for complex power supply systems, it is less universal than the logic–probabilistic approach. Nevertheless, within a comprehensive framework, it can complement other methods, especially in failure scenario analysis and criticality localization [37,41].

4.2. Logic–Probabilistic Method

The logic–probabilistic method differs from other structural approaches because it formalizes the operability criterion of the system as a whole, rather than only a single top event [43,44]. Its basic idea is to describe a complex system via a Boolean function of element states. This logical model is then transformed into a probabilistic form using the reliability characteristics of individual components [45,46]. For power supply systems, this distinction is important because operability depends not only on the absence of failures but also on admissible supply paths, redundancy criteria, and the functional connectivity of the network [42,47].

Studies on complex technical systems and power supply systems use the logic–probabilistic method to describe the topology and operating logic of objects with switching, redundancy, and structural reserve [3,44]. The method has a clear advantage in separating model structure from element parameters. The system operability scheme is defined logically, while probabilistic characteristics of elements can be updated without restructuring the structural function [45,48]. This feature is useful when element reliability must be corrected with regard to technical condition [11,49].

Another development direction is improved computational efficiency. Modern studies use binary decision diagrams and related algorithms to analyze system reliability and avoid the direct combinatorial expansion of logical expressions [43,48]. For power supply systems with complex topologies, this algorithmic support is important because dimensionality growth and numerous operable-state combinations remain major limitations of classical structural methods [44,47].

A major strength of the logic–probabilistic method in power supply applications is the rigorous formalization of the operability criterion. Fault tree analysis focuses on the top event and the paths leading to it, whereas the logic–probabilistic method focuses on the set of admissible system states. This distinction is important for distribution and industrial schemes, where alternative supply routes, sectionalizing, and redundancy create many combinations in which the system remains operable after some element failures [3,42]. This approach is consistent with operating practice because the failure of an individual element does not always mean the failure of the system.

The logic–probabilistic method can also be used to calculate reliability indices for power supply systems with integrated energy storage and other modern structural elements. Studies of such systems show the need to retain a structural representation of the network while updating the characteristics of elements and subsystems [7,47]. In industrial power engineering, electrical complexes often combine in-house sources, reserves, and power electronics. For such objects, this approach provides a flexible structural basis for reliability assessment [8].

The logic–probabilistic method also has limitations. The most widely discussed limitation is the assumption of independent element failures in the basic formulation [43,48]. For power supply systems, this assumption may be restrictive because some failures have common causes, including unfavorable operating conditions, overloads, personnel errors, organizational deficiencies, external impacts, and cascading effects [3,4]. The computational complexity may also increase sharply if the direct logical form becomes too cumbersome, especially when algorithmic simplification tools are not used [45,46].

Despite these limitations, the logic–probabilistic method remains a strong candidate for the core structural layer in the reliability assessment of power supply systems with complex topologies. It provides a rigorous description of the operability criterion, accounts for redundancy, and separates the model structure from element parameters. This separation is important when technical condition data are incorporated into the model [43,48]. Within an integrated reliability architecture, the method therefore serves not as a universal solution but as the basic layer of system-level structural calculation [3,47].

5. Markov Models for Reliability Assessment

Markov models are used to describe the time evolution of an object through transitions between a finite set of states [50,51]. In power supply reliability studies, this allows operable, partially degraded, failed, and restored states of elements or subsystems to be represented explicitly. These states are linked through failure and restoration rates [50,51].

Markov models are especially useful when reliability must be analyzed together with maintainability. Studies emphasize that, for power supply systems, the failure event itself is not the only relevant factor. The ability of an object to return to an operable state, with time-dependent probabilities rather than only static coefficients, is also important [50,51]. This distinguishes Markov models from purely structural schemes.

In distribution networks and microgrids, Markov models are used to assess reliability while accounting for the network configuration, distributed generation, subcomponent failures, and load restoration [6,52]. In systems based on renewable energy sources, they can represent discrete states of sources, reserves, switching devices, and service subsystems [5,53]. Their advantage in these applications is that the reliability model can be linked to the time process of maintenance and restoration.

When technical condition is considered, Markov models are useful because equipment degradation levels can be treated as chain states [50,53]. This allows transitions from a healthy state to several deterioration levels and then to failure to be described. Restoration after repair or technical intervention can also be included. For this reason, studies on electrical equipment condition models propose a multi-level representation of the object state rather than only the binary scheme “operable–failed” [11,54].

Applying Markov models to large-scale power supply systems leads to combinatorial growth in the number of states [51,55]. If a large system is represented as a single Markov object, the transition matrix can become excessive even with a moderate number of elements and degradation levels. Recent studies identify this limitation as one of the main barriers to scaling the Markov method [51,56].

A more scalable approach is to use Markov models mainly at the level of individual elements or limited subsystems, rather than as a universal model of the entire system [50,57]. Parameters obtained from the Markov analysis of degradation and restoration can then be transferred to the system level—for example, to a logic–probabilistic or simulation model. This multi-level use preserves the strengths of Markov modeling while avoiding explosive dimensional growth [57,58].

Another problem is the identification of transition rates. Even if the state scheme is appropriate, transition parameters must be based on statistical data or physical justification. Otherwise, a detailed model loses verifiability [11,59]. For some objects, especially in industrial power engineering, data on transitions between degradation levels remain limited. As a result, some parameters are still assigned on an expert basis [54,59]. This does not eliminate the value of Markov models, but it necessitates caution when results are interpreted.

Markov models can also be combined with other methods. For example, repair scheduling for microsystem equipment can combine Markov chain logic with Monte Carlo simulation [32,57]. In operational reliability and network configuration optimization, Markov processes can describe state dynamics, while efficiency and risk are assessed within a broader calculation framework [55,58].

Markov models therefore do not replace structural methods. Instead, they complement them by describing time-dependent degradation, repair, and restoration. They are most effective at the level of an individual element or subsystem, where degradation, repair, and state transitions must be represented over time [31,50]. Within an integrated reliability architecture, the Markov block can update element parameters dynamically, including parameters related to technical condition [11,53].

6. Monte Carlo-Based Simulation

Simulation methods, especially Monte Carlo simulation, are widely used in the reliability assessment of electric power systems because they can represent the random nature of failures, restoration, loads, operating constraints, and external disturbances [32,60]. Their main strength is the ability to avoid overly rigid analytical simplifications and examine many possible system trajectories over time [61,62].

In distribution networks, Monte Carlo simulation is used to assess risks to consumer power supply, account for demand uncertainty, and analyze the distribution of failure consequences [28,60]. In systems with distributed generation, integrated energy resources, and network reconfiguration, it can jointly represent failures, restoration options, and changing operating modes [63,64]. Studies on active distribution networks and multi-source data show that simulation-based formulations are often well suited to representing complex combinations of uncertainties [65,66].

A separate direction involves sequential and quasi-sequential Monte Carlo simulation. These approaches model the chronology of events and analyze not only failure frequencies but also the time consequences for load supply [62,67]. In generation adequacy and combined generation–transmission assessment, they can be complemented by rare event simulation algorithms and procedures for determining maximum and minimum loadability limits [27,32]. This is important when reliability depends not only on the network element condition but also on the generation capacity, repairs, and peak demand dynamics [25,68].

From a methodological point of view, Monte Carlo simulation provides a practical tool for moving from structural system representations to adequacy indices [20,22]. Studies on the power deficit probability, generation inadequacy, and adequacy reliability show that simulation can represent complex combinations of failures, reserves, repairs, and operating constraints without excessive analytical assumptions [21,24]. On this basis, the probability of deficit, expected energy not supplied, frequency of insufficient supply, and other integrated reliability measures can be calculated [23,26].

For distribution networks and cable infrastructure, Monte Carlo simulation can represent damage and restoration processes in greater detail. Examples include sequential simulation for underground cables, reliability index assessment for investment decision-making, and distribution network reliability calculations that account for feeder restoration and network reconfiguration [18,69]. These works show that simulation models are useful when reliability depends on the sequence of restoration actions and on changes in network configuration over time [64,69].

Monte Carlo simulation is not a universal solution to all reliability assessment problems. Its computational burden may be high, especially in rare event analysis and large systems [27,68]. Result accuracy depends on the sample size and input parameter quality. Therefore, simulation flexibility does not eliminate the problem of parameterizing failures and restoration [70,71]. If the initial characteristics of elements are poorly specified, detailed simulation will propagate uncertainty rather than remove it [66,72].

Monte Carlo simulation is therefore better interpreted as a computational layer for consequence analysis, rather than as a complete description of system structure and degradation [32,67]. The structural logic of the system, element failure probabilities and intensities, and technical condition models must be specified at earlier methodological levels. The simulation model can then transform this information into adequacy and operational reliability indices [28,62].

Hybrid schemes that combine Monte Carlo simulation with analytical methods are also important. For example, a sequential analytical reliability assessment method based on continuous-time Markov chains combines Markovian state description with a sequential analytical framework [50,56]. Other works supplement the statistical generation of random electric power system states with regression and clustering procedures [26,71]. These studies indicate a broader trend. Monte Carlo simulation increasingly functions not as an isolated technique but as part of a multi-level reliability assessment framework [15,66].

7. Incorporation of Technical Condition into Reliability Assessment

Moving from averaged normative element characteristics to parameters that reflect actual technical condition is a major direction in modern reliability assessment [11,16]. In power supply systems, the actual failure probability depends not only on the element type and normative service life but also on thermal regimes, environmental conditions, maintenance quality, defects, diagnostic results, and repair history [17,54].

The literature distinguishes three main approaches to incorporating technical condition. The first uses correction factors that modify normative reliability indices. The second relies on integrated health indices. The third links reliability with consumed life and physically interpretable degradation factors [13,16]. These approaches are not mutually exclusive. They differ in physical substantiation, data requirements, and ease of integration into system-level calculations.

The correction factor approach is practically convenient because it can adapt basic failure parameters to the current equipment condition even when the available information is limited. It uses diagnostic results, inspections, and expert evaluation [11,49]. This approach is compatible with system-level methods, including fault trees, the logic–probabilistic method, and Monte Carlo simulation. Its main weakness is that the correction scale and its link to failure probability often retain a substantial expert component [17,30]. Statistical calibration becomes more difficult as the object complexity increases and empirical failure data become less available.

In the correction factor approach, a coefficient reflecting the assessed equipment condition multiplies the baseline failure rate. If the condition is worse than the reference state, this coefficient increases the baseline value and produces a condition-adjusted failure rate. This affects the resulting probability of failure-free operation. The approach is simple and convenient for system-level models. However, in many studies, the coefficient is still assigned using expert scales, diagnostic classes, or limited empirical rules rather than stable failure statistics [11,49]. Direct data-based estimation of the actual failure rate requires large and consistent datasets that include operating modes, diagnostic measurements, defects, repairs, and confirmed failures. In practice, such information is rarely available in a unified form, which makes statistical calibration difficult [10,59].

The health index approach follows a different logic. A health index is an integrated condition indicator calculated from the diagnostic and operating characteristics of equipment. It should not be interpreted as a failure probability or as a correction factor. In reliability assessment, the health index needs a calculation rule that relates it to the consumed life, equivalent operating time, or another reliability-related parameter. Consumed life can also be determined without a health index, using analytical models based on measurable operating factors such as thermal loading, operating conditions, and degradation mechanisms. In both cases, the baseline failure rate may remain unchanged. The time argument of the reliability function is instead replaced by the consumed life or equivalent operating time. This makes the resource-based approach physically interpretable, although its practical use still requires calibration against diagnostic, operating, and failure data [11,16].

Health indices are widely used in operating practice because they provide a unified scale for ranking equipment and prioritizing technical actions [13,19]. For transformers, cable assets, and network assets, the condition index often serves as a link between diagnostics and operation [10,13]. However, its use in quantitative reliability assessment depends on the availability of a calculation rule that connects the index value with the consumed life, equivalent operating time, or another reliability-related parameter. Without such a rule, the health index mainly supports the comparison, ranking, and prioritization of equipment. With such a rule, it can serve as an input to resource-based reliability parameterization [10,13].

Consumed life models provide a related and physically interpretable direction. Their basic idea is that the element condition is determined by the accumulated effect of operational factors over time, while reliability change is described through the equivalent operating time or consumed life. This consumed life may be calculated directly from measurable operating factors or obtained through a calculation rule based on the health index [11,16]. This formulation is close to the physics of aging and allows non-stationary reliability effects to be represented. If operating conditions are more severe than nominal, the element consumes life faster. If restorative actions have been performed, part of the life may be recovered [12,54].

Specifying the failure intensity as a time-dependent function remains a methodologically important issue. An analytical study of distribution networks shows that abandoning stationary failure rates changes the character of reliability calculations and requires a different view of the relation between time, condition, and failure parameters [12,31]. For enterprise power supply systems, this issue is important because actual operating modes and loads often differ from normative assumptions. A stationary model may therefore systematically distort the result [2,54].

The practical implementation of condition-based reliability assessment still faces several barriers. The first is the shortage of high-quality and comparable data. Reviews of data and asset management problems in power supply systems show that, at many facilities, even basic information on defects, operating time, repairs, and diagnostics is either not unified or not linked to reliability assessment within a single digital environment [10,73]. The second barrier concerns statistical verification. Even when data are available, separating the influence of a particular element’s condition from organizational factors, operational errors, and external effects remains difficult [59,74].

The third barrier concerns economic interpretation. Monitoring and diagnostics require expenditure, so their implementation must be justified not only by improved assessment accuracy but also by reduced risks, damage, and repair costs [19,75]. Therefore, technical condition accounting should not be treated as an automatic transition to a more complex model. It should be treated as a balance between model detail, parameter verifiability, and decision-making value [17,74].

A practical scheme is to assess technical condition at the level of an individual element and then transform it into reliability parameters suitable for a system-level model [11,49]. In this logic, the health index can support equipment ranking and, when a calculation rule is available, the estimation of the consumed life or equivalent operating time. Consumed life models provide physically interpretable parameterization, whereas correction factors remain a practical integration mechanism under incomplete data [13,16]. At the system level, this parameterization can be embedded into the logic–probabilistic method, fault trees, Markov models, or simulation models, depending on the task [12,30].

8. Comparative Analysis of Reliability Assessment Methods

The considered methods differ not only in their mathematical apparatus but also in the role that they play within the overall reliability assessment process. Fault tree analysis is useful for the cause-and-effect analysis of a specific top event and for identifying critical combinations of failures. The logic–probabilistic method is suited to formalizing the operability criterion of the system as a whole. Markov models describe state dynamics, degradation, and restoration. Monte Carlo simulation translates component failure events into consequence, deficit, and operational reliability indices [14,15].

When methods are compared in terms of complex topology representation, structural approaches have the strongest position, especially the logic–probabilistic method [43,48]. Fault tree analysis can be informative for individual functions and subsystems. However, as the dimensionality and the number of reserve links grow, tree updating becomes increasingly labor-intensive [37,41]. In this respect, the logic–probabilistic method is more flexible because it uses a logical operability function and can account for alternative supply paths and complex criteria for preserving system function [44,47].

If the restoration of elements is the main focus, Markov models become more appropriate [31,50]. Unlike structural methods, they directly represent state transitions and can calculate the time dynamics of probabilities. However, this advantage is accompanied by dimensional growth and by difficulties in identifying transition parameters [51,56]. Therefore, scaling Markov models to the whole system is usually inefficient, whereas their use at the level of elements and subsystems is more appropriate [55,57].

For adequacy consequences and deficit-related indices, simulation methods have a strong position [20,32]. They can combine structural, operational, and probabilistic aspects within a single computational framework. This makes them useful for adequacy indices and rare event analysis [23,27]. At the same time, Monte Carlo simulation is less constrained by a specific model structure, but it is highly dependent on input parameter quality [27,65].

Accounting for technical condition does not form an independent class of system-level reliability calculation. It acts as a parameterization layer [11,16]. From this viewpoint, the key issue is to separate model structure from element parameters. Structural methods describe the network and the operability criterion. Markov models describe degradation and restoration at the element level. Simulation methods calculate consequences and deficit indices. Technical condition approaches update element parameters using operational and diagnostic data [10,11]. Based on the literature discussed in Section 4, Section 5, Section 6 and Section 7,

Table 2. Comparative characteristics of reliability assessment methods for power supply systems.

Method	Object and Logic of Modeling	Main Advantages	Main Limitations	Typical Field of Application
Fault Tree Analysis	Top event and its cause-and-effect decomposition	Transparency of cause-and-effect relationships, identification of minimal cut sets, analysis of critical combinations of failures	Combinatorial growth of structure, labor-intensive updating, limited ability to represent time dynamics	Cause-and-effect analysis of the failure of a specified function and identification of critical combinations of failures
Logic–Probabilistic Method	Operability criterion and structural relationships of the system	Rigorous representation of complex topology, redundancy, and switching; separation of system structure from element parameters	Assumption of independent failures in the basic formulation; growth of computational complexity in direct logical expansion	Structural reliability assessment of systems with complex topology, redundancy, and switching logic
Markov Models	Transitions between states of elements or subsystems over time	Representation of restoration, multi-level degradation, and time-dependent state dynamics	Explosion in the number of states; difficulties in calibration of transition parameters	Description of degradation, restoration, and state transitions at the level of elements and limited subsystems
Monte Carlo Method	Stochastic scenarios of system operation over time	Flexibility in representing uncertainty; ability to calculate adequacy indices, power deficits, and consequences of failures	High computational burden; strong dependence on the quality of input data	Assessment of adequacy indices, failure consequences, and power deficits under stochastic conditions
Technical-Condition Accounting	Reliability parameters of individual elements	Ability to account for actual equipment condition, diagnostic results, operating conditions, and maintenance and operating history	Expert component in some approaches, shortage and heterogeneity of data, difficulties of statistical verification	Parameterization of reliability models on the basis of diagnostic, operational, and resource-related data

compares the considered methods by modeling their objects, advantages, limitations, and fields of application.

Figure 2 summarizes the distribution of the reviewed methods across the main problem levels considered in this article. The columns correspond to element-level parameterization, the structural representation of operability, state dynamics, operational consequences, and decision support. Filled and open circles denote core and auxiliary applicability, respectively. The applicability map is derived from the comparative findings in Table 2 and from the method-specific discussions in Section 4, Section 5, Section 6 and Section 7. It shows the distribution of core and auxiliary roles across problem levels, rather than a universal ranking of methods.

Figure 2 shows that no single method covers all problem levels equally well. The clearest division of roles is observed between technical condition approaches at the element level, structural methods at the system structure level, Markov models in state dynamics, and Monte Carlo simulation in operational consequence assessment.

The comparative analysis leads to a clear methodological conclusion. Methods should not be compared on a binary scale in search of a single best approach, because they address different levels of the same problem [12,14]. A method that performs well in one aspect may be limited in another. Structural approaches are strong in topology description but weaker in state dynamics. Markov models represent restoration and state transitions, but their scalability is limited. Monte Carlo simulation is flexible, but it depends strongly on input parameterization. Health indices are useful in operating practice, but they do not provide a system-level reliability model by themselves [17,74].

9. Multi-Level Framework for Comprehensive Reliability Assessment

The literature analysis supports the synthesis of complementary methods within a multi-level reliability assessment framework, rather than competition among individual approaches [11,15]. This framework is based on the division of functions between model levels. At the element level, the model identifies the current condition, consumed life, degradation rate, and expected failure parameters. At the network structure level, it determines which combinations of element states correspond to operability. At the operational analysis level, it assesses the consequences of failures for load supply and consumers [20,32]. Figure 3 illustrates the conceptual organization of the proposed framework.

Figure 3 reflects the division of roles between element-level parameterization, structural representation, operational consequence assessment, and decision support within the proposed framework.

The interaction between the layers can be illustrated without introducing a separate case study. At the element level, technical condition information is converted into the probability of the operable state of each element for a given calculation horizon. In resource-based formulations, the consumed life can be interpreted as the equivalent operating time. It is used as the time argument of the reliability function instead of the calendar operating time alone. Therefore, the baseline failure rate does not necessarily have to change. The effect of technical condition is reflected through the effective amount of resources consumed by the element.

The logic–probabilistic layer then formalizes the system operability function. It determines which combinations of element states correspond to successful power supply. If adequacy-related indicators are required, the Monte Carlo layer can use these element-state probabilities together with the structural operability function. In each simulation trial, random element states are generated according to their probabilities of being operable. The resulting state vector is checked against the operability function. If the criterion is not satisfied, the simulation layer can calculate disturbance consequences, including power deficits, expected energy not supplied, and other adequacy-related indicators. Thus, Monte Carlo simulation does not replace the structural logic–probabilistic calculation. It extends the range of calculated indices from system operability probability to deficit and consequence indicators. Technical condition affects these indicators through the element-state probabilities supplied to the simulation model.

The non-stationarity in the proposed framework does not necessitate the continuous reconstruction of the structural logic–probabilistic model. The logic–probabilistic layer describes the Boolean structure of system operability. Time dependence is introduced through the element-state probabilities supplied to this structure [43,48]. These probabilities can be obtained from condition-based parameterization, including correction factors, health index-based resource estimates, and consumed life models [11,16]. They can also be obtained from element-level state models, especially when degradation and restoration are described by Markov-type representations [50,51].

In the simplest homogeneous Markov formulation, transition rates are assumed to be constant within a selected state or calculation interval. If operating conditions, loads, maintenance cycles, or degradation rates vary over time, the parameters should be updated for selected time horizons, operating scenarios, condition classes, or maintenance states. They should not be treated as fixed for the entire service interval [12,31]. Where appropriate, element-state probabilities may also be calculated using non-exponential lifetime distributions or time-varying failure rates. Computational tractability is preserved because the time-dependent degradation process is handled at the element or scenario level, while the system-level Boolean structure remains unchanged [51,57].

At the element level, the framework is based on models that account for technical condition. Depending on the equipment type and data availability, these models may include correction factors, health indices, consumed life models, and Markov degradation models [13,16]. For equipment with pronounced staged deterioration, the Markov approach is useful because it links diagnostics with transition probabilities between states [11,49]. For objects with limited data, correction factors and condition indices can be used, preferably with subsequent calibration against actual failures and defects [30,59].

At the system level, a structural method is needed to describe complex topologies. Based on the reviewed literature, the logic–probabilistic method is well suited to this function [43,48]. It can represent the real power supply scheme, redundancy, and switching logic. It can also accept updated probabilistic parameters from the element level without restructuring the entire model [44,47]. In this role, the logic–probabilistic method serves as the core structural layer of a comprehensive assessment system, rather than as a universal reliability method [3,11].

For the analysis of failure consequences and the calculation of adequacy indices, the resulting structural and element-level parameters can be used in a simulation layer [20,32]. It is precisely the Monte Carlo method, combined with load, generation, repair, and network constraint models, that makes it possible to move from the probability of network failure to the probability of a power deficit, expected energy not supplied, and other practically significant indices [23,28]. Where necessary, this layer may be supplemented with rare event methods, sequential simulation, and computational cost reduction algorithms [27,66].

Table 3. Distribution of functions among methods in the proposed multi-level reliability assessment framework for power supply systems.

Level of Reliability Analysis	Main Task	Preferred Methodological Apparatus	Main Result
Element	Assessment of current technical condition, consumed life, degradation level, and restoration	Health index, correction factors, consumed life models, Markov models	Failure intensities, transition probabilities, equivalent operating time
System Structure	Formalization of the operability criterion, redundancy, and structural relationships	Logic–probabilistic method; for specific scenarios—fault tree analysis	Probability of failure-free operation, importance of structural elements, critical combinations
Operating Mode and Consequences	Assessment of failure consequences taking into account load regime, generation, constraints, and energy not supplied	Monte Carlo method, sequential simulation	Probability of power deficit, expected energy not supplied, indices characterizing outage consequences for consumers
Asset Management	Prioritization of technical actions and maintenance measures	Risk-oriented models, maintenance management systems	Asset priority rankings, technical action programs, optimized maintenance plans

summarizes the proposed distribution of functions based on the reviewed literature and the comparative analysis presented in Section 8 and Section 9.

The decision support level can be interpreted as a hierarchy of actions formed from the outputs of different model layers. At the element level, technical condition indicators can justify local actions for a specific item of equipment, including additional diagnostics, intensified monitoring, repair, or the replacement of a degraded unit. At the structural level, the logic–probabilistic model identifies elements whose failure has the greatest influence on system operability. Such elements may require higher diagnostic priority even when their current condition is not the worst, because their structural importance strongly affects the probability of successful power supply.

At the consequence assessment level, Monte Carlo simulation adds another decision basis by identifying scenarios that lead to power deficits, expected energy not supplied, long interruption durations, or high consumer-related damage. These outputs do not define a maintenance schedule directly. They become actionable when candidate technical measures are compared by their expected effects on risk and consequences. In practice, maintenance or renewal actions can be ranked by the expected reduction in power deficits, energy not supplied, or outage damage. Asset condition, structural importance, repair cost, accessibility, workforce, and allowable outage windows should also be considered. Thus, the proposed framework supports decision-making at several levels, including local condition-based actions, diagnostics for structurally important elements, and consequence-oriented maintenance priorities based on risk reduction.

Such integration of methods should be understood as the division of the model into verifiable layers, not as a mechanical combination of algorithms [10,15]. If structure, state dynamics, and failure consequences are mixed within one undifferentiated formalism, model verification and calibration become difficult. By contrast, a stepwise framework allows element-level condition models, the structural system scheme, and disturbance consequence calculations to be verified separately [12,73].

10. Unresolved Problems and Future Research Directions

Despite considerable progress, the literature still leaves several unresolved problems. The first concerns the high share of expert judgment in technical condition accounting [13,30]. Even when condition indices and diagnostic scales are available, the transition from these indicators to failure probability parameters often remains weakly formalized. The key challenge is therefore not to develop another integral indicator but to build a substantiated link between observable degradation signs and reliability parameters [11,16].

The second problem is the stationarity of reliability parameters in many models [12,31]. For real electrical equipment, the failure intensity often varies over time under the influence of the load, temperature, environment, defect history, and technical interventions. Further methodological development should therefore focus on time-varying element parameters and the joint consideration of consumed life and restoration [49,59].

The third problem is the insufficient consideration of common-cause failures, dependent damage, and organizational factors [4,48]. In enterprise power supply systems and distribution networks, failure is often not an independent random event of a single element. It may result from a combination of overload, personnel errors, repair defects, weather impacts, and operating factors. Basic structural models reflect these effects only partially. They therefore require further development toward dependent events and physically coupled failures [12,48].

This limitation can be reduced without enumerating all dependent combinations of element failures in the system state space. A more tractable approach is to introduce shared-cause events or scenario variables. In the structural layer, common causes can be represented as additional logical events that simultaneously affect several elements or groups of elements. In the simulation layer, weather conditions, overload scenarios, maintenance quality states, or organizational factors can be generated as scenario variables. These variables modify the probabilities of several related failures. This preserves the structural operability function and represents dependent damage at the level of grouped causes or operating scenarios. This approach does not fully solve the problem of dependent failures, but it is more realistic than assuming complete independence or constructing an excessively large state-space model.

The fourth problem is digital and organizational. Studies on data collection problems and predictive maintenance show that the main limitation is not the absence of sensors but data fragmentation, heterogeneous formats, and weak links between operational information systems, diagnostics, and reliability assessment systems [10,74]. Without a unified procedure for data collection and description, even a mathematically sophisticated model cannot provide reproducible results [73,75,76].

Before industrial adoption, the proposed multi-level framework should be validated by model layer rather than as a single black-box calculation. At the element level, the main task is to check whether equipment with worse diagnostic indicators, higher consumed life, or larger correction factors shows a higher frequency of defects, repairs, or confirmed failures. At the structural level, the operability function should be verified against real single-line diagrams, switching schemes, reserve supply logic, and known emergency restoration scenarios. This check confirms whether the logic–probabilistic model correctly distinguishes operable and inoperable system states.

At the simulation level, deficit-related indices should be compared with available interruption statistics, outage durations, energy-not-supplied records, and expert-confirmed disturbance scenarios. At the decision support level, calculated priority rankings should be tested retrospectively against actual maintenance actions, defect histories, and operating experience. When complete datasets are unavailable, sensitivity analysis and expert cross-validation can identify the input parameters that most strongly influence the final reliability and adequacy indices. This staged validation is more realistic for industrial power supply systems than the simultaneous calibration of all model layers.

The fifth problem concerns the interpretation of calculation results. Correct reliability assessment alone is not sufficient for operational decision-making. Probability values and indices must be translated into managerial actions, including asset renewal prioritization, maintenance program formation, monitoring selection, and redundancy solutions [19,73,77]. Studies on risk-oriented maintenance and the techno-economic efficiency of monitoring show that the link between reliability assessment and maintenance practice is a key criterion for model usefulness [75,78].

Recent studies on metaheuristic and hybrid optimization algorithms also indicate that moving from reliability or risk indices to practical decisions requires a clearly defined objective function, reproducible constraints, and careful algorithm selection. Such algorithms may help to calibrate correction factors, health index-based resource models, or maintenance priorities. However, they cannot compensate for the absence of reliable diagnostic, operating, and failure data [79,80].

Table 4. Main scientific problems in reliability assessment of power supply systems and promising directions for their solution.

Scientific Problem	Problem Description	Promising Direction of Development
High share of expert judgment in accounting for technical condition	Insufficient calibration of indices and correction factors against actual failures and defects	Calibration of indices and correction factors against actual failures, defects, and resource-related data
Stationarity of element reliability parameters	Insufficient account of time-varying failure intensities under the influence of operating mode and equipment condition	Reliability assessment accounting for technical condition and time-varying element parameters
Independence of failures	Incomplete representation of common-cause failures, dependent damage, and cascading effects	Development of hybrid structural–probabilistic models with physical and operational dependencies
Data scarcity	Non-unified databases of defects, repairs, and monitoring results	Formation of a unified digital data contour for defects, repairs, monitoring, and reliability indices
Insufficient linkage between calculated reliability indices and technical decision-making	Difficulty in translating reliability indices into decisions	Risk-oriented integration of reliability assessment with maintenance, repair, and asset management programs

summarizes the main scientific problems in the reliability assessment of power supply systems and promising directions for their solution.

The foregoing discussion indicates that the further development of reliability assessment methodologies for power supply systems should proceed in three interrelated dimensions. The first dimension is mathematical and concerns models that account for non-stationarity and failure dependence. The second is informational and concerns verifiable datasets on condition, failures, and repairs. The third is managerial and concerns the integration of calculation results into risk-oriented decisions on maintenance, monitoring, and modernization [11,19].

11. Conclusions

This review demonstrates that reliability assessment methods for power supply systems have complementary areas of applicability. Fault tree analysis remains useful for cause-and-effect analysis and for identifying critical combinations of events [33,34]. The logic–probabilistic method is well suited to the structural layers of systems with complex topologies, redundancy, and switching logic [43,48]. Markov models describe degradation, restoration, and time-dependent state transitions, but they are most effective at the level of individual elements and subsystems [50,53]. Monte Carlo simulation links structural system descriptions with adequacy indices and supply interruption consequences [28,32].

Equipment condition does not form an independent class of system-level reliability methods. It acts as a parametric layer that updates element characteristics using diagnostic, operational, and resource-related data [13,16]. Health indices, resource-based parameterization, and Markov degradation models are not interchangeable descriptions of the same process. Their selection depends on the purpose of the model and on data availability and quality [11,12].

For power supply systems, the review supports the synthesis of complementary approaches within a multi-level reliability assessment framework rather than a search for a single universal method [10,14]. In this framework, the logic–probabilistic method serves as the core structural tool, Markov models provide the dynamic layer for degradation and restoration, Monte Carlo simulation supports adequacy and consequence assessment, and condition-based models update element parameters [11,57]. Further research can develop this integrated framework for methodological studies and practical applications in power supply system reliability.

From a practical point of view, the proposed framework supports decision-making in operating power supply systems. It helps to identify elements that need detailed diagnostics because their condition strongly influences system operability. It also separates tasks where structural reliability assessment is sufficient from tasks that require adequacy or consequence simulation. Finally, it provides a methodological basis for linking condition monitoring, reliability calculation, and risk-oriented maintenance planning. The practical value of the framework therefore lies not only in calculating reliability indices but also in transforming them into priorities for monitoring, maintenance, renewal, and technical actions.