Abstract Medium-voltage distribution networks are an important element of the global power system, being responsible for the distribution of electrical energy from transformer stations to local points of delivery and to transformer stations of lower voltage levels. The reliability of the operation of these networks has a direct impact on the continuity of energy supply and the level of unmet energy demand in the power system. The article presents a reliability analysis of a selected segment of a medium-voltage distribution network located in northern Poland. In this study, a probabilistic model of the operation process based on an eight-state graph describing successive levels of technical degradation of the analyzed network was applied. Transitions between the states of the model were described by failure intensities and restoration intensities of the system elements. On the basis of the Kolmogorov–Chapman state equations, the probabilities of the system being in particular operational states were determined. The results obtained were then used to assess the energy-related consequences of failures by linking state probabilities with the share of unmet energy demand. The analysis conducted enabled the identification of the most critical elements of the analyzed network structure and the determination of their impact on the energy supply capability of the distribution network. The obtained results may constitute a basis for planning operational activities, maintenance strategies, and modernization processes of medium-voltage distribution networks. 1. Introduction The issue of the reliability of technical and power systems is widely described in the literature. It includes both general models of the availability and reliability of complex systems [ 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], as well as operational analyses of technical objects using probabilistic models and Markov processes [ 31, 32, 33]. In relation to power systems, models that take into account the quality of system operation, disturbances and recovery conditions after failures are also being developed [ 34, 35]. At the same time, selected sections of medium-voltage distribution networks considered as local structures with a defined topology are relatively less frequently analyzed; for these systems, it is possible to simultaneously determine the probabilities of operational states and the corresponding energy-related consequences [ 36, 37]. There is therefore a research gap in the insufficient recognition of the impact of failures of selected sections of medium-voltage distribution networks on the level of unmet energy demand, while simultaneously providing a probabilistic description of the system degradation process. In particular, there is a need for analyses that combine a model of operational states with the actual structure of a network segment and with an assessment of energy-related consequences of failures of individual sections. The novelty of the proposed approach lies not only in the application of a probabilistic multi-state model itself but primarily in linking the operational state probabilities of a real medium-voltage distribution network segment with the corresponding levels of unmet energy demand. In contrast to classical Markov-based reliability analyses focused mainly on availability indices or failure frequencies, the proposed framework combines degradation states, network topology, and energy-related operational consequences within a unified reliability model. The article undertakes a reliability analysis of a selected segment of a medium-voltage distribution network located in northern Poland, in the area of Bytów County [ 38]. The aim of the study is to determine the probabilities of individual operational states and to analyze the effect of failures in selected network sections on unmet energy demand levels. For this purpose, an eight-state operational model was applied, in which the system states were associated with successive levels of degradation in the energy supply capability of the analyzed network segment. 2. Literature Review Despite the dynamic development of analytical methods, classical probabilistic reliability models still play an important role in the analysis of the operation processes of complex technical systems. In particular, probabilistic state models based on the Kolmogorov–Chapman equations are widely used, as they enable a description of transitions between the operational states of a system and determination of the probabilities of the system being in particular states as a function of time. This approach is commonly applied in the reliability analysis of technical systems and critical infrastructure [ 61]. In particular, probabilistic state models based on the Kolmogorov–Chapman equations enable the description of transitions between degradation and restoration states of complex infrastructure systems. For this reason, such models are widely used in the reliability analysis of critical infrastructure, where the continuity of operation and the consequences of failures must be assessed under changing operational conditions. The approaches presented in the literature indicate that the reliability analysis of power systems constitutes a complex issue that includes both the modeling of operational processes of power infrastructure and an assessment of the consequences of failures for the functioning of the power supply system. In particular, probabilistic state models based on the Kolmogorov–Chapman equations constitute an effective tool that enables analysis of transitions between the operational states of technical systems and determination of their reliability characteristics. 3. Methodology of a Reliability Analysis of the Selected Medium-Voltage Distribution Network A reliability analysis of technical infrastructure systems, including power systems, requires the use of mathematical models enabling a description of changes in the operational states of a system over time. In the case of medium-voltage distribution networks, the operation process is characterized by the occurrence of various states related to proper system operation, gradual degradation of infrastructure elements, and failure states. Conducting reliability studies of technical objects requires the acquisition of input data that describes the actual operation process of the system being analyzed. These data may originate from observations of the real operation process of the object, from the operational documentation of the power system, or from a properly designed simulation experiment. The basic quantities characterizing the operation process of a technical object include, among others, the usage time of the object, the time required to repair failures, and the time spent performing preventive activities. One of the fundamental measures used in the reliability analysis of technical systems is the system availability function K g ( t ) , defined as the probability of the object being in an operable state at time t . K g ( t ) = P [ S ( t ) ] (1) where K g ( t ) —the system availability function, P [ S ( t ) ] —the probability of the object being in an operable state at time t . The second fundamental characteristic used in reliability theory is the object reliability function R ( t ) , defined as the probability of the correct operation of the object for a time not shorter than t . R ( t ) = P ( τ ≥ t ) (2) where R ( t ) —the reliability function of the object, τ —the random time to failure, t —the time of operation of the object. The procedure for reliability studies of the analyzed technical object’s operational process is presented in the form of an algorithm shown in Figure 1. This algorithm presents successive stages of conducting simulation studies, starting from the input of data, followed by the construction of the operation process model, and concluding with a qualitative and comparative analysis of the results obtained. In order to perform a reliability analysis of the selected section of the medium-voltage distribution network, a probabilistic multi-state model based on the Kolmogorov–Chapman state equations was employed. The first step of this procedure is to define the set of operational states of the analyzed distribution network. In this study, a finite set of states denoted by symbols S0, S1, S2, S3, S4, S5, S6 and S7 was adopted, representing successive levels of technical degradation of the network from full operability to unserviceability. Next, the possible transitions between the states and the corresponding intensities of the degradation and restoration processes are determined. The next stage of the algorithm is the construction of a state graph describing the structure of transitions between the operational states of the analyzed distribution network. Based on a graph defined in this manner, it is possible to formulate a system of differential equations resulting from the Kolmogorov–Chapman equations, which describe changes over time in the probabilities of the network being in particular operational states. The application of the Kolmogorov–Chapman formalism enables a probabilistic description of transitions between successive degradation and restoration states of the analyzed network. In comparison with classical static reliability indices, this approach allows the operational process to be represented dynamically and enables the simultaneous assessment of operational state probabilities and their corresponding energy-related consequences. In the subsequent steps of the algorithm, the state probability functions and reliability characteristics of the analyzed section of the medium-voltage distribution network are determined. A detailed technical interpretation of individual operational states and the construction of the state graph for the analyzed network section are presented in the next section of the article. 4. Reliability Analysis of the Selected Medium-Voltage Distribution Network 4.1. Operating Conditions of the Selected Medium-Voltage Distribution Network The power system in Poland is controlled by the NLD, i.e., the National Load Dispatch Center. The subordinate units are as follows: ALDC—Area Load Dispatch Center, CLDC—Central Load Dispatch Center (central in the sense of a local distribution system operator), RLDC—Regional Load Dispatch Center. Every day, forecasts are collected from energy producers regarding the amount of electrical energy planned to be introduced into the power system on the following day. Additionally, weather information and its impact on energy generation are analyzed, including generation from sources for which there is no obligation to provide reports and forecasts. The amount of energy to be consumed from the system is also determined on the basis of historical data. The NLD also takes into account energy demand in neighboring countries and in interconnected power systems. All of the above information leads to decisions regarding the amount of energy required from stable energy sources (such as coal, gas, and hydro power plants) and the possibility of accepting energy from unstable sources (wind and solar power plants). The NLD provides subordinate units with guidelines concerning the necessity to generate energy or to limit its generation. Information about the need to introduce limitations is continuously monitored and adjusted, and decisions concerning, for example, a shutdown or reduction in power from sources are communicated as late as possible but in time to allow their implementation without any adverse effects on the system and equipment. This usually occurs with a lead time of 1–2 h. In cases where the anticipated scope of generation limitations proves insufficient, the NLD makes decisions on so-called supplementary activations. In such situations, the implementation time of actions is defined as immediate, which enables a dynamic response of the system to changing operating conditions. These actions are carried out using remote control systems that enable changes in generated power setpoints or direct control of circuit breakers in power plants operating within the distribution network area. The process of transitions from one state to another under the influence of failure and restoration streams for the analyzed system is presented by means of a graph in Figure 2. The operating conditions of the selected medium-voltage distribution network are characterized by local system responses to changes in power demand. At the main substation level (the main transformer and switching stations), these responses are implemented when variations in power demand result in voltage changes at the transformer terminals. The basic method is the application of automatic voltage regulation (AVR) control. This is an automatic system built into HV/HV, HV/MV and MV/MV transformers, implemented by changing transformer tap positions. The response time to a deviation from the set voltage level is, for example, 120 s for the smallest difference, which is naturally shorter when the difference between the measured and the set voltage is greater. In the case of a sudden deficit of supplied power relative to the required power in the system, a frequency drop may occur. This situation is dangerous, as it may lead to emergency disconnection of generators by frequency protection systems and to cascading system failures, potentially resulting in a blackout. The primary protection in such situations is UFLS automatic control. The response time resulting from the guidelines of the power system operator is immediate. A slight delay is permissible in the case of transformer and switching stations that have not yet been modernized. The operation of the UFLS automatic system consists of disconnecting groups of consumers (usually several medium-voltage lines) in order to relieve the power system. The UFLS typically has two stages, that is, two groups of disconnected consumers. If the disconnection of the first stage does not restore the frequency to normal, the second stage is activated. Currently, due to the large number of distributed generation sources connected to medium-voltage lines, the operation of the UFLS automatic control requires analysis and correction, as such operation may often be detrimental due to the disconnection of generation connected along medium-voltage lines. In cases where the generated power is too high relative to the demand, as a result of an increase in frequency, regulation should be introduced as in items 4 and 5, and sudden changes should result in an emergency shutdown of generation by overfrequency protection systems. The diagram of the medium-voltage distribution network selected for the reliability analysis in the studied area is presented in Figure 2. 4.2. Operational Process Model of the Selected Medium-Voltage Distribution Network (MVDN) The distribution network selected for the reliability analysis consists of grouped buses numbered 5, 6, and 8. The remaining sections in the distribution network diagram shown in Figure 1, i.e., those numbered {1,2,3,4,7}, are not significant for the operation of the distribution network in this area. Therefore, only the distribution network sections marked with numbers 5, 6, and 8 were included in this functional and reliability analysis of the MVDN. The adopted eight-state model does not represent all possible topological switching configurations of the distribution system. In practical distribution networks, the number of potential switching configurations may increase significantly depending on the network topology and the number of switching elements. In the proposed approach, operationally equivalent configurations were aggregated into generalized degradation states associated with the energy supply capability of the analyzed network segment. Such an approach reduces the computational complexity of the model while preserving the engineering interpretation of the system degradation process and its operational consequences. The investigated distribution network presented in Figure 2 is subjected during operation both to internal and external influences. Over time, these circumstances may lead to a complete failure of the system. The selected distribution network may be subjected to very small (insignificant) or large (significant) failures. Each failure of the distribution network has consequences for the future operation of this system. Such a state directly and negatively affects the capability of transmitting electrical energy through this network. According to the studies, all failures of the distribution network can be divided into insignificant or significant (critical) failures. A significant (parametric) failure is a technical term describing a case in which the selected distribution network can only partially perform the required function in the transmission of electrical energy. A failure of a technical object that develops gradually during its operation is referred to as a non-critical failure. This type of failure is most often caused by natural deterioration resulting from aging and the influence of environmental conditions (such as temperature and pressure). Numbered vertices correspond to possible states. Arrows indicate transitions from one state to another. The individual states are presented as follows: S0 is the operable state—all elements of the system (MVDN) are functional. S1 is the state of partial operability of the distribution network. This state occurs when line no. 5 (i.e., due to trees and branches) is damaged—this is an insignificant failure. S2 is the state of partial operability of the distribution network. This state occurs when line no. 6 is damaged (weakening of electrical insulation)—this is an insignificant failure. S3 is the state of reduced operability of the distribution network. This state occurs when line no. 8 is significantly damaged (i.e., aging, material fatigue)—this is a significant failure. S4 is the state of reduced operability of the distribution network; this state occurs when line nos. 5 and 6 are damaged (i.e., due to trees and branches, as well as weakening of the electrical insulation of the line)—this is an insignificant failure. S5 is the state of significantly reduced operability of the distribution network. This state occurs when line nos. 5 and 8 are damaged (trees and branches and additionally aging and material fatigue)—this is a moderately significant failure. S6 is the state of clearly critical operability of the distribution network. This state occurs when line nos. 6 and 8 are damaged (i.e., weakening of the electrical insulation and additionally aging and material fatigue)—this is a significant failure. S7 is the state of unserviceability of the distribution network (MVDN). This state occurs when all the lines in the distribution network are damaged. It may be caused by the influence of trees and branches, weakening of electrical insulation, as well as aging and material fatigue processes. This corresponds to a supercritical failure leading to a complete loss of the energy supply capability of the system (blackout). The graphical representation of the operation process of the considered distribution network, described by the eight-state model, is presented in Figure 3. Transitions denoted by symbols λ i correspond to the processes of technical degradation of the power network, leading to a deterioration of its operational state, whereas transitions denoted by symbols μ i represent restoration processes associated with the repair or replacement of the damaged network elements. Descriptions of the transition relationships ( Figure 4) between the states in the operation process model are as follows: – λ 1 —transition from state S0 to state S1, – λ 2 —transition from state S0 to state S2, – λ 3 —transition from state S0 to state S3, – λ 4 —transition from state S1 to state S4, – λ 5 —transition from state S2 to state S5, – λ 6 —transition from state S3 to state S6, – λ 7 —transition from state S4 to state S7, – λ 8 —transition from state S5 to state S7, – λ 9 —transition from state S6 to state S7, – μ 1 —transition from state S1 to state S0, – μ 2 —transition from state S2 to state S0, – μ 3 —transition from state S3 to state S0, – μ 4 —transition from state S4 to state S0, – μ 5 —transition from state S5 to state S0, – μ 6 —transition from state S6 to state S0, – μ 7 —transition from state S7 to state S0. The system presented is characterized by the following Kolmogorov–Chapman equations: R S 0 ′ ( t ) = − λ 1 ⋅ R S 0 ( t ) + μ 1 ⋅ P S 1 ( t ) − λ 2 ⋅ R S 0 ( t ) + μ 2 ⋅ P S 2 ( t ) − λ 3 ⋅ R S 0 ( t ) + μ 3 ⋅ P S 3 ( t ) + μ 4 ⋅ P S 4 ( t ) + μ 5 ⋅ P S 5 ( t ) + μ 6 ⋅ P S 6 ( t ) + μ 7 ⋅ P S 7 ( t ) P S 1 ′ ( t ) = λ 1 ⋅ R S 0 ( t ) − μ 1 ⋅ P S 1 ( t ) − λ 4 ⋅ P S 1 ( t ) P S 2 ′ ( t ) = λ 2 ⋅ R S 0 ( t ) − μ 2 ⋅ P S 2 ( t ) − λ 5 ⋅ P S 2 ( t ) P S 3 ′ ( t ) = λ 3 ⋅ R S 0 ( t ) − μ 3 ⋅ P S 3 ( t ) − λ 6 ⋅ P S 3 ( t ) P S 4 ′ ( t ) = λ 4 ⋅ P S 1 ( t ) − μ 4 ⋅ P S 4 ( t ) − λ 7 ⋅ P S 4 ( t ) P S 5 ′ ( t ) = λ 5 ⋅ P S 2 ( t ) − μ 5 ⋅ P S 5 ( t ) − λ 8 ⋅ P S 5 ( t ) P S 6 ′ ( t ) = λ 6 ⋅ P S 3 ( t ) − μ 6 ⋅ P S 6 ( t ) − λ 9 ⋅ P S 6 ( t ) P S 7 ′ ( t ) = λ 7 ⋅ P S 4 ( t ) + λ 8 ⋅ P S 5 ( t ) + λ 9 ⋅ P S 6 ( t ) − μ 7 ⋅ P S 7 ( t ) (3) Given the starting conditions: R S 0 ( 0 ) = 1 P S 1 ( 0 ) = P S 2 ( 0 ) = P S 3 ( 0 ) = P S 4 ( 0 ) = P S 5 ( 0 ) = P S 6 ( 0 ) = P S 7 ( 0 ) = 0 (4) By utilizing the Laplace transformation, the subsequent system of linear equations is obtained: s ⋅ R S 0 * ( s ) − 1 = − λ 1 ⋅ R S 0 * ( s ) + μ 1 ⋅ P S 1 * ( s ) − λ 2 ⋅ R S 0 * ( s ) + μ 2 ⋅ P S 2 * ( s ) − λ 3 R S 0 * ( s ) + μ 3 ⋅ P S 3 * ( s ) + μ 4 ⋅ P S 4 * ( s ) + μ 5 ⋅ P S 5 * ( s ) + μ 6 ⋅ P S 6 * ( s ) + μ 7 ⋅ P S 7 * ( s ) s ⋅ P S 1 * ( s ) = λ 1 ⋅ R S 0 * ( s ) − μ 1 ⋅ P S 1 * ( s ) − λ 4 ⋅ P S 1 * ( s ) s ⋅ P S 2 * ( s ) = λ 2 ⋅ R S 0 * ( s ) − μ 2 ⋅ P S 2 * ( s ) − λ 5 ⋅ P S 2 * ( s ) s ⋅ P S 3 * ( s ) = λ 3 ⋅ R S 0 * ( s ) − μ 3 ⋅ P S 3 * ( s ) − λ 6 ⋅ P S 3 * ( s ) s ⋅ P S 4 * ( s ) = λ 4 ⋅ P S 1 * ( s ) − μ 4 ⋅ P S 4 * ( s ) − λ 7 ⋅ P S 4 * ( s ) s ⋅ P S 5 * ( s ) = λ 5 ⋅ P S 2 * ( s ) − μ 5 ⋅ P S 5 * ( s ) − λ 8 ⋅ P S 5 * ( s ) s ⋅ P S 6 * ( s ) = λ 6 ⋅ P S 3 * ( s ) − μ 6 ⋅ P S 6 * ( s ) − λ 9 ⋅ P S 6 * ( s ) s ⋅ P S 7 * ( s ) = λ 7 ⋅ P S 4 * ( s ) + λ 8 ⋅ P S 5 * ( s ) + λ 9 ⋅ P S 6 * ( s ) − μ 7 ⋅ P S 7 * ( s ) (5) 5. Results of the Study The reliability analysis was carried out for a selected segment of a medium-voltage distribution network located in northern Poland, in the area of Bytów County. The considered system constitutes part of a distribution network serving an area of a mixed urban-rural character. The analysis was performed for a time horizon corresponding to one year of operation, that is, t = 8760 [h]. The structure of the considered fragment of the distribution network is presented in Figure 5. This diagram serves as a reference point for further interpretation of the results and enables the linking of the probabilistic model with the actual topology of the distribution network. The figure presents a segment of a medium-voltage distribution network. The network with the considered sections of overhead cable lines (line section nos. 1–8) in the normal operating configuration is supplied from Main Power Station 1. In cases of unserviceability of a selected network section, after closing the sectionalizing disconnectors (the line break point), it is possible to supply the remaining operable sections of the network from Main Power Stations 2 to 5. The overhead line poles marked in the figure correspond to the locations of the division of the considered network sections and define the boundaries of the sections used in the further reliability analysis. The remaining infrastructure elements, not being relevant from the point of view of the adopted model, were omitted in order to maintain the clarity of the diagram. The presented topological structure constitutes a direct basis for the construction of the operation model of the MVDN system. In particular, the identification of key network sections and possible power switching scenarios makes it possible to define a set of operational states and the admissible transitions between them. The operation process of the MVDN system was described using an eight-state model, S0–S7, in which transitions between states are determined by failure intensities λ i and restoration intensities μ i . The values of these parameters were determined on the basis of the operational data obtained from power system operators. A summary of the calculated failure and restoration intensity parameters adopted in the reliability analysis is provided in Table 2. The failure and restoration intensities used in the probabilistic model were determined on the basis of aggregated operational data assigned to the corresponding operational transitions. For each network section, the outage and restoration times associated with all operational areas were summed prior to the determination of the intensity parameters. The intensity parameters were determined using the relationships: λ i = 1 t a (6) μ i = 1 t n (7) where t a denotes the aggregated outage-related time and t n denotes the aggregated restoration time associated with the considered operational transition. For example, for network section no. 5, the aggregated outage-related time was determined as t a = 14.55 + 0.95 + 5.39 + 0.42 = 21.31 [ h ] (8) and the corresponding failure intensity was calculated as λ 1 = 1 21.31 = 0.04693 [ 1 / h ] (9) Analogously, the intensity parameters corresponding to multi-section degradation states were determined as reciprocal values of the aggregated outage or restoration times associated with the considered operational transitions.” Based on the state model, the probabilities of the system being in particular operational states after a period of one year were determined. The following values were obtained: The probability that the MVDN system will remain in the state of full operability S0 after one year is equal to R S 0 = 0.3446 (10) The probability that the MVDN system will be in the state of partial operability S1 after one year is equal to P S 1 = 0.1347 (11) The probability that the MVDN system will be in state S2 after one year is equal to P S 2 = 0.1636 (12) The probability that the MVDN system will be in critical state S3 after one year is equal to P S 3 = 0.1192 (13) The probability of the system being in state S4 after one year is equal to P S 4 = 0.0607 (14) The probability of the system being in state S5 after one year is equal to P S 5 = 0.0563 (15) The probability of the system being in state S6 after one year is equal to P S 6 = 0.0414 (16) The probability of the system being in an unserviceable state S7 after one year is equal to P S 7 = 0.0792 (17) The obtained probability values satisfy the normalization condition: R S 0 + P S 1 + P S 2 + Q S 3 + Q S 4 + Q S 5 + Q S 6 + Q S 7 = 1 (18) In order to quantitatively assess the impact of individual states on the level of undelivered energy, the probabilities of the state occurrence were combined with the corresponding share of undelivered energy. The results are presented in Table 3. In order to illustrate the relationship between the probability of the occurrence of operational states and the corresponding level of undelivered energy, a graphical interpretation of the results is presented in the form of a plot ( Figure 6) of the dependence of U i on probability p i . The size of the symbols corresponds to the value of the contribution p i ⋅ U i , which allows a direct assessment of the significance of the individual states in the structure of undelivered energy. In order to determine the states that have the greatest impact on the level of undelivered energy in the system, a Pareto analysis was carried out for the values of contribution p i ⋅ U i . The results are presented in Figure 7. The analysis presented demonstrates that the largest contribution to the expected value of undelivered energy in the system comes from states S2, S3, S1, and S7, which together account for approximately 73.2% of the total contribution to the value of undelivered energy in the operation process considered. After including state S5, this share increases to approximately 85.9%. In order to better illustrate the impact of the degradation of successive network sections on the energy supply capability of the system, a schematic interpretation of changes in the energy supply capability of the considered power supply path is presented. Figure 8 does not present direct results of probabilistic calculations but rather a graphical illustration of the degradation mechanism of the system’s energy supply capability resulting from the topology of the considered distribution network. The analysis of the network structure presented in Figure 2 made it possible to determine the impact of failures of individual sections on the energy supply capability of the considered segment of the system. The analysis of the system topology indicates that the key role in energy delivery is played by the network sections marked with numbers 5, 6 and 8. These sections are located at successive nodes of the energy supply path, and their degradation leads to a progressive limitation of the system’s energy supply capability. Failure of network section no. 5 results in a loss of approximately 63% of the energy flow in the considered part of the system. This means that only about 37% of the original energy supply capability remains in the further part of the energy supply path. In the case of further system degradation and a failure of network section no. 6, the limitation of energy flow increases to approximately 84%. This section is located at the next node of the network structure; therefore, its failure causes a further decrease in the system’s energy supply capability. The most critical element of the considered network segment is section no. 8. Its failure leads to a complete loss of the ability to supply energy to the considered area, which corresponds to 100% of undelivered energy. As a result, the system degradation may be interpreted as a sequential transition from a limitation of the energy flow at the level of approximately 63%, through a level of approximately 84%, up to a complete loss of the system’s energy supply capability. In this interpretation, state S1 corresponds to the first significant limitation of the energy flow resulting from the degradation of network section no. 5. State S2 is associated with a further deterioration of system operating conditions caused by the degradation of section no. 6. In turn, state S3 and the subsequent states lead to a situation in which the system loses the ability to supply energy to the considered supply area. In order to determine the annual energy consequences resulting from the failures of the analyzed feeders, the share of undelivered energy was determined for the entire feeders 5, 6, and 8, taking into account all associated areas. For feeder 5, the annual share of undelivered energy is equal to 0.63 ⋅ Q r e f ≈ 0.001533 (19) which corresponds to approximately 0.1533% of the annual undelivered energy in the considered segment of the distribution network. For network section no. 6, the share of undelivered energy is equal to 0.84 ⋅ Q r e f ≈ 0.001622 (20) which corresponds to approximately 0.1622% of the annual unmet energy demand in the considered segment of the distribution network. In turn, for network section no. 8, a failure leads to a complete loss of the energy supply capability of the considered segment of the distribution network, which can be expressed as 1.00 ⋅ Q r e f = 0.015732 (21) which corresponds to approximately 1.5732% of the annual unmet energy demand in the considered segment of the distribution network. In order to quantitatively illustrate the annual energy effect resulting from failures of the key sections of the distribution network, a summary of the annual unmet energy demand associated with failures of section nos. 5, 6, and 8 is presented. These values were determined on the basis of the previously defined reference energy Q r e f and the shares of energy supply capability loss assigned to the individual elements of the network structure. The results obtained are presented graphically in Figure 9. The results obtained indicate that network section nos. 5, 6 and 8 have the greatest impact on the level of unmet energy demand in the considered segment of the distribution network, with the most significant energy consequences associated with the failure of section no. 8, leading to a complete loss of the system’s energy supply capability. 6. Discussion The results obtained in Section 5 enable an in-depth interpretation of the reliability properties of the analyzed segment of the medium-voltage distribution network. The applied eight-state operational model allows the simultaneous representation of both the probabilistic behavior of the system and the energy consequences of failures occurring in individual network elements. This approach enables not only the determination of the probabilities of the system being in particular operational states but also a quantitative assessment of the operational effects associated with these states, significantly extending the practical applicability of classical reliability models in the analysis of distribution networks. The state probabilities determined for the annual operation horizon indicate that the analyzed system remains in the state of full operability S0 with a probability equal to 0.3446. This means that, for approximately one-third of the analyzed operation period, the system is capable of supplying electrical energy without limitations. From an operational point of view, this value indicates that the analyzed network segment is exposed to a relatively high frequency of degraded operating conditions during the annual operation horizon. This result confirms that failures of the critical sections significantly affect the continuity of the electricity supply and may justify intensified maintenance and modernization activities within the analyzed network structure. An important element of the analysis is the relationship between the probability of occurrence of operational states and the corresponding share of unmet energy demand. The graphical interpretation presented in Figure 6 indicates that the impact of individual operational states on the system energy balance is not determined solely by their probability of occurrence but rather by the combined effect described by the product p i ⋅ U i . This quantity represents the contribution of individual operational states to the expected value of unmet energy demand in the system. The Pareto analysis presented in Figure 7 shows that the majority of the operational effects associated with system degradation are concentrated in a limited number of operational states. In particular, states S2, S3, S1, and S7 jointly account for approximately 73.9% of the total expected unmet energy demand in the analyzed operation process. After additionally including state S5, this share increases to approximately 83.8%. This indicates that the majority of the operational consequences of the analyzed network segment are associated with a relatively small number of operational states. A particularly important observation concerns the role of state S7, corresponding to the complete unserviceability of the system. Despite the relatively low probability of occurrence of this state, it generates a significant contribution to the expected value of unmet energy demand due to the complete loss of the system’s energy supply capability. Consequently, even relatively infrequent occurrences of this state may lead to significant operational consequences. The structural interpretation of the analyzed network segment is presented in Figure 8. The analysis of the topology of the power supply path indicates that network section nos. 5, 6, and 8 form a sequential structure in which successive failures lead to a gradual limitation of the system’s energy supply capability. Failure of section no. 5 results in a reduction in the energy supply capability to approximately 63% of the nominal value, while further degradation associated with section no. 6 limits the energy supply capability to approximately 16% of the initial value. Failure of section no. 8 results in a complete loss of the ability to supply energy. The annual operational effects of failures of the analyzed sections are presented in Figure 9. The results obtained on the basis of the updated data indicate that the greatest impact on the level of unmet energy demand is associated with section no. 8, for which the annual share of unmet energy demand is approximately 1.57% of the reference energy. Section nos. 6 and 5 generate lower, but still significant, levels of unmet energy demand amounting to approximately 0.162% and 0.153%, respectively. From an operational point of view, the obtained results indicate that network section nos. 5, 6, and 8 constitute the most critical elements of the analyzed segment of the medium-voltage distribution network. Their technical condition directly affects the energy supply capability of the system and the level of unmet energy demand experienced by consumers. Consequently, these elements should be subject to increased technical supervision, including more frequent inspections, preventive actions, and infrastructure modernization. The presented approach, combining a probabilistic state model with an assessment of the operational consequences of failures, constitutes a useful analytical tool enabling the identification of critical elements of the distribution network and supporting decision-making processes related to operational planning and improvement of the reliability of electricity supply. The analyzed network operates formally in a radial configuration from both the protection and operational points of view. However, the presence of sectionalizing disconnectors and alternative supply paths enables partial reconfiguration of the supply structure under selected operating conditions. Therefore, the proposed approach may support both operational reliability improvement in existing radial systems and future modernization planning aimed at increasing network flexibility and resilience. In this sense, the model may be treated not only as a tool supporting current operation but also as a framework supporting long-term reliability-oriented development of medium-voltage distribution systems. 7. Conclusions The studies conducted confirmed the usefulness of the eight-state probabilistic model in the reliability analysis of medium-voltage cable distribution networks and in the quantitative assessment of the energy-related consequences of failures. The results obtained showed that the distribution of unmet energy demand is highly non-uniform, and the largest share is associated with states S2, S3, S1 and S7, which together account for approximately 73.9% of the total unmet energy demand. This indicates that the majority of energy losses are concentrated in a limited number of operational states. The structural analysis indicated that network section nos. 5, 6 and 8 have the greatest impact on the energy supply capability of the analyzed network segment, with the failure of the terminal section leading to a complete loss of the ability to supply energy to consumers. The results obtained indicate the validity of focusing operational activities on selected network elements and the need to consider topological dependencies in the process of a reliability assessment of power supply systems. These results may provide direct support for decision-making processes related to power balancing in the power system, particularly in determining the demand for energy from stable sources and the permissible share of unstable sources. In practice, this means the possibility of using the results of the analysis to formulate operational guidelines concerning generation and power limitations, implemented with an appropriate time advance or in an intervention mode. In further research, it is advisable to extend the model to include variability in operating and environmental conditions, as well as to consider dependencies between failures of network elements, which will allow for a more realistic representation of degradation processes and an improvement in the accuracy of reliability forecasts. Abbreviations Symbol/Acronym Description MVDN Medium-voltage distribution network MV Medium voltage MPS Main power station ANR Automatic network reconfiguration LBP Line break point (sectionalizing point) PDC Power dispatch control CPD Central power dispatch RPD Regional power dispatch APD Area power dispatch S ii-th operational state of the system, where i = 0 , 1 , … , 7 S0State of full serviceability S1– S6Intermediate degradation states S7State of unserviceability λ iFailure intensity (transition to a degraded state) μ iRepair intensity (transition to an improved state) P i(t) Probability of the system being in state S i at time t t Time of operation t = 8760 h One-year operation horizon E Reference energy in the analyzed period E iUndelivered energy assigned to state S i u iShare of undelivered energy for state S i [%] C iContribution of state S i to expected undelivered energy Algorithm for modeling the reliability of the operational process of the analyzed section of the medium-voltage distribution network. Algorithm for modeling the reliability of the operational process of the analyzed section of the medium-voltage distribution network. Diagram of the medium-voltage distribution network for the selected area. Diagram of the medium-voltage distribution network for the selected area. State transition graph of the eight-state operational model for the considered medium-voltage distribution network. State transition graph of the eight-state operational model for the considered medium-voltage distribution network. Example trajectory of state process S ( t ) for the eight-state operational model of the considered medium-voltage distribution network. Example trajectory of state process S ( t ) for the eight-state operational model of the considered medium-voltage distribution network. Selected section of the medium-voltage distribution network considered in the reliability analysis. Red lines indicate the network sections included in the reliability analysis, whereas black dashed lines represent adjacent network sections and connections that were not considered in the present study. Selected section of the medium-voltage distribution network considered in the reliability analysis. Red lines indicate the network sections included in the reliability analysis, whereas black dashed lines represent adjacent network sections and connections that were not considered in the present study. Relationship between the probability of operational states and the corresponding share of undelivered energy in the MVDN system. The bubble size represents contribution p i ⋅ U i . Relationship between the probability of operational states and the corresponding share of undelivered energy in the MVDN system. The bubble size represents contribution p i ⋅ U i . Pareto analysis of the contribution of operational states to the expected undelivered energy in the MVCN system. Pareto analysis of the contribution of operational states to the expected undelivered energy in the MVCN system. Conceptual illustration of the degradation of the energy supply capability of the distribution network resulting from failures of network section nos. 5, 6 and 8. Conceptual illustration of the degradation of the energy supply capability of the distribution network resulting from failures of network section nos. 5, 6 and 8. Relative contribution of network sections (grouped buses) 5, 6 and 8 to the undelivered energy. The values are normalized to the maximum value corresponding to section 8. Relative contribution of network sections (grouped buses) 5, 6 and 8 to the undelivered energy. The values are normalized to the maximum value corresponding to section 8. Table 1. Outage durations and repair times for the analyzed medium-voltage distribution network in 2023. Table 1. Outage durations and repair times for the analyzed medium-voltage distribution network in 2023. Grouped Buses Line Cause of Damage Unpowered Time [h] Repair Time [h] 2023 2023 5 Area I Total 14.55 7.275 Trees and branches 13.36 6.68 Local weakening of electrical insulation 0 0 Aging, material fatigue 1.19 0.595 Area II Aging, material fatigue 0.95 0.475 Area III Aging, material fatigue 5.39 2.695 Area IV Extension of planned work time 0.42 0.21 6 Area I Total 6.74 3.37 Trees and branches 1.28 0.64 Extension of planned work time 5.46 2.73 Aging, material fatigue 0 0 Area II Total 10.16 5.08 Trees and branches 4.4 2.2 Aging, material fatigue 5.76 2.88 Area III Aging, material fatigue 0 0 8 Area I Total 90.17 45.085 Trees and branches 64.61 32.305 Aging, material fatigue 25.56 12.78 Area II Total 26.52 13.26 Trees and branches 14.54 7.27 Extension of planned work time 7.23 3.615 Aging, material fatigue 4.75 2.375 Area III Total 21.12 10.56 Trees and branches 0.06 0.03 Extension of planned work time 3.71 1.855 Aging, material fatigue 17.35 8.675 Note: Values shown in bold represent aggregated (“Total”) outage and repair times used in the reliability calculations presented in this study. Table 2. Parameters for the system’s reliability. Table 2. Parameters for the system’s reliability. Parameter Value [1/h] λ 10.046926326 λ 20.059171598 λ 30.007256367 λ 40.026171159 λ 50.006284565 λ 60.006463706 λ 70.005681173 λ 80.005681173 λ 90.005681173 μ 10.09385265 μ 20.11834320 μ 30.01451273 μ 40.05234232 μ 50.01256913 μ 60.01292741 μ 70.01136235 Table 3. State probabilities and corresponding values of undelivered energy in the PSNS system. Table 3. State probabilities and corresponding values of undelivered energy in the PSNS system. State Number State Probability p i Undelivered Energy U i [ % Q n ] p i ⋅ U i 0 0.3446 0 0.0000 1 0.1347 63 8.4861 2 0.1636 84 13.7424 3 0.1192 100 11.9200 4 0.0607 84 5.0988 5 0.0563 100 5.6300 6 0.0414 100 4.1400 7 0.0792 100 7.9200 p i —the probability of the occurrence of the i -th operational state of the system, Q n —the reference energy of the system in the analyzed period, U i —the share of undelivered energy assigned to the i -th state, expressed as a percentage of reference energy Q n , p i ⋅ U i —the contribution of the i -th state to the expected value of undelivered energy in the system. 2. Literature Review Despite the dynamic development of analytical methods, classical probabilistic reliability models still play an important role in the analysis of the operation processes of complex technical systems. In particular, probabilistic state models based on the Kolmogorov–Chapman equations are widely used, as they enable a description of transitions between the operational states of a system and determination of the probabilities of the system being in particular states as a function of time. This approach is commonly applied in the reliability analysis of technical systems and critical infrastructure [ 61]. In particular, probabilistic state models based on the Kolmogorov–Chapman equations enable the description of transitions between degradation and restoration states of complex infrastructure systems. For this reason, such models are widely used in the reliability analysis of critical infrastructure, where the continuity of operation and the consequences of failures must be assessed under changing operational conditions. The approaches presented in the literature indicate that the reliability analysis of power systems constitutes a complex issue that includes both the modeling of operational processes of power infrastructure and an assessment of the consequences of failures for the functioning of the power supply system. In particular, probabilistic state models based on the Kolmogorov–Chapman equations constitute an effective tool that enables analysis of transitions between the operational states of technical systems and determination of their reliability characteristics. 3. Methodology of a Reliability Analysis of the Selected Medium-Voltage Distribution Network A reliability analysis of technical infrastructure
