Simulation Model of a Unified Energy System for Different Scenarios of Planned Disturbances

Abstract

The study established that the application of graph theory enables the creation of a model of a country’s power system structure in the form of a tiered graph. This allows complex structural elements of the system, such as generating units, electrical substations, and power transmission lines, to be represented as nodes and edges in simulation models that can be used for analysis, dispatch control, and optimization of system operation. A simulation model of the unified power system has been developed to analyze operational efficiency and performance under various planned disturbance scenarios. To solve the given task, it is necessary to develop a model of the power system in the form of a tiered graph, where the nodes are generating equipment stations, transmission system substations with voltages from 330 kV to 750 kV, and distribution system substations with voltages from 110 kV to 220 kV, and the edges are power transmission lines with voltages from 110 kV to 750 kV. The model takes into account the generated and transmitted power, the nominal capacity and the number of transformers at the substations, the cross-section and maximum throughput of the power transmission lines, which made it possible to determine complex interconnections between its nodes and integrate the equipment into a unified power system for efficiency and performance analysis.

1. Introduction

In today’s high-tech world, reliable and uninterrupted power supply is becoming critically important across various areas of life [1,2]. Studying models and methods for analyzing the efficiency and operability of electric power systems is of utmost significance [3,4]. The management of fuel element cladding lifetime under variable loading conditions in nuclear reactors presents challenges that are particularly relevant in the context of energy system reliability. The complexity of ensuring a safe and prolonged operation of nuclear fuel cladding is underscored by studies on VVER-1000 reactors, which outline principles for controlling cladding lifetime in fluctuating operational environments [5]. Additionally, effective predictive models are crucial for managing the risk of cladding failure, especially in scenarios involving multiple cyclic reactor power changes, which can impact the overall reliability and safety of power systems [6]. This research underscores the need for robust simulation models that can accurately predict the operational efficiency and reliability of power systems under diverse disturbance scenarios. The substantial number of electricity consumers and the ever-growing demand for new technologies put serious pressure on the energy infrastructure [7]. This pressure necessitates continuous adaptation to new challenges, ensuring the highest levels of efficiency and operability.

The advancement of modern technologies in the energy sector demands constant refinement of analytical and predictive methods [8,9]. This enables prompt responses to changes within the system and maximizes resource management efficiency to meet consumer needs [10,11,12]. Such an approach helps ensure the stability of the power supply, reduces the likelihood of outages, and promotes efficient use of energy resources within environmental dimensions.

The development and implementation of advanced technologies and equipment, as well as socio-economic and political transformations aimed at achieving sustainable development, lead to gradual changes in the methods and means of managing energy systems. Moving away from centralized or vertically hierarchical models, energy systems are transitioning to decentralized or multi-level approaches. Under these conditions, the issue of effective management and ensuring reliability in the operation of energy systems at the distribution and transmission levels becomes particularly relevant [13]. The application of modern management methods and technologies allows for optimizing energy transmission systems, enhancing productivity, ensuring stability, and reducing the risk of emergency situations.

A literature review on the topic revealed a scientific–technical contradiction: current energy systems incorporate models and methods that regulate the amount of electricity generated during normal operating modes or planned disturbances [14,15,16,17]. The arguments in such models include the amount of electricity generated, electricity quality, transmission and distribution losses, and technological expenses [18,19,20,21]. However, current energy system models [22,23,24,25] and methods [26,27,28] do not account for scenarios in which, under consumer management conditions, the consumer orders only the necessary volume of electricity [22,29,30], or the unified energy system is subject to random disturbances [17,28,31,32]. Recent advancements in modeling disturbed power systems include the use of Markov Jump Systems to model and control inverter-fed weak grids [33] and distributed secondary control for AC microgrids under non-uniform delays [34]. While these approaches focus on specific challenges, such as control optimization and handling delays, the proposed graph-based model emphasizes structural analysis, providing insights into reliability, operability, and energy flow optimization under varying disturbance scenarios.

This contradiction lies in the discrepancy between the supply and consumption volume needed for critical infrastructure operation and the inability to adapt to various random disturbances. Resolving this contradiction is possible by developing a simulation model and methods for assessing the efficiency and operability of the unified energy system.

Thus, the purpose of this study is to develop a simulation model of a unified power system that can effectively analyze operational efficiency and performance under various planned disturbance scenarios. The goal is to understand the interconnections and dependencies between different system components, from generation units to distribution substations, and to optimize system responses during disturbances.

To achieve this goal, it is necessary to consistently solve the following tasks:

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develop a simulation model of the unified power system in the form of a layered graph.

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identify the nodes representing generation stations, transmission substations with voltages ranging from 330 kV to 750 kV, and distribution substations with voltages from 110 kV to 220 kV.

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analyze the model to assess operational efficiency and performance under planned disturbance scenarios.

2. Materials and Methods

Graph-based modeling is the process of creating mathematical models that represent graphs, where nodes (vertices) are objects and edges are connections between them [35,36]. Such models can be used to analyze and forecast various systems and processes where objects and their interconnections are significant. Examples of the application of graph-based modeling include modeling social networks, road networks, genetic networks, and many others [37,38,39,40].

Graph theory provides tools for analyzing graph models, such as measuring vertex centrality, finding the shortest path between two vertices, and detecting communities in a graph [39]. These tools can be used to predict system behavior and evaluate the effect of changing parameters on the system.

Various approaches can be used to create graph models, including topological, stochastic, probabilistic, and others. Graph models can be constructed as either static or dynamic, where connections between vertices change over time [41,42,43].

Graph-based modeling has many applications in the power industry. Graph theory is an important tool for analyzing and modeling power supply systems. Graphs help represent the complex network of power system components, such as generators, transformers, transmission lines, and consumers, in the form of nodes (vertices) and connections (edges) between them.

The main concepts of graph theory applied to power supply systems include the following:

• Network Representation. The power supply system can be represented as a complex network of various elements that interact to transmit electricity. Graph theory helps depict this network as a graph, where nodes represent different elements of the system, and edges represent the connections between them. The main elements of the power supply system that can be represented as graph vertices include:

  (a)

  Power plants. Graph vertices can represent various types of power plants, such as thermal, hydroelectric, wind, or solar power plants. Each vertex corresponds to a specific power plant and contains information about its electricity production.

  (b)

  Substations. Substations used for the transmission of electricity between different system elements can also be represented as graph vertices. These may include substations of different voltage levels that provide connections between power plants, transformers, and consumers.

  (c)

  Transformers. Graph vertices can represent transformers used to change the voltage level between different sections of the power supply system. Transformers are usually used to step down the voltage during electricity transmission from production to consumers and to step up the voltage during transmission from substations to remote sections.

  (d)

  Transmission lines. Graph edges represent transmission lines that connect different elements of the power supply system. These can include high-voltage transmission lines, cables, towers, and other means of electricity transmission.

2.

Connection Analysis. Connection analysis in power supply systems using graph theory involves identifying existing connections between different elements and determining their dependencies. For this purpose, breadth-first search (BFS) and depth-first search (DFS) algorithms are used, which help determine power transmission paths and the hierarchy of dependencies among system elements. Main tasks include:

(a)

Finding power transmission paths. Applying BFS or DFS to a power supply system graph allows for finding all possible electricity transmission paths from production sources (power plants) to consumers. The algorithms traverse graph vertices, exploring all possible paths from one vertex to another. This helps determine which system elements are connected to power plants and how electricity is transmitted through the system.

(b)

Determining dependencies between elements. BFS and DFS algorithms also help identify dependencies among different power supply system elements. For instance, when applying BFS from a particular vertex, all vertices that can be reached from that vertex will have a distance of 1 from it. Thus, it is possible to determine which elements are directly dependent on a specific element. DFS also helps reveal the depth of dependency among elements, as it explores the graph deeply.

Connection analysis helps understand how the elements of the power supply system interact and how changes in one element can affect others. This helps solve issues related to network optimization, identifying weak points, and developing management and maintenance strategies for the power supply system.

3.

Reliability Analysis. Reliability analysis of power supply systems using graph theory involves analyzing reliability and identifying critical elements that impact the continuity of electricity supply. Graph representation allows for modeling the disconnection or failure of individual system elements by removing graph vertices or edges. Key tasks include:

(a)

Modeling disconnections. Graph representation of the power supply system allows for the isolation of individual graph vertices or edges corresponding to elements that may fail or be disconnected. Removing a vertex means disconnecting the respective element, such as a power plant, substation, or transformer, from the system. Removing an edge reflects disconnecting a transmission line or link between elements.

(b)

Impact analysis of disconnections. After modeling disconnections, the impact on the power supply system can be analyzed. This may include determining elements dependent on disconnected vertices or edges and identifying critical power transmission paths that may be severed in the event of disconnections.

(c)

Determining reliability and critical elements. Graph theory helps determine the reliability of the power supply system and identify critical elements that have the greatest impact on the continuity of electricity supply. Critical elements may be vertices whose disconnection leads to an interruption in power supply to key consumers or edges that represent vulnerable transmission lines.

Reliability analysis of the power supply system using graph theory helps solve issues related to preventing disconnections, planning redundancy, and improving the system’s structure to ensure uninterrupted power supply.

4.

Network Optimization. Power supply network optimization using graph theory involves applying various algorithms to find optimal power transmission routes, minimize energy losses, and ensure efficient system operation. The main algorithms that can be used include shortest path search, maximum flow, and the traveling salesman problem. Main tasks include:

(a)

Finding shortest paths: Shortest path search algorithms, such as Dijkstra’s algorithm or the Bellman–Ford algorithm, help find the shortest routes for power transmission between different system elements. This helps optimize energy transmission paths, reduce energy losses, and ensure more efficient electricity distribution.

(b)

Maximum flow: Maximum flow algorithms, such as the Ford–Fulkerson algorithm or the Edmonds–Karp algorithm, help determine the maximum volume of electricity that can be transmitted through the power supply system. This allows for identifying overloaded sections of the network and finding an optimal operating mode for the system with maximum use of available capacity.

(c)

Traveling salesman problem: The traveling salesman problem involves finding the shortest path that passes through all elements of the power supply system and returns to the starting point. Applying algorithms that solve this problem helps optimize the sequence of traversing network elements, reducing time and energy consumption for movement.

Using these algorithms and graph theory methods ensures optimal functioning of the power supply system, reduces energy losses, improves reliability, and ensures efficient use of resources. Network optimization is a crucial step to increase the productivity and stability of the power supply system.

5.

Development Planning. Power supply system development planning using graph theory allows for analyzing the network structure and identifying opportunities to improve its efficiency and reliability. Key aspects that can be considered in the context of development planning include identifying weak points, and redundant elements, and assessing the impact of new technologies. Main tasks include:

(a)

Identifying weak points. Graph structure analysis helps identify weak points in the power supply system, such as areas with high energy losses, insufficient capacity, or limited throughput. Identifying these weak points helps plan network expansion or improve existing elements to enhance system efficiency and reliability.

(b)

Redundant elements. Graph analysis also helps identify redundant elements in the power supply system, such as alternative power transmission routes or backup power sources. Using redundant elements ensures reliability and reduces the risk of system failure in the event of problems in one of the elements.

(c)

Impact assessment of new technologies. Applying graph theory allows for assessing the impact of introducing new technologies into the power supply system. New technologies, such as renewable energy sources, energy-efficient solutions, or smart grids, may require changes in the network structure and operation. Graph analysis helps determine optimal locations for implementing new technologies and assess their impact on system performance and reliability.

Planning the development of the power supply system using graph theory allows for making informed decisions about network expansion, implementing new technologies, and improving system efficiency, thereby ensuring reliable power supply.

Graphs are an effective tool for visualizing the structure of power grids, providing a clear representation of the network topology, as well as the relationships between nodes and transmission lines. However, their application extends far beyond visual analysis. Leveraging the mathematical foundation of graphs enables the development of sophisticated models for analyzing and forecasting network performance.

An adjacency matrix is used to describe the connections between nodes in the network, such as substations or distribution points, indicating the presence of transmission lines between them. An incidence matrix further links nodes with transmission lines, reflecting the network’s topological features. These data structures, combined with the characteristics of nodes (e.g., load, generation, degree of connectivity) and lines (e.g., resistance, capacity, length), serve as the foundation for constructing stochastic models.

Stochastic modeling of power grids accounts for random changes and uncertainties, such as daily, seasonal, or random fluctuations due to disturbances, emergency line outages, changes in generation capacity, or the unpredictable behavior of renewable energy sources. This approach enables detailed analysis of system behavior under various scenarios and the development of measures to enhance its reliability and efficiency.

The simulation model of the power system can be represented as a layered graph. A layered graph (also known as a hierarchical or level graph) is a type of graph in which nodes are grouped into levels or layers based on their hierarchical or structural relationships. Each level represents a certain degree of importance or detail of the system components.

This graph structure allows for a visual representation of the system’s hierarchy and organization and reveals its complex interrelationships. A layered graph is often used to model hierarchical structures, such as organizational structures, information systems, technical networks, management systems, power supply systems, etc.

One of the main types of layered graphs is the top-down graph, where nodes are arranged in layers from top to bottom, from the upper layer to the lower layer. The opposite type is the bottom-up graph, where nodes are arranged in layers from bottom to top, from the lower layer to the upper one. Figure 1 shows an example of a layered graph.

Figure 1. Example of a layered graph.

The layered graph has its advantages and disadvantages that should be considered when applying it, namely:

• a layered graph allows for the visual representation of the hierarchical structure of a system or organization. This makes it easy to distinguish different levels of importance and dependencies between components.
• the graphical representation as a layered graph is highly intuitive and easy to comprehend. It helps to quickly navigate the structure of the system and the relationships between its components.
• a layered graph conveniently facilitates tracking and analyzing interrelationships between system components.
• the layered graph can be easily extended or modified by adding new nodes or levels, allowing for convenient consideration of the system’s development or changes in its structure.
• for very large systems with a significant number of components and interrelationships, a layered graph may become difficult to understand and analyze due to the large number of nodes and connections.
• a layered graph provides a general overview of the system’s structure but may be limited in accurately depicting details and relationships between components.
• the layered graph may be less flexible compared to other types of graphs since it is constrained by a specific hierarchical structure and dependencies between levels.
• a large number of components and connections in the system may lead to the loss of some detail when represented in a layered graph.

3. Research Results

For creating the simulation model of the power system, a layered graph $G=(V, E)$ was used, which included:

Set $V=\{v1, v2,\dots ,vn\}$—the set of vertices;

Set $E=\left\{e\mathrm{1,2},e\mathrm{2,3},\dots ,ei,j\right\}$—the set of edges or arcs of the graph.

At the top layer, “generation level,” are the vertices that represent the main power-generating stations or energy sources, such as thermal power plants, cogeneration plants, nuclear power plants, hydroelectric power plants, and pumped-storage power plants.

Table 1 provides detailed information about the vertices at the “generation level”. The properties of each vertex at this level include an identifier, name, generating capacity, and type of power plant.

Table 1. Set of vertices V at the top layer “generation level”.

Vertex Identifier	Vertex Name	Generating Capacity (MW)	Type of Power Plant
V1	Gen1	3000	Nuclear
V2	Gen2	6000	Nuclear
V3	Gen3	2000	Nuclear
V4	Gen4	2835	Nuclear
V5	Gen5	1800	Thermal
V6	Gen6	302	Pumped-storage
V7	Gen7	500	Thermal
V8	Gen8	700	Thermal
V9	Gen9	1825	Thermal
V10	Gen10	972	Pumped-storage
V11	Gen11	702	Hydro
V12	Gen12	482.5	Hydro
V13	Gen13	2351	Thermal
V14	Gen14	510	Thermal
V15	Gen15	636.2	Thermal
V16	Gen16	2079	Thermal
V17	Gen17	910	Thermal
V18	Gen18	1532	Thermal
V19	Gen19	2850	Thermal
V20	Gen20	2265	Thermal
V21	Gen21	3600	Thermal
V22	Gen22	1270	Thermal

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 750 kV transmission substations. Table 2 provides detailed information about the vertices at the “750 kV transmission substations” level. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 2. Set of vertices V at the top layer “750 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V30	PS_750_1	3	333	125,999
V31	PS_750_2	3	250	600
V32	PS_750_3	2	999	1598
V33	PS_750_4	2	1250	3600
V34	PS_750_5	2	999	1598
V35	PS_750_6	2	250	600
V36	PS_750_7	1	999	799
V37	PS_750_8	2	999	1500
V38	PS_750_9	3	999	2398
V39	PS_750_10	Foreign Substation	-	-

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 500 kV transmission substations. Table 3 provides detailed information about the vertices at the “500 kV transmission substations” level. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 3. Set of vertices V at the top layer “500 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V40	PS_500_1	2	200	320
V41	PS_500_2	2	200	400

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 400 kV transmission substations. Table 4 provides detailed information about the vertices at the “400 kV transmission substations” level. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 4. Set of vertices V at the top layer “400 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V42	PS_400_1	2	200	980

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 330 kV transmission substations. Table 5 provides a fragment of detailed information about the vertices at the “330 kV transmission substations” level. The complete list of vertices at this layer is provided in Appendix A. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 5. Set of vertices V at the top layer “330 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V43	PS_330_1	3	250	600
V44	PS_330_2	2	125	200
V45	PS_330_3	Transit	-	-
V46	PS_330_4	2	325	488
V47	PS_330_5	2	200	320
V48	PS_330_6	3	200	500
V49	PS_330_7	2	125	200
V50	PS_330_8	2	250	600
V51	PS_330_9	1	200	160
V52	PS_330_10	2	125	360
V53	PS_330_11	2	125	200
V54	PS_330_12	3	125	300

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 220 kV transmission substations. Table 6 provides a fragment of detailed information about the vertices at the “220 kV transmission substations” level. The complete list of vertices at this layer is provided in Appendix A. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 6. Set of vertices V at the top layer “220 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V132	PS_220_1	2	63	101
V133	PS_220_2	2	63	105
V134	PS_220_3	2	125	200
V135	PS_220_4	2	125	200
V136	PS_220_5	2	125	200
V137	PS_220_6	2	63	101
V138	PS_220_7	2	200	320
V139	PS_220_8	2	125	200
V140	PS_220_9	2	125	200
V141	PS_220_10	2	200	320

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 150 kV distribution substations. Table 7 provides a fragment of detailed information about the vertices at the “150 kV distribution substations” level. The complete list of vertices at this layer is provided in Appendix A. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 7. Set of vertices V at the top layer “150 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V167	PS_150_1	1	16	12
V168	PS_150_2	1	25	19
V169	PS_150_3	2	40	168
V170	PS_150_4	2	25	40
V171	PS_150_5	1	200	160
V172	PS_150_6	2	25	40
V173	PS_150_7	2	25	40
V174	PS_150_8	2	40	64
V175	PS_150_9	2	40	64
V176	PS_150_10	2	40	64

Source: created by the authors based on data [44,45].

The next layer contains the vertices representing 110 kV distribution substations. Table 8 provides a fragment of detailed information about the vertices at the “110 kV distribution substations” level. The complete list of vertices at this layer is provided in Appendix A. The properties of each vertex at this level include an identifier, name, number of transformers, nominal transformer capacity, and substation load.

Table 8. Set of vertices V at the top layer “110 kV transmission substations”.

Vertex Identifier	Vertex Name	Number of Transformers	Nominal Transformer Capacity (MW)	Substation Load (kW)
V200	PS_110_1	2	16	26
V201	PS_110_2	2	25	40
V202	PS_110_3	2	10	17
V203	PS_110_4	2	32	50
V204	PS_110_5	2	25	40
V205	PS_110_6	2	32	48
V206	PS_110_7	3	10	25
V207	PS_110_8	1	40	53
V208	PS_110_9	2	25	40
V209	PS_110_10	2	40	64

Source: created by the authors based on data [44,45].

After defining the graph’s vertices, it is necessary to define the connections or dependencies between them, i.e., determine the graph’s edges. The edges represent physical or functional connections between the components of the power supply system, which allows for revealing various aspects of the system’s functioning.

Physical connections between components include the transmission of electricity through cable or overhead power lines, which connect generating stations, substations, and consumers. Identifying the graph’s edges allows for the creation of a complete simulation model of the power supply system, in which each graph edge represents important connections between components.

Table 9 presents detailed information and properties of the set of edges EEE of the layered graph. The properties of each edge include an identifier, length, redundancy, maximum line losses, cost per kilometer, and line voltage.

Table 9. Set of edges E of the layered graph.

Edge Identifier	Length (km)	Cross-Section	Redundancy	Maximum Losses (kW)	Cost per km (UAH/km)	Line Voltage (kV)
e (36,37)	126	4 × AC-500/64	–	7410	4 × 270,000	750
e (36,37)	162	2 × AC-300/39	330 kV backup	164,023	2 × 220,000	330
e (3,36)	261	5 × AC-400/51	–	4339	5 × 230,000	750
e (33,40)	198	2 × AC-400	–	2975	4 × 230,000	500
e (13,42)	197	2 × ACO-500	–	34,954	4 × 270,000	400
e (5,43)	132	2 × AC-300/39	–	21,468	2 × 220,000	330

Source: created by the authors based on data [44,45].

Figure 2 shows a fragment of the layered graph of the country’s power supply system.

Figure 2. Fragment of the layered graph of the power system of the country (constructed by the authors). Legend: green lines: 150 kV power lines; blue lines: 220 kV power lines; magenta lines: 330 kV power lines.

The complete graph consists of various levels, with each level corresponding to a specific voltage class and hierarchy.

-

Level 0 represents power generation stations.

-

Levels 1–5 correspond to transmission system operator substations with voltage levels ranging from 220 kV to 750 kV.

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Levels 6–7 represent distribution system operator substations with voltage levels of 110–150 kV.

Each node in the graph represents either a power generation station or an individual substation of various voltage classes.

Overall, the simulation model in the form of a layered graph consists of 385 vertices and 626 connections (edges).

A circular layered graph is a type of graph that has specific properties. It consists of vertices and edges, where each vertex is connected to two neighboring vertices by an edge, and each pair of vertices is also connected by an edge. The feature of the circular layered graph is that it can be represented as a circle, where the vertices are positioned along the circumference, and the edges are like radii connecting the center of the circle to the vertices.

The main properties of the circular graph include:

• The circular layered graph consists of vertices and edges. The vertices are represented by points on the circle, and the edges are segments connecting these vertices.
• Each vertex in the graph is connected by an edge to two neighboring vertices. This creates a closed loop encompassing all the vertices of the graph.
• Since the vertices are positioned on a circle, geometric properties of the circle can be used for analyzing and studying the graph.
• Circular layered graphs can be used to model various situations, such as electronic circuits, where vertices can represent components and edges can represent connections between them.

The main advantages of the circular graph include:

• A simple and intuitive structure, which makes it easy to study and analyze.
• Due to the simple structure and closed loop of the circular layered graph, analysis of its properties, such as diameter, radius, and other metrics, can be performed quite easily.
• The ability to represent a circular layered graph as a circle allows for easy visualization of its structure and interconnections between vertices.

The main disadvantages of the circular graph include:

• The number of edges in a circular layered graph increases proportionally to the number of vertices, which can lead to increased data volume and processing complexity.
• In cases where the graph has a large number of vertices and edges, the circular layered graph may require significant memory to store its structure.
• In some cases, the structure of a circular layered graph may be too simple for adequately modeling complex systems or analyzing intricate interrelationships.

Figure 3 shows an example of a circular graph.

Figure 3. Example of a circular graph of a system (constructed by the authors).

Figure 4 shows a fragment of the circular layered graph of the power system simulation model.

Figure 4. Fragment of the circular layered graph of the power system simulation model (constructed by the authors). Legend: red lines (110 kV) represent lower voltage transmission lines; green lines (150 kV) represent medium voltage lines at the distribution level; blue lines (220 kV) are transmission lines operating at a higher voltage; magenta lines (330 kV) represent high voltage transmission lines.

Another way to represent a graph is in the form of an adjacency matrix AG. This method is used when working with a graph on a computer. The simulation model consists of 385 vertices, so the adjacency matrix AG will be of size 385 × 385, containing the set of elements AG = (aij). In this matrix, the indices i, j vary from one to n, and the value aij is one if there is an edge between the corresponding vertices Vi, Vj, or zero if no such edge exists.

$$a_{i,j}=\left[\left(v_i,v_j\right)\in E\right]=\left\{\begin{array}{c}1,\left(v_i,v_j\right)\in E\\ 0,\left(v_i,v_j\right)\notin E\end{array}\right..$$

(1)

To automate the construction of the adjacency matrix, a VBA program code was developed and embedded in Microsoft Excel (Algorithm 1). The input data for constructing the adjacency matrix is the database of edges, which is provided in Appendix A.

Algorithm 1 VBA code for constructing the adjacency matrix

Sub matrix_sm()         Dim v As Integer, e As Integer, i As Integer, j As Integer         Worksheets("v").Cells.Clear         v = Worksheets("e").Cells(2, 12)         MsgBox ("Number of graph vertices–" and v)         e = Worksheets("e").Cells(3, 12)         MsgBox ("Number of graph edges–" and e)         Worksheets("v").Cells(2, 2) = "V"         For i = 1 To v                 For j = 1 To v                         Worksheets("v").Cells(2, 2 + i) = i                         Worksheets("v").Cells(2 + i, 2) = i                         Worksheets("v").Cells(2 + j, 2 + i) = 0                 Next j         Next i         For i = 1 To e                 e1 = Worksheets("e").Cells(1 + i, 2)                 e2 = Worksheets("e").Cells(1 + i, 3)                 Worksheets("v").Cells(2 + e1, 2 + e2) = Worksheets("v").Cells(2 + e1, 2 + e2) + 1                 Worksheets("v").Cells(2 + e2, 2 + e1) = Worksheets("v").Cells(2 + e2, 2 + e1) + 1         Next i End Sub

The main disadvantages of the incidence matrix include the fact that for graphs with a large number of vertices and edges, the incidence matrix can require a significant amount of memory, especially in the case of sparse graphs where most edges are absent. The incidence matrix can also be challenging to read and edit, particularly for large graphs, as it contains many zero values, which complicates determining the connections between vertices and edges.

Figure 5 shows a fragment of the adjacency matrix of the simulation model.

Figure 5. Fragment of the incidence matrix AG = (aij) (constructed by the authors).

The incidence matrix BG connects the vertices and edges in graph G. Like the adjacency matrix, this method is used when working with graphs on a computer. Each element of the matrix BG = (bij) takes the value 1 if vertex vi is an endpoint of edge eij, and takes the value 0 if this condition is not met.

To automate the construction of the incidence matrix, a VBA program code was developed and embedded in Microsoft Excel (Algorithm 2). The input data for constructing the incidence matrix is the database of edges, provided in Appendix B.

Algorithm 2 VBA code for constructing the incidence matrix

Sub matrix_inc()         Dim v As Integer, e As Integer, i As Integer, j As Integer         Dim e1 As String, e2 As String         v = Worksheets("e").Cells(2, 12)         MsgBox ("Number of graph vertices–" and v)         e = Worksheets("e").Cells(3, 12)         MsgBox ("Number of graph edges–" and e)         Worksheets("inc").Cells(2, 2) = "V"         For i = 1 To v                 For j = 1 To e                         Worksheets("inc").Cells(2 + i, 2) = i                         e1 = Worksheets("e").Cells(1 + j, 2)                         e2 = Worksheets("e").Cells(1 + j, 3)                         Worksheets("inc").Cells(2, 2 + j) = e1 and "-" and e2                         Worksheets("inc").Cells(2 + i, 2 + j) = 0                 Next j         Next i         For i = 1 To e                 e1 = Worksheets("e").Cells(1 + i, 2)                 e2 = Worksheets("e").Cells(1 + i, 3)                 Worksheets("inc").Cells(2 + e1, 2 + i) = 1                 Worksheets("inc").Cells(2 + e2, 2 + i) = 1         Next i End Sub

The main disadvantage of the incidence matrix is that for graphs with a large number of vertices and edges, the incidence matrix can require a significant amount of memory, especially in the case of sparse graphs where most edges are absent. The incidence matrix can also be difficult to read and edit, particularly for large graphs, since it contains many zero values, which complicates identifying the connections between vertices and edges.

Figure 6 shows a fragment of the incidence matrix for the simulation model.

Figure 6. Fragment of the incidence matrix BG = (b_i,j) (constructed by the authors).

4. Discussion and Conclusions

Based on the conducted research, the scientific novelty of the study can be formulated as the proposed simulation model of the unified power system in the form of a layered graph. The nodes of this graph represent the generation equipment stations, transmission system substations with voltage levels from 330 kV to 750 kV, and distribution system substations with voltage levels from 110 kV to 220 kV. The edges represent power transmission lines with voltage levels from 110 kV to 750 kV. The model takes into account the generated and transmitted power, the nominal capacity, and the number of transformers at the substations, as well as the cross-section and maximum transmission capacity of the power lines. This approach to the results obtained allowed for the identification of complex interconnections between its nodes and the integration of equipment into a unified power system for analyzing operational efficiency and performance.

The proposed layered graph model differs significantly from recent graph-theoretic approaches in its scope and application. For instance, Biswas et al. (2021) [17] introduced a graph-theoretic method for identifying saturated cut-sets in meshed power networks during multiple outages. This approach focuses on real-time vulnerability assessment and situational awareness, optimizing solution time for critical contingency management. In contrast, our model emphasizes the structural analysis of hierarchical power systems, enabling the evaluation of energy transmission paths and reliability under a wide range of disturbance scenarios. While the method by [17] provides actionable insights for operational decision-making in real-time, our model offers a more holistic representation, capable of simulating energy flow disruptions across multiple voltage tiers and identifying optimal pathways for energy delivery. This distinction highlights our model’s utility in long-term planning and analysis of system-wide resilience.

Werho et al. (2016) [21] developed a network flow algorithm to monitor power system connectivity in real-time, which determines the maximum flow between two nodes in a directed graph. Their method identifies system vulnerabilities by analyzing the minimum number of branches required to disconnect specific nodes. This approach is particularly useful for operational decision-making during major disturbances, such as the 2008 island formation in the Entergy power system. In contrast, our model emphasizes long-term structural analysis of hierarchical power systems. While Werho et al.’s method offers critical insights for real-time network visualization and situational awareness, our model focuses on evaluating system-wide reliability and operability under a variety of scenarios, such as stochastic disturbances or planned outages. Moreover, the layered graph structure of our model enables multi-tiered analysis of voltage classes, which provides a broader scope for planning and optimization in large-scale energy systems.

In contrast to [25], our model focuses on a layered graph representation, which captures the hierarchical structure of power systems and provides a detailed analysis of energy transmission paths. While Zhang et al.’s approach excels in evaluating immediate vulnerabilities and resilience at the topological level, our model offers a broader scope for long-term planning, including stochastic disturbance scenarios and optimization of energy delivery across multiple voltage tiers. These complementary approaches highlight the diversity in applying graph theory to power system analysis, with our model particularly suited for planning and system-wide structural analysis.

In comparison to [28], the proposed layered graph model focuses on a multi-tier representation of energy systems, analyzing energy transmission paths and system operability under disturbances. While Beyza et al.’s method excels in evaluating immediate structural vulnerabilities and cascading failures, our model offers a broader scope, incorporating voltage tier hierarchies and enabling long-term planning for reliability and optimization. These methods complement each other, with our model designed for system-wide analysis and strategic decision-making, while Beyza et al.’s approach emphasizes rapid assessment in operational contexts.

The proposed layered graph model differs from existing methods in its focus on structural analysis of hierarchical power systems. Unlike the Markov Jump System approach [33], which models weak grids and addresses specific control challenges, our model provides a holistic representation of energy transmission paths, enabling the analysis of system-wide reliability and efficiency. Similarly, while the distributed secondary control method [34] is tailored to microgrids, the proposed model is designed for larger power systems, accommodating multiple voltage tiers and their interconnections.

The newest research [46] introduced a data-driven graph modeling approach for electric power transmission networks (EPTNs) using synchrophasor measurements. Their method constructs graph models based on real-time data from substations, bypassing the need for prior knowledge of network connectivity. The approach leverages an exhaustive search algorithm and evaluates the accuracy of Transmission Network Graph Models (TNGMs) under varying power plant operating conditions, achieving an average error margin within 2%.

In contrast, the proposed layered graph model focuses on the hierarchical representation of energy systems, analyzing structural reliability and energy transmission paths under disturbances. While Venayagamoorthy et al.‘s method excels in providing real-time situational awareness and connectivity analysis, our model offers a broader scope for long-term strategic planning and optimization. These methods complement each other, with the data-driven approach suited for operational decision-making and our model tailored for planning and system-wide analysis.

Unlike these methods, our model integrates a hierarchical representation of power systems, enabling detailed evaluation of energy transmission reliability across multiple voltage tiers. This long-term focus positions our model as a strategic planning tool, complementary to operational approaches like those by [46], which are optimized for real-time monitoring and network reconfiguration.