Modeling Nuclear-Centric Scenarios for Ukraine’s Low-Carbon Energy Transition Using Diffusion and Regression Techniques

Abstract

This study presents a mathematical model for forecasting the development of Ukraine’s Integrated Power System (IPS) until 2040, with a specific focus on the expansion of nuclear energy as a cornerstone of the nation’s low-carbon transition. The model is an extension of Frank Bass’s mixed influence diffusion model, incorporating both economic and technological factors. These factors are treated as stochastic variables essential for accurately predicting the evolution of an integrated energy system, particularly in the context of rapid renewable energy sources (RES) growth. The model employs regression techniques using generalized logistic curves, improving forecasting efficiency by aligning modeling parameters with experimental data. The study’s results indicate the potential for optimizing IPS components, including nuclear and thermal power generation, through the model’s application. The model is distinguished by its inclusion of economic and technological impacts, such as state matrices, control actions, and external influence matrices, which enhance the accuracy of simulations and predictions. The validation of the model, based on scenarios of electricity consumption and generation, shows significant alignment with observed trends, confirming the model’s reliability. The findings suggest that this model is an effective tool for developing and refining energy system scenarios, with nuclear energy playing a pivotal role in Ukraine’s sustainable energy future.

1. Introduction

The global shift towards sustainable development has underscored the need for energy systems that prioritize low-carbon emissions and energy security. While renewable energy sources (RES) such as wind, solar, and hydropower have been at the forefront of this transition, there is growing recognition of the complementary role that nuclear energy can play alongside RES. Studies have shown that, while RES are vital for reducing carbon emissions, their intermittent nature presents challenges for grid stability and continuous energy supply. Nuclear energy, with its ability to provide consistent baseload power, addresses these challenges by ensuring a reliable energy supply that compensates for fluctuations in renewable generation. This synergy allows for a more stable and resilient energy system, supporting both economic growth and environmental sustainability [1,2].

In recent years, several countries have explored and implemented hybrid nuclear–renewable energy systems to enhance their energy security and reduce dependence on fossil fuels. These integrated approaches leverage the strengths of both energy sources: the low-carbon footprint and consistent output of nuclear power and the flexibility and low emissions of renewables. For example, research has highlighted that hybrid systems can significantly lower the overall cost of energy production by optimizing resource use and reducing the need for extensive energy storage solutions. Moreover, nuclear energy’s role in providing thermal energy for industrial processes, such as hydrogen production, further expands the potential of these systems to contribute to decarbonization efforts across multiple sectors [3,4,5].

While the global focus remains on expanding the share of renewables, hybrid scenarios—like those proposed in this study that combine nuclear and thermal energy—deserve closer attention and study. These scenarios can play a crucial role, especially in countries where achieving a complete shift to renewables may face economic, technical, or geographic constraints. For instance, nuclear and natural gas are already being integrated in several regions to create a balanced energy mix that ensures stability and scalability. This approach allows countries to reduce their carbon footprints gradually without compromising energy security or economic viability. Therefore, further exploration of these hybrid systems, alongside more conventional renewable-focused strategies, can provide valuable insights and contribute to the broader goals of sustainable energy development [6,7,8].

The first application of the concept of “Sustainable Development” (SD) [9] dates back to the 1987 Brundtland Report published by the World Commission on Environment and Development [10]. At that time, the first materials were presented, which testified to the negative consequences of globalization associated with industrialization and uncontrolled population growth. Since then, the concept of SD has evolved significantly, encompassing 17 Sustainable Development Goals (SDGs) that are indirectly related to sustainable energy development. Among these goals, Goal 7, which emphasizes ensuring access to clean and affordable energy, is particularly crucial as it underpins the development of various sectors, including agriculture, business, communications, education, healthcare, and transportation [11,12].

The analysis of sustainable development policies, as measured by the World Energy Trilemma Index (WETI) 2020, typically revolves around three main concepts—energy security, energy justice, and environmental sustainability [9]. However, there is a growing recognition of the need for a more in-depth exploration of these concepts, especially in the context of energy transitions. The rapid development of renewable energy technologies has created new opportunities while also presenting challenges related to climate change, which is now widely acknowledged as a reality, despite ongoing debates regarding its causes and consequences [13]. As the global momentum shifts away from fossil fuels towards sustainable green energy, the importance of addressing these challenges becomes ever more pressing [13].

Within the European context, sustainable development has become a fundamental priority for energy policy. Over recent years, European energy policy has evolved to address new geo-economic, geopolitical, and geostrategic realities. The main priorities of this policy include ensuring energy supply under market conditions, enhancing energy efficiency and conservation, and minimizing the negative impact on the environment [14]. Consequently, there is increased attention on the interconnections between sustainable development, energy resource availability, and energy security, particularly within the framework of globalization [14].

Existing approaches to modeling energy systems, particularly those involving the integration of nuclear or renewable energy, have been extensively explored in the literature [15,16,17,18,19,20,21,22,23,24,25]. These models typically focus on the deployment of different technologies, addressing issues such as grid stability, energy storage, and the economic feasibility of large-scale RES integration. The complexity of these models often requires interdisciplinary methods that account for various technical, economic, and environmental factors. However, these models sometimes overlook the unique challenges and opportunities presented by specific national contexts, such as those faced by Ukraine.

Despite significant global efforts to promote RES, contradictions between human development and environmental preservation have become increasingly apparent [26]. These contradictions underscore the necessity of sustainable development as an essential goal for the future. RES, which are cleaner compared to fossil fuels, play a pivotal role in this transition. Their advantages include a reduced environmental impact, increased fuel diversity, a stable energy supply, and the promotion of regional economic development [27]. As a result, countries around the world have set ambitious targets and implemented relevant regulations and policies to foster the growth of RES [27].

At the same time, modern trends in the expansion of nuclear energy offer valuable lessons from other countries that have successfully integrated nuclear power into their low-carbon strategies [28]. These examples demonstrate how nuclear energy can serve as a reliable and low-carbon backbone of the energy system, providing a stable supply of electricity while supporting the transition to RES. The experiences of these countries highlight the importance of combining nuclear energy with RES to achieve a balanced and sustainable energy mix, particularly in nations with energy profiles similar to Ukraine.

However, the successful integration of renewable energy into energy systems is a complex challenge that requires interdisciplinary and transdisciplinary approaches [25]. The literature documents various unintended consequences of technical and policy interventions aimed at promoting renewable energy, including economic, environmental, and social impacts [25,28,29,30]. Addressing these challenges requires a comprehensive approach that spans behavior and organizational actions from local to global scales [28,30].

A significant gap in the existing research is the lack of comprehensive studies focused on modeling scenarios that incorporate both nuclear energy expansion and the strategic use of thermal power as a backup within the context of a low-carbon transition in Ukraine [28]. While numerous studies have explored various aspects of Ukraine’s energy systems, including the development of generating power capacities [31,32,33], the integration of RES [34,35,36,37,38], and the optimization of power grid operations [39,40], there remains a notable absence of a detailed analysis of nuclear-centric scenarios. This gap is particularly critical as Ukraine seeks to ensure a sustainable and resilient energy future, where nuclear energy could play a pivotal role [25,28].

In light of these complexities and the unique challenges facing Ukraine’s energy sector, this study aims to address these gaps by developing and analyzing nuclear-centric scenarios for Ukraine’s IPS until 2040. By doing so, the study will provide valuable insights into the strategic planning of Ukraine’s energy sector, emphasizing the role of nuclear energy as a cornerstone of the nation’s low-carbon transition.

2. Materials and Methods

A review of the above works allows us to identify some essential conditions that are recognized by the majority of researchers as follows:

• The fulfillment of both the economic development and the ecological balance requirements;
• Compliance with these requirements over an indefinite, or at least very long, period;
• A need for hierarchical sustainability management that reconciles divergent interests while meeting key requirements.

To formalize the concept of sustainable development and solve the problems of its management, it is advisable to use mathematical models. Based on the analysis of the concept of sustainable development of ecological–economic systems, the following general requirements for the sustainable development of a quasi-dynamic energy system can be identified: homeostasis, compromise, and dynamic coherence.

It can be argued that, individually, the following conditions are necessary and, taken together, sufficient conditions for the sustainable development of any dynamic system with the participation of people:

• The homeostatic condition expresses the basic requirements for all aspects of the system’s functioning;
• The condition of compromise ensures the adequacy of the influence of all associated entities in it while taking into account their interests in a compromise;
• The condition of dynamic consistency refers to the consistency of the short-term and long-term criteria of the optimality of actors and, thus, the disadvantage for them to deviate from the agreed compromise solution over time.

Energy systems are a separate case of major economic systems, forecasting the development of which is based on the following three main theoretical areas:

• The theory of economic balance, which is based on research [41] parameters of sustainable states, the reasons for their violation, and the recovery mechanisms;
• The theory of Walt Rostow’s [42] stages of economic growth, the subject of which is to determine the conditions of sustainable economic growth, the equilibrium of sustainable growth, and development;
• The theory of economic cycles or the theory of conjuncture [43] that explains the fluctuations in the economic activity of society over time.

All three of these theories do not contradict but, instead, complement each other.

The theory of cycles is a system theory that explores the regularities of the cyclic structure formation in the various processes types of scientific directions. In the systems development, they occur as cyclic processes. A period of several years or decades of growth is accompanied by a reduction phase. At the same time, despite fluctuations, the overall trend is characterized by growth. The theory of cycles, also called Kondratieff’s theory of long waves, is the basis for innovation and numerous innovative models of the new technologies’ diffusion and parameters of their life cycle.

Innovation Diffusion Fundamental Model

The efficiency of technology (e.g., the installed capacity of an energy-generating technology) or the return on capital invested in it [44] is uneven. The general rule is that efficiency gradually increases, peaking somewhere in the middle of the life cycle, and then decreases, so dynamics resemble a graph of normal distribution (Figure 1).

Lower returns at the beginning of the period are associated with the natural constraints inherent in any new resource and its initial development, and at the end, with the process of physical obsolescence and competition from new products and technologies.

The founder of the innovative theory of cycles, Kondratieff, investigated the general course of the dynamics of cumulative quantities—for capital and population and their connection with the development of technology. As he states [43], he managed to prove that the law of change of cumulative elements is expressed by a differential equation, which was subsequently reformulated in terms of the introduction of innovative technologies and is considered a fundamental model of innovative diffusion [45]:

$${\displaystyle \frac{dN\left(t\right)}{dt}}=g\left(t\right)\left(M-N\left(t\right)\right),$$

where N(t) is the cumulative number of users at time t who have adopted and are using the technology; M is the maximum possible number of potential users; g(t) is the diffusion coefficient (rate), which reflects the probability that potential users will accept innovations in a small period around time t. The value of g(t) depends on such characteristics of the diffusion process as the type of innovation, communication channels, time, and the characteristics of the social system.

Depending on the formula for the diffusion coefficient g(t), three models of innovative diffusion have been proposed [46]:

• The external influence model [47], where the diffusion coefficient g(t) is the coefficient or diffusion rate of the innovation p;
• The internal influence model [48], where the diffusion coefficient is g(t) = qN(t). The specific form of this model is represented by the well-known Gompertz function for use in predicting the development of new technology [49,50];
• The mixed influence model developed by Bass [51] combines both previous models. For the mixed influence model, the diffusion coefficient is g(t) = p + qN(t). Due to their great commonality and due to the consideration of internal and external influences, mixed influence models are most often used in research [46,52,53]. The mixed influence model can be expressed using the following equation:

$${\displaystyle \frac{dN\left(t\right)}{dt}}=\left(p+{\displaystyle \frac{q}{M}}N\left(t\right)\right)\left(M-N\left(t\right)\right).$$

(1)

In the special case where the coefficient or diffusion rate of an innovation p is zero, the Bass model is simplified to the following equation:

$${\displaystyle \frac{dN\left(t\right)}{dt}}={\displaystyle \frac{q}{M}}N\left(t\right)\left(M-N\left(t\right)\right).$$

This model is called the logistic model.

Both the Bass model and the logistic model give S-curves of the total number of users. By definition, the diffusion S-shape (Figure 1) initially increases at an increasing rate, with the cumulative number of users increasing over time. Over time, the curve reaches an inflection point, and the diffusion rate begins to decrease. Finally, diffusion reaches saturation level M.

All of these models assume a constant maximum possible number of potential users M. But in reality, M is not a constant. Therefore, we need models in which the number of potential users is a function of time M(t). For this, the diffusion coefficient g(t) in the fundamental model of innovation diffusion (1) changes as follows:

$$g\left(t\right)=K_{m}F\left[PPF\left(t\right),ET\left(t\right),FCF\left(t\right),EGR\left(t\right)\right]\left\{p+{\displaystyle \frac{q}{M\left(t\right)}}N\left(t\right)\right\},$$

(2)

where at time t:

$K_m$—scaling factor;

$F\left[PPF\left(t\right),ET\left(t\right),FCF\left(t\right),EGR\left(t\right)\right]$—functional, that depends on the following stochastic quasi-dynamical functions:

PPF(t)—the region purchasing power for which the scenario for the development of innovative technology is calculated;

ET(t)—the efficiency of the innovative technology under study;

FCF(t)—the average final cost to the consumer of a system using innovative technology;

EGR(t)—the annual economic growth rate.

The functional is calculated using the following formula:

$$F\left[PPF\left(t\right),ET\left(t\right),FCF\left(t\right),EGR\left(t\right)\right]=F\left(t\right)={\displaystyle \frac{PPF\left(t\right)/PPF\left(t_0\right)ET\left(t\right)EGR\left(t\right)}{FCF\left(t\right)/FCF\left(t_0\right)}},$$

(3)

where t₀ is the starting point, for example, the year 2000. Thus, F(t) is a dimensionless quantity.

The particles included in F(t) − PPF(t)/PPF(t₀) and FCF(t)/FCF(t₀) are denoted by:

KPPF(t) = PPF(t)/PPF(t₀)—the purchasing power ratio of the region;

KFCF(t) = FCF(t)/FCF(t₀)—the coefficient of the average final cost to the consumer.

The stochastic quasi-dynamical functions included in the functional F(t) will be called the coefficients of economic and technological influence and expressed as a percentage. In this case, the values are equal to 100% at the time t₀. Therefore,

$$F\left(t\right)={\displaystyle \frac{KPPF\left(t\right)ET\left(t\right)EGR\left(t\right)}{KFCF\left(t\right)}}.$$

(4)

Accounting for the above, the equation for the model of economic and technological influence takes the following form:

$${\displaystyle \frac{dN\left(t\right)}{dt}}=K_{m}F\left(t\right)\left(p+{\displaystyle \frac{q}{M\left(t\right)}}N\left(t\right)\right)\left(M\left(t\right)-N\left(t\right)\right).$$

(5)

In turn, the cumulative number of users or, for example, the total installed capacity of the technology, at time t + 1 is determined by the following equation:

$$N(t+1)=N\left(t\right)+K_{m}F\left(t\right)\left(p+{\displaystyle \frac{q}{M\left(t\right)}}N\left(t\right)\right)\left(M\left(t\right)-N\left(t\right)\right).$$

(6)

This model explicitly accounts for the influence of economic and technological factors, and as a model of mixed influence, reflects the development of innovative technology in the form of the logistics curve. To show a life cycle, which has two or more upswings, in the form of a generalized logistics curve that is the sum of several logistics curves, the model (7) proposed is

$$N^{\Sigma }(t+1)=N^{\Sigma }\left(t\right)+K_m{\sum }_{j=1}^{J}F(t,j)\left(p_j+{\displaystyle \frac{q_j}{M_j\left(t\right)}}{N}_j\left(t\right)\right)\left(M_j\left(t\right)-N_j\left(t\right)\right).$$

(7)

In the model (7), j—the number of the logistic curve, the total number of which is J.

In general, just as J can be different, the parameters pj, qj, and Mj(t) can be different for each individual j, but it is assumed that pj and qj are constant during the simulation period.

The equations presented in this section form the backbone of our scenario modeling for Ukraine’s IPS. Specifically, Equation (1) describes the diffusion of energy technologies, allowing us to model the adoption rates of different power generation sources, including nuclear and thermal energy. By adjusting parameters within the diffusion coefficient g(t)g(t)g(t), we can simulate various scenarios reflecting changes in policy, economic incentives, or technological advancements. Furthermore, Equation (5) integrates economic and technological factors as stochastic variables, which allows us to predict long-term energy trends and assess the stability of the energy system under different conditions. For instance, we use this equation to forecast how changes in economic growth (modeled through EGR(t)EGR(t)EGR(t)) impact the adoption of new nuclear technologies. Similarly, Equation (7) provides a way to evaluate the cumulative effects of different logistic growth patterns, enabling a detailed comparison of optimistic and pessimistic scenarios for energy expansion.

Using statistical data, the dynamics forecast of the listed economic and technological coefficients was calculated [54]. The regression calculation method for the forecast indicators is generally the same and differs only with the formulas that are used to extrapolate the statistical data number. The form and formulas of functions are selected. Then, using the least squares method, the parameters of the extrapolating function are determined.

In recent years, numerous studies have focused on modeling energy systems with integrated RES, utilizing a variety of approaches that consider both technological and economic factors. These models often address the stochastic nature of RES, which significantly impacts grid stability and the overall economic viability of energy systems. For instance, as detailed in [25], advanced regression models and diffusion techniques have been employed to predict the adoption and impact of RES technologies within complex energy systems. These approaches not only provide valuable insights into the potential growth of RES but also highlight the challenges associated with their integration, such as variability in energy supply and the need for flexible grid management strategies.

The modern development of energy technologies offers a wide range of choices between different energy sources. However, the efficiency and economic feasibility of their application depend on several factors, such as capital investment, service life, environmental performance, and the possibility of integration with existing systems.

Table 1. Key indicators of energy technologies considered in the model calculations.

Technology	Efficiency (%)	Investment Cost ($/kW)	Lifespan (Years)	Environmental Impact	Scalability
Nuclear Energy	90%	5000	60	Low CO2 emissions	High
Wind Power	35–45%	1500	25	Low emissions, land use	Moderate
Solar Power	20–25%	1000	20	Low emissions, high land use	High
Coal-fired TPP	35–40%	2000	45	High CO2 emissions	Limited
Gas-fired TPP	50–60%	1000	20	Medium CO2 emissions	Limited

presents the key characteristics of the main energy technologies considered in this study.

In Table 1, the parameters used for different energy technologies were derived from multiple authoritative sources and validated using statistical methods to ensure their accuracy for the Ukrainian context. The specific parameters are as follows:

Efficiency and Lifespan: The efficiency values and lifespan estimates for each technology (e.g., nuclear, solar, and wind) were gathered from international energy agencies and manufacturers’ reports and cross-referenced with operational data from comparable systems in Ukraine and neighboring countries.

Investment Costs: The investment cost values were obtained from recent studies and market data from national and international energy reports, reflecting the average cost of installation per kilowatt (kW) in 2024. For nuclear and renewable technologies, these figures were adjusted to account for projected inflation and local economic conditions.

Environmental Impact: This parameter is based on the lifecycle assessments (LCA) available in the literature, focusing on CO₂ emissions and other pollutants relative to energy production over the operational lifetime of each technology.

The choice of these parameters was guided by their relevance to the energy transition scenarios analyzed in this study. For example, the lifespan of PV panels in this study is assumed to be 20 years, reflecting the industry-standard warranty period. However, it is acknowledged that many PV panels can continue to operate efficiently beyond this period, with efficiency levels often remaining above 80% after 20 years, depending on the technology and maintenance conditions. While this assumption provides a conservative estimate for scenario planning, future iterations of the model could be adjusted to account for extended operational lifespans of PV panels. This adjustment would allow for a more accurate reflection of the long-term benefits and cost-efficiency of solar energy within the integrated power system.

In particular, the focus is on the optimal combination of renewable energy sources and energy storage technologies, which will allow for high efficiency and resilience of the energy system

It is clear, that in addition to the above, specific models should also account for other, no less significant indicators, which are influential factors both for the economy as a whole and for specific progressive and innovative technologies, in particular.

Long-Term Technological Renewal Model of the Energy System Structure

The long-term technological renewal model of the energy system structure [55] is intended for the study of long-term scenarios. The main difference of the proposed model is the explicit consideration of the influence of economic and technological indicators of development of the national economy and production presented in the form of quasi-dynamic functions with discrete stochastic variables. The actual and predicted values of these variables were used in the calculations at the corresponding step of the modeling horizon.

The model [55] aimed at studying the directions and optimal parameters of technological renewal of the components of the energy system in the long term.

The model of long-term technological renewal of the structure of consumption and generating capacity of the IPS of Ukraine [55] is presented as a hierarchy of scenarios. At the top level of the hierarchy, aggregated energy supply technologies are involved. The system state matrix $\mathsf{\Omega }\left(k,\tau \right)$ reflects the structure of energy generation, supply, and consumption volumes at the step τ, τ = 0, 1, 2, ..., T of the modeling horizon. All the technologies used k = 1, ..., K are involved in ensuring the balance of energy supplied $E_{S\tau }^{\sum }$ and consumed $E_{C\tau }^{\sum }={\sum }_{k=1}^K{E}_{k\tau }^{Consum}$ at each step τ. The main constraints of the model are to maintain a balance of total power $P_{\tau }^{\sum }$, maneuvering power $P_{m\tau }^{\sum }$, and volumes of generated $E_{G\tau }^{\sum }={\sum }_{k=1}^K{E}_{G\tau }^k$, supplied $E_{S\tau }^{\sum }$ and consumed $E_{C\tau }^{\sum }$ energy, provided that all parameters belong to a set of possible states. A measure ${\mu }_{\tau }$ of inconsistency between the vectors of supplied, and consumed energy has been introduced. The initial information for modeling is as follows:

• The target sequence of annual energy consumption $E_{C\tau }^{\sum }={\sum }_{k=1}^K{E}_{k\tau }^{Consum}$;
• The initial state of the total supply vector $E_{S0}^{\sum }$;
• The net benefit vector $N{B}_k=L_k\left(LAC{E}_k-LCO{E}_k\right)$—integral differences in parameters for each of the aggregated technologies involved in the calculation;
• The functional economic and technological influence

  $$F\left(k,\tau \right)={\displaystyle \frac{KPPF\left(\tau \right)ET\left(k,\tau \right)EGR\left(\tau \right)}{KFCF\left(k,\tau \right)}};$$
• The projected cost of technology components: $C_{k\tau }^{cap}$—investment, $C_{k\tau }^{FO}$—fixed, and $C_{k\tau }^{VO}$—variable operating costs.

A set of admissible states ${\Phi }_{k\tau }\left\{N{B}_k,F\left(k,\tau \right)\right\}$ for the trajectory development of the aggregated technology k is calculated. The problem of calculating such a supply vector $E_{S\tau }^{\sum }\left({\Phi }_{k\tau }\right)$ that minimizes the measure ${\mu }^{\sum }$ of the total inconsistency of the vectors of supplied and consumed energy is solved by

$${\mu }^{\sum }={\sum }_{\tau =1}^T\left.\left(E_{S\tau }^{\sum }-\right.E_{C\tau }^{\sum }\right)\forall u\left(\tau ,k\right),\xi \left(\tau ,k\right))\to min,$$

(8)

where, at each step τ = 0, 1, 2, ..., T, for each aggregated technology k: $E_{S\tau }^{\sum }$, $E_{C\tau }^{\sum }$—total volumes of energy supplied and consumed; $u\left(\tau ,k\right)$—vector of control actions; $\xi \left(\tau ,k\right)$—vector of random external perturbations. The optimal target trajectory of the supply vector $E_{S\tau }^{\sum \_OPT}\left({\Phi }_{k\tau }\right)$ is determined on the basis of forecasts of the economy’s general development and related consumption structure $E_{C\tau }^{\sum }={\sum }_{k=1}^K{E}_{C\tau }^k$.

The values of the total supply volumes obtained at the upper level and the values of the vector components $E_{S\tau }^{\sum }={\sum }_{k=1}^K{E}_{S\tau }^k$ for each of the aggregated technologies are the initial data for calculations: the required balance $P_{\tau }^{\sum }$ of total power, the required structure $P_{m\tau }^{\sum }$ of maneuvering capacities, the volume of total and component-by-component generation $E_{G\tau }^{\sum }={\sum }_{k=1}^K{E}_{G\tau }^k$, the volume of required investments $C_{k\tau }^{\sum cap}={\sum }_{k=1}^K{C}_{k\tau }^{cap}$, the volume of fixed operating costs $C_{k\tau }^{\sum co}={\sum }_{k=1}^K{C}_{k\tau }^{co}$, and the amount of variable operating costs $C_{k\tau }^{\sum vo}={\sum }_{k=1}^K{C}_{k\tau }^{vo}$.

At the next levels of modeling, similar calculations are performed within each of the aggregated technologies, i.e., the components of the scenarios for the development of the corresponding vectors of each of the aggregated technologies are calculated.

3. Results

With the help of the presented models, the forecast of the structure of electricity consumption and generating capacities of the IPS of Ukraine until 2040 was calculated. The forecast is calculated in three stages:

1.

First, with the help of the logistic model (7), the pessimistic and optimistic forecast of the consumption for the Ukrainian IPS up to 2040 (Figure 2) have been developed with the Formula (9):

$$E_C^{\sum }\left(\tau +1\right)=E_C^{\sum }\left(\tau \right)+K_m{\sum }_{j=1}^{J}F\left(\tau \right)\left\{p_j+{\displaystyle \frac{q_j}{E_{Cj}^{\sum }\left(\tau \right)}}{E}_{Cj}\left(\tau \right)\right\}\left(E_{Cj}^{\sum }\left(\tau \right)-E_{Cj}\left(\tau \right)\right).$$

(9)

2.

In the second stage, using the modified formula of net benefit $N{B}_k=K_m(LAC{E}_k-LCO{E}_k)$ [36] and the formula to minimize the aggregate cost of generation $C_{S\tau }^{\sum }={\sum }_{k=1}^{K}N{B}_k\cdot E_{S\tau }^k\to min$, the ratios of the required capacity and volume of the total annual generation of aggregated technologies are calculated;

3.

At the third stage, the problem of minimization of the inconsistency of the total volume of generation and consumption of electric power (8) was solved (Figure 3 and Figure 4).

To enhance the economic feasibility of the scenarios analyzed, we incorporated cost minimization models, which have been effectively utilized in previous studies to optimize energy system configurations. These models are particularly relevant for scenarios involving the expansion of nuclear energy, where cost efficiency is a critical factor. By applying these models, we were able to identify the most cost-effective strategies for energy generation and distribution, ensuring that the scenarios not only meet environmental goals but are also economically viable.

The pricing of CO₂ emissions plays a critical role in shaping the economics of energy production. As carbon pricing mechanisms are implemented or increased, energy production from coal and gas faces rising costs, while renewable energy sources gain competitive advantages. This shift has significant implications for both energy policy and the operational dynamics of energy systems. For instance, coal-fired power plants, traditionally a staple in many energy grids, are increasingly challenged by stricter CO₂ regulations, while natural gas serves as a transitional fuel, offering lower emissions but still contributing to the overall carbon footprint. Meanwhile, renewable energy sources such as wind and solar benefit from these policies but face their own set of challenges, including variability and storage needs.

To assess the long-term efficiency of different technologies, it is important to consider various scenarios for the development of the energy system. Depending on the chosen path, each scenario will have different economic and environmental consequences.

Table 2. Scenarios for energy system development.

Scenario	Total Capacity (GW)	Cost ($ Billion)	CO2 Emissions (Mt)	Share of RES	Nuclear Share (%)
Nuclear-centric Scenario	75	30	200	25%	50%
Renewable-heavy Scenario	90	35	150	60%	30%
Balanced Scenario	80	33	175	40%	40%
Business-as-usual Scenario	65	28	300	20%	30%

presents three main scenarios for the development of the energy system, taking into account the integration of renewable energy sources and storage technologies.

Each of the considered scenarios has its own advantages and challenges. The choice of the most optimal path will depend on the strategic goals of the country, the availability of funding, and the ability to adapt to new market conditions and climate changes. The presented analysis helps to determine the most rational approach to modernizing the energy system.

This comparison underscores the importance of accounting for economic variables such as CO₂ pricing when planning long-term energy strategies, particularly in the context of transitioning to a low-carbon energy system. The use of such optimization techniques allows for a comprehensive assessment of the trade-offs between different energy technologies and their long-term impacts on the overall energy system.

To further contextualize our findings, it is important to compare the results of our renewable energy scenarios with those from other studies. For instance, several models have been employed in recent years to forecast the development of RES in national energy systems. These models typically account for the stochastic nature of RES and the challenges of grid integration. When comparing our results with those obtained in [25], it becomes evident that while the general trend of increasing RES capacity is consistent, the specific growth rates and integration challenges vary significantly due to differences in model assumptions and regional factors. This comparison highlights the adaptability of the existing models to the unique circumstances of Ukraine’s energy system, suggesting that further refinement of these models could improve the accuracy of long-term energy forecasts in the Ukrainian context.

In order to check the adequacy of the model, test calculations of the supply structure of the IPS of Ukraine and their comparison with numerous scenarios of the structure of world energy supply from various researchers presented in [56,57] were carried out. The results of the comparative test calculation of the supply structure of the IPS of Ukraine and the world energy supply are presented in Figure 5. The red arrow highlights the scenario developed by the General Energy Institute of NAS of Ukraine (GEI), based on calculations performed within the framework of scientific research conducted at our institute. The green shadow represents the global energy demand forecast for 2050, serving as a benchmark for comparison.

A critical component of our analysis involves the role of nuclear energy in Ukraine’s energy future. According to recent studies, expanding nuclear power capacity is seen as a crucial strategy for ensuring energy security and reducing carbon emissions in line with international climate commitments. Our results indicate that incorporating nuclear energy into the IPS by 2040 could provide a stable and reliable energy source that complements the variability of RES.

The economic and technological implications of various energy diffusion models highlight the complexities of integrating new technologies into energy systems. Each model presents distinct economic challenges, with some requiring significant initial investments and others offering more moderate cost-saving potential. Technologically, the pace of adoption and improvement varies across models, impacting the speed and efficiency of energy system upgrades. For example, the diffusion model requires substantial upfront investments but leads to rapid adoption once the initial barriers are overcome. The logistic regression model provides more gradual improvements, offering cost savings over time with steady adoption rates. Meanwhile, the mixed influence model, though involving high upfront costs, delivers quicker technological advancements and a faster adoption trajectory. These economic and technological factors will determine the projected growth of each model from 2020 to 2040, shaping the future landscape of energy systems.

That is why it is important to consider the impact of each model on the overall project cost, its efficiency, and its ability to adapt in the future.

Table 3. Technological and economic characteristics of energy system development models.

Model Type	Economic Impact	Technological Impact	Adoption Rate (%)	Projected Growth (2020–2040)
Diffusion Model (Bass)	High investment needed	Slow start, rapid uptake	40–60%	Steady growth after 2030
Logistic Regression Model	Moderate cost savings	Gradual improvement	50–70%	Gradual growth throughout
Mixed Influence Model	Significant upfront cost	Moderate technological improvement	70–85%	Fast growth by 2025

provides the key economic and technological indicators for each of the models under consideration.

As seen from the presented data, economic factors have a significant influence on the choice of a particular model. Models based on renewable energy sources require substantial initial investments but offer considerable long-term advantages. These findings highlight the importance of strategic planning and the correct combination of technologies to ensure the resilience and sustainability of the energy system.

By comparing these scenarios with those from other countries that have successfully integrated nuclear power into their energy mix, we can observe that the inclusion of nuclear energy not only enhances grid stability but also reduces the overall cost of the energy transition. This supports the conclusion that nuclear energy could play a central role in Ukraine’s low-carbon transition.

4. Discussion

Based on the analysis of the concept of sustainable development of ecological and economic systems, mathematical models are proposed for formalizing and solving the control and development forecasting problems of the IPS of Ukraine.

A three-stage algorithm was proposed to achieve the following outcomes::

At the first stage, using a generalized logistics model, a pessimistic and optimistic consumption forecast for the IPS of Ukraine until 2040 was developed;

At the second stage, using the modified net benefit formula and the formula for minimizing the total cost of generation, the ratio of the required installed capacity and the volume of the total annual generation of aggregated technologies was calculated;

At the third stage, using the least squares method, the problem of minimizing the discrepancy between the projected volume of electricity consumption and the required total volume of generation was solved.

The mathematical models and equations presented in this study serve as the foundation for analyzing various energy scenarios for Ukraine’s IPS. By utilizing diffusion models, we were able to simulate the adoption rates of different power generation technologies, including nuclear and thermal energy. The integration of economic and technological factors as stochastic variables allowed us to account for uncertainties and predict long-term energy trends. These models enabled a detailed assessment of how different scenarios could unfold, providing insights into the stability, sustainability, and economic viability of nuclear-centric energy solutions.

A key distinction of the model developed in this study is its incorporation of economic and technological influences that are specific to Ukraine’s energy landscape. Unlike conventional models, which may generalize or overlook local nuances, our approach integrates state matrices, control actions, and external influence matrices. This allows the model to account for the unique factors that impact energy generation and consumption in Ukraine, such as government policies, economic conditions, and regional energy demands. By including these elements, the model enhances the accuracy of simulations and provides more reliable forecasts. This comprehensive approach enables the development of scenarios that not only explore the expansion of nuclear energy but also strategically utilize thermal power as a reserve, addressing the challenges of grid stability and energy security during the low-carbon transition.

While the proposed scenarios demonstrate a clear pathway towards achieving a low-carbon energy future for Ukraine, several economic and technological barriers could impede their implementation. As discussed above, the integration of RES often faces significant challenges, including high initial capital costs, the need for advanced grid infrastructure, and the unpredictability of RES supply. These factors can lead to increased operational costs and necessitate the development of sophisticated energy management systems. Moreover, the economic feasibility of expanding nuclear energy, a key component of the proposed scenarios, is contingent on securing substantial investments and overcoming public opposition, both of which could delay or even derail the implementation of these plans. Addressing these barriers will require concerted efforts from policymakers, industry stakeholders, and international partners to ensure the successful realization of Ukraine’s low-carbon transition.

To comprehensively understand the multifaceted challenges in Ukraine’s low-carbon energy transition,

Table 4. Summary of barriers to energy sources’ integration in Ukraine’s low-carbon transition.

Energy Source	Economic Barriers	Technological Barriers	Social and Political Barriers
Nuclear Energy	- High capital costs	- Infrastructure limitations	- Public acceptance and perception
	- Financing and investment risks	- Need for specialized technology	- Strict regulatory and policy frameworks
	- Operational and maintenance costs	- Technical expertise and workforce requirements	- International cooperation and compliance
	- Market fluctuations	- Safety and security concerns
Renewable Energy Sources	- High initial investment	- Grid integration challenges	- Environmental impact concerns
(Solar, Wind, Hydro, etc.)	- Financing and investment risks	- Energy storage solutions needed	- Public acceptance and perception
	- Market fluctuations	- Technology maturity and availability	- Evolving regulatory and policy frameworks
		- Infrastructure limitations
Thermal Energy	- Operational and maintenance costs	- Infrastructure limitations	- Public opposition due to environmental concerns
(Coal, Gas)	- Market fluctuations	- Technology obsolescence	- Stringent emissions regulations
	- High fuel costs (coal and gas)	- High emissions requiring mitigation technology
	- Cost competitiveness compared to nuclear and RES

summarizes the intricate relationships between different energy sources and the associated economic, technological, social, and political barriers.

As depicted in Table 4, addressing these interconnected barriers requires a holistic and strategic approach, involving coordinated efforts across multiple sectors and stakeholders to ensure a successful and sustainable energy transition.

To further contextualize the findings, it is useful to compare Ukraine’s nuclear energy expansion plans with similar efforts in other countries. As highlighted earlier, several nations have successfully integrated nuclear power into their energy systems as part of a broader strategy to reduce carbon emissions [58,59]. For example, countries like France and Finland have relied heavily on nuclear energy to maintain grid stability while reducing their dependence on fossil fuels. These examples illustrate the potential benefits of nuclear energy in achieving energy security and environmental sustainability. However, they also underscore the importance of robust regulatory frameworks, public engagement, and long-term financial planning. By learning from these global experiences, Ukraine can better navigate the challenges associated with nuclear energy expansion and align its energy strategy with international best practices.

At the same time, the joint development of nuclear and thermal energy systems has proven to be an effective strategy for expanding energy capacity and meeting long-term carbon reduction goals. In several countries, like India, collaborative ventures have been established to pool resources, such as financial investment, technological expertise, and project management capabilities [60,61]. These partnerships often leverage the strengths of nuclear power as a reliable, low-carbon energy source while using thermal energy to support grid stability and manage energy demand fluctuations. Such collaborations enable accelerated nuclear capacity growth, integrating nuclear energy into a broader energy mix that includes thermal power for flexible energy production. This approach not only ensures energy security but also facilitates the transition to a low-carbon economy by capitalizing on the complementary roles of nuclear and thermal energy systems. The global experience demonstrates that these joint efforts can pave the way for sustainable energy development, ensuring balanced energy portfolios that support both environmental and economic objectives in the long term. These initiatives contribute to the ambitious global targets of achieving net-zero emissions, while accommodating the complexities of national energy systems.

While this study primarily focuses on nuclear-centric scenarios, it is crucial to acknowledge the significant roles that natural gas and hydrogen will play in Ukraine’s energy transition. Both energy sources provide essential transitional solutions that complement renewable energy integration. Natural gas, with its lower CO₂ emissions compared to coal, serves as a flexible and lower-emission bridge fuel, especially useful in scenarios where renewable deployment is not immediate. Meanwhile, hydrogen, as a versatile energy carrier, holds potential for long-term decarbonization across multiple sectors, particularly in industry and transport.

As highlighted in recent studies [62,63], natural gas helps maintain grid stability by compensating for renewable energy fluctuations, while hydrogen offers promising storage solutions and resilience in IES. Its integration supports both the decarbonization of power generation and the energy-intensive sectors that are difficult to electrify. Therefore, future energy scenarios in Ukraine should incorporate these complementary roles to ensure a balanced and sustainable energy transition.

The obtained results allow us to conclude that comparison of the calculated supply structure of the IPS of Ukraine with numerous scenarios for the structure of global energy supply from different researchers allow us to conclude that the compared scenarios are sufficiently similar, which confirms the adequacy of the proposed model and the possibility of using it to develop scenarios for the development and renewal of energy systems.

5. Conclusions

The mathematical model developed in this study provides a comprehensive framework for forecasting the evolution of Ukraine’s Integrated Power System through 2040, particularly under nuclear-centric scenarios aimed at facilitating a low-carbon transition. By incorporating both economic and technological factors as stochastic variables, the model enhances the accuracy of long-term energy system predictions, addressing the unique challenges posed by rapid renewable energy sources (RES) expansion.

Key findings from the application of the model include the identification of optimal configurations for nuclear and thermal power generation, highlighting their critical roles in maintaining grid stability while reducing overall carbon emissions. The integration of generalized logistic regression models further improved the forecasting process, ensuring that the modeling parameters align closely with empirical data.

While the model presents a viable pathway for Ukraine’s energy transition, it also underscores the significant economic and technological barriers that must be addressed. The need for substantial capital investments, advanced infrastructure, and robust regulatory frameworks is paramount, particularly in light of the stochastic nature of RES and the high costs associated with nuclear energy development. These challenges suggest that a cautious and well-planned approach is necessary to achieve the desired outcomes within the proposed timeline.

Moreover, international comparisons with countries like France, Finland or India, which have successfully integrated nuclear power into their energy systems, offer valuable lessons for Ukraine. These examples demonstrate that a balanced energy mix combining nuclear energy with RES is essential to achieving both energy security and sustainability. However, the success of such a strategy in Ukraine will depend on the country’s ability to navigate its unique economic and political landscape as well as to implement effective risk management strategies.

In conclusion, the study provides a robust methodological foundation for developing and refining energy system scenarios that prioritize low-carbon energy sources. The model’s flexibility allows for continuous adaptation to emerging technological trends and economic conditions, making it a vital tool for policymakers and energy planners. Moving forward, the focus should be on addressing the identified barriers and leveraging international best practices to ensure a successful and sustainable energy transition for Ukraine.