Mathematical Model for Assessing New, Non-Fossil Fuel Technological Products (Li-Ion Batteries and Electric Vehicle)

Abstract

Since private cars and vans accounted for more than 25% of global oil consumption and about 10% of energy-related CO₂ emissions in 2022, increasing the share of electric vehicle (EV) ownership is considered an important solution for reducing CO₂ emissions. At the same time, reducing emissions entails certain economic losses for those countries whose exports are largely covered by the oil trade. The explosive growth of the EV segment over the past 15 years has given rise to overly optimistic forecasts for global EV penetration by 2050. One of the major obstacles to such a development scenario is the limited availability of resources, especially critical materials. This paper proposes a mathematical model to predict the global EV fleet based on the limited availability of critical materials such as lithium, one of the key elements for battery production. The proposed model has three distinctive features. First, it shows that the classical logistic function, due to the specificity of its structure, cannot correctly describe market saturation in the case of using resources with limited serves. Second, even the use of a special multiplier that describes the market saturation process taking into account the depletion (finiteness) of the used resource does not obtain satisfactory economic results because of the “high speed” depletion of this resource. Third, the analytical solution of the final model indicates the point in time at which changes in saturation rate occur. The latter situation allows us to determine the tracking of market saturation, which is more similar to the process that is actually occurring. We believe that this model can also be validated to estimate the production of wind turbines that use rare earth elements such as neodymium and dysprosium (for the production of powerful and permanent magnets for wind turbines). These results also suggest the need for oil-exporting countries to technologically diversify their economies to minimize losses in the transition to a low-carbon economy.

1. Introduction

Looking back at the industrial development of the 20th century, it is obvious that the energy resources that drove it were hydrocarbon raw materials—coal, oil and gas—and that this trend is still dominant in the 21st century. According to the latest International Energy Agency (IEA) data and estimates [1], oil and gas operations amounted to 5.1 billion tons (Gt) CO₂-eq in 2022. Global energy-related GHG emissions in 2022 were about 40 Gt CO₂-eq, meaning that the oil and gas industry is directly responsible for nearly 15% of energy GHG emissions [1]. Considering that global primary energy consumption equaled 1092 Mtoe in 1900, and 9242 Mtoe in 2000 (that is, it increased almost 9 times), by the end of the 20th century, oil accounted for 38%, coal for 23%, and gas for 22% of the world’s energy consumption [2]. Between 1960 and 2000, global daily oil production more than tripled, increasing from 21 million barrels to 65.7 million barrels [3]. In 2023, global average daily oil production reached 96.4 million barrels, of which 30.4% came from the Persian Gulf countries, Iran, and Iraq, 27.2% from North America (USA, Canada, and Mexico), 13.8% from the former USSR (Russia, Azerbaijan, and Kazakhstan), 7.2% from Africa, and 7.0% from Latin America (Brazil and Venezuela). China and Europe accounted for 4.3% and 3.3% of global oil production, respectively [4]. According to OPEC, approximately 95% of this production was provided by 42 countries with a high level of per capita GDP (OECD member countries and the Persian Gulf countries), accounting for 33.7 million barrels, giving 35% of the total average daily production [3].

An important feature of the development of the global oil sector over the past 65 years has been its significant geographic diversification. While in 1960, seven countries (the USSR, Saudi Arabia, the USA, Iran, Iraq, Kuwait and Venezuela) accounted for 80.8% of the world’s daily oil production, by 2020, the country map of world oil production had changed significantly. In 2020, the share of the above countries decreased to 48%, while a group of 12 other countries began oil production on an industrial scale (Angola, Algeria, Libya, Nigeria, UAE, Brazil, China, Kazakhstan, Mexico, Canada, Norway and the United Kingdom), and this accounted for about a third of the world’s daily oil production [3]. For developed countries, oil production and exports are not critical for raising national income, but for several developing economies, oil is the main export commodity, with percentages varying from 43% (Russia) and 59% (Kazakhstan) to 98% (Venezuela), with oil accounting for 9% of global merchandise exports [5]. It is clear that countries that are highly dependent on oil exports may be the biggest losers in the transition to low-carbon technologies and the broader decarbonization of their economy. The potential economic losses for these countries could be significant if the automotive industry, one of the key sectors of global industry, undergoes a major transition to electricity use based primarily on the use of lithium-ion batteries over the next 25 years.

The postwar global economic recovery period of the mid-20th century triggered long-term economic growth in Europe, North America and Southeast Asia. Its distinctive feature was the industry’s leading role in saturating the market with new products. The favorable conditions of the oil market, with its low prices, were a critical factor in the rapid growth of the global automotive industry. Over twenty years, from 1950 to 1970, the annual production of passenger cars tripled from 8 to 25 million, and by 2007, the annual production further doubled, reaching more than 50 million cars [6]. Passenger car production peaked in 2017 (73.4 million); then, it began to decline, and in 2023, global passenger car production dropped to 67.1 million units. However, the highest ever production volume for passenger cars and commercial vehicles was reached in the same year, totaling 93.5 million vehicles [7]. The second half of the 20th century in the global automotive industry was characterized by US automakers, which accounted for 80% of world production in 1950, and German and Japanese automakers, which accounted for 12.7% and 27.4% of world production in 1990. The situation changed radically in the 21st century, with China’s rapid industrial development making it the world’s automotive industry leader in terms of production volume in 2023, with a share of 33.5% of global production, just below the combined share of North American and European production [8]. Today, the global automotive industry is an example of the combination of high technology and deep cooperation, with a complex supply chain. The global automotive industry spends approximately USD 100 billion annually on R&D, equivalent to 3% of total output. Automobile companies account for a quarter of the R&D expenditure of the world’s 20 largest corporations, and more than 10% of the total number of engineers, technicians, and researchers in the workforce [9].

Due to the rapid pace of motorization, the global car fleet in 2023 reached 1.47 billion units [10], while in 2022, global CO₂ emissions from the transportation sector increased by more than 250 million tons to almost 8 billion tons of CO₂, with petroleum products accounting for almost 91% of the sector’s final energy consumption [11]. According to the same report, about 1 billion tons of CO₂ emissions came from sport utility vehicles (SUVs).

The UN’s Paris Climate Agreement aims to limit global warming to within 1.5 to 2 °C. To achieve this goal, energy-related greenhouse gas emissions must be reduced threefold from 2019 levels of 33.3 billion tons of CO₂. This will be possible if by 2040–2050 there is an energy transition from fossil hydrocarbon fuels (coal, oil, and natural gas), which are currently the mainstream, to renewable energy sources, and if the share of renewable energy in the overall energy balance reaches at least 40% [12].

The global trend toward decarbonization in many areas of industrial activity has triggered the development of electric vehicles, and EV production is a new trend in the global automotive industry in the 21st century. Based on IEA data on the dynamics of EV sales in 2012, global sales were only 100 thousand units (all in the US); then, five years later, sales volumes amounted to 1.2 million cars, half of which were sold in China, and in 2022, sales were 10.3 million units, more than half of which (6 million cars) were sold in China, 2.7 million in Europe and 1 million in the United States [13]. The growth rate has been impressive, with sales volume increasing approximately tenfold every five years. With these high sales rates, the total number of EVs approached 40 million in 2023, but still represented only 2.7% of the world’s total vehicle fleet, and by 2024, global EV sales are expected to reach 16.6 million units [13].

All new technologies developed in the past twenty–thirty years (personal computers, smartphones, electric cars, solar panels, wind turbines, data centers, and now the much-discussed artificial intelligence technologies) depend on the large-scale exploitation of certain natural resources that are considered “critical materials”. This concept has gained wide acceptance in recent years, and has been the subject of many studies on the availability of materials, such as Graedel et al. [14], Nassar et al. [15], OECD [16], Pitron [17], Gielen [18], Gielen and Papa [19], CSS-UMichigan [20], IEA [21], Devezas et al. [22], etc.

The rapid global proliferation of EVs would not be possible without the evolution of battery technology, and the advent of lithium-ion batteries in the 21st century has brought about significant changes in battery technology for EVs. Of the four main cathode materials—lithium, cobalt, manganese, and nickel—lithium and cobalt have been identified as the primary concerns [23].

While technology, investment, and market mechanisms can be influenced to some extent by economic actors and institutions, the presence or absence of certain critical materials is determined entirely by the nature of geological processes. As is well known from economic history, the role and importance of certain materials increases sharply at certain times. Lithium currently plays such an important role in the development of EVs. Since lithium-ion battery production technology is currently dominant, it is the production and reserves of lithium that will serve as a potential limiter on EV production.

Taking into account the above, the main objective of the study is to assess the potential global EV fleet given the limited global reserves of lithium. To achieve this goal, it is necessary to (1) analyze the dynamics of EV production over the past 20 years and the main drivers of its high growth rate, including economic incentive measures, both at the industry and the national level; (2) to develop a mathematical model that can estimate the trajectory of global EV production considering the limited amount of global lithium reserves; (3) to compare the results obtained in the course of model calculations with existing scenarios of EV industry development; and (4) to evaluate the results obtained in the course of model calculations for consistency or inconsistency with the emerging trend of slowing energy transition in the broader context.

The obtained results are expected to be helpful in discussing both the duration and unevenness of the energy transition, as well as factors such as the characteristics of national economies, geopolitical tensions, and the growing fragmentation of the global market [24,25,26,27,28]. Based on the above issues, the following research question is formulated:

RQ. 

How could the scarcity and availability of critical source material influence and hinder the dynamics of the production of EVs in the future?

The present paper presents the current state of the EV industry and the available minerals needed for the EV market, followed by the development, presentation, and proposal of a mathematical model showing how the limited minerals have a significant impact on the expected growth rate and trends of EV penetration, and indirectly on CO₂ emissions and sustainability. After the presentation of the results and findings, a discussion is provided, and the paper closes with a conclusion.

2. Current State of the EV Industry

EVs were very common at the beginning of the 20th century, but lost out to competition from classic cars powered by internal combustion engines, ending their almost 100-year history. In 2005, the global EV fleet numbered less than 1.5 thousand vehicles, but by 2010 it exceeded 20 thousand [29]. Until 2015, the United States was the EV fleet world leader, with over 400 thousand EVs on its roads, but since 2016, China has taken over the lead, with an EV fleet approaching 650 thousand, while the global EV fleet exceeded 2 million that year [30]. Thus, the global EV fleet increased more than 1300-fold in 10 years. This exponential growth is the result of several factors.

2.1. Investments in the New Production Segment

First of all, we might consider the significant investment in the new production sector by both the private sector and the state, which amounted to USD 13.1 billion during the period 2008–2012, of which 8.7 billion was spent on R&D, 3.4 billion on financial incentives for producers and consumers, and 1 billion on infrastructure development [29]. The volume of investment has since increased, and in 2018–2023, about EUR 170 billion were invested in the production of batteries for EVs alone (

Table 1. Global Investment in the Battery Industry by Investors Type, in Billions of Euros (2018–2023).

Investor Type	2018	2019	2020	2021	2022	2023	Total
Vehicle Manufacture	5.95	9.66	8.73	12.50	19.00	14.70	70.54
Private Equity	15.20	2.50	5.20	7.70	11.20	6.90	48.70
Venture Capital	1.91	1.75	2.88	8.32	6.47	5.00	26.33
Others (Public investments, Pension Funds, etc.)	2.50	3.30	3.50	4.40	4.80	5.50	24.00
Total	25.56	17.21	20.31	32.92	41.47	32.10	169.57

Source: [31].

).

Significant investments have led to a significant drop in battery prices, falling more than fivefold from USD 780/kWh to USD 139/kWh between 2013 and 2023, while simultaneously increasing the range of the battery without charging to 380 km, thus greatly expanding the potential mass market [32]. Some experts consider the USD 100/kWh price mark to be the “tipping point” at which battery EVs (BEVs) become price–equivalent to internal combustion engine (ICE) vehicles [33]; as of April 2019, EU automakers have invested EUR 145.8 billion in EV and battery production [34]. Global investments announced for 2022–2023 amount to USD 275 billion for EVs and USD 195 billion for batteries, of which about USD 190 billion has already been invested in various projects [13]. In 2022, the world’s 36 largest automakers announced their plans to invest about USD 1.2 trillion (roughly the same amount) in EV and battery production by 2030, with the ultimate goal of increasing annual EV production to 50 million vehicles (including 20 million by Tesla) and total battery production capacity to 5.819 Twh [35].

However, as the example of the implementation of investment projects in the EV and battery production sector in the United States shows, about 50% of the announced intentions have been allocated [36]. Assuming that this trend is typical for projects at the global level, the amount of investment would still be important for the development of the industry.

2.2. The System of Economic Incentives

Another important factor in the development of the industry is the different types of economic incentives schemes adopted in many countries to support producers and consumers. In the early stages, these included direct subsidies for EV purchases (up to CNY 60 thousand, or up to USD 10 thousand in China) and exemptions from purchase and excise taxes (from USD 5000 to 8500), with the Government’s intention to reduce EV subsidies by 20% from 2017. In Japan, the subsidy covers 50% of the price difference between the purchased EV and a gasoline-powered vehicle (but not more than CNY 1 million). In the USA there is a tax credit of up to USD 7.5 thousand for purchased EVs, depending on the battery capacity [29,30,37]. A recently published study on the impact of end-user subsidies on EVs in 13 countries that account for 95% of global EV sales between 2013 and 2020 found that EVs benefit from subsidies up to 70–80%, with this figure being higher for tax incentives than direct consumer subsidies [38].

European countries have introduced a variety of incentives to encourage the widespread adoption of EVs. These incentives, which vary from country to country, aim to make EVs more affordable, accessible, and environmentally friendly [39].

Norway, a pioneer in EV adoption, offers substantial tax breaks, including exemptions from VAT (25%) and purchase taxes [39]. Additionally, EV owners benefit from reduced fees for ferries, toll roads, and public parking (50%). Germany provides generous subsidies for EV purchases, especially for lower-priced models (EUR 9 thousand for a purchase price up to EUR 40 thousand and EUR 7.5 thousand for a purchase price up to EUR 65 thousand). These subsidies can be further increased when older vehicles are scrapped. France offers similar subsidies and tax benefits, with additional incentives for commercial EV purchases (e.g., 50% discount or complete exemption from registration tax [39]).

Spain and Italy also contribute to the EV push with various incentives. Spain provides significant subsidies for EV purchases, particularly for lower-priced models, and offers tax breaks for both private and commercial EV owners. Italy, on the other hand, focuses on tax benefits and support for charging infrastructure. Private individuals and companies can deduct a portion of the costs for installing charging points from their taxes [39].

By offering these incentives, these European countries are actively promoting the transition to electric mobility, reducing carbon emissions, and improving air quality. These measures, coupled with advances in EV technology and increasing charging infrastructure, are driving the growth of the EV market in Europe [39].

As several studies have shown, various forms of subsidies also play an important role in increasing production volumes. For example, rebates ranging from USD 1000 to 3000 contributed to a 12–17% increase in sales [40], and USD 1000 tax incentives for hybrid EVs resulted in a 3–5% increase in sales [41].

2.3. National Regulations

Once fiscal incentives allowed automakers to scale up EVs and battery production, the next phase of the new product expansion was national regulations aimed at supporting the electrification of road transport; the European Union’s Net Zero Industry Act and “Fit for 55” package, the U.S. Inflation Reduction Act, India’s Faster Adoption and Manufacture of Hybrid and Electric Vehicles Scheme, and China’s New Energy Vehicle policies have been adopted. At the same time, stricter fuel economy and CO₂ emission regulations played a major role in promoting EV sales. To be fair, some of the Inflation Control Act regulations were intended to limit the supply of Chinese EVs and batteries to the US market, limiting tax credits if the battery-powered vehicles contain components manufactured or assembled by foreign companies (including China) of concern. Governments around the world have also been supporting the development of EV charging infrastructure through measures such as direct investment in the installation of public chargers and incentives for EV owners to install charging stations at their homes. For example, the Chinese government announced a USD 1.4 trillion public spending program for digital infrastructure, which includes funding for EV charging stations. The U.S. Charging Infrastructure Development Plan provides government grants and incentives to build 500,000 chargers, and the Canadian government has approved a USD 112 million zero-emission vehicle (iZEV) infrastructure program [37]. Government support for manufacturing companies has become another area of national policy. For example, in 2023, the Canadian government announced a 10-year USD 43.6 billion government support program for battery production by Northvolt, Volkswagen, and Stellantis-LGES [42]. As part of the Inflation Reduction Act, the U.S. Department of Energy announced USD 52 billion worth of loans for the period 2022–2031 to implement 17 new projects for EV battery production [43].

As the published data show, significant government support for transportation electrification programs is concentrated in two segments: battery production and charging infrastructure. However, global venture capital investment in clean energy startups, including EVs and batteries, declined significantly in 2023 compared to 2022. According to the IEA, early-stage investment in EV and battery technology startups fell by 20% to USD 1.4 billion, while growth-stage investment was down 35% to USD 10.1 billion [13]. There are several reasons for this, but geopolitical tensions, supply chain disruptions, high energy prices, and rising inflation and interest rates are limiting the availability of higher-risk capital [13].

2.4. Market Competition as Key Factor

The EV production volume in 2023 was 14.4 million units, indicating that the industry has reached a certain level of maturity, and market competition is becoming one of the key factors in pricing and the market entry of new manufacturers. Growing competition has led to a rapid drop in EV prices. The prices of compact EVs and SUVs fell by up to 10% in 2023 compared to 2022. In the first quarter of 2024, Tesla reduced prices in the Chinese market by up to 6%, and BYD, China’s largest manufacturer, has continued to reduce its prices by 10–20%. However, EVs are still 10% to 50% more expensive than their internal combustion engine counterparts in Europe and the US, depending on the country and vehicle segment [13]. The result of nearly 15 years of rapid EV growth is a new structure of global vehicle ownership, as shown in

Table 2. Key trends in the global car fleet (Internal Combustion Engine (ICE) and EV) in 2022.

Indicators	ICE Vehicle	(EV)
Global car fleet	1430 M. vehicle	26.6 M. vehicle
Total light-duty vehicles (LDV) sails (65.3 M. cars)	86% (56.2 M.)	14% (9.1 M.), from which battery EVs (BEV)—2/3 plug-in hybrids (PHEV)—1/3
Share of small and large Sport Utility Vehicles (SUV) in sails	51%	49%
Share of medium car	13%	5%
Share of large car	12%	25%
Share of small car	24%	21%
Energy consumption	6.9 LGE/100 km	0.64 KWh/km The average range of: small cars—150 km; medium-sized cars and SUVs—380 km.
CO2 emission	150 gCO2/km	-
Average weight of LDV	1530 kg	1530 plus 13%
Average battery capacity	–	BEV—60 KWh PHEV—40 KWh On average, adding 100 km of range adds 330 kg to vehicle weight for a BEV
Purchase price difference (per vehicle)	100%	Plus 10–50%

Prepared by authors based on source [1,13,44].

.

2.5. EV Market Saturation Due to Critical Material Scarcity

For a more detailed analysis, it is useful to examine two trends: (1) the saturation rate of the electric vehicle market, and (2) the increasing role of critical materials in shaping the global market for this new product. Since 2011, when EVs began to widely penetrate global markets, the saturation rate has been steadily declining, as Figure 1 shows.

As Figure 1 shows, at the global level, the saturation rates declined from 3.3 times in 2011 to 1.5 times in 2023, but more significantly in some selected countries, such as China, the US and Germany (from 3.65× to 1.54× in China, from 5.7× to 1.59× in the USA and from 7.56× to 1.34× in Germany). In Norway the drop rate declined moderately, from 1.66 to 1.14 between 2011 and 2023. In France, where data are available from 2013, the saturation rate fell from 8.25 in 2023 to 1.49 in 2023. In parallel, the natural limits of the critical materials underlying the production of all batteries need to be considered and analyzed. In 2023, the total capacity of EV batteries exceeded 750 GWh, which is well below the 2035 target defined in the three different scenarios: the Stated Policies Scenario (STEPS) assumes a total capacity of 5.5 TWh, the Announced Pledges Scenario (APS) assumes 7.0 TWh, and the Net Zero Emissions by 2050 Scenario (NZE) assumes 9.0 TWh, where the goal is to increase battery production capacity by 8 to 13 times [47]. Clearly, such an increase in production would require a similar multiplier growth rate for the use of critical materials, at least within the framework of the technologies currently in use.

2.6. Critical Materials for Battery Production

Until very recently, industrial applications of lithium and its compounds were limited, mainly as oxide components in heat-resistant glass and ceramics, lithium grease lubricants, flux additives for iron and steel manufacturing, and more recently, the development of advanced lithium–aluminum alloys for the aviation industry. However, the situation changed dramatically with the introduction of lithium-ion batteries to laptop computers, cell phones, and electric vehicles. Today, 71% of global production is directed towards the battery sector (57% of which is for the automotive industry), 14% to ceramics and glass, 4% to lubricants and greases, 2% to continuous casting processes, 2% to polymers additives, and the remaining 7% to others industries [48]. The U.S. Geological Survey estimates global lithium reserves at around 28 million tons [49]. Figure 2 presents the global production of lithium in the time span 1985–2023.

As can be seen, global lithium production has increased spectacularly since 2015, mainly due to the explosive expansion in the EV market. This trend exceeds the exponential trend forecast (see Figure 2): production in 2023 (170,800 metric tons) is 31.3% higher than in 2022, with Australia (46.7%) being the largest global producer, followed by Chile (23.8%), China (18%), Argentina (5%), Brazil (2.6%), Canada and Zimbabwe (1.8% each). According to GlobalData, lithium production is expected to increase at a CAGR (Compound Annual Growth Rate) of about 13% through to 2030 [52].

Australia’s production comes primarily from hard rock mines (greenbush pegmatites), while the other producers rely primarily on lithium extraction from brine. The U.S. Geological Survey [51] estimates global lithium reserves in 2021 to be around 20 million tons, but points out that such estimates are very difficult to calculate. This is mainly due to the fact that most lithium classification schemes were developed for mineral ore deposits, whereas brine is a fluid that cannot be treated with the same classification scheme due its large concentrations. Pegmatite ores are much richer in lithium (ca. 2.4%) than brine, which has very low concentrations (typically less than 1%).

It is also important to note that there are many very optimistic estimates of the immense potential reserves of hard rock ores in the Democratic Republic of Congo, Zimbabwe, and Afghanistan, as well as brine in the so-called “Lithium Triangle”, which includes Chile, Bolivia, and Argentina, with high-quality salt flats. Although brine resources are more abundant than hard rock ores, the technology currently used to extract lithium from continental brines relies on open-air evaporation to concentrate the salt, and this method loses large amounts of water containing lithium carbonate, raising concerns about the sustainability of the entire process. Furthermore, because the process is very time-consuming (10 to 24 months), brine exploration cannot account for short-term changes in demand, which would explain the sharp decline in production around 2020.

Figure 3 shows the close relationship between lithium prices and production/demand. Lithium prices began to rise after 2015 as demand increased, fell in tandem with production around 2020, and have recently skyrocketed in response to the enormous demand for electric vehicles and lithium-ion batteries in general.

Lead-acid batteries, which have long dominated the global automotive industry, showed limited energy density and could not be recharged while driving to provide an acceptable range for consumers. The 21st century has seen a major shift in EV battery technology with the advent of lithium-ion batteries. New technologies have been found for this type of battery that can provide high energy density, stability, and low self-discharge rates. To achieve a balanced energy density and power density, the cathode of a Li-ion battery can be made of lithium cobalt oxide (LiCoO₂) for home appliances, lithium iron phosphate (LiFePO₄) for EVs, and nickel cobalt manganese oxide (NCM) or nickel cobalt aluminum oxide lithium (NCA), as well as many other materials (see

Table 3. Material values used for each type of Li-ion battery.

Battery Type	Kilograms per kWh				Kilograms per 75 kWh Battery
Cathode Chemistry	Lithium	Cobalt	Nickel	Manganese	Lithium	Cobalt	Nickel	Manganese	Total/75 kWh
LCO	0.113	0.959			8.48	71.93			80.40
NCA	0.100–0.106	0.117–0.130	0.618–0.670		7.50–7.95	8.77–9.75	46.35–50.25		62.62–67.95
NMC-111	0.118–0.139	0.313–0.400	0.312–0.400	0.292–0.370	8.85–10.42	23.47–30.00	23.40–30.00	21.90–27.75	77.62–98.17
NMC-622	0.100–0.130	0.170–0.190	0.508–0.610	0.159–0.200	7.50–9.75	12.75–14.25	38.10–45.75	11.92–15.00	70.27–84.75
NMC-811	0.090–0.100	0.076–0.090	0.608–0.750	0.071–0.090	6.75–7.50	5.70–6.75	45.60–56.25	5.32–6.75	63.37–77.25
LMO	0.097			0.103	7.27			7.72	14.99
LFP	0.087–0.100				6.52–7.50				6.52–7.50

Notes: LCO—lithium cobalt oxide; NCA—nickel cobalt aluminum oxide; NMC—lithium nickel manganese cobalt oxide; LFP—lithium iron phosphate oxide; LMO—lithium manganese oxide. Prepared by authors base on source: [23,53] (p. 84).

).

These minerals and cathode materials are critical components of lithium-ion batteries, and they affect their performance and safety. NCA has high energy and power density, but suffers from safety issues and high cost. NMC has a good balance of energy and power density, but its stability may be compromised. LFP, on the other hand, prioritizes safety with its inherent stability and resistance to abuse, but at the expense of energy density. The choice of cathode material depends on the requirements of the specific application, balancing factors such as energy density, power output, safety, and cost [54] (p. 6). Lithium-cobalt compounds are predominantly used in batteries for cell phones, laptop computers, and digital cameras, but their relatively short service life and low thermal stability limit their use. Given the high cost of cobalt, experts believe that the market share of this product is almost stable [55]. The study noted that lithium–aluminum (NCA) cathodes have the highest capacity (>240 Wh/kg), but lithium–aluminum (LMO) and lithium–phosphate (LFP) compounds are preferred in terms of specific power and thermal stability. Lithium–titanate (LTO) has a smaller capacity, but outperforms most other batteries in terms of lifetime, and has a better performance at low temperatures. According to the authors of this study, safety and service life are the dominant characteristics in the transition to electric powertrains [55]. According to the estimates of several research institutes, by 2040–2050, only lithium-based batteries (NMC—lithium nickel manganese cobalt oxide [54]) will dominate the global market. Table 3, shown earlier, is useful to the mathematical modeling work because it provides the basis for calculating the lithium consumption per battery, and thus the potential production of EVs.

3. Research Methodology and the Mathematical Model

First of all, attention should be paid to individual studies that have estimated EV fleets based on individual mathematical models using various variants of the logistic function. One such study by Fuchs et al. [56] examined the distribution process of battery EVs (BEV) in the European Union (EU) and beyond. The model used in this paper is based on the assumption that EV diffusion rates vary with the use of public financial incentives. Financial incentives determine the upper limit of the EV adoption process. In another study, the Italian National Energy and Climate Plan (NECP) scenario of 4.3 million EVs by 2030 was analyzed. Based on the Bass model and logistic functions, a “business as usual” scenario based solely on historical trends and an alternative scenario that would accelerate diffusion were considered. The modeling results indicate that incentive schemes and decarbonization strategies need to be seriously reviewed in order to achieve the net-zero emissions goals [57]. A study of EV penetration in the US market over a time horizon to 2035 using a symmetric logistic function is detailed in [58]. This paper includes two unique approaches: the first uses the same growth rate for each US state, but with different initial conditions based on sales volumes in 2021. The second takes into account the limited supply of lithium-ion batteries. In the baseline scenario, the share of EVs in the US vehicle fleet reaches 11.5% in 2030 and 30.8% in 2035. This translates to approximately 31 million EV cars and 47 million EV trucks on the road in 2035. This EV fleet would require approximately 7.5 TWh of batteries, which is roughly equivalent to the total battery production capacity if all US lithium reserves were used. The group of researchers estimated the replacement rate of ICE vehicles with EVs in the Mexican market with a logistic function, and estimated the tax losses (gasoline tax) associated with the introduction of EVs [59]. By 2032, EV-based motorization will reach a saturation level of 50%, and on a cumulative basis until 2050, Mexico will suffer revenue losses of USD 70 and 105 billion to the national treasury. In particular, new electricity taxes, registration taxes, and corresponding consumption taxes would need to be introduced to offset these losses.

Many of the various scenarios used to estimate the global EV fleet are based on specific future emission reduction targets, such as 2030, 2040, or 2050. Typically, these scenarios provide a general estimate of the share of EVs in the global vehicle fleet. For example, in the rapid adoption scenario, EVs will account for 100% of new vehicle sales and more than 50% of the global light-duty vehicle fleet by 2040 [60]. Another study estimates that by 2040, 54% of new vehicle sales and 33% of the global vehicle fleet will be electric by 2040, as falling battery prices make price-competitive EVs available in all major passenger car segments by 2030 [61]. There are also estimates for up to nine different scenarios with different growth rates for the total number of passenger cars and EVs. According to the scenario with the highest growth rate in EV sales, by 2050, their share will be at best just under 50%, with the number of EVs ranging from 611 million to 1.11 billion units [62]. More conservative projections for the growth of EV ownership also exist, with the share of EVs by 2050 not exceeding 31%, with a total of 672 million vehicles [63]. Some scenarios and estimates are based on a sales target of 100% zero-emission vehicles and a phase-out of ICE vehicles by 2050. By that time, it is assumed that annual vehicle sales will have increased to 160 million, and the global fleet to 2.3 billion, with annual EV sales approaching 33 million [64,65]. Some studies evaluate the development prospects of EVs in individual large markets. Thus, the assessment of the European transportation sector is based on a set of scenarios covering a number of assumptions for the measurement of transportation technology and climate policy, allowing for a slight increase in the total number of passenger cars to 270 million units by 2050, with EVs accounting for 90% of new car sales [66]. Thus, many studies are based on “scenario” principles rather than the “model” principle.

By detailing the current state of the EV industry, including incentive schemes, national regulations, market competition, and the fact that the market itself is saturated with new products while there is a clear lack of critical materials, including batteries, this study aims to demonstrate the diversity of factors that influence the development of the EV sector. The diversity of factors is particularly important for countries. Given the diversity of factors, it appears that building a multi-factor model is not the best solution, especially given the difficulty of collecting information on such factors at the country level.

For this reason, on the one hand, we propose a model based on a logistic function, but on the other hand we introduce a constraint in the form of lithium reserves, one of the key minerals. This constraint played an important role in determining the evolution of the logistic curve over the forecast period, as will be discussed later in the analytical part of the model.

Another important aspect of the proposed model is that the analytical approach yields a solution that is confirmed by calculations relative to the point in time when the growth mode changes (in our calculations, this is mid-2021). In essence, the exponential growth mode for this segment changed three years ago, and future growth rates will be more moderate.

This study describes and predicts the trends in EV production and lithium exploration, applies a quantitative approach to find the correlations between these two factors, and shows how the shortage and scarcity of critical materials can affect trends and significantly influence the expected EV production trends. The objective of the study is to propose a mathematical model that can determine the optimal time span for forecasting EV production/demand dynamics, taking into account lithium availability constraints. The behavior of the mathematical model and its functions will be evaluated using different time spans. Furthermore, the objective is to explore the benefits and drawbacks of the developed mathematical model with respect to its forecasting nature.

As already mentioned, in the developed scenarios, EV production is directly dependent on the production of batteries for EVs. Moreover, lithium is a chemical element used in all types of batteries, and its availability (production and reserves) will determine the future production of EVs (at least within the framework of the most common battery production technologies today) to a large extent. Based on the above, a novel mathematical model is developed and proposed to evaluate new technological products (lithium-ion batteries and EVs) that do not use fossil fuel, and to take into account the limits of lithium reserves.

3.1. Initial Data

There are two key parameters in the original data. The first parameter is lithium reserves. As discussed in Section 2.6, until very recently, lithium and its compounds had limited industrial use. Today, 71% of global lithium production is directed towards the battery sector (57% to the automotive industry). The U.S. Geological Survey estimates global lithium reserves at around 28 million tons [49]. Based on these data, we will estimate the lithium reserves for EV production of EVs at 11.8 million tons.

The second parameter relates to the amount of lithium used per battery. Table 3 shows the variation in lithium usage for different types of lithium-ion batteries. For simplicity of calculation, 8 kg of lithium per battery is used as a set parameter.

3.2. Model Based on Logistic Function

In the literature, logistic functions of the form (1) (logistic) are often used to construct predictions of process evolution over time;

$$N=N\left(t\right)={\displaystyle \frac{a}{1+e^{-b(t-T_0)}}}$$

(1)

which is a solution to the differential equation

$${\displaystyle \frac{dN}{dt}}=kN(t)\left(a-N\left(t\right)\right)$$

(2)

where $k$ is the intensity or growth rate of the logistic function $N\left(t\right)$, and a is the maximum asymptotic value of the logistic function.

Let us consider a simple but, as proven below, not entirely effective approach to approximating EV production data using the logistic function (1) without considering the limited supply of lithium.

Let $M_L$ be the lithium reserves, and $m_L$ the mass of lithium in an average lithium-ion battery. Then, the maximum number of EVs that can be produced, $N_{\max }$, is calculated using a simple formula,

$$N_{\max }={\displaystyle \frac{M_L}{m_L}}$$

(3)

Therefore, it is possible to first approximate the available data on the number of EVs ${\left(t_k,N_k\right)}_{k=1,2,\dots ,n}$ using the logistic function (1), where $t_k=2010,\dots ,2023$, and then determine the year of lithium reserves’ depletion based on the following condition

$${\displaystyle \sum _{k=1}^M}N\left(t_k\right)=N_{\max }$$

(4)

When fitting the observed data up to 2024 with the logistic function (1) without the constraint (4), there are practically equally good approximations for the different values of the parameters, $b,T_0$, according to the SSE (Sum Squared Error, i.e., the minimum of (2)) and R² criteria. The two logistic approximations are shown in Figure 4.

However, when the logistic data are plotted to, for example, 2060, the trajectories are very different from each other, and the approximations are highly variable and drastically different, leading to a completely implausible result, as shown in Figure 5.

Based on the above results, the least squares method (LSM) seems advisable for use to obtain the parameters $a$, $b$ and $T_0$ of the logistic function (1):

$$\underset{a,b,T_0}{\min }F\left(a,b,T_0\right)=\underset{a,b,T_0}{\min }{\displaystyle \sum _{k=1}^n}{\left(N\left(t_k\right)-N_k\right)}^2$$

(5)

That is, to impose an additional constraint in the form of lithium reserves (4) when solving the problem of minimizing the sum of squares of the errors. Problems (4)–(5) show a nonlinear optimization problem, and are solved for a discrete set of predicted horizon values $M$ (or $t_M$). As a next step, the smallest of the minimum values $F\left(a,b,T_0\right)$ found for each $M$. is selected. The corresponding parameters $a,b,T_0$ and $M$ are considered optimal and are used in constructing EV production forecasts using (1) until the time $t_M$ when the lithium is exhausted.

The results obtained using the approach described above are presented below. Problems (4)–(5) were solved using the Sequential Quadratic Programming (SQP) method [67]. Figure 6 shows the dependences of the SSE, R² and errors in constraint criteria on $t_M$.

From the graphs in Figure 6, it can be seen that for 2035 to 2047, all approximations are of approximately the same quality. For example, the two forecasts corresponding to $t_M=2035$ and $t_M=2041$ are shown in Figure 7.

The forecast corresponding to the best approximation given by the SSE criterion is shown in Figure 8.

As Figure 8 displays, the annual production volumes of batteries (and therefore EVs) are constantly growing, reaching about 80 million units by 2035. However, such nonstop and rapid growth, due to the factor $kN\left(t\right)$ in (2) and the actual absence of a constraint on the asymptotic value corresponding to the factor $\left(a-N\left(t\right)\right)$ in (2), is unlikely from an economic perspective, given the long-term consumer nature of the product. Based on these considerations, a modified logistic equation is proposed.

3.3. Modification of the Logistic Equation

A modification of the logistic Equation (2) when resources are limited is proposed. The multiplier $\left(a-N\left(t\right)\right)$ in the logistic Equation (2) corresponds to the asymptotic saturation of the market, which is not known a priori under resource constraints, but instead it is proposed to introduce a multiplier that slows down the growth rate of $N\left(t\right)$ as it approaches resource exhaustion. This multiplier can be constructed as follows:

$$k=k\left(t\right)={\left[N_{\max }-\underset{t_0}{\overset{t}{{\displaystyle \int }}}N\left(\tau \right)d\tau \right]}^{\alpha }$$

(6)

where the integral

$$\underset{t_0}{\overset{t}{{\displaystyle \int }}}N\left(\tau \right)d\tau$$

(7)

is the number of EVs produced by the time $t$, starting from the time point $t_0$.

Thus, instead of the logistic Equation (2), we obtain the integro-differential equation

$${\displaystyle \frac{dN}{dt}}=k_0{\left[N_{\max }-\underset{t_0}{\overset{t}{{\displaystyle \int }}}N\left(\tau \right)d\tau \right]}^{\beta }N\left(t\right),\hspace{1em}N\left(t_0\right)=N_0$$

(8)

in the solution of which $N=N\left(t;k_0,\beta ,N_0\right)$ it is necessary to select the parameters $k_0,\beta ,N_0$ so as to approximate the available data on the number of EVs using the LSM.

$$\underset{k_0,\beta ,N_0}{\min }F\left(k_0,\beta ,N_0\right)=\underset{k_0,\beta ,N_0}{\min }{\displaystyle \sum }_{k=1}^n{\left(N\left(t_k;k_0,\beta ,N_0\right)-N_k\right)}^2$$

(9)

The meanings of the parameters in Equation (8) are assumed to be as follows:

(a)

Parameter $k_0$ characterizes the exponential growth rate of the number of EVs under conditions of sufficient resources;

(b)

Parameter $\beta$ characterizes the decrease in the production rate as the resources approach exhaustion.

Equation (8) is unlikely to have an analytical solution for the direct application of the least squares method (9), so two options are possible.

Option 1 is the application of a numerical scheme for solving (8). For example, if the explicit Euler method is used to solve the differential equation, and the integral (7) is calculated using the trapezoidal method on a uniform grid $t_0,t_1,t_2,\dots$ with a constant step $h$, then the following numerical scheme is obtained as a result:

$${\tilde{N}}_{i+1}={\tilde{N}}_i+h{k}_1{\tilde{N}}_i{\left[N_{\max }-\left({\displaystyle \frac{h}{2}}\left({\tilde{N}}_0+{\tilde{N}}_i\right)+h{\displaystyle \sum _{j=0}^{i-1}}{\tilde{N}}_j\right)\right]}^{\beta }$$

(10)

where ${\tilde{N}}_i$ is the approximate value of the solution at time $t_i$, $i=0,1,2,\dots$

Under option 2, if using the replacement

$$N\left(t\right)={\displaystyle \frac{dV}{dt}},V\left(t_0\right)=0,$$

(11)

then Equation (8) is reduced to the following system of ordinary differential equations:

$$\left\{\begin{array}{l}{\displaystyle \frac{dN}{dt}}=k_0{\left[N_{\max }-V\left(t\right)\right]}^{\beta }N\left(t\right)\hfill \\ {\displaystyle \frac{dV}{dt}}=N\left(t\right)\hfill \end{array},N\left(t_0\right)=N_0,V\left(t_0\right)=0.\right.$$

(12)

The equation system (12) can be solved using numerical methods, for instance, the Runge–Kutta method.

In any of the variants, the numerical scheme forms constraints in the minimization problem (9), and the grid of the numerical scheme is assumed to include time instances t_k, at which there are data $N_k$, $k=1,2,\dots ,n$. The numerical integration with adaptive step selection to ensure a given accuracy continues up to the time instance T, when the right-hand side of Equation (8) equals zero with a given accuracy, which indicates resource depletion. This time instance $T$ is not known a priori, so the integration interval can be chosen to be large enough, and once an approximation of the data is found, only the part corresponding to $t\le T$ can be used.

For simplicity, instead of solving the optimization problems (9)–(10), for instance, we can find the minimum in (9) on the grid in the cube $\left[{\underset{\_}{k}}_0,{\overline{k}}_0\right]\times \left[\underset{\_}{\beta },\overline{\beta }\right]\times \left[{\underset{\_}{N}}_0,{\overline{N}}_0\right]$, where ${\underset{\_}{k}}_0<{\overline{k}}_0$, $\underset{\_}{\beta }<\overline{\beta }$, ${\underset{\_}{N}}_0<{\overline{N}}_0$ are the given values. The results thus obtained are shown in Figure 9. In this case, both of the above variants of the numerical solution of Equation (8) were used, and the same results were obtained with the established solution accuracy.

The solutions obtained within the introduced framework of restrictions, as can be seen in Figure 9, are not satisfactory, as there is a clear dominance of the multiplier $N\left(t\right)$ on the right side of Equation (8), which results in strong exponential growth, while the multiplier (6), which is responsible for the slow-down in growth under conditions of resource depletion, has a weak effect on the growth rate. This is why, in the next section, we will consider an alternative approach to modifying the logistic Equation (2).

3.4. Model with Separation of Factors of Initial Exponential Production Growth and of Subsequent Resource-Related Slowdown

We assume that the initial phase of production is characterized by exponential growth, while the next stage faces a slowdown in the growth rate of production as lithium resources are gradually depleted. Thus, the modified integro-differential logistic equation takes the form:

$${\displaystyle \frac{dN}{dt}}=\min \left\{f^{\left(1\right)}\left(N\right),f^{\left(2\right)}\left(t,N\right)\right\},\hspace{1em}N\left(t_0\right)=N_0$$

(13)

where

$$f^{\left(1\right)}\left(N\right)=k_{1}N\left(t\right),\hspace{1em}{k}_1>0$$

(14)

characterizes the exponential growth of production, and

$$f^{\left(2\right)}\left(t,N\right)=k_2\left[N_{\max }-\underset{t_0}{\overset{t}{{\displaystyle \int }}}N\left(\tau \right)d\tau \right],\hspace{1em}{k}_2>0$$

(15)

models the slowdown in production rate to zero as lithium reserves are exhausted. Equation (13) is solved up to time $T$, which in turn is the solution of the equation

$$f^{\left(2\right)}\left(T,N\right)=0.$$

(16)

As will be shown later, the switch between $f^{\left(1\right)}\left(N\right)$) and $f^{\left(2\right)}\left(t,N\right)$) on the right-hand side of (13) occurs at a time determined by the coefficients $k_1,k_2$ and $N_0$, which in turn are found using the least squares method (5) from the original data. Therefore, the original data completely determine the behavior of the solution $N\left(t\right)$ of Equation (13) for all $t$, and the switch between $f^{\left(1\right)}\left(N\right)$ and $f^{\left(2\right)}\left(t,N\right)$ is not artificial.

The solution to Equation (13) can be approximated using numerical integration formulas, as in the case of the model described earlier. For example, using a combination of the explicit Euler method with the trapezoid method to calculate the definite integral in (13) for a uniform grid $t_0,t_1,t_2,\dots$ with a constant step $h$, the following numerical scheme is derived:

$${\tilde{N}}_{i+1}={\tilde{N}}_i+h\min \left\{k_1{\tilde{N}}_i,k_2\left[N_{max}-\left({\displaystyle \frac{h}{2}}\left({\tilde{N}}_0+{\tilde{N}}_i\right)+h{\displaystyle \sum _{j=0}^{i-1}}{\tilde{N}}_j\right)\right]\right\},$$

(17)

where is the approximate solution value at time $t_i$, $i=0,1,2,\dots$.

Calculations according to Formula (17) stop at $i=i_{\max }$, which is determined from the condition

$$N_{\max }-\left({\displaystyle \frac{h}{2}}\left({\tilde{N}}_0+{\tilde{N}}_{i_{\max }}\right)+h{\displaystyle \sum _{j=0}^{i_{\max }-1}}{\tilde{N}}_j\right)\le 0.$$

(18)

Let us show that the solution of Equation (13) can be obtained analytically. For simplicity, without loss of generality, consider the case $t_0=0$. At the initial instances of time $f^{\left(1\right)}\left(N\right)<f^{\left(2\right)}\left(t,N\right)$, this corresponds to the case of the beginning of production and removal when the resource is depleted. Then, we can infer that Equation (13) is equivalent to the equation

$${\displaystyle \frac{dN}{dt}}=k_{1}N\left(t\right),N\left(t_0\right)=N_0,$$

(19)

the solution of which is

$$N^{\left(1\right)}=N_0{e}^{k_{1}t}.$$

(20)

Switching between stages occurs at time $T_s$, determined by the condition

$$k_1{N}^{\left(1\right)}\left(T_s\right)=k_2\left[N_{\max }-{\displaystyle \underset{0}{\overset{T_s}{\int }}}{N}^{\left(1\right)}\left(\tau \right)d\tau \right].$$

(21)

Substituting (20) into (21) leads to the following equation:

$$k_1{N}_0{e}^{k_1{T}_s}=k_2\left[N_{\max }-{\displaystyle \underset{0}{\overset{T_s}{\int }}}{N}_0{e}^{k_{1}t}d\tau \right].$$

(22)

The integral on the right-hand side of Equation (22) is easily calculated, which leads to the equation

$$k_1{N}_0{e}^{k_1{T}_s}=k_2\left[N_{\max }-{\displaystyle \frac{N_0}{k_1}}\left(e^{k_1{T}_s}-1\right)\right].$$

(23)

The solution to (23) is the time instance $T_s$, which determines switching between $f^{\left(1\right)}\left(N\right)$ and $f^{\left(2\right)}\left(t,N\right)$ in (13), and is calculated as follows:

$$T_s={\displaystyle \frac{1}{k_1}}\ln {\displaystyle \frac{k_2\left(k_1{N}_{\max }+N_0\right)}{N_0\left(k_1^2+k_2\right)}}.$$

(24)

Now, knowing the time $T_s$ and that for $t<T_s$, the solution to Equation (13) has the form (20), it is easy to find the solution for $t\ge T_s$, which is determined by the following equation:

$$\begin{gathered} {\displaystyle \frac{dN}{dt}}=k_2\left[N_{\max }-{\displaystyle \underset{0}{\overset{t}{\int }}}N\left(\tau \right)d\tau \right]=k_2\left[N_{\max }-{\displaystyle \underset{0}{\overset{T_s}{\int }}}{N}^{\left(1\right)}\left(\tau \right)d\tau -{\displaystyle \underset{T_s}{\overset{t}{\int }}}N\left(\tau \right)d\tau \right]= \\ =k_2\left[N_{\max }-{\displaystyle \frac{N_0}{k_1}}\left(e^{k_1{T}_s}-1\right)-{\displaystyle \underset{T_s}{\overset{t}{\int }}}N\left(\tau \right)d\tau \right], \end{gathered}$$

(25)

with the initial condition of

$$N\left(T_s\right)=N_0{e}^{k_1{T}_s}.$$

(26)

If we consider

$${\displaystyle \frac{dv}{dt}}=N\left(t\right),\mathrm{and}$$

(27)

$$v\left(T_s\right)=0,$$

(28)

then (25) takes the following form:

$${\displaystyle \frac{dN}{dt}}=k_2\left[N_{\max }-{\displaystyle \frac{N_0}{k_1}}\left(e^{k_1{T}_s}-1\right)-v\left(t\right)\right].$$

(29)

The systems of Equations (27) and (29) with initial conditions (28) and (26), respectively, are linear, and by solving them, $N^{\left(2\right)}=N^{\left(2\right)}\left(t\right)$ can be easily determined,

$$N^{\left(2\right)}=\sqrt{k_2}\left[N_{\max }-{\displaystyle \frac{N_0}{k_1}}\left(e^{k_1{T}_s}-1\right)\right]\sin \left(\sqrt{k_2}\left(t-T_s\right)\right)+N_0{e}^{k_1{T}_s}\cos \left(\sqrt{k_2}\left(t-T_s\right)\right).$$

(30)

The last instance of time $T$ corresponding to resource depletion is defined as the root of the equation (the integral of from $N^{\left(2\right)}$ is easily computed)

$$N_{\max }-{\displaystyle \frac{N_0}{k_1}}\left(e^{k_1{T}_s}-1\right)-{\displaystyle \underset{T_s}{\overset{T}{\int }}}{N}^{\left(2\right)}\left(\tau \right)d\tau =0.$$

(31)

The corresponding expression for $T$ is rather cumbersome, and is not provided here.

To summarize, the final solution of Equation (13) is written as follows:

$$N=\left\{\begin{array}{cc}{N}^{\left(1\right)}\left(t\right),\hfill & \hfill t<T_s\\ N^{\left(2\right)}\left(t\right),\hfill & \hfill T_s\le t\le T\end{array}\right.,$$

(32)

where $N^{\left(1\right)}$ is determined by Equation (20), $N^{\left(2\right)}$ is determined by Equation (30), and $T_s$ is determined by (24).

For example, Figure 10 shows graphs of solutions to Equation (13): the analytical solution using Equation (32) and the numerical solution given by (17)–(18). These graphs correspond to the following parameter values: $N_0=10,N_{\max }={10}^3,k_1=0.1$, and $k_2=5·{10}^{-3}$. A grid of $5·{10}^4$ equally spaced nodes was used for the numerical solution, and the round markers on the graph correspond to the values of the numerical solution on a more rarefied grid.

Equation (13) was used to forecast for EV production numbers. The results are shown in Figure 11. The stage changeover time is $T_s\approx 2021.5$. Thereafter, the exponential growth in the number of EVs changes to growth with a decreasing rate due to the limitation in lithium resources. According to this projection, lithium reserves will be depleted at time T ≈ 2051.

In a very concise form, we would like to highlight three important steps in constructing the final model that distinguish our proposed approaches from previously published works.

First of all, we note that the model is based on the widely used logistic function (Equation (1)), which is also presented as a differential Equation (33):

$${\displaystyle \frac{dN}{dt}}=kN\left(t\right)\left(a-N\left(t\right)\right)$$

(33)

Unfortunately, the multiplier $\left(a-N\left(t\right)\right)$, which is responsible for the natural saturation of the market in the case of unlimited resources, does not make any economic sense in the case of resource constraints, which seems evident in Figure 4 and Figure 5. Figure 4 shows that there is a good initial approximation (justified by the highly acceptable SSE criteria, R², which is a constraint error in Figure 6). Figure 5, however, proves that the final predicted results are quite different. For this reason, we abandoned the use of the model in the form of Equation (33).

The modified novel model is reflected in the form of integral–differential Equation (8), where the multiplier ${\left[N_{\max }-{\int }_{t_0}^{t}N\left(\tau \right)d\tau \right]}^{\beta }$ shows the depletion (finiteness) of the used resource (lithium). However, as depicted in Figure 9, this model demonstrates a “very fast” depletion of resources (as the steepness and the saturation of the graph depicts, all lithium reserves will be depleted by 2035 with an annual production volume of 160 million EVs), which is, of course, economically untenable.

In its final form, the model takes the form of Equation (13), the parameters of which are found using the least squares method based on real data. An analytical solution of Equation (13) has proven that there is a turning point of time (mid-2021) when the exponential growth changes to a slower saturation mode. Although the initial series of real data on the volume of global EV production is short (since 2011), we have obtained a model by use of two successive approximations with an analytical solution. As for the validation of the model, after 2021, there are only two points (2022 and 2023), which is not enough. However, as the data are highly up-to-date, we need to wait until at least 2027 to derive more data points.

In our opinion, what distinguishes this model from all other scenarios and models built is the analytical solution obtained with this model. The second feature of this model is the consideration of lithium, an important critical resource, as an obvious limiting factor for the distribution of new products.

Of course, the model could become more complex. For example, options for replenishing lithium reserves could be introduced, as well as lithium reuse based on effective disposal technologies for used batteries (which have yet to be solved technologically and economically). The challenge for our research group was to develop a new type of model that could extend the tools for studying technological and economic processes. The scope of application could be tested to evaluate the production volumes of wind turbines that use rare earth elements such as neodymium and dysprosium (for the production of powerful permanent magnets for wind turbines).

As can be seen, after some modifications, we obtained a model that adequately describes the diffusion process of new technological products.

4. Discussion

Estimates of the global EV fleet for 2030–2035 obtained based on the proposed model indicate that the EV stock will approach 229 million units in 2030, and reach approximately 458 million units in 2035. If these estimates are confirmed in the future, EVs will account for more than 60% of the world’s passenger cars in 2050. These results differ somewhat from all known scenarios for the decarbonization of transport by 2035, especially those presented in the study [13].

To compare the results obtained based on the proposed model with already-published estimates of the global EV potential,

Table 4. Global EV stock projection (million units).

Scenario	2030	2035	2040	2050	Source
STEPS	251.6	523.7	-	-	[13]
APS	263.0	585.1	-	-
NZE	300.0	790.0	-	-
ETS	194.8	383.8	730.0	-	[68,69]
Proposed model	229.1	458.6	750.7	1447.9	-

Note: STEPS—Stated Policies scenario; APS—Announced Pledges scenario; NZE—Net Zero Emissions by 2050 scenario, ETS—Economic Transition scenario. Based on sources: [13,68,69].

summarizes the global EV stock forecasts.

First of all, through 2030 and 2035, the developed model gives lower EV fleet projections than the STEPS, APS, and NZE scenarios. Only the ETS scenario has lower projections than the proposed model. For the outlook through to 2040, the proposed model and the ETS scenario’s EV fleet numbers are very close. Unlike the other scenarios given in Table 4, the model also estimates a potential EV fleet of 1.44 billion units in 2050. Assuming a global vehicle fleet of 2.3 billion in 2050 [64], the EV share would be about 62%, which, of course, seems a very optimistic estimate. In the following, we discuss the extent to which the decarbonization of the transportation sector is realistic.

The proposed model, although based on the logistic function, has a significant limitation that concerns lithium reserves. Given that lithium-ion battery production technologies will be dominant until 2050, lithium reserves are a natural limitation in battery production and, as a result, in EV production. The expansion of the EV fleet is also expected to require a significant increase in investment in mining projects. At this point, three problems arise.

The first is of a purely environmental nature, because lithium mining involves greater environmental damage than is currently estimated. Therefore, the study by Schenker et al. [70] points out that the data available in the literature underestimate the environmental impact of ${\mathrm{Li}}_2{\mathrm{CO}}_3$ production from brine. Integrating the obtained life cycle inventories (LCl) of lithium-ion batteries demonstrates that the overall impact on climate change increases by up to 19%.

The second issue is linked to the size of investment that is necessary for developing new lithium deposits, regardless of their nature, as these could be pegmatites, geothermal brine, or seawater. Our modeling results suggest that in order to reach 1.44 billion EVs by 2050, approximately 11.7 million tons of lithium reserves would be necessary for battery production alone. In 2023, global lithium production was around 180,000 tons, of which 75,400 tons were used for EV batteries (estimated by authors, based on Petavratzi and Josso [48]). Further calculations show that over the next 25 years, annual lithium production for lithium-ion batteries alone would need to be around 460,000 tons, but these calculations do not take into account the potential for battery recycling and reusing. Some studies have already suggested that 15–18% of cobalt, 9–11% of lithium, and 15–17% of nickel demand could be reused by 2035 [71]. Even if this level of lithium recycling is achieved, annual production volumes of 400,000 tones appear significant.

The second component of the investment relates to building new battery production capacities. Based on predictions, global lithium-ion battery production capacity should reach 2400 GWh by 2030 [72]. With an average battery capacity of 75 kWh, such installed capacity would be enough to produce approximately 32 million EVs per year. Considering that global capacity in 2020 did not exceed 750 GWh, it is clear that significant investments will be required to more than triple such capacities in just 10 years. According to conservative estimates, up to 2050, 27 TWh of batteries or equivalent to 2.9 million tones (Mt) of lithium, 10.7 Mt of nickel, 0.8 Mt of cobalt and 5.5 Mt of manganese need to be produced in Europe alone [73].

The third issue is the use of “green” energy for all EVs after 2037. The number of EVs on the road is growing much faster than the amount of “green” energy produced. By 2037, the number of EVs on the road is projected to require as much “green” electricity as all the “green” electricity produced at that time. Thus, it will not be possible for EVs introduced to the market after 2037 to operate in a “green” way [74]. Thus, there could be a serious obstacle here to the widespread adoption of electric transportation in the form of a lack of “green” power generation capacity.

The most likely option to overcome the first two challenges is the development of new battery technologies, although most estimates suggest that lithium-ion technologies will be dominant in the period up to 2030 [47]. After 2030, solid-state batteries (SSBs) are considered promising, which are based on the use of solid or quasi-solid electrolytes instead of conventional liquid electrolytes, thus increasing cell energy density and operation safety. Nonetheless, BloombergNEF estimates that solid-state batteries would require 45% to 130% more lithium at the cell level (for the electrolyte and separator) than traditional batteries [75]. Lithium sulfur batteries also have development potential, as they can theoretically use both liquid and solid electrolytes and offer high energy density; furthermore, the use of cathodes based on abundant and inexpensive sulfur makes them economically competitive [47]. Technologies based on the use of iron (LFP, LMFP) and sodium (sodium-ion batteries) are also promising. These technologies do not use lithium, which makes them a good solution to mitigate lithium supply constraints and price fluctuations [73]. Currently, major cell manufacturers and automotive OEMs are increasing their investments in second-generation (Gen 2) silicon anode materials and production, mainly due to the fact that silicon anodes offer significantly higher energy density than traditional graphite anodes, resulting in EVs with a longer range and faster charging times—potentially as little as 11 min to charge from 20% to 80% of capacity. Another advantage of Gen 2 silicon anodes is their compatibility with current lithium-ion cell manufacturing processes [76]. Considering the third problem, the development of such energy sources as hydrogen, nuclear energy based on SMRs (small modular nuclear reactors), and, in the future, fusion energy seems a real option to overcome these hurdles.

5. Conclusions

It seems that achieving in practice the quantitative estimates of the global electric vehicle fleet obtained based on the proposed model depends on many factors. The most significant factors lie in the field of battery production technologies and the materials from which they are produced. The currently dominant technologies rely on lithium, cobalt, nickel and manganese. If new technologies are developed, for example, based on silicon, sulfur, sodium or iron, this transportation sector’s development potential may change significantly. The picture may also be significantly transformed if effective technologies for the reuse of lithium-ion batteries or lithium extraction from used and expired batteries are developed.

Other factors lie in the field of decarbonization policies, which are highly dependent on the economic characteristics of each country as well as geopolitical changes. We are witnessing a significant fragmentation of global trade and great geopolitical tensions that could slow down the global pace of emissions reduction, which in turn could affect the level of electric transportation use.

Competition is becoming an important factor in the expansion of battery and electric vehicle production. As the recent bankruptcy of Northvolt, Europe’s largest battery manufacturer, shows, even a USD 5.3 billion investment could not guarantee the company’s survival in the face of fierce competition from Chinese manufacturers [77]. Northvolt’s bankruptcy shows that Europe lacks the necessary stamina and leadership to realize its battery ambitions.

As new technological products, electric vehicles contribute to the reduced consumption of fossil fuels, and oi- and gas-exporting countries may face substantial revenue declines in the medium term. This scenario could be particularly painful for those countries wherein oil and gas exports represent a significant portion of foreign trade revenues. Diversifying the economies of such countries, both industrially and technologically, may be the most desirable way to overcome their excessive dependence on oil and gas.

The model proposed in this paper can be tested to evaluate the production of wind turbines that use rare earth elements such as neodymium and dysprosium (for the production of powerful and permanent magnets for wind turbines). From a mathematical point of view, the analytical solution for determining the turning point of the growth mode is universal, and can be used to describe the market saturation trajectory of a product whose production depends on the use of two or three critical materials.