An application of the graph approach to life-cycle optimisation of vehicle electrification

Although durable goods with low energy consumption are being promoted to achieve a decarbonised society, from the perspective of life-cycle assessment, the choice of new durable goods may increase CO2 emissions. To address this problem, research has been conducted on product replacement based on life-cycle optimisation (LCO), a method for identifying a replacement life span that minimises life-cycle CO2 emissions. However, several additional assumptions complicate the analysis of replacement patterns of products and conditional formulas because cumulative emissions do not increase linearly when considering energy mix and technology improvement, and it is difficult to extend the model to optimisation methods in previous LCO studies. This study developed a new LCO approach by applying the shortest path problem to graph theory. Our methodology can contribute to the following: (i) it is computationally inexpensive; (ii) it is intuitively easy to add complex conditions, such as various policy scenarios and parameter changes; and (iii) once the graph of replacement patterns is defined, the optimal solution can be derived using existing solution methods, such as the Dijkstra algorithm. As a case study, we focused on vehicle replacement, which is a major source of CO2 emissions and is being electrified. In particular, we identified vehicle switching paths that minimise life-cycle CO2 emissions by considering changes in Japan’s energy mix and alternative fuel vehicle (AFV) characteristics. We determined that the optimal vehicle replacement path method to reduce CO2 emissions is to switch first to plug-in hybrid electric vehicles (PHEVs) and then to battery electric vehicles (BEVs). Thus, we suggest that the transition to electric vehicles requires a step-by-step process. This methodology is not only conducive to AFV deployment for decarbonisation but can also be applied to other products, such as air conditioners and lighting. Thus, various transition policies could be formulated using our methodology.


Introduction
The spread of durable goods with low energy consumption is promoted to achieve a decarbonised society [1].For example, Japan offers subsidies for capital investment in energy-efficient refrigeration equipment, air conditioning, and controlled lighting for business establishments (energy-conservation investment promotion support programs [2]).These measures are particularly prominent in transportation, a major source of greenhouse gas emissions.The transition to battery electric vehicles (BEVs) is progressing worldwide [3].In general, BEV deployment scenarios have been developed to reduce tailpipe greenhouse gas emissions during driving (tank-to-wheel emissions) by eliminating internal combustion engine vehicles (ICEVs) and replacing them with BEVs; however, emissions from manufacturing BEVs and the production of electricity for driving (well-to-tank emissions) are significant under current energy mixes and could increase life-cycle emissions [4,5].CO 2 emissions associated with well-to-tank and manufacturing of hybrid electric vehicles (HEVs), PHEVs, and BEVs (in this study, these three are considered alternative fuel vehicles; AFVs) are larger than those of ICEVs in regions that are highly dependent on fossil fuels (e.g., India [4], Japan [5,6]).In addition, batteries with high manufacturing emissions when used in AFVs for long periods, must be replaced.Depending on the magnitude of the reduction in tank-to-wheel emissions and the magnitude of the increases in emissions, the blanket promotion of automobile electrification will not contribute to life-cycle emission reductions [4][5][6].
Previous studies have shown that life-cycle emissions from the long-term use of older vehicles with greater tank-to-wheel emissions are less than that of switching to newer, more fuel-efficient vehicles [7][8][9].However, tank-to-wheel reduction is necessary to achieve zero emissions in the future, and it is important to reduce wellto-tank emissions during manufacturing through energy mix and technological innovation while promoting the transition to electric vehicles [10].To this end, climate mitigation policies should consider an AFV deployment plan that minimises the cumulative GHG emissions relative to the policy target emission parameters at each lifecycle stage.Thus, the transition to new durable goods with lower greenhouse gas emissions requires elaborate analysis and detailed discussion.
Research on product replacement based on life-cycle optimisation (LCO) using dynamic programming (e.g.Chung et al [11]) has been conducted.Kim et al [12] explicitly identified a replacement span for newer vehicles with lower life-cycle emissions that minimised each of the four environmental burdens.Spitzley et al [13] optimised for both financial cost for consumers and environmental burden and recommended a mid-range between long-term use that achieves the lowest consumer cost and CO 2 emissions and short-term use that minimises NOx and other emissions.Spitzley et al [13], Kim et al [12], and Kim et al [14] showed that the optimal period of use varies depending on the objectives to be optimised, particularly the minimisation of consumer costs, air pollution, and life-cycle CO 2 emissions.However, Bakker et al [15] analysed LCO for refrigerators and laptops.These papers showed that a longer product lifetime results in lower life-cycle CO 2 emissions.
The optimal lifetime differs significantly for durable goods through product innovations.For example, lighting and automobiles require different model constructions because replacement with a new product (e.g., switching from ICEV to BEV for automobiles or from incandescent lamps to light-emitting diodes for lighting) with significantly different emissions during use and manufacturing must be considered.Liu et al [16] and Dzombak et al [17] conducted LCO for lighting.Liu et al [16] considered changes in energy mix and improvements in technology, and several types of lighting, including incandescent lamps and light-emitting diodes, were considered to optimise consumer costs and energy use.Mizuno et al [18] considered ICEVs and HEVs in automobile LCO.Contrary to previous studies [12][13][14][15], these studies suggested that switching to new products earlier minimises life-cycle CO 2 emissions.
Especially in the case of automobiles, the factors to be considered vary greatly depending on the type of fuel, such as tank-to-wheel, battery replacement, installation of home chargers, and vehicle inspections.Therefore, a simple and versatile framework is required.Previous studies have been based on a specific objective function for the optimisation problem.Additional functions are needed to consider the replacement patterns of each fuel type, the variation in technological factors, such as battery replacement, the reduction of emissions from manufacturing, and the incorporation of policy factors, such as energy mix, taxation, and subsidies.Therefore, it is difficult to extend a model based on optimisation methods in previous studies of LCO.The reasons are as follows.First, the objective function for minimisation is complicated by several additional assumptions (e.g., battery replacement required for long-term use and vehicle inspections).Second, the calculation patterns and computation time increases as fuel types and study periods increase.
To address this problem, this study proposes an LCO framework that represents vehicle replacement patterns as weighted directed graphs and applies Dijkstra's algorithm [19] to graph theory.A graph in this context comprises nodes (circles 1-6 in figure 1) connected by edges.Specifically, by assigning consumer choices to edges and emissions from consumer choices to the weight of the edges, the replacement strategy with the smallest cumulative emissions can be interpreted as the path with the smallest weight.As shown in figure 1, path 1-2-4-6 has the lowest total emissions of 25 and is the optimal strategy.
The optimisation method using graphs to derive the shortest path can contribute to LCO in that it can efficiently compute optimisation problems with many replacement patterns for the following reasons: (i).The increase in computational complexity is small compared with the increase in the number of replacement patterns, making it possible to analyse the replacement patterns for each company or vehicle model over a long study period.
(ii).It is easy to intuitively add complex conditionalities, such as the irregular addition of conditions (e.g.battery replacement) and parameter changes, allowing us to respond to future restraint changes, set various policy scenarios, and extend the model to various durable goods, such as home appliances (e.g.air conditioners and lighting).
(iii).Once the graph for vehicle replacement patterns is defined, existing solution methods, such as Dijkstra's and the k-best algorithms [20,21], can be used to derive the optimal solution.
This case study identifies vehicle switching paths that minimise life-cycle CO 2 emissions in Japan, considering changes in energy mix and vehicle manufacturing technologies.

Defining a transition graph
This study identified an optimal transition strategy that minimised emissions using k fuel types for Y years.Let }be a set of car types.The variables are summarized in table 1. Battery replacement was also considered.Our method formulates the optimal transition strategy problem as a well-known network optimisation called the shortest path problem.Intuitively, a route in the input graph corresponds to a strategy.
In this formulation, node v t l y , , represents a state in which one uses a car of type t for l years as of year y.Each link from a node is connected to a node corresponding to a possible strategy implemented the following year.For example, consider a situation in which one uses a car of type t for l years in year y.If he/she keeps using the car, a link from the corresponding node v y t l , , is connected to node + + v , y t l 1, , 1 representing that he/she uses a car of type t for + l 1 years as of year + y 1.The edge weight (distance) is defined as the annual emissions of a type t car.This value can be computed in advance.In the constructed network, each node in the shortest path represents a strategy for each year to minimise emissions.The shortest path problem can be solved efficiently using Dijkstra's algorithm [19].Therefore, it is necessary to construct a network.In the following section, we explain the construction of the network in detail.

Nodes
For year y (   y Y 0 ), car type Î t T, and usage period l (   l L 0 ), we define a node v , , , which denotes a state one uses a car of type t for l years as of the year y.Moreover, we created two special nodes, s and g.Node s  is the starting point and g is the goal, the shortest path between nodes s and g is the optimal replacement choice, and its distance (cumulative weight) is the minimum cumulative emission.Thus, the network contains nodes.Note that redundant nodes exist in the network (e.g.node v t 1, ,2 represents a state in which one uses a car of type t for 2 years as of year 1.This node is invalid since the usage period is 2 years even though it was only purchased for 1 year.).Such invalid cases are deleted.

Edges
In each year, the replacement strategy involves + k 1 different strategies, including continuing to use the current vehicle or switching to another car of type ¢ Î t T. Let us consider links from node v .
[Case: Continuing to use the current vehicle] In this case, as of the next year, the current car usage period will be extended by 1 year.Thus, we connect node v y t l , , to + + v , y t l 1, , 1 which represents the state in which one uses a type t car for + l 1 years in year + y 1.For the edge 1, , 1 the weight is defined by the amount of emissions that a car of type t emits in the year + y 1 when the car is used for + l 1 years.We also considered that battery replacement would be required every 8 years.Therefore, we add the emissions induced by battery production to the weight of 1, , if l is a multiple of 8 [5].We assume that the replaced cars are new (not used cars).Moreover, cars used for 14 years-that is, cars with 15 years of usage-were forcibly scrapped [22].
[Case: Switching to another car of type ¢ Î t T] If we replace the current car of type t with a new car of type ¢ t , we will use the car with zero years of use in year + y 1.If it occurs, we assume that car replacement is implemented at the beginning of a year.Thus, we connect node v y t l , , to + ¢ v , y t 1, ,0 which represents the state in which a new car of type ¢ t is used in year + y 1.The weight of the edge 1, ,0 is defined by the amount of CO 2 emissions produced during the manufacturing of a car of type ¢ t and the emissions when the new car is used for 1 year.Finally, we add to the network edges of weight zero from s to v y t l , , with = y 0 and edges of weight zero from v y t l , , with = y Y to t.The total weight (total distance) of the shortest path corresponds to the amount of emissions from the optimal replacement strategy.

Example
Figure 2 shows an example of a network for determining the optimal strategy with two types of cars over 3 years.The red path represents the path corresponding to the strategy.On this path, the owner buys a new Type 1 car in year 0.Then, he/she keeps the car every year 1.In other words, cars were used for 1 year.At the beginning of year 1, the Type 1 car was replaced by a new Type 2 car.Therefore, the emissions are from Type 1 car disposal, Type 2 car manufacturing, and Type 2 car driving emissions.These emissions are the weight of edge 2, 2,0 Thus, the total emissions are the driving emissions of Type 1 cars for 1 year between years 0 and 1, the emissions due to switching cars at the beginning of year 1, and those of Type 2 cars for 1 year between years 1 and 2; this corresponds to the sum of the weights of the edges on the red path.We attempted to determine the shortest path corresponding to a strategy that minimises the emissions in such networks.
In this study, we used the k-best algorithm to display the top k optimal path and suboptimal paths, which was modified to allow for flexible determination of the optimal timing of transfers and vehicle types.We have included Python code for the shortest path problem and k-best algorithm, and graph example in the Supplementary Information (SI1).
2.5.Calculation of weight of edges 2.5.1.Annual CO 2 emissions 2.5.1.1.Fuel consumption and direct emissions in the tank-to-wheel phase The tank-to-wheel emissions (tailpipe emissions) of a type t vehicle (ICEV, HEV, and PHEV) in the manufacturing year m, T W m 2 , t ( ) can be calculated by the CO 2 emissions from gasoline combustion and the fuel efficiency of each fuel type t.However, the tank-to-wheel emissions of BEV were assumed to be 0. In this study, PHEV are assumed to run 40% on gasoline and 60% on electricity.The driving distance was assumed to be 10000 km per year.See SI (tables S2-1-S2-5, and SI3) for further data, such as fuel efficiency and details of the calculation.

Estimation of the carbon footprint of automobiles
Life-cycle emissions were calculated by combining the input-output table for the analysis of the next-generation energy system (IONGES) [23], which covers inventories for each fuel type (ICEV, HEV, PHEV, and BEV) and environmental extended input-output analysis [24,25].We used the life-cycle inventory data for vehicles using each fuel type from Input-Output table for Analysis of Next-Generation Energy System [26].Owing to data limitations, this study assumes that the input-output structure for the target years (2020-2050) is constant for 2015.
Combining the inventory data and input-output analysis, we can obtain the carbon footprint of each life stage of each fuel type vehicle.The carbon footprint of the manufacturing of a fuel type t vehicle in year y is denoted by mfg y t ( ) where the annual reduction in manufacturing emissions is assumed to be 1.5% [27].In addition, the CO 2 emissions from the disposal of fuel type t vehicles is described by disposal .)is defined as the well-to-tank emissions in year y of the fuel type t vehicle manufactured in year m, CO 2 emissions for well-to-wheel is calculated as: Finally, the carbon footprint at battery replacement, battery l y , , t ( ) and vehicle inspection, insp l , t ( ) for vehicle type t can be defined.This study assumes that the CO 2 emissions from battery replacement will decrease annually similar to that of manufacturing.The details of the calculations are in the Supplementary Information.

Weight of edges
Here, we define the weights of the edges in the replacement network.When choosing to switch vehicle t owned for l years in period y, the cost of disposing of the vehicle owned in the previous period, manufacturing emissions, and driving emissions of vehicle type ¢ t (vehicle switched to) are incurred, and the emissions are obtained as follows: In addition, if vehicle t continues to be driven, there will be driving emissions for vehicle t in years l.As an additional constraint in this case study, we assumed that battery replacement occurs in year l if it is a multiple of 8 [5], and vehicle inspection occurs every 2 years after vehicle purchase.Repair and maintenance occur annually, and an EV home charger is installed when switching from an ICEV or HEV to a PHEV or BEV.
Here, m r& t and charge are the CO 2 emissions associated with battery production, car inspection, repair and maintenance, and the installation of an EV home charger, respectively.
Note that we identified the shortest paths that minimised the emissions from 2020 to 2055 and analysed the period 2020-2050 of the identified paths.Dijkstra's algorithm determines the optimal path for year y based on the optimal paths in year - y 1. Basically, this is the case when the optimal path in year y and the optimal path in year + y 1 do not change significantly.However, the choices at the end of the period may change.For example, replacement with an ICEV or HEV, which are vehicles with low manufacturing emissions in 2049 if the vehicle owned has a lifespan in 2048 (details in SI5).This is because there is no time for the higher manufacturing emissions to be compensated by lower operating emissions.Thus, we recommend using a longer period than our study period.

Results
Figure 3 shows the cumulative CO 2 emissions from the top five optimal paths and baseline path.Note that the baseline assumes longer vehicle use and switching to BEV before 2035, and its emission is 45.6 tons.In this simplified case study, the optimal emission reduction path would be 42.8 tons with initial PHEV ownership (Opt_4).It is 2.8 tons (6%) less than the baseline path's emissions.The replacement sequence is as follows: 1st car: PHEV driven for 14 years (until lifetime) → 2nd car: PHEV driven for 10 years → 3rd car: BEV driven for 7 years (figure 3).Path Opt_1 is to drive an HEV initially and replace PHEV as the second vehicle (in 2028) and the BEV as the third vehicle (in 2042).The length of ownership of the initial vehicle is slightly different from other paths with initial PHEV ownership.In addition, all BEV replacements in the optimal paths were after 2042.The key factors affecting a choice are the manufacturing and driving phase (See SI4 for CO 2 emissions at each life stage of each vehicle type).Specifically, well-to-tank emissions of BEVs are higher than that of ICEVs until 2042.Transition to clean energy urgently needs to be promoted to encourage BEV use.The paths of switching vehicle before battery replacement are not at the top of the list.In other words, although battery replacement is crucial in life-cycle emissions, it did not have much impact on the timing of vehicle replacement.
Table 2 shows ownership period and cumulative CO 2 emissions of vehicles replaced in 2020, 2030, and 2040.The green highlighted part represents the fuel type with the lowest cumulative emissions in the year.For HEV, the emissions from manufacturing can be absorbed in 2 years by a reduction in the driving phase, but the manufacturing emissions cannot be recouped for BEVs manufactured in 2020 and 2030.The cumulative emissions of ownership of BEVs manufactured in 2030 will not be less than even the emissions of HEVs until 14 years ownership.Considering Japan's average ownership period of vehicles, the choice of BEVs would be appropriate after 2040.

Discussion
Case studies were conducted for four fuel types (ICEV, HEV, PHEV, and BEV).From the simplified case study, we can suggest that, given the current levels of technological innovation and emission improvements, the broad dissemination of BEVs after 2042 would be appropriate and vehicle electrification should proceed gradually from GVs and HEVs to PHEVs and then to BEVs.Notably, the attainment using our novel approach of an optimal strategy comparable to that found by other methods [12][13][14][15][16][17][18] supports the validity of this replacement strategy.For example, as can be seen from the results of optimal paths, Opt_1 and Opt_2-4, the choice of vehicle type makes a difference in the ownership period.The strategies that use PHEVs over their lifetime (Opt_2-4) [12][13][14][15] and switch from HEVs to PHEVs early before their lifespan (Opt_1) [16][17][18] are consistent with previous research.Based on our findings, we can discuss optimal transition policies that can maximize the contribution of vehicle electrification to decarbonisation.Our analysis is a simple, versatile, and low computation time framework because it performs complex calculations for emissions from each life stage in advance and then identifies optimal strategies from the computation amount ) All calculations were done using Dell Precision 3450 (11th Gen Core i7-111700, 2.5 GHz, 16MB cache, 32GB memory) and MATLAB 2022a.In this case study, the calculation results were obtained for the top five paths in approximately 12 s, using fuel type = k 4, study period = y 36, and vehicle lifespan = L 14 as parameters.The increase in computation time is small for an increase in these parameters.Therefore, the calculation can be performed without difficulty even if the study period is increased, light-duty vehicles are considered, or lifespan is changed for inter-regional comparisons.In addition, changes in annual distance and annual reduction ratio of manufacturing emissions affect the graph construction but does not affect graph optimisation.Basically, different driving patterns can be handled depending on how the graph is created.However, Dijkstra's algorithm determines the optimal path for year y based on the optimal paths in year y 1, and it is difficult to define the path after year y in such a way that the choice depends on previous choices before years - y 2 (e.g., the second car will have an extended life only if the first car is replaced early).The calculation effort for the methodology in this study is equivalent to the conventional approach.This study used input-output life-cycle assessment (IO-LCA) for the calculation of the carbon footprint of each life stage and construction of the vehicle replacement graph.IO-LCA is a major LCA tool, and calculation is easy, with low computation time.Input-output databases can be obtained easily and freely, and environmental data, such as CO 2 emissions are also available.However, the construction of the graph does not depend on how it is calculated.Depending on the country, type of car, and durable goods, the life-cycle inventory that has been constructed previously may be used (e.g.Nakamoto et al [5] calculated and published emissions for each life stage of ICEVs and AFVs for 2020-2050).
The results of this study should be interpreted flexibly.For example, Opt_3-4 has small differences in both paths and emissions.It is recommended to examine the top paths and make policy decisions flexibly.Because there is not only one optimal solution, multiple optimal solutions can be calculated.Thus, it is important to comprehensively evaluate various perspectives for feasible policy making.In addition, a comprehensive assessment including consumer costs and another environmental indicator (e.g., NO x ) in the optimal paths should also be conducted.
It should be noted that our framework has some limitations.First, depending on the time period, the optimal paths may change somewhat, and a less appropriate strategy may emerge (e.g.switching to HEV as the third car in 2049).In such cases, it is necessary to set a longer study period or remove the less appropriate path from the multiple optimal paths.Alternatively, because computation time does not change much even if Y is changed, it is possible to set Y from 1 to 100 and determine the optimal path for each Y .
Second, this case study is based on strong assumptions.For global comparison or scenario analysis, sensitivity analysis is important because results can vary widely depending on annual driving distance, lifespan, battery replacement span, emissions during manufacturing, and energy mix.In addition, data collection should be adapted to regional and national characteristics, as they vary greatly depending on the conditions of each assumption.Third, we cannot optimise the emissions and budgets simultaneously.In this case, it is necessary to use multi-objective optimisation, but it would be a large computational load.Our methodology, identifying multiple suboptimal paths could offer policymakers and product users a more practical selection, such as a path expected to yield substantial emission reduction with less financial cost and more feasibility than the optimal replacement trajectory in previous studies.

Conclusion
This study developed the formulation of graphs for life-cycle optimisation.The methodology in this study is flexible for various fuel types and conditions (e.g.additional taxation, additional emissions, and events in a given year), and the computation can be accomplished quickly.In this study, we calculated the replacement paths for four fuel types over 30 years, considering energy mix scenarios, technological progress, and some other parameters.
In this simplified case study, the optimal way to reduce CO 2 emissions would be to switch first to PHEVs and then to BEVs.The transition to electric vehicles requires a step-by-step process.Reducing CO 2 emissions from manufacturing vehicles and driving through the green electricity generation are significant goals for increasing the adoption rate of BEVs in the short-term.
The method used in this study is useful because the calculations can be performed only by updating the data and adjusting the operations on the target edges for various assumptions.Because there are many libraries for computing graph optimisation problems (e.g.The Boost Graph Library in C ++, Networkx in Python, Graph and Network Algorithms in MATLAB), we can easily solve the shortest path problem.Owing to its augmentability and reduced computation time, exhaustive calculations can be performed in response to changes in various parameters, constraints, and variables related to the product under analysis.This methodology is not only conducive to AFV deployment for decarbonisation but can also be applied to products, such as air conditioners and lighting, which have been analysed in previous studies.Furthermore, identifying multiple suboptimal paths could offer policymakers and product users a more practical selection, such as a path expected to yield substantial emission reduction with less financial cost and greater feasibility than the optimal replacement trajectory in previous studies.

Figure 2 .
Figure 2. Example of the network for determining the optimal strategy using two types of cars for 3 years.

Figure 3 .
Figure 3. Cumulative CO 2 emissions in the top five optimal paths and baseline path.

Table 1 .
Variables and definitions.
y Year t Fuel type l Usage period  l y ( ) Y Study period k Number of fuel type L Maximum usage period v y t l , ,Node (state in which one uses a car of type t for l years as of year y) ¢ e v vEdge (choice of move to v' at state v)

Table 2 .
Ownership period and cumulative CO 2 emissions of vehicles replaced in 2020, 2030, and 2040.Bold numbers indicate reversal of the fuel type with the lowest cumulative emissions, and the green highlighted numbers represent the fuel type with the lowest cumulative emissions in the year.