Rolling resistance force (Froll) is a resistive force that opposes the motion of a rolling body: (1)Froll=μrmg,where the rolling friction coefficient (μr) typically has a value of 0.01–0.015 and it depends on the tire type and the pressure inside of the tire [19]. The value 0.012 is chosen in this paper. The value for free fall acceleration is 9.81m/s2 [20].

Total mass (m) is defined as a sum of the empty vehicle mass (m0), battery mass (mbat) that is proportional to its capacity and the load mass (mload). In this study, different load types (such as passengers, luggage, or commercial payload) are represented in an aggregated manner through the additional load mass mload. This approach assumes that the primary influence of these loads on vehicle dynamics and energy consumption is captured via the total added mass, rather than through secondary effects like changes in the center of gravity or aerodynamics. Consequently, the load type is not explicitly modeled as a separate variable but is parameterized through mload to maintain model simplicity and computational efficiency. In this way, the variable mload indirectly affects the whole energy consumption and range of a vehicle. Renault Zoe has a curb weight of 1577 kg [21], and a battery that weights 326 kg [22], whereas Tesla Model 3 has a curb weight of 1760 kg [23], and a battery that weights 480 kg [24].

Aerodynamic drag (Faero) is a loss that is generated because of the friction when an object moves through the air. It depends on the shape of an object [20] as follows: (2)Faero=0.5ρCdAv2(t),where ρ is the density of air (1.295kg/m3) [25], Cd is the aerodynamic drag factor that increases when a vehicle has a poorly designed aerodynamic shape (for regular cars Cd has a value of 0.3 [19]), A is the frontal area of a vehicle and v is vehicle velocity.

Note that the frontal area of Renault Zoe is approximately 2.79 m2 [21], while Tesla Model 3 has a frontal area of 2.67 m2 [23].

Hill climbing force (Fhc) is a force that is needed to drive the vehicle up a slope [19] as follows: (3)Fhc=mgsin⁡θ,where sinθ corresponds to the slope grade which is defined as a ratio between changed height and changed distance and can be changed depending on the desired hill slope. For small angles sin θ≈tan⁡θ≈grade.

In the basic simulation grade is equal to zero to avoid big energy results and too big consumption. Instead, hill climbing force is contained in the requirement to drive uphill through the “performance point”. At a given speed and a given slope, the power of the drive is calculated then penalized. That approach keeps the consumption and the whole simulation on a realistic level and at the same time satisfies demanded grade.

As the vehicle speed changes over time, inertial force (Finert) can be calculated based on the vehicle acceleration a(t) as follows: (4)Finert=ma(t).

Traction force (Ftrac) needs to overcome all aforementioned resistances to motion (Eqs. (1)–(3)) and the inertial force (Eq. (4)) during driving, and is therefore defined as [19] (5)Ftrac(t)=Froll+Faero+Fgrade+Finert.

The required driveline mechanical power (Pmech) which needs to be provided must by the propulsion system can be calculated as [20]: (6)Pmech(t)=Ftrac(t)v(t),where v is vehicle velocity in (m/s).

Electric vehicle motor torque T_m can be reconstructed from mechanical power and motor angular speed as follows [20]: (7)Tm(t)=Pmech(t)wm,where wm is the angular velocity that is equal to ratio of the vehicle velocity (v(t)) and the size of the tire, which is 0.33 m according to [26].

All the losses within the electric motor need to be accounted for. The following equation was adjusted and simplified, so the variables like torque can be directly used and minimized by replacing constants with coefficients [27] as follows: (8)Ploss=kfric+kCuTm2+kFewm2,where kfric are the constant mechanical losses [W] representing friction in bearings, transmission and has a value of 400 [W], then kCuTm2 are the copper (ohmic) losses in the windings and the higher the torque, the higher the losses. The coefficient kCu has a value of 2⋅10−3 W. The term kFewm2 represents the iron losses that depend on the speed of rotation because they are caused by the change in magnetic flux in the cores. The coefficient kFe has a value of 2⋅10−5 W. Coefficients (kfric, kCu and kFe) are assumed parameters that physically correspond to different types of losses (friction, copper, iron). They are used because they provide a sufficiently precise, yet simple model for analyzing motor performance. These values are chosen so that the total losses (expressed in Watts) are within a realistic range.

Output drive power delivered to the inverter (equal to the sum of mechanical power and losses, limited by the maximum drive power) is given by: (9)Pout=min(Pmech(t)+Ploss(t),Pmax⋅1000).

This considers that the drive system cannot deliver more power than its peak (rated) value [28]. DC power that battery needs to deliver to the inverter according to [20] is: (10)Pdc(t)=Pout(t)ηinv,where Pout(t) is the output power (defined in Eq. (9)), and ηinv is the efficiency of the inverter. The efficiency of the inverter is usually above 95% [29]. In this paper the value for the efficiency of the inverter is selected to be 0.97.

The battery model is based on a simplified calculation of energy flow and state of charge (SoC). Momentary battery current (I(t)) (with auxiliary consumption (Paux) that compensates all electric consumers that are not directly involved in drivetrain) is given by the derivative of the main formula for electrical power at DC bus [30]: (11)I(t)=Pdc(t)+PauxVnom,where Pdc(t) is an DC power output with auxiliary consumption (Paux) that compensates all electric consumers that are not directly involved in drivetrain. Auxiliary consumption (Paux) has a value of 500 W [31]. Nominal voltage of the battery Vnom for Renault Zoe is 350 V, and for Tesla Model 3 is 357 V [32].

State of charge (SoC) is calculated iteratively for each simulation step (Coulomb Counting) [33] with the negative sign indicating the discharge regime of the EV battery: (12)SOCk+1=SOCk−I(t)ΔtQ,where Q is the capacity of the battery, and k is the time step (time increment, with the simulation time step Δt = 1 s used in this work). The gradual discharge of the battery during the driving cycle is modelled in this way. The conversion of battery energy to capacity in ampere-hours (Ah) is based on the standard relationship Ah=Wh/V [34]. Battery charge capacity Q is defined in this study in the following way: (13)Q=Ebat⋅fusable⋅1000Vnom,where Ebat[kWh]is a nominal capacity of the battery, fusable is the usable share of the battery charge (energy) and the value of the usable share is 0.9 [35]. Total battery capacity of Renault Zoe is 52.0 kWh, while Tesla Model 3 battery has a total battery capacity of 78.1 kWh [32].

The total energy drawn from the battery during the simulation (Ebatt,out) in Watt-hours (Wh) is calculated as the sum of momentary values of DC power: (14)Ebatt,out=∑kmax(Pdc,kΔt3600,0),where Pdc,k is the current power that battery delivers at simulation step k. It includes the power required for the motor and inverter, motor and inverter losses and auxiliary consumption. The factor 1/3600 converts [Ws] to [Wh]. Max operator ensures that only discharge energy accumulates, while regenerative breaking is not considered (conservative assumption). In standardized driving cycles such as WLTC, regenerative braking can significantly reduce net energy consumption, particularly in urban and transient phases characterized by frequent decelerations. The inclusion of regenerative braking would primarily reduce the total energy drawn from the battery and, consequently, increase the achievable driving range for a given battery capacity. As a result, incorporating regenerative braking into the model would likely shift the optimization results toward lower optimal battery capacities, especially for smaller vehicles. While this simplification may reduce quantitative accuracy when compared to real-world operation, it ensures conservative sizing and computational efficiency in early stages of vehicle design.

The total energy consumption during the driving cycle (Econs) is calculated as the sum of the momentary DC power delivered by the battery to the inverter: (15)Econs=∑tPdc(t)Δt3600,where Pdc(t) is the instantaneous DC power delivered by the battery to the inverter. Δt is the simulation time step in seconds. While Ebatt,out gives a realistic estimate of how much energy is actually drawn from the battery (without returning energy), Econs represents the total energy required by the system to move the vehicle during the driving cycle.

The travelled distance (s) is calculated as the integral of the vehicle velocity over time: (16)s=∫0Tv(t)dt.

Specific consumption is defined based on the total energy consumption and travelled distance [36] as follows: (17)spec=Econss.

Specific consumption in Wh/km indicates how much energy a vehicle consumes per kilometre, which allows comparison of different scenarios and configurations.

Finally, estimated range of the vehicle is defined as the amount of usable energy in the battery divided by specific consumption (spec). A realistic estimate of the driving range based on the driving cycle is derived from the relationship between the range (in km) and the specific energy consumption: (18)range=Ebat⋅fusable⋅1000spec,where Ebat is the nominal battery capacity [kWh], fusable is the usable share of the battery energy, and the factor 1000 converts [kWh] to [Wh].

The optimization algorithm used in this study is the so-called DIRECT (Dividing RECTangles). DIRECT, which is a deterministic sampling method that is designed for bound-constrained non-smooth problems in a small number of variables. It is applicable to engineering design problems, in which complicated simulators are used to construct the objective function. Sampling occurs at the centers of hyperrectangles. In each iteration, the method divides existing hyperrectangles and then evaluates the objective function at the centers of the newly formed sub-hyperrectangles [37].

In this study, the DIRECT method is implemented in the MATLAB environment using WLTC 3b driving cycle data [38]. The maximum number of objective function evaluations and the number of iterations is limited to ensure reasonable computational complexity and simulation runtime. This method ensures that the found solution is sufficiently close to the global optimum, which is important considering the complexity of the electric vehicle model and the multiple conditions (range, power, charge/discharge rate (C-rate), SoC limits, performance point) that must be met.

In order to optimize parameters of an electric vehicle, it is necessary to define an objective function (J) which the algorithm minimizes. The objective function includes capital expenses (CAPEX), operating expenses (OPEX) and penalties that ensure that the vehicle meets the given conditions: (19)J=CAPEX+OPEX+penalties.

Vector of variables that are being optimized is defined as follows:(20)x=[Ebat[kWh],Pmax[kW],mload[kg]],where m_load is the useful vehicle payload.

Capital expenses depend on the battery capacity, the peak motor power and their prices:(21)CAPEX=cbatEbat+cmotorPmax,where cbat stands for the battery price, the price that is used in simulation is 120 EUR/kWh [39], and cmotor stands for the motor and inverter prices. The whole electric drive module costs around 570 EUR/kW [40]. The latter is a very conservative cost estimate, but realistic for the electrical motor plus inverter according to [40].

Operating expenses are calculated based on the estimated specific energy consumption during the cycle and the price of electricity:(22)OPEX=cel(Wh/km1000)N,where cel represents the price of electricity from the main utility grid, with the selected value in this study to be 0.2 EUR/kWh [41], and N is the reference distance in kilometers.

As shown above, CAPEX is linear, while OPEX represents the cost of energy at a given distance. The unit cost values are expressed in [EUR/kWh], [EUR/kW], [EUR/kWh_el]. They are in a realistic range, and they enable comparison between different configurations. The objective function is expressed in euros, but this does not represent the real market prices of the vehicle but serves as a comparative measure that equalizes the different contributions (battery, motor, energy consumption).

If the solution does not meet any of the conditions, the objective function is increased, adding a penalty and searching for a configuration that meets all the conditions. The conditions that are penalized are: (a)

Insufficient range (if less than the specified minimum)

The penalty terms used in the optimization are detailed in Table 1. To ensure a robust search process, quadratic penalization is applied to range violations to strongly discourage configurations that do not meet the primary mission requirement. Meanwhile, linear penalties are employed for power, C-rate, SoC, and performance constraints. This formulation is designed to preserve the smoothness of the objective function landscape, facilitating the convergence of the DIRECT algorithm while ensuring that all performance constraints are strictly respected.

By combining all mentioned expenses and penalties, the overall objective function takes the form:(23)J=CAPEX+OPEX+penrange+penpower+pencrate+pensoc+penperf,where pen_range, pen_power, pen_crate, pen_soc and pen_perf are penalty factors related to insufficient range, insufficient motor power, C-rate factor limitations, SoC limit and performance point, as discussed above. Such an approach allows the optimization algorithm to search for the battery, motor and mass configuration that provides the best compromise between cost, energy efficiency and performance.

The optimization framework shown in Fig. 1 follows a sequential information flow starting from the driving cycle input, which provides the vehicle speed profile as a function of time. Together with the optimization variables and fixed vehicle parameters, the driving cycle is processed by the vehicle dynamics model to calculate the traction force and mechanical power demand. This power demand is then passed to the electric motor and inverter model, where electrical losses and power limitations are considered. The resulting DC power demand is supplied by the battery model, which computes the State of Charge (SoC) evolution and total extracted energy. Based on these results, the specific energy consumption and driving range are calculated. Finally, all relevant cost components and penalty terms are combined within the objective function, the value of which is minimized using the DIRECT optimization algorithm.

For the Renault Zoe, the optimizer converged to Ebat=52 kWh, Pmax = 110 kW and mload = 114 kg with an objective function value of J = 17,246 EUR. The WLTC simulation yields a specific consumption of 196 Wh/km and an estimated range of 238 km based on the usable battery energy (90% of nominal). The peak drivetrain output during the cycle is 49.6 kW, well below the installed Pmax, indicating adequate performance margin and no power-limit penalty. The state of charge decreases from 0.95 to 0.85, remaining safely above the SoC minimum.

Fig. 2 shows the WLTC speed profile for Renault Zoe operating under WLTC driving cycle conditions. Fig. 3 shows the local sensitivity of the objective function J to three optimization variables for Renault Zoe: battery capacity (Ebat),peak motor power (Pmax) and payload (mload). It is visible that the battery capacity has the greatest impact on the value of the objective function, which is expected because it directly affects the investment cost, but it also directly affects the specific consumption and range of the vehicle. The peak power of the electric motor has a smaller impact, since the values of its maximum power are rarely required during the WLTC cycle. The smallest impact is shown by the additional load, although the increase in mass increases energy consumption and reduces the range.

Fig. 4 shows the contours of the objective function J for Renault Zoe as a function of the battery capacity (Ebat) and the peak power of the electric motor (Pmax), with the load mass (mload) fixed at the optimal value. The yellow dot indicates the optimal solution. It is noticed that the optimum is located to the left of the red line, which means that the vehicle in this case does not meet the minimum range requirement (300 km). However, this result is expected given all the losses included in the model (mechanical, electrical and auxiliary), which realistically reduce the effective range of the vehicle. This confirms the importance of introducing penalty conditions into the objective function, because the algorithm still finds the configuration that best balances costs and performance, even if some requirements (such as range) are not fully met.

Fig. 5 contains 3 graphs. The first graph (a) shows the vehicle speed over time, which corresponds to the driving cycle profile and varies from a standstill to approximately 130 km/h. The second graph (b) shows the motor output power (Poutpeak). During acceleration and driving at higher speeds, there are significant power spikes, while at rest and at low speeds, the demands are minimal. The third graph (c) shows the change over time in the battery state of charge (SoC). A gradual and almost linear discharge of the battery is visible over the duration of the cycle. The final SoC value at the end of the simulation is 0.85, confirming that the vehicle had sufficient energy available to meet the power demands throughout the cycle.

In the case of Tesla Model 3 vehicle, the optimizer converged to Ebat=131.9 kWh, Pmax = 128 kW and mload = 120.939 kg with an objective function value of J = 27,041 EUR. The WLTC simulation yields a specific consumption of 227 Wh/km and an estimated range of 523 km based on the usable battery energy (90% of nominal). The peak drivetrain output during the cycle is 59.6 kW, well below the installed Pmax, indicating adequate performance margin and no power-limit penalty. The state of charge decreases from 0.95 to 0.91, remaining safely above the SoC minimum.

Fig. 6 shows the local sensitivity of the objective function J to three optimization variables for Tesla Model 3: battery capacity (Ebat),peak motor power (Pmax) and payload (mload). It is visible that the peak power of electric motor (Pmax) has the greatest impact on the value of the objective function. This means that the choice of peak motor power plays a crucial role in the balance between price, performance and energy consumption. On the other hand, the additional load (mload) has a medium level of influence on the objective function, which makes sense because the increase in mass directly affects the energy consumption and thus the range. The smallest impact is shown by the battery capacity (Ebat), because the optimization is already in the area where an additional increase in capacity does not bring the proportional benefits.

Fig. 7 shows the contours of the objective function J for Tesla Model 3 as a function of the battery capacity (Ebat) and the peak power of the electric motor (Pmax), with the load mass (mload) fixed at the optimal value. The yellow dot indicates the optimal solution. It is noticed that the optimum is located to the left of the red line, which means that the vehicle in this case does not meet the minimum range requirement (550 km).

However, this result is expected given all the losses included in the model (mechanical, electrical and auxiliary), which realistically reduce the effective range of the vehicle. This confirms the importance of introducing penalty conditions into the objective function, because the algorithm still finds the configuration that best balances costs and performance, even if some requirements (such as range) are not fully met.

Fig. 8 again contains 3 graphs. The first graph (a) shows the vehicle speed over time, which closely follows the driving cycle profile and varies from a standstill to approximately 130 km/h. The second graph (b) shows the motor output power (Poutpeak). During acceleration and driving at higher speeds, there are significant power spikes, while at rest and at low speeds, the demands are minimal. The third graph (c) shows the change over time in the battery state of charge (SoC). A gradual and almost linear discharge of the battery is visible over the duration of the cycle. The final SoC value at the end of the simulation is approximately 0.91, confirming that the vehicle had sufficient energy available to meet the power demands throughout the cycle.