In the evolving landscape of electric mobility, the development of range-extended electric vehicles (EREVs) represents a critical bridge between pure battery electric vehicles and conventional hybrids. As an engineer focused on optimizing powertrain systems, I recognize that the core challenge lies in efficiently matching energy sources—the power battery and the range extender—to meet diverse performance targets without redundancy. Traditional approaches often involve iterative simulations and validation loops, leading to prolonged development cycles and potentially over-sized components. This inefficiency is particularly pronounced in the context of battery electric vehicle architectures, where the battery pack is the primary energy source, and its integration with a range extender introduces complex interactions. To address this, I have developed a parametric matching model using MATLAB, which transforms the energy source selection problem into a computationally efficient interpolation task. This article details the methodology, from parameterizing vehicle demands and energy source characteristics to constructing a full-factorial model and applying it to a practical case. The goal is to provide a systematic framework that enhances design precision, reduces weight and cost, and accelerates the development process for EREVs, all while ensuring that the fundamentals of battery electric vehicle performance are preserved or enhanced through strategic hybridization.
The essence of an EREV is its dual-energy-source system: a high-voltage battery pack that functions as the primary energy source, similar to a pure battery electric vehicle, and a range extender—typically an internal combustion engine coupled to a generator—that acts as an auxiliary power unit to extend driving range. This configuration allows the vehicle to operate in pure electric mode for daily commutes, leveraging the cost and environmental benefits of a battery electric vehicle, while mitigating range anxiety through the onboard generator. However, the integration poses a significant matching problem: the battery and range extender must be sized to collectively satisfy vehicle performance requirements across various driving conditions, including acceleration, gradeability, and endurance, without unnecessary oversizing. In traditional development flows, the range extender is often sized based on peak sustained power demands in charge-sustaining mode, and the battery is then selected to meet all-electric range targets. This sequential approach can lead to suboptimal outcomes, especially with the trend toward downsized range extenders, where the battery must compensate more actively for power deficits. Consequently, the matching process becomes iterative, requiring repeated simulations to verify performance under dual-source operation, which is time-consuming and may result in redundant battery capacity. My objective was to streamline this by creating a parametric model that directly links design parameters to performance outcomes, thereby enabling rapid exploration of the design space and identification of minimal-sufficient solutions.
To achieve this, I first parameterized the vehicle’s matching requirements, which are derived from driving conditions. The energy source must supply sufficient electrical power to the drive motor to overcome vehicle resistances: rolling resistance, aerodynamic drag, grade resistance, and acceleration inertia. The required motor input power, \(P_{in,MO}\), is calculated based on these forces. The total driving resistance, \(F_t\), is given by the sum of individual resistances, each expressed as functions of vehicle speed, \(V\), road grade, \(i\), and acceleration, \(a\). For computational efficiency, rolling and aerodynamic resistances are often combined into a road load equation derived from coast-down tests:
$$F_R = A + B \times V + C \times V^2$$
where \(F_R\) is the total road load force in Newtons, \(V\) is the speed in km/h, and \(A\), \(B\), and \(C\) are coast-down coefficients specific to the vehicle. Grade resistance, \(F_i\), depends on the vehicle mass, \(m\), gravitational acceleration, \(g\), and road grade, \(i\):
$$F_i = m \times g \times \sin(\arctan i)$$
Acceleration resistance, \(F_j\), accounts for translational and rotational inertia through a mass factor, \(\delta\):
$$F_j = \delta \times m \times a$$
Thus, the total force at the wheels is \(F_t = F_R + F_i + F_j\), and the required motor output torque, \(T_{out,MO}\), and input power, \(P_{in,MO}\), are:
$$T_{out,MO} = \frac{(F_R + F_i + F_j) \times r_w}{\eta_T \times i_T}$$
$$P_{in,MO} = \frac{T_{out,MO} \times n_{MO} \times \pi}{\eta_{MO} \times 30000}$$
where \(r_w\) is the dynamic wheel radius in meters, \(\eta_T\) is the transmission efficiency, \(i_T\) is the gear ratio, \(n_{MO}\) is the motor speed in rpm, and \(\eta_{MO}\) is the motor efficiency. For modeling purposes, I discretized driving conditions into two types: non-time-varying fixed conditions (constant speed and grade) and time-varying cycle conditions (standardized driving cycles). The parameters for these are summarized in Table 1, which outlines the discrete intervals used to create a sample space for model inputs. This parameterization allows the model to cover a wide range of operational scenarios, from highway cruising to aggressive hill climbs, ensuring that the energy source matching is robust across real-world usage patterns, much like those encountered in a battery electric vehicle but with the added complexity of range extender engagement.
| Condition Type | Parameter | Lower Bound | Upper Bound | Step Size |
|---|---|---|---|---|
| Non-Time-Varying | Vehicle Speed, \(V\) (km/h) | 5 | 200 | 2 |
| Road Grade, \(i\) (%) | 0 | 10 | 0.1 | |
| Travel Time, \(t\) (min) | 10 | 500 | 10 | |
| Time-Varying | Cycle Type (e.g., WLTC, Highland High-Speed) | — | — | — |
| Number of Cycles, \(n\) | 1 | 10 | 0.1 |
Next, I parameterized the energy source selection itself. The power battery, a key component inherited from battery electric vehicle technology, is modeled using an internal resistance equivalent circuit. Each cell is characterized by its open-circuit voltage, \(U_{open}\), and internal resistance, \(R_0\), both of which vary with the state of charge (SOC). The battery pack is assembled by connecting cells in series, and its properties scale with the number of series cells, \(N_s\). For a given cell type, the pack’s open-circuit voltage, internal resistance, maximum charge power, \(P_{maxCha,BA}\), and maximum discharge power, \(P_{maxDis,BA}\), are:
$$U_{open,BA} = N_s \times U_{open}$$
$$R_{BA} = N_s \times R_0$$
$$P_{maxCha,BA} = N_s \times P_{maxCha,cell}$$
$$P_{maxDis,BA} = N_s \times P_{maxDis,cell}$$
The battery capacity, \(Q_{BA}\), in ampere-hours, is equal to the cell capacity if no parallel connections are used. The range extender, on the other hand, is simplified to its maximum sustained discharge power, \(P_{maxin,RE}\), assuming that engine-generator matching and efficiency variations are encapsulated within this parameter. This reduction in variables helps avoid dimensionality issues in the model. The selection parameters are discretized as shown in Table 2, enabling a comprehensive sweep of the design space. By treating the battery’s series cell count and the range extender’s power as continuous variables within bounds, the model can explore trade-offs between the two energy sources, akin to optimizing a hybridized battery electric vehicle system for cost and performance.
| Component | Parameter | Lower Bound | Upper Bound | Step Size |
|---|---|---|---|---|
| Power Battery | Cell Series Count, \(N_s\) | 0 | 400 | 2 |
| Range Extender | Max Sustained Discharge Power, \(P_{maxin,RE}\) (kW) | 0 | 60 | 0.06 |
With the driving conditions and energy source parameters defined, I constructed a full-factorial matching model in MATLAB. The model’s independent variables are the driving condition parameters (e.g., \(V\), \(i\), \(t\), or cycle type and \(n\)) and the selection parameters (\(N_s\) and \(P_{maxin,RE}\)). The dependent variable is the battery’s remaining SOC at the end of the driving period, \(SOC_{end}\), which serves as a proxy for whether the energy sources can sustain the required performance without depleting the battery below a target threshold. The core of the model is an iterative calculation that updates the SOC over time based on the power balance at the battery terminals. At each time step, the battery terminal power, \(P_{end}(t)\), is determined by the difference between the range extender’s output, the motor’s demand, and auxiliary loads:
$$P_{end}(t) = P_{in,RE}(t) – P_{in,MO}(t) – P_{in,AC}(t)$$
Here, \(P_{in,RE}(t)\) is set to \(P_{maxin,RE}\) to assess peak capability, \(P_{in,MO}(t)\) is derived from the driving condition using the earlier equations, and \(P_{in,AC}(t)\) is a constant accessory load. The battery current, \(I(t)\), is then calculated from the equivalent circuit model:
$$I(t) = \frac{-U_{open}(t) + \sqrt{U_{open}^2(t) + 4 \times R_{BA}(t) \times P_{end}(t)}}{2 \times R_{BA}(t)}$$
provided that \(P_{end}(t)\) meets the battery’s power limits. The change in SOC over a time increment \(\Delta t\) is:
$$\Delta SOC(t) = \frac{I(t) \times \Delta t}{Q_{BA} \times 3600}$$
and the SOC is updated as \(SOC_{end}(t + \Delta t) = SOC_{end}(t) + \Delta SOC(t)\), starting from an initial SOC, \(SOC_0\). Constraints are applied to handle cases where power demands exceed battery limits, assigning invalid outputs (NaN) to ensure model robustness. This formulation results in two model functions: one for non-time-varying conditions, \(SOC_{Con,end} = f(N_s, P_{maxin,RE}, V, i, t)\), and another for time-varying cycles, \(SOC_{Var,end} = f(N_s, P_{maxin,RE}, x, n)\), where \(x\) denotes the cycle type. The model essentially generates high-dimensional response surfaces that map design parameters to performance outcomes, enabling rapid interpolation for any given set of inputs.
The parametric model reveals insightful trends about energy source interactions in EREVs. For fixed driving conditions, the battery’s ending SOC is positively correlated with both \(N_s\) and \(P_{maxin,RE}\), but as travel time accumulates, the dependence on the range extender’s power becomes dominant, highlighting that sustained performance is ultimately limited by the auxiliary source. This mirrors a key consideration in battery electric vehicle design, where the battery alone must meet all demands, but here the range extender can offset battery depletion. For cyclic conditions, such as WLTC or highland drives, the model shows that milder cycles may be achievable with the battery alone, while severe cycles necessitate combined power from both sources. These insights underscore the importance of co-optimizing battery and range extender sizes, rather than treating them sequentially, to avoid oversizing the battery—a common pitfall in attempts to hybridize a battery electric vehicle platform.
To demonstrate the model’s utility, I applied it to a compact EREV case study. The vehicle had a target of maintaining performance with a downsized range extender (\(P_{maxin,RE} = 30\) kW) and a battery sized minimally to achieve a 150 km all-electric range on the WLTC cycle, akin to a typical battery electric vehicle range expectation. The matching requirements included: (1) sustained cruising at 135 km/h on level road for 1000 km, (2) climbing a 5% grade at 90 km/h for 30 minutes, (3) WLTC all-electric range of 150 km, (4) completion of a highland high-speed cycle, and (5) completion of a highland hill-climb cycle. All targets were translated into a desired ending SOC of 10% (i.e., 90% depth of discharge) from an initial SOC of 90%. Using the parametric model, I interpolated the response surfaces for each condition to find the curve of \(N_s\) versus \(P_{maxin,RE}\) that yields \(SOC_{end} = 10\%\). For the WLTC all-electric range requirement, \(P_{maxin,RE}\) is set to 0, simulating pure battery electric vehicle operation, which gave a minimum \(N_s\) of 206 cells. For the other conditions at \(P_{maxin,RE} = 30\) kW, the required \(N_s\) values were lower, such as 116 cells for the cruising case. However, to account for low initial SOC scenarios—critical for real-world robustness—I repeated the analysis with \(SOC_0 = 50\%\) and \(SOC_0 = 20\%\). The results, summarized in Table 3, show that as the starting SOC decreases, the required battery size increases, with the hill-climb condition demanding up to 256 cells at \(SOC_0 = 20\%\). This indicates that to ensure performance across all charge states, the battery should be sized at 256 cells, even though the all-electric range target alone would suggest 206 cells. This holistic view prevents under-sizing and ensures that the EREV performs reliably, much like a well-designed battery electric vehicle, but with extended versatility due to the range extender.
| Initial SOC, \(SOC_0\) | Matching Condition | Required \(N_s\) at \(P_{maxin,RE} = 30\) kW | Notes |
|---|---|---|---|
| 90% | Sustained Cruise (135 km/h, 0%) | 116 | Below all-electric range requirement |
| Grade Climb (90 km/h, 5%) | 81 | Minimal demand | |
| WLTC All-Electric Range | 206 | With \(P_{maxin,RE} = 0\) kW | |
| Highland High-Speed Cycle | 32 | Easily met | |
| Highland Hill-Climb Cycle | 5 | Trivial requirement | |
| 50% | Sustained Cruise (135 km/h, 0%) | 130 | Increased from 90% SOC case |
| Grade Climb (90 km/h, 5%) | 89 | Higher demand due to lower SOC | |
| WLTC All-Electric Range | 206 | Unchanged (pure electric mode) | |
| Highland High-Speed Cycle | 64 | Moderate increase | |
| Highland Hill-Climb Cycle | 5 | Still minimal | |
| 20% | Sustained Cruise (135 km/h, 0%) | 256 | Substantial increase, critical sizing factor |
| Grade Climb (90 km/h, 5%) | 200 | Second most demanding | |
| WLTC All-Electric Range | 206 | Unchanged (pure electric mode) | |
| Highland High-Speed Cycle | 142 | Significant rise | |
| Highland Hill-Climb Cycle | 5 | Remains low |
The parametric approach offers several advantages over conventional methods. First, it drastically reduces the time for energy source selection by replacing iterative simulations with interpolation on pre-computed response surfaces. Second, it provides a clear visualization of trade-offs between battery size and range extender power, facilitating informed decisions that minimize cost and weight. Third, it ensures non-redundant matching by identifying the exact parameter combinations that meet performance thresholds, avoiding the over-sizing common in traditional battery electric vehicle derivations where safety margins are often added arbitrarily. Importantly, the model maintains flexibility; by switching the discrete variables (e.g., cell type or vehicle platform), it can be adapted to different EREV projects, making it a versatile tool in the development toolkit for electrified vehicles.
In conclusion, the parametric matching model developed here represents a significant step forward in the design of range-extended electric vehicles. By parameterizing both driving conditions and energy source characteristics, and by leveraging MATLAB for efficient computation, the model transforms a complex, iterative selection problem into a straightforward interpolation task. The application to a compact EREV demonstrated its effectiveness in identifying minimal-sufficient battery and range extender sizes that meet diverse performance targets, including those stemming from battery electric vehicle heritage like all-electric range. This methodology not only enhances development efficiency but also contributes to lighter, more cost-effective vehicles, supporting the broader adoption of electrified transportation. As the automotive industry continues to evolve toward electrification, tools like this will be crucial for optimizing hybrid architectures that balance the purity of a battery electric vehicle with the practicality of extended range, ultimately benefiting consumers and the environment alike.