As a clean and efficient means of personal mobility, the electric car has drawn global attention in the context of energy shortage and environmental concern. Compared with conventional internal combustion engine automobiles, the electric car is driven by a motor powered by a battery pack, and it offers a promising route to reduce petroleum consumption and greenhouse gas emissions. However, the limited driving range of the electric car remains one of the most critical barriers to its wide adoption. To extend the driving range without increasing the size and weight of the battery pack, regenerative braking has become a vital technology. When the driver of an electric car presses the brake pedal, the kinetic energy of the vehicle can be partially converted back into electrical energy by operating the traction motor as a generator. Recovering the braking energy directly improves the overall energy efficiency of the electric car and thereby increases its mileage. In this article, we focus on a series regenerative braking control strategy for a pure electric car. We combine the ideal braking force distribution curve (I-curve) with the ECE regulation curve (M-curve) to develop a new front-and-rear axle braking force allocation scheme. Then, we design a fuzzy controller that takes battery state of charge (SOC), vehicle speed at the onset of braking, and braking intensity as inputs, and outputs the regenerative braking force distribution coefficient. We employ a co-simulation approach using MATLAB/Simulink and AVL Cruise. The simulation results under the New European Driving Cycle (NEDC) confirm that the proposed fuzzy control strategy can achieve a higher regenerative energy recovery ratio than a conventional fixed front-and-rear braking force distribution strategy.
The significance of braking energy recovery for the electric car cannot be overstated. During ordinary city driving, frequent braking events provide a substantial opportunity for energy recuperation. Studies have shown that up to about fifteen percent of the energy consumed in an electric car driving cycle can be recovered through a well-designed regenerative braking system. In a typical urban route, the driver continuously decelerates and stops at intersections, traffic lights, and congested sections. Without regenerative braking, the kinetic energy of the electric car is turned into heat by friction brakes and dissipated to the ambient air, wasting a large amount of usable energy. If that energy is successfully captured and stored in the battery, the state of charge of the battery declines more slowly, and thus the total driving range of the electric car grows. This is especially important for small battery electric cars that have a modest battery capacity. Moreover, reducing the usage of mechanical brake pads also reduces particulate emissions and maintenance costs, which contributes to the sustainability of the electric car ecosystem.
Many prior investigations have explored different aspects of regenerative braking. Some researchers developed an integrated control strategy that combines anti-lock braking protection with regenerative braking so that the recovered energy can be maximized while the wheels remain stable. Other authors carried out theoretical analyses for a parallel regenerative braking system in a pure electric car and designed a control model that respects the driver’s braking feeling. Their simulation results indicated that the model can effectively extend the driving range. More recently, a group considered the road adhesion coefficient and studied a compound braking strategy for a dual-motor pure electric car, which increased the braking energy recovery proportion. Another line of study used particle swarm optimization together with fuzzy logic to optimize the distribution between the front and rear axle braking forces as well as the ratio of regenerative braking to mechanical braking for an electric car. In addition, some works have focused on the rate of change of the motor torque demand based on fuzzy control, considering the battery state as a constraint, and verified through experiments that the braking energy recovery can be effectively improved. Furthermore, for hybrid power systems using fuel cells and supercapacitors, a strategy based on Pontryagin’s minimum principle was proposed to maximize regenerative energy recovery while minimizing hydrogen consumption. These studies demonstrate the importance of an appropriate control strategy for maximizing the energy recovery capability of an electric car. In this article, we contribute to this field by proposing a series regenerative braking allocation strategy for a pure electric car and a fuzzy controller to adapt the regenerative braking fraction in real time.
The organization of the paper is as follows. In Section 2, we classify regenerative braking systems and describe the architecture of a series regenerative braking system. Section 3 establishes the vehicle dynamic model, including the vertical load forces, the ideal braking force distribution, and the ECE regulations. Section 4 introduces the proposed front-and-rear axle braking force distribution strategy. Section 5 describes the design of the fuzzy controller, including membership functions and fuzzy rules. Section 6 shows the arrangement of the co-simulation and gives vehicle parameters. Section 7 presents and discusses the NEDC simulation results, comparing three control models. Finally, Section 8 draws conclusions from this research.
2. Regenerative Braking System Architecture
Regenerative braking systems for an electric car can be roughly divided into parallel and series types. In a parallel regenerative braking system, the mechanical braking force and the regenerative braking force act independently. In other words, the mechanical brake force is always present during a braking action, and its magnitude is determined by the conventional brake pipeline and brake pedal ratio. When the electric car brakes, the motor applies a regenerative drag torque to the wheels which is added on top of the mechanical braking torque. The parallel structure is relatively simple and inexpensive, requiring only minor modifications to the existing hydraulic brake system. However, because the regenerative braking contribution is not managed in a coordinated way, the typical amount of recovered energy is rather low. The motor is often not used at its optimal operating region, and the mechanical brakes may consume a large portion of the kinetic energy unnecessarily.
In contrast, a series regenerative braking system gives priority to the regenerative braking torque. Whenever the driver requests a deceleration, the controller first determines the maximum torque that the traction motor can generate in the generating mode under the current battery SOC, temperature, and vehicle speed. If the motor torque can satisfy the entire required braking torque, only regenerative braking is applied and the mechanical brakes are disconnected. If the available motor torque is lower than the required torque, the mechanical brakes are then coupled to make up the deficiency. This arrangement permits a significantly larger amount of the vehicle kinetic energy to be converted into electricity, and therefore the energy recovery efficiency is higher than that of a parallel arrangement. The drawback is that a series system needs a more complex control unit and an accurate communication network between the motor inverter, the battery management system, the hydraulic brake module, and the vehicle controller. Nevertheless, due to the increasing demand for long-range electric cars, series regenerative braking systems are becoming more prevalent in modern electric car designs.
A typical series regenerative braking system for a pure electric car is composed of several key components: the motor, the motor controller or power electronics unit (PEU), the battery pack, the brake pedal position sensor, the wheel speed sensors, and the vehicle control unit (VCU). During a braking request, the VCU calculates the total required braking force based on the displacement and force applied to the brake pedal. Depending on the control strategy, the VCU then splits the required braking force into two paths: one path is sent to the electric motor as a negative torque command, and the other path is converted into a hydraulic pressure command for the front and rear wheel brake cylinders. The motor, operating as a generator, produces a drag torque on the drivetrain and simultaneously charges the battery through the PEU. The mechanical braking system is activated only when the motor’s maximum regenerative torque is insufficient to meet the required braking force, or when safety conditions, such as emergency braking or high battery SOC, force the controller to limit or disable the regenerative function. Such architecture ensures that the regenerative braking operation is always consistent with the current status of the electric car and the road conditions.
3. Vehicle Dynamics and Brake Force Distribution
To design an effective braking force distribution for the electric car, we first derive the longitudinal dynamic equations during braking. The analysis treats the electric car as a rigid body moving in a straight line on a flat road. For simplicity, we neglect the rolling resistance from the wheels, the aerodynamic drag, and the inertial resistances caused by rotating components. The front and rear axle vertical loads can be expressed by the following equilibrium equations around the rear contact point and the front contact point respectively:
$$F_{z1} = \frac{G \left( b + z h_g \right)}{L}, \qquad F_{z2} = \frac{G \left( a – z h_g \right)}{L}$$
In these equations, \(F_{z1}\) and \(F_{z2}\) are the normal reactions on the front axle and rear axle, respectively; \(G\) is the total gravitational force of the electric car; \(a\) is the horizontal distance from the center of gravity to the front wheel center, \(b\) is the horizontal distance from the center of gravity to the rear wheel center; \(h_g\) is the height of the center of gravity above the road; \(L\) is the wheelbase; and \(z\) is the braking intensity, defined as the ratio of the deceleration to the gravitational acceleration \(g\). The braking intensity \(z\) is positive for normal braking and can be written as:
$$z = \frac{a_{\text{dec}}}{g}$$
where \(a_{\text{dec}}\) is the magnitude of the vehicle deceleration. The maximum ground braking force on each wheel is limited by the tire-road adhesion coefficient \(\varphi\). If both front and rear wheels are on the point of locking, the ground braking forces become equal to the available adhesion forces:
$$F_{xb1} = \varphi F_{z1}, \qquad F_{xb2} = \varphi F_{z2}$$
Here \(F_{xb1}\) and \(F_{xb2}\) denote the ground braking forces of the front and rear axle, respectively. In an ideal braking condition, the front wheels and rear wheels lock simultaneously. Such a condition ensures both high braking efficiency and good directional stability. From the above equations, the relationship between the front and rear axle braking forces can be derived to yield the so-called ideal braking force distribution curve, or I-curve:
$$F_{xb2} = \frac{1}{2} \frac{G}{h_g} \sqrt{b^2 + \frac{4 h_g L}{G} F_{xb1}} – \left( \frac{G b}{2 h_g} – F_{xb1} \right)$$
The I-curve is plotted in the coordinate system where the horizontal axis is the front axle braking force and the vertical axis is the rear axle braking force. Any point lying on the I-curve represents a distribution of the tandem braking forces that, under a given road adhesion coefficient, causes both axles to lock at the same time. If the actual braking force distribution is always under the I-curve, then the front wheels will lock before the rear wheels, which is considered a stable condition because the vehicle has a tendency to go straight rather than spin out. If the distribution goes above the I-curve, then the rear wheels tend to lock first, which is dangerous due to potential vehicle instability. Consequently, an effective braking force distribution strategy for an electric car must keep the operating point as close as possible to the I-curve without surpassing it, while also respecting the laws and regulations.
Beside the I-curve, another important boundary is the ECE R13 braking regulation curve, sometimes denoted as the M-curve. The ECE regulation specifies that for passenger cars with a braking intensity \(z\) in the range between 0.2 and 0.8, the road adhesion utilization curves must satisfy the following inequality:
$$z \ge 0.1 + 0.85\left( \varphi – 0.2 \right)$$
The combination of this regulation with the vehicle geometric and inertial parameters gives a boundary line known as the M-curve. The M-curve represents the minimum allowable braking force that can be contributed by the rear axle for a given front axle braking force, so that the vehicle can meet the legal braking performance requirements. In the conventional plot, the region between the M-curve and the I-curve is regarded as the safe area where the brake force distribution may be chosen while still satisfying both the braking stability and statutory requirements. For a pure electric car with regenerative braking, the controller should allocate the total braking force in such a way that the resultant point lies inside this safe region. Moreover, because the front axle usually bears more load during braking due to the load transfer, it is often advisable to utilize the front motor regenerative braking torque to recover the maximum amount of energy for an electric car with front-wheel drive.
4. Proposed Front-and-Rear Axle Braking Distribution Strategy
The braking demand of an electric car varies with the driver’s operation and the traffic environment. Normally, the driver perceives a light braking action while slowing down smoothly, and a high braking intensity during an emergency. To recover more braking energy, the front axle should be favored because the dynamic load on the front axle increases markedly during braking. Therefore, in this article we put forward a distribution strategy based on the braking intensity \(z\) as follows.
Case 1: For very light braking, when \(z < 0.1\), the total braking force demanded by the driver is relatively small. Since a modern electric car’s traction motor is capable of producing a sufficiently large torque in the generating mode, the entire braking force can be supplied by the front axle regenerative brake. In this situation, the mechanical brakes on both the front and rear axles are completely inactive. The front motor then provides the solely braking torque, charging the battery while decelerating the electric car. This choice maximizes the regenerative energy recovery for low-intensity braking conditions and ensures a smooth response.
Case 2: For moderate braking, when \(0.1 \le z < 0.7\), the total required braking force is beyond the maximum torque capacity of the motor or beyond the acceptable charging power limit of the battery. In this range, the mechanical braking system has to participate. To improve the energy recovery efficiency, we purposely allocate more braking torque to the front axle because the front axle has a larger vertical load under deceleration. Specifically, the braking force distribution follows a polygonal line defined by the points A, B, and C in the coordinate system of front axle force and rear axle force. From the origin to point A, the front axle force increases while the rear axle force remains zero. After point A is reached, the rear axle force increases rapidly to point B, whereas the front axle force is maintained at a constant allowable level. Once the distribution arrives at point B, the subsequent distribution follows a constant braking ratio \(\beta\) which is predetermined for the mechanical brakes. In this study, we set \(\beta = 0.65\), meaning that the front axle mechanical brake force accounts for 65% of the total mechanical braking force, while the rear axle accounts for the remaining 35%. The regeneration controller proportionally adjusts the front axle regenerative torque and mechanical torque such that the actual front-to-total force ratio does not violate the safety boundaries. Actually, in this moderate region, the algorithm first calculates the total front axle braking force according to the point on the chosen borderline, then the electronic control unit splits the front axle braking torque between the motor and the front friction brake. The regenerative part is limited by the motor capability and the allowed charging current, and the difference is supplemented by the front mechanical brake. Meanwhile, the rear axle is only operated by the hydraulic mechanical brake. This distribution ensures that the front motor can recover as much braking energy as possible during the common urban deceleration processes of the electric car.
Case 3: For emergency braking, when \(z \ge 0.7\), the driving safety becomes the dominant concern. In this case, the regenerative braking may be switched off entirely in order to avoid any adverse effect on the anti-lock braking system and on the directional stability of the electric car. The braking force distribution should then follow the ideal braking force distribution curve to make full use of the adhesion between the tires and the road. The motor stops generating torque and the entire braking force is handled by the conventional hydraulic brakes. This conservative choice is commonly made in the literature because modern electric cars give top priority to safety when the deceleration demand is high.
The proposed distribution strategy is illustrated in the brake-force plane. The region between the M-curve and I-curve defines a feasible envelope for safe braking. Our strategy lies inside this envelope between point A and point B, and then transitions to the constant \(\beta\) line. By choosing the borderlines precisely, we guarantee that the front wheels do not lock prematurely and that the rear axle does not lose stability even under low-adhesion road surfaces. Through this design, the electric car is able to retain excellent braking safety while achieving a higher regenerative energy recovery fraction than the conventional fixed-ratio strategy.
5. Fuzzy Control Strategy for Regenerative Braking
Even though a static allocation rule can improve energy recovery compared to a fixed-ratio distribution, the actually optimal regenerative braking fraction for an electric car changes with the battery SOC, the vehicle speed, and the braking intensity. For example, when the battery is fully charged, accepting electricity from regenerative braking is not allowed because it would overcharge the cells and damage the battery. Similarly, if the vehicle speed is very low, the kinetic energy that can be recovered is small and the motor-generator may not operate efficiently, so it might be more effective to reduce the regenerative braking fraction. If the braking intensity is extremely high, the stability of the electric car demands that the motor contribute less and friction brakes provide the majority of the torque. Hence a real-time adaptive strategy is required. Fuzzy control is well suited for this purpose: it is a nonlinear and model-free control method that uses linguistic variables and rule-based reasoning, exhibiting strong robustness and adaptability to uncertain and time-varying systems. Consequently, we design a fuzzy controller with three inputs and one output.
The three inputs are the battery state of charge (SOC), the vehicle speed at braking onset (V), and the braking intensity (z). The output is the regenerative braking distribution coefficient \(k\), which represents the fraction of the total front axle braking torque that is supplied by the regenerative motor. Equivalently, \(k\) can be interpreted as the percentage of total required braking force that is produced by the electric motor; a larger \(k\) implies greater energy recovery. The fuzzy controller determines \(k\) according to the current operating state of the electric car.
For the input variable SOC, its membership functions are represented by trapezoidal functions. The fuzzy sets are divided into three linguistic labels: L (low), M (medium), and H (high). The universe of discourse is [0,1]. When the SOC is smaller than 30% or larger than 90%, the system should limit the regenerative braking proportion to protect the battery life. More specifically, if the SOC is too low, the battery may not safely accept a large charging power; if the SOC is too high, the battery cannot store additional energy without risking overcharge. Thus, SOC is one of the principal inputs to limit the value of \(k\).
For the input variable vehicle speed \(V\), the universe is [0,120] km/h and the membership functions are also of the trapezoidal form, with three labels: L, M, H. The kinetic energy of the electric car is proportional to the square of the vehicle speed; thus, a higher braking speed means a greater amount of energy is available to be regenerated. At the same time, the motor’s back electromotive force and its maximum power capability determine the maximum recoverable power. At higher vehicle speeds, it is often feasible to apply a larger regenerative braking torque while still remaining within the motor’s electric safety limits. Therefore, in general, a larger coefficient \(k\) can be applied at high V than at low V, unless other limitations apply.
For the input variable braking intensity \(z\), its universe is [0,1], meaning from no deceleration to the maximum possible braking force. We use the labels L, M and H to represent small, medium and large braking intensity. According to our earlier allocation strategy, a light braking operation (\(z<0.1\)) can be completely handled by regenerative braking, but the fuzzy controller should smooth the transition and reduce \(k\) when \(z\) approaches zero because the motor’s energy efficiency at very low torque is low. For moderate braking intensities, a higher fraction can be regenerated. For an emergency braking event, \(k\) should fall to a very low value to ensure the safety of the electric car. The membership function of \(z\) is also constructed with trapezoidal shapes.
The output variable \(k\) is assumed to range between 0 and 1. Its fuzzy subsets are composed of five linguistic values: VL (very low), L (low), M (medium), H (high), and VH (very high). Triangular membership functions are used for simplicity. In the actual control implementation, the center-of-gravity defuzzification method is adopted to obtain the crisp value of \(k\).
To build the fuzzy rule table, we apply the following knowledge: when the battery SOC is high (H), the charging ability is limited, so the regenerative braking coefficient is set to VL regardless of the vehicle speed and braking intensity. When the battery SOC is low (L), the battery is willing to accept a large amount of energy, but the actual regenerative force must still be restricted at high braking intensities or at very low vehicle speeds to maintain comfortable and stable braking. Therefore, for SOC=L and brake intensity L, the coefficient \(k\) becomes medium to high when the vehicle speed is high; however, for an emergency braking operation, \(k\) should be low or very low. When the SOC is medium (M), the system can make full use of the available charging capability; the value of \(k\) is mostly determined by the vehicle speed and braking intensity. In general, larger \(k\) is selected for large V and moderate z, while smaller \(k\) is selected for low V and high z.
Based on the aforementioned principles, the fuzzy control rules are summarized in the following table. The inputs SOC, z, and V each have three labels, giving 27 rules. The output label is selected as VL, L, M, H, or VH.
| No. | SOC | z | V | k |
|---|---|---|---|---|
| 1 | L | L | L | L |
| 2 | L | M | L | L |
| 3 | L | H | L | VL |
| 4 | L | L | M | M |
| 5 | L | M | M | H |
| 6 | L | H | M | M |
| 7 | L | L | H | H |
| 8 | L | M | H | VH |
| 9 | L | H | H | M |
| 10 | M | L | L | H |
| 11 | M | M | L | M |
| 12 | M | H | L | L |
| 13 | M | L | M | VH |
| 14 | M | M | M | H |
| 15 | M | H | M | H |
| 16 | M | L | H | VH |
| 17 | M | M | H | VH |
| 18 | M | H | H | H |
| 19 | H | L | L | VL |
| 20 | H | M | L | VL |
| 21 | H | H | L | VL |
| 22 | H | L | M | VL |
| 23 | H | M | M | VL |
| 24 | H | H | M | VL |
| 25 | H | L | H | VL |
| 26 | H | M | H | VL |
| 27 | H | H | H | VL |
In the above table, the SOC label L represents a low state of charge (below about 30%), M means a medium state (between 30% and 90%), and H means a high state (above 90%). Notice that for the high SOC condition, the output \(k\) is always VL because the battery cannot absorb appreciable energy. For SOC = L and a large braking intensity H, the output \(k\) is VL or M to avoid relying on a high motor torque that may not be possible or safe. For SOC = M and small braking intensity, the coefficient can be VH so that during gentle braking process the electric car can recover the majority of its kinetic energy. The proposed rules ensure that the regenerative braking function of the electric car is always adapted to the external operating conditions and to the internal limitations of the battery and the motor. This yields a balance between recovery efficiency and braking stability.
6. Co-Simulation Platform and Vehicle Parameters
To verify the efficacy of the fuzzy-based regenerative braking strategy, we carry out a co-simulation between AVL Cruise and MATLAB/Simulink. AVL Cruise is a powerful simulation environment that can model an entire vehicle driveline, including the electric machine, battery, gearbox, wheels, and driver model. It is particularly suitable for calculating fuel consumption, emissions and driving range for various powertrains. MATLAB/Simulink allows us to implement the fuzzy controller easily using the Fuzzy Logic Toolbox and to generate a dynamic link library (DLL) that can be integrated into AVL Cruise. The co-simulation workflow is described as follows. First, we establish the electric car vehicle model in AVL Cruise by selecting the proper components and connecting them according to the power flow. The mechanical signals are represented by blue lines, and electrical signals are represented by red lines. The signal buses are configured to pass the relevant quantities between the vehicle model and the controller. Then, we build the fuzzy control algorithm in MATLAB/Simulink, compile it into a DLL file, and import it into the MATLAB DLL block inside AVL Cruise. During the simulation, AVL Cruise calculates the vehicle states and sends the current battery SOC, vehicle speed, and braking intensity to the Simulink model. The model computes the regenerative braking coefficient \(k\) and then returns it to AVL Cruise, where the actual brake torque distribution is applied to the wheels and the motor. This closed-loop interaction takes place at each simulation time step, allowing the controller to react in real time to the changing driving cycle.
The vehicle selected for the simulations is an existing pure electric car model from the AVL Cruise library. The main parameters used for the modeling are listed in the following table. Since this research is focused on a small urban electric car, a curb mass of 1210 kg and a wheelbase of 2467 mm are reasonable. The motor is able to deliver a maximum torque of 240 N·m and a peak power of 77 kW. The battery pack is specified with a capacity of 10 A·h and a rated voltage of 320 V. The drivetrain efficiency is set to 96%, and the final drive ratio is 6.06. The wheel rolling radius is 301 mm.
| System | Parameter | Value |
|---|---|---|
| Vehicle | Curb mass \(m\) / kg | 1210 |
| Vehicle | Wheelbase \(L\) / mm | 2467 |
| Vehicle | Distance from CG to front axle \(a\) / mm | 1200 |
| Driveline | Final drive ratio | 6.06 |
| Driveline | Transmission efficiency | 0.96 |
| Wheel | Tire rolling radius \(r\) / mm | 301 |
| Motor | Maximum torque \(T_{\max}\) / N·m | 240 |
| Motor | Maximum power \(P_{\max}\) / kW | 77 |
| Battery | Capacity \(C\) / A·h | 10 |
| Battery | Rated voltage \(U\) / V | 320 |
After constructing the vehicle model in AVL Cruise, we set up a simulation task based on the New European Driving Cycle (NEDC). The NEDC consists of four repeated urban driving cycles and one extra-urban driving cycle, with a total distance of about 11.04 km and a duration of 1180 seconds. In many battery electric car tests, the NEDC has traditionally been used to evaluate the energy consumption and driving range. For our simulation, we set the initial battery SOC to 85% and the vehicle loading condition to the driver-only scenario, while the ambient temperature and other environmental conditions are fixed at standard values as defined in the simulation. We then perform three simulations to compare three different control models: a no-regenerative-braking model, a model with a fixed front-rear braking force ratio (i.e. 65% front / 35% rear) without fuzzy control, and our proposed fuzzy control model. All other parameters and conditions remain the same among the three simulations so that the effect of the regenerative braking controller can be isolated.
7. NEDC Simulation Results and Discussion
The three vehicle models are all run over one complete NEDC cycle. Figure shows the SOC evolution of the battery over the cycle for the three cases. It has been observed that during the initial moments of the cycle, the SOC curves for the three control models almost overlap with each other. This is because the proportion of regenerative braking is relatively small early in the driving schedule, or the vehicle has not yet encountered a significant stopping event; therefore, the accumulated recovered energy difference is still small. As the driving progresses, the difference among the curves becomes apparent. The no-regeneration model always has the lowest SOC because the only energy consumed is the traction energy and no energy is fed back into the battery. The fixed-ratio model has a higher SOC than the no-regeneration model because the regenerative brake recovers some amount of braking energy whenever the total braking force is distributed with a fixed ratio and the motor is allowed to generate electricity. The fuzzy control model achieves the highest SOC at the end of the cycle due to its adaptive adjustment of the regenerative braking coefficient.
At the end of a single NEDC driving cycle, the no-regenerative-braking control model has a remaining SOC of 74.76%, the fixed-ratio strategy model yields a remaining SOC of 75.77%, and the fuzzy control model yields a remaining SOC of 76.13%. Thus, compared to the no-regeneration case, the fixed-ratio strategy increases the remaining SOC by 1.01 percentage points, while the fuzzy strategy increases it by 1.37 percentage points. In comparison with the fixed-ratio model, the fuzzy model improves the terminal SOC by 0.36 percentage point. Although this absolute SOC difference after one driving cycle seems rather small, it is actually meaningful in terms of energy recovery because the NEDC contains many stop-and-go segments and the total energy recovered per cycle is only a small fraction of the battery energy capacity. The accumulated benefit would become much larger if many cycles are considered, as in everyday driving.
In order to evaluate the energy saving benefit in a more rigorous way, we introduce an indicator called the relative energy-saving contribution \(S\). This indicator is defined as the normalized difference between the SOC reduction value without regenerative braking and the SOC reduction value with a certain regenerative braking strategy. Let \(\Delta \text{SOC}_{\text{noregen}}\) be the total decrease in SOC during the driving cycle when no regenerative braking is used, and let \(\Delta \text{SOC}_{\text{strategy}}\) be the total decrease of SOC with a particular regenerative braking strategy. Then the relative energy-saving contribution is expressed by:
$$S = \frac{\Delta \text{SOC}_{\text{noregen}} – \Delta \text{SOC}_{\text{strategy}}}{\Delta \text{SOC}_{\text{noregen}}} \times 100\%$$
The measured SOC variations and the computed relative energy-saving contributions are given in Table 3. From the table, we observe that the no-regeneration model has a SOC reduction of 10.24 percentage points. The fixed-ratio strategy reduces the SOC consumption to 9.23 percentage points, giving a relative energy-saving contribution of 9.86%. The fuzzy control strategy reduces the SOC consumption to 8.87 percentage points, leading to a relative energy-saving contribution of 13.38%. Therefore, the proposed fuzzy control strategy increases the relative energy-saving contribution by 3.52% compared with the conventional fixed-ratio strategy. This result clearly shows that a fuzzy adaptive controller can harvest more braking energy for the electric car while maintaining the same driving cycle.
| Strategy | Initial SOC (%) | Terminal SOC (%) | SOC decrease (%) | Relative energy-saving contribution (%) |
|---|---|---|---|---|
| No regenerative braking | 85 | 74.76 | 10.24 | — |
| Fixed front-rear ratio strategy | 85 | 75.77 | 9.23 | 9.86 |
| Fuzzy control strategy | 85 | 76.13 | 8.87 | 13.38 |
Another meaningful comparison is made by calculating the equivalent driving range for the electric car when the state of charge is allowed to deplete from 85% down to 20%. This is a common range test procedure because a battery electric car cannot normally be fully discharged under regular driving. We run repeated NEDC cycles for each model until the terminal SOC reaches 20%. The maximum iteration number is set to 10, and the initial SOC is 85%. The resulting driving ranges are listed in Table 4. As shown, the no-regeneration model can travel a total distance of 67.79 km before the SOC drops to 20%. The fixed-ratio strategy model can travel about 73.84 km, while the fuzzy control model reaches about 77.99 km. Therefore, the proposed fuzzy strategy increases the driving range by about 10.2 km compared to the case without any regenerative braking, and by about 4.15 km compared to the fixed-ratio strategy. This demonstrates the tangible benefit of regenerative braking for an electric car: the daily driving radius is extended by several kilometers merely by recovering the kinetic energy that would otherwise be dissipated as heat.
| Strategy | SOC range (%) | Driving range (km) | Relative range contribution (%) |
|---|---|---|---|
| No regenerative braking | 85–20 | 67.79 | — |
| Fixed front-rear ratio strategy | 85–20 | 73.84 | 8.92 |
| Fuzzy control strategy | 85–20 | 77.99 | 15.05 |
To quantify the improvement in terms of driving range, we define the relative range contribution \(C_R\) by:
$$C_R = \frac{R_{\text{strategy}} – R_{\text{noregen}}}{R_{\text{noregen}}} \times 100\%$$
where \(R_{\text{noregen}}\) is the driving range of the electric car without regenerative braking, and \(R_{\text{strategy}}\) is the range with the concerned regenerative braking strategy. From the range values in Table 4, the relative range contribution of the fixed ratio strategy is 8.92%, while that of the fuzzy control strategy is 15.05%. Hence, the fuzzy fuzzy control strategy improves the relative range contribution by 6.13% compared with the fixed front-rear ratio strategy. All of these results consistently show that applying a fuzzy controller to modulate the regenerative braking coefficient can meaningfully increase the recovered braking energy and thereby extend the total driving range of the electric car.
The reason for this improvement lies in the fuzzy controller’s ability to adapt the regenerative braking fraction to the variable conditions during the NEDC. In the fixed-ratio strategy, a constant portion of the front axle braking torque is fed back through the motor regardless of the battery state and braking speed. Consequently, the system may sometimes recover excessive energy when the battery cannot accept it, forcing the controller to derate the charging power and waste some braking energy; in other situations the fixed ratio may be too conservative, leaving some recoverable energy unused. The fuzzy controller avoids such limitations by observing the current battery SOC in real time, by considering the vehicle speed to assess the available kinetic energy, and by taking the braking intensity as a measure of the safety constraint. With those inputs, the fuzzy rules choose a larger \(k\) only when the electric car is in an appropriate state to safely absorb the recuperated energy. On the contrary, \(k\) is reduced to a small value when the battery SOC is high, when the speed is too low to permit efficient regeneration, or when an emergency braking event has already appeared.
One additional point deserves attention. The electric car front motor is here used for regenerative braking on the driven axle, which is the front axle. Since the front axle dynamic load increases with braking, the front tires can withstand a larger braking force without locking, which is beneficial to energy recovery. The fuzzy logic strategy tends to allocate greater regenerative torque to the front axle within the safe boundaries of the braking force distribution, rather than simply using a fixed proportionality on the whole vehicle. This is one of the main reasons for the superiority of the fuzzy strategy relative to a non-adaptive fixed-ratio strategy. Moreover, the fuzzy controller is robust to parameter variations because it does not require an accurate analytical model of the nonlinear electric car powertrain. The shape and rules of the fuzzy controller can be tuned to accommodate different vehicle types and battery chemistries. Therefore, the present control strategy can serve as a useful reference for the design of regenerative braking controllers in future electric car models.
8. Conclusion
In this paper, we have presented a comprehensive study on regenerative braking energy recovery control for a pure electric car. The main conclusions of this research are listed as follows.
1. Under the premise of satisfying the I-curve and the ECE regulation, a new series-type regenerative braking force distribution strategy is proposed. According to the braking intensity \(z\), the controller chooses different ways to distribute the front and rear axle braking forces: for very light braking, only regenerative braking is activated; for moderate braking, a front-biased distribution is applied with the mechanical brakes on the rear axle and a hybrid regenerative plus mechanical brake on the front axle; and for emergency braking, regenerative braking is disabled and the braking force follows the ideal curve for safety.
2. A fuzzy controller with three inputs—battery SOC, vehicle speed at the onset of braking, and braking intensity—has been developed to adaptively adjust the regenerative braking force distribution coefficient. The controller is implemented in MATLAB/Simulink and integrated with AVL Cruise through a DLL-based co-simulation platform. The membership functions and fuzzy rules are presented in detail.
3. The co-simulation results under the NEDC show that after a single driving cycle with an initial SOC of 85%, the fuzzy control model gives a terminal SOC of 76.13%, which is 1.37% higher than the no-regenerative-braking model and 0.36% higher than the fixed front-rear braking ratio model. Based on the relative energy-saving contribution index, the fuzzy control strategy achieves a 13.38% contribution, which is 3.52% higher than the fixed-ratio strategy.
4. When the SOC decreases from 85% to 20% over repeated NEDC cycles, the fuzzy control model offers a driving range of 77.99 km, which is about 10.2 km more than the no-regeneration model and about 4.15 km more than the fixed-ratio model. The relative range contribution of the fuzzy control model reaches 15.05%, which is 6.13% higher than the fixed-ratio model.
In summary, the proposed fuzzy control-based regenerative braking strategy can notably enhance the energy recovery ability of a pure electric car without compromising braking safety. Future studies may extend this approach to four-wheel drive electric cars, incorporate road slope information and traffic signal information, and combine fuzzy logic with other intelligent algorithms such as particle swarm optimization or model predictive control for even better performance. Moreover, the influence of regenerative braking on the brake feeling and anti-lock braking system should be further investigated with hardware-in-the-loop tests and real-road experiments for commercial electric car applications.