A comparative study of energy management systems under connected driving: cooperative car-following case

2.1. Internal Combustion Engine Model

Since the internal combustion engine is a complex system that includes many components, an experimental dataset as a function of engine speed and engine torque is used in this study. The engine speed and torque express the engine fuel consumption ratio by Equations (1), and Equation (2) denotes the engine torque as follows.

where $$n_e(t)$$ denotes the engine speed, $$T_e(t)$$ denotes the engine torque, $$\alpha (t)$$ is the engine throttle, and $$T_{emax}(n_e(t))$$ is the engine maximum torque at the current engine speed.

2.2. Electric Motor Model

The purpose of electric motor (EM) modeling is to obtain the motor power based on motor speed. By ignoring the effect of the dynamic properties of the EM, Equation (3) expresses the efficiency of the motor as a function of the motor speed and motor torque. Then, the required engine power is defined by Equation (4).

where $$n_m(t)$$ denotes the motor speed, $$T_m(t)$$ denotes the motor torque, $${\eta }_m$$ is the motor efficiency, and $$P_m(t)$$ is the required motor power (kW).

2.3. Control-Oriented Powertrain Model

Taking advantage of the mixed powertrain configuration (both series and parallel) ^[42], a power-split HEV model of Toyota Prius is adopted in this study. This HEV powertrain model has been successfully used and commercialized as it has the advantages of the mixed configuration^{[43, 44]}. The power-split HEV model is shown in Figure 2.

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Figure 2. Structure of power-split hybrid electric vehicles ^[14].

The whole powertrain components of the power-split HEV include one engine, generator/motor (M/G1 and M/G2). The planetary gear set implementation achieves power splitting functionality where the engine is connected to the planet carrier, M/G1 is connected to the sun gear, and a torque coupler is used to combine the ring gear with the M/G2 to power the final drive ^{[45, 46]}. Figure 3 shows the structure of the planetary gear system. The kinematic equation of the gear system can be derived as the angular velocities of the sun gear, ring gear, and planet carrier.

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Figure 3. Structure and lever diagram of the planetary gear system equipped in a power-split HEV.

where the radius of the sun gear and the ring gear are denoted by $$S$$ and $$R$$ , respectively. Angular velocities of sun, ring, and carrier are given by $${\omega }_S(t)$$ , $${\omega }_R(t)$$ , and $${\omega }_C(t)$$ , respectively. If we assume that all powertrain shafts are rigid and neglect the pinion gears inertia, the powertrain dynamics can be expressed as follows:

where the inertias of the generator, engine, and motor are denoted by $$I_{M/G1}$$ , $$I_{eng}$$ , and $$I_{M/G2}$$ , respectively; the engine torque is $$T_{eng}(t)=T_C(t)$$ ; $$M/G1$$ torque is $$T_{M/G1}(t)=T_S(t)$$ ; and $$M/G2$$ torque is $$T_{M/G2}(t)=T_R(t)$$ . The internal force on pinion gears is $$F$$ , the gear ratio of the final drive is $$g_f$$ , and the produced torque from powertrain on the drive axle is $$T_{axle}(t)$$ . To benefit of the control-oriented model in energy optimization, the steady-state values are used of the left-hand in Equations (6)–(8), leading to the following $$M/G2$$ torque and vehicle velocity equations

where $$R_{wheel}$$ is the radius of wheel, $$V$$ denotes the longitudinal velocity of the vehicle, the vehicle mass is denoted by $$m$$ , the friction brake torque is $$T_{brake}(t)$$ , $$g$$ is the gravitational acceleration, and $$\theta (t)$$ is the road grade, which is assumed to be zero in this study. $$\frac{1}{2}\rho A{C}_d{V}^2$$ denotes the aerodynamic drag resistance and $$C_d$$ is the rolling resistance coefficient. A well-designed EMS seeks to compute optimal power split between the internal combustion engine and electric motor/generator to minimize energy consumption at each time instant of the solution. Assuming that the engine is in optimal operating condition and the dynamic characteristics are ignored, then the engine fuel rate $${\dot{m}}_{fuel}$$ and operational efficiencies of $$M/G1$$ and $$M/G2$$ ($$\eta {}_{M/G1}$$ and $$\eta {}_{M/G2}$$ ) are extracted from empirical data as functions of angular velocities and torques as follows:

where empirical data for the engine, generator, and motor are $${\mathrm{\Psi }}_{eng}$$ , $${\mathrm{\Psi }}_{M/G1}$$ , and $${\mathrm{\Psi }}_{M/G2}$$ , respectively. A battery in a power-split HEV is used to supply or recover energy from electricity via an inverter. A fundamental battery resistance model is utilized to describe the battery ^{[47, 48]}. Then, the battery charge sustainability, i.e., state of charge (SOC), is calculated as:

where $$I_{batt}\left(t\right)$$ denotes the battery current, $$Q_{max}$$ is the maximum capacity of battery, $$P_{batt}\left(t\right)$$ denotes the battery power, $$R_{batt}$$ is the internal resistance, and $$V_{oc}$$ is the the open circuit voltage. Then, the following equation expresses the terminal battery power requirement:

where $$P_{M/G1}(t)$$ and $$P_{M/G2}(t)$$ are shaft powers and $${\eta }_{inv}$$ is the inverter efficiency.

Equations (5)–(17) explains the energy management-oriented model used in this paper. The main parameters of the power-split hybrid electric vehicle are given in Table 1.

2.4. Equivalent consumption minimization

Taking advantage of a real-time energy optimization strategy, ECMS does not require entire driving cycle information in advance, and, by converting electricity consumption to equivalent fuel consumption as well as considering the constraints in engine, motor, generator, and battery, instantaneous equivalent fuel consumption is minimized. To this end, the equivalent factor (EF) is required to convert the electricity consumption into equivalent fuel consumption. The general formulation for the above-mentioned problem is given below.

subject to

where $$a(t)$$ is the control input, $${\dot{m}}_{eqv}(t)$$ denotes the equivalent fuel consumption, $${\dot{m}}_f(t)$$ is the engine fuel consumption (kg/s), $${\dot{m}}_e(t)$$ is the equivalent fuel consumption (kg/s), $$s(t)$$ is the equivalent factor, $${\omega }_{eng}(t)$$ is the engine angular speed (rpm) with minimum $${\omega }_{eng}^{min}$$ and $${\omega }_{eng}^{min}$$ values, $${\omega }_{M/G1}(t)$$ is the $$M/G1$$ angular speed (rpm) with minimum $${\omega }_{M/G1}^{min}$$ and maximum $${\omega }_{M/G1}^{max}$$ values, $${\omega }_{M/G2}(t)$$ is the $$M/G2$$ angular speed (rpm) with minimum $${\omega }_{M/G2}^{min}$$ and maximum $${\omega }_{M/G2}^{max}$$ values, $$T_{eng}(t)$$ is the engine torque (Nm) with minimum $$T_{eng}^{min}$$ and maximum $$T_{eng}^{max}$$ values, $$T_{M/G1}(t)$$ is the $$M/G1$$ torque (Nm) with minimum $$T_{M/G1}^{min}$$ and maximum $$T_{M/G1}^{max}$$ values, $$T_{M/G2}$$ is the $$M/G2$$ torque (Nm) with minimum $$T_{M/G2}^{min}$$ and maximum $$T_{M/G2}^{max}$$ values, $$P_{batt}(t)$$ is the battery power (kW) with minimum $$P_{batt}^{min}$$ and maximum $$P_{batt}^{max}$$ values, and $$I_{batt}(t)$$ is the battery current with minimum $$I_{batt}^{min}$$ and $$I_{batt}^{min}$$ maximum values. At each time step of the simulation, the cost function in Equation (18) is minimized by satisfying the constraints in Equations (20)–(28).

3.1. Intelligent Driver Model

The Intelligent Driver Model (IDM) was developed by Treiber, Hennecke, and Helbing in 2000, which is a type of adaptive cruise control system designed to set the desired longitudinal speed and safety time interval of the driver. It is a type of car-following model that can adjust the driver's behavior according to the changing traffic. To this end, in the IDM, the follower vehicle acceleration varies depending on the distance and speed of the preceding vehicle. In the IDM, the $$n$$ th vehicle acceleration at time $$t$$ is defined below ^[30].

where $${\ddot{x}}_n(t)$$ is the acceleration of the $$n$$ th vehicle, $${\dot{x}}_n(t)$$ is the speed of the $$n$$ th vehicle, $$a$$ is the follower vehicle's maximum acceleration, $$\delta$$ is the exponent for vehicle's acceleration, $${\dot{x}}_0$$ is the desired velocity of the $$n$$ th vehicle, $$\mathrm{\Delta }{\dot{x}}_n(t)$$ is the relative velocity of the follower vehicle with its preceding vehicle (i.e., $$\mathrm{\Delta }{\dot{x}}_n\left(t\right)={\dot{x}}_{n-1}\left(t\right)-{\dot{x}}_n\left(t\right)$$ ), and $$s_n$$ is the distance gap (m) distance between $$n$$ th and $$(n-1)$$ th vehicle, defined as

where $$\mathrm{\Delta }{x}_n(t)$$ is the distance between the follower vehicle and the vehicle in front of it, i.e., $$\mathrm{\Delta }{x}_n\left(t\right)=x_{n-1}\left(t\right)-x_n\left(t\right)$$ , and $$l_n$$ is vehicle length. The desired minimum gap of the $$n$$ th vehicle, $$s_n^{\ast }$$ , is given by

where $$b$$ denotes the $$n$$ th vehicle maximum deceleration and $$T$$ is the safe time headway (s). $$s_0$$ denotes the minimum space headway between $$n$$ th vehicle and $$(n-1)$$ th vehicle (ahead of the $$n$$ th vehicle). In the IDM, acceleration divides the vehicle into two parts. In the model, (a) is the maximum acceleration that the vehicle can achieve with free flow velocity and (b) is the comfortable braking deceleration. In this context, $${\ddot{x}}_n\left(t\right)$$ can be divided into sections; the first part $$(1-{\left(\frac{{\dot{x}}_n}{{\dot{x}}_0}\right)}^{\delta })$$ indicates the required acceleration depending on the desired speed, while the second part $${\left(\frac{s^{\ast }({\dot{x}}_n,{\mathrm{\Delta }\dot{x}}_n)}{s_n}\right)}^2$$ indicates the required deceleration depending on the desired gap between the $$n$$ th vehicle and $$(n-1)$$ th vehicle. The second part is valid if the distance $$\mathrm{\Delta }{x}_n$$ between $$n$$ th vehicle and $$(n-1)$$ th vehicle is less than the desired distance.

The desired distance $$s^{\ast }$$ between the $$n$$ th vehicle and $$(n-1)$$ th vehicle consists of the minimum stopping distance $$s_0$$ and the speed-dependent distance $${\dot{x}}_{n}T$$ . This corresponds to preceding vehicle in front of the follower vehicle in flowing traffic conditions in the desired $$T$$ time interval. In this braking situation, the vehicle can be decelerated comfortably depending on the maximum deceleration $$b$$ .

3.2. Gazis–Herman–Rothery (GHR) Model

The GHR model, also known as the General Motors (GM) model, is developed by Gazis–Herman–Rothery in the 1950s. The GHR model is a stimulus-based car following model ^[31]. GHR model assumes that the following vehicle responds to arbitrarily small changes in relative speed. GHR model also considers that the follower vehicle responds to the actions of the preceding vehicle, even though the distance to the preceding vehicle is very large, and the follower vehicle's response vanishes as the relative velocity is zero. In the GHR model, the acceleration of the follower vehicle, i.e., $$n$$ th vehicle, is proportional to the speed of the preceding vehicle, i.e., $$(n-1)$$ th, the speed between the $$n$$ th vehicle and the $$(n-1)$$ th vehicle, and the distance between them ^[53]. According to the GHR model, the $$n$$ th vehicle acceleration at time $$t$$ is calculated below ^[31].

where $${\ddot{x}}_n(t)$$ denotes the acceleration of the $$n$$ th vehicle, $${\dot{x}}_n(t)$$ denotes $$n$$ th vehicle speed, $$x_n(t)$$ is the position of the $$n$$ th vehicle, $$\mathrm{\Delta }{\dot{x}}_n(t)$$ is the speed difference between the $$n$$ th vehicle and the $$(n-1)$$ th vehicle, $$\mathrm{\Delta }{x}_n(t)$$ is the distance between the following vehicle and the preceding vehicle, and $$T$$ is the vehicle's reaction time. $$c$$ , $$m$$ , and $$l$$ are model control parameters. Coefficient $$m$$ shows the extent of the speed of $$n$$ th vehicle. This can affect the acceleration applied by the driver of the nth vehicle at time $$t$$ . The constant $$l$$ indicates how much the distance $$\mathrm{\Delta }{x}_n$$ between the follower and the followed vehicles contributes to the acceleration. Moreover, $$T$$ reaction time is related with $$c$$ sensitivity constant. These parameters were obtained as a result of experimental studies.

3.3. Optimal Velocity Model

The Optimal Velocity Model (OVM) is a dynamic equation-based car-following model developed by Bando, Hasebe, Nakayama, and Shibata in 1955. According to the OVM, the movement of the vehicle is controlled by an optimal speed. Therefore, in the OVM, the acceleration of the vehicle is calculated based on the difference between the optimal speed and the speed of the vehicle. In the OVM, the acceleration of the $$n$$ th vehicle at time $$t$$ is defined by the formula below ^[32].

where $${\ddot{x}}_n(t)$$ denotes the $$n$$ th vehicle acceleration, $$\mathrm{\Delta }{x}_n(t)$$ is the space headway between the follower vehicle and the followed (preceding) vehicle (i.e., $$\mathrm{\Delta }{x}_n(t)=x_{n-1}(t)-x_n(t)$$ ), $${\dot{x}}_n(t)$$ is the speed of the $$n$$ th vehicle, $$k$$ is the driver sensitivity and is given by inverse of the delay time, and $$V\left(\mathrm{\Delta }{x}_n\right(t))$$ is the optimal speed function of the $$n$$ th vehicle, which is given by

where $$l_n$$ is the vehicle length, $$V_1$$ and $$V_2$$ are independent variables, and $$C_1$$ and $$C_2$$ are constants. These parameters are obtained as a result of experimental studies. In the OVM car-following model, the optimal speed ($$V(\mathrm{\Delta }{x}_n(t))$$ ) is a function of the space headway that denotes the distance between the follower vehicle and the vehicle ahead of it. Based on this function, the acceleration value of the OVM is adjusted to avoid collisions between vehicles.

The values of the above-mentioned models are shown in Table 2^{[54, 55]}.

4.1. IDM

In the IDM, the follower vehicle acceleration depends on the speed and position of the vehicle in front of it. With the help of the following quadratic ordinary differential equation formulations, the IDM car-following model is converted to a CACC-based model for $$N$$ vehicles.

where

that is

where

$$\ddot{x}=({\ddot{x}}_1,{\ddot{x}}_2,{\ddot{x}}_3...{\ddot{x}}_N{)}^T$$

$$\dot{x}=({\dot{x}}_1,{\dot{x}}_2,{\dot{x}}_3...{\dot{x}}_N{)}^T$$

$$x=(x_1,x_2,x_3...x_N{)}^T$$

where $$p$$ is the IDM parameter vector. Equation (38) describes a system of ordinary differential equations for a platoon of $$N$$ vehicles where all the CACC vehicles' motion can be captured with the use of the IDM car-following behavior.

4.2. GHR

In the GHR, the acceleration of the follower vehicle depends on the speed and position of the vehicle in front of it. With the help of the following quadratic ordinary differential equation formulations, the OVM car-following model is converted to a CACC-based model for $$N$$ vehicles as follows.

Then, the system of differential equation of $$N$$ vehicles in the CACC platoon becomes

where

$$x=(x_1,x_2,x_3...x_N{)}^T$$

where $$p$$ is the GHR parameter vector. In Equation (40), $$\dot{x}$$ and $$x$$ present the velocities and positions of the CACC vehicles, as well as with the $$p$$ parameter vector of GHR, respectively.

4.3. OVM

In the OVM, the acceleration of the follower vehicle depends on the speed and position of the vehicle in front of it. With the help of the following quadratic ordinary differential equation formulations, the OVM car-following model is converted to a CACC-based model for $$N$$ vehicles.

where

The system of differential equations of $$N$$ vehicles in a CACC platoon becomes

where

$$x=(x_1,x_2,x_3...x_N{)}^T$$

$$p$$ is the OVM parameter vector. Similar to the previous derivations, the vector of positions and speeds of $$N$$ vehicles CACC platoon are expressed in Equation (42). With the current format, all of the car-following models are packed into a set of differential equations, allowing to develop a system-level model of traffic dynamics.

5.1. Driving performance verification results of CACC-ECMS scheme under NEDC, WLTP, HWFET, and HIL driving cycles

In this section, the effectiveness of the CACC-ECMS driving performance, which includes reference speed deviation of the follower vehicles with respect to the preceding vehicles, is assessed. Since the speed trajectory following is an important evaluation index in the platoon, we aim to minimize the deviations of the velocity between vehicles, therefore ensuring string stability. Figures 7–10 show the reference speed trajectory deviations of the vehicles in the platoon under NEDC, WLTP, HWFET, and HIL driving cycles. In Figure 7, we can observe the velocity fluctuations of the follower vehicles with respect to their leader vehicles using three different car-following models. One observation made is that the GHR-based CACC-ECMS model performs better under NEDC cooperative driving cycle than the OVM- and IDM-based CACC-ECMS schemes. However, the GHR- and OVM-based CACC-ECMS methods put more effort than the IDM-based CACC-ECMS method into car-following; thus, HEVs consume more fuel in the platoon, as shown in Table 3. Similarly results are seen in Figures 8 and 9. In Figure 8, the GHR- and OVM-based CACC-ECMS schemes perform better in the car-following mode than that of the IDM-based CACC-ECMS method, as zoomed in the figure. The tradeoff between cooperative driving performance and fuel economy exists under WLTP cycle as well. To show this, the CACC-ECMS scheme results for each car-following model are illustrated under HWFET cycle. In Figure 9, the GHR- and OVM-based CACC-ECMS schemes perform relatively well compared to the IDM-based CACC-ECMS case, where its fuel economy performance is the best in this case as well. The design is also tested in a driving simulator, dSPACE software, which is a high-fidelity driving simulator to test and validate the proposed approach. To this end, a human-in-the-loop driving simulator is an immediate and economical solution to explore the energy-saving potentials of a cooperative hybrid electric vehicle platoon under the presence of a leader vehicle driver style. The simulator is demonstrated in Figure 5. The driver controls the steering wheel, the throttle pedal, and the braking pedal. In the environment, a straight road cooperative driving condition is created, where the human driver controls the leader vehicle and six HEV followers are in the platoon for a driving test. As shown in Figure 10, the GHR- and OVM-based CACC-ECMS schemes perform better in the car-following mode than that of the IDM-based CACC-ECMS method, as zoomed in the same figure. It is seen that driver's speed profile is fluctuating; thus, the follower HEVs cannot effectively track the velocity trajectory of the preceding HEV using the IDM. It is worth noting that the tradeoff between cooperative driving performance and fuel economy exists under this cycle as well.

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Figure 7. Speed profiles of car following models under NEDC driving cycle.

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Figure 8. Speed profiles of car following models under WLTP driving cycle.

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Figure 9. Speed profiles of car following models under HWFET driving cycle.

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Figure 10. Speed profiles of car following models under HIL driving cycle.

Some conclusions can be drawn as follows: (i) The tradeoff between reference velocity following of followers versus the fuel consumption of the platoon shows that the driving performance metric is conflicted with the consumed fuel in the platoon; (ii) Even though the GHR-based CACC-ECMS method represents the best driving performance in terms of the reference speed trajectory following, we cannot state its fuel economy is the worst because the fuel economy is also affected by the different model parameters of the proposed scheme.

5.2. Energy saving performance verification results of CACC-ECMS scheme under NEDC, WLTP, HWFET, HIL driving cycles

The effectiveness of the CACC-ECMS scheme is evaluated for energy-saving potential in the platoon in this section. Figures 11–14 show the SOC trajectory deviations of the vehicles in the platoon under NEDC, WLTP, and HWFET driving cycles. The SOC initial value, which is 60%, is to be sustained over the entire driving cycle. Figure 11 shows that the SOC levels with GHR and OVM models are closer to the reference level for each HEV under NEDC cycle in the platoon. We can also observe end-of-cycle SOC values in Table 4. This is indeed a good indicator of the proposed scheme, especially in terms of predictability of its trajectory using GHR and OVM models under NEDC drive cycle, so that the battery works in low resistance. One drawback of this approach is that the fuel economy is deteriorated for the members of the platoon as compared with using IDM under the same drive cycle, as seen in Table 3. The platoon fuel economy per 100 km (L/100 km) is 16.59 L using IDM, while it is 18.09 and 18.53 L using GHR and OVM, respectively. Figure 12 represents SOC findings using the CACC-ECMS scheme under WLPT drive cycle. We can observe that, similar to the NEDC case, the GHR and OVM models exhibit close values to the reference SOC level, while SOC level fluctuates around the reference level using IDM. However, the IDM-based CACC-ECMS scheme presents better fuel-saving, i.e., 17.33 L/100 km, while it is 18.73 and 19.47 L /100 km using GHR and OVM, respectively, in the platoon. Table 3 presents each HEV fuel consumption using different car-following models, subsequently global fuel-savings in the platoon. We can also observe end-of-cycle SOC values using the GHR-based CACC-ECMS scheme in Table 4. SOC trajectories of the proposed CACC-ECMS scheme under HWFET drive cycle are illustrated in Figure 13. Since HWFET cycle is a standard highway, we expect the battery SOC level to be maintained closer to the reference value, as seen in the figure. Even though all car-following models present a similar pattern, only the IDM-based SOC level deviates around the reference level for each HEV in the platoon. Similar to the previous cases, the fuel economy result of the IDM-based CACC-ECMS scheme shows improvements among the HEVs and the platoon relative to the other cases. For this driving cycle, the platoon fuel economy per 100 km (L/100 km) is 9.403 L using IDM, while it is 9.756 and 9.817 L using GHR and OVM, respectively. Lastly, SOC trajectories of the proposed CACC-ECMS scheme under HIL drive cycle are illustrated in Figure 14. The figure shows that the SOC levels deviate around the reference level for each HEV in the platoon only in the OVM case. The fluctuations in the speed profile deteriorate the string stability and energy consumption of HEVs in the platoon. In this aspect, for this driving cycle, the platoon fuel economy per 100 km (L/100 km) is 15.22 L using IDM, while it is 17.11 and 18.62 L using GHR and OVM, respectively.

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Figure 11. NEDC drive cycle SOC graph.

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Figure 12. WLTP drive cycle SOC graph.

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Figure 13. HWFET drive cycle SOC graph.

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Figure 14. HIL drive cycle SOC graph.

One main conclusion can be drawn for all cases: the IDM-based CACC-ECMS scheme provides the best fuel economy, while there are tradeoffs for the SOC reference trajectory following performance under various drive cycles. The GHR-based CACC-ECMS scheme performs relatively better than the OVM-based scheme for energy-saving, and both methods present a similar SOC trajectory following performance under NEDC and HWFET drive cycles.

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments.

Authors' contributions

Made substantial contributions to conception, manuscript preparation, experimental studies, and manuscript editing/revision: Yazar O, Coskun S

Made substantial contributions to reviewing/editing and technical notes: Zhang F, Li L

Availability of data and materials

Not applicable.

Financial support and sponsorship

This study is supported by the Scientific and Technological Research Council of Turkey with Project No. 121E260 under the grant name CAREER.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2022.