Economic Potential for Hybrid Electric Vehicles in Urban Signal-free Intersections with Decentralized MPC

Rapid urbanization causes an increasing number of vehicles and intensifies traffic congestion, especially in urban districts. This phenomenon can lead to many problems, such as potential safety hazards and increased energy and time consumption, particularly in urban intersections, where the interaction between different traffic flows is extremely complex. Although some research focuses on designing novel traffic light systems to alleviate these problems (Yang et al., 2017; Xiang et al., 2018), the efficiency of the traffic flow at an intersection and the energy consumption is still not fully optimized (Namazi et al., 2019). To maximize urban transportation efficiency, a smoother and environmentally friendly scheme needs to be proposed. Nowadays, the development of information technology enables vehicle-to-everything (V2X) communication for connected and autonomous vehicles (CAVs) to be deployed on real-world road networks (Zhao and Malikopoulos, 2022). Therefore, it is an attainable goal to realize a signal-free intersection control to optimally coordinate a group of CAVs crossing the intersection for safety, mobility, and energy efficiency objectives (Zhong et al., 2021).

There have been numerous efforts of autonomous intersection control algorithms designed based on CAVs reported in the literature, which can be divided into two categories, centralized and decentralized, respectively (Čakija et al., 2019; Gholamhosseinian and Seitz, 2022). The centralized framework coordinates the vehicles’ movement using one central intersection controller (IC) (Riegger et al., 2016; Mihály et al., 2020; Liu et al., 2020), while the decentralized framework relies on local controllers on each vehicle (Chalaki and Malikopoulos, 2022b, a; Hadjigeorgiou and Timotheou, 2022). Some existing literature focuses on battery electrical vehicles (BEVs) as the target CAVs for urban intersection crossing problems, instead of conventional vehicles (CVs), as BEVs are the current trend of vehicle development offering the advantage of reduced operating cost and waste gas emissions, although they also suffer from low power and long charging time (Hult et al., 2018; Pan et al., 2022a, b). In (Hult et al., 2018), the energy consumption is modeled by electric motors efficiency maps and optimized by an economic model predictive control (MPC) method. Recent research efforts in (Pan et al., 2022a, b) develop hierarchical and convex optimization approaches to find the optimal trade-offs of battery energy consumption and travel time.

However, the global electricity price has increased sharply recently. For instance, the UK electricity price has risen by 64% from about $226.59$ (GBP/MWh) to $372.27$ (GBP/MWh) during the last 12 months, compared with the lower rate increase of fuel price by 18% from 138.64 (pence/liter) to 163.62 (pence/liter) (TRADING ECONOMICS, 2022; Department for Business, Energy & Industrial Strategy, 2022). This phenomenon significantly influences the operational cost of the BEV, and it may promote hybrid electric vehicles (HEVs), which are powered by hybrid energy sources. This paper aims to investigate the energy-saving potential of HEVs in comparison with BEVs in the context of a road intersection, which is the main bottleneck of urban traffic. The work focuses on the series HEV (sHEV) architecture, which is a common arrangement for modern HEVs and involves a number of products in the market, such as the Nissan Note e-Power, and numerous other extended-range electric vehicles Chen et al. (2019). Some recent literature that considers sHEVs as a CAV option for urban signalized intersection crossing problems is presented in Tang et al. (2021); Jian et al. (2021). In Tang et al. (2021), a multi-objective hierarchical optimal strategy is proposed to optimize fuel consumption and riding comfort in signalized intersections. A two-level cooperative control method is designed in Jian et al. (2021) to improve the travel time and energy consumption of sHEV in a signalized intersection. However, to the best knowledge of the authors, existing researches lack investigating convex optimization-based autonomous intersection management approaches, particularly for sHEVs. This paper works on bridging the gap by proposing a convex optimization framework of sHEVs for the signal-free intersection crossing problem. The resulting problem is then solved by a decentralized model predictive control (DMPC) approach, rather than centralized MPC, that strikes the trade-off between optimality and computational complexity (Pan et al., 2022a). The contributions of this paper are therefore: 1) development of a new convex modeling framework for eco-driving control of an sHEV, 2) utilization of the contributed convex modeling framework of a single sHEV to develop a decentralized convex model predictive control for autonomous intersection coordination of a fleet of sHEVs, and 3) numerical investigation of the energy-saving potential for sHEVs in urban signal-free intersections; in the context of the UK’s electricity price, the running cost of the sHEV in the urban intersection is compared with the CV and BEV scenarios.

The rest of this paper is organized as follows: Section 2 provides the signal-free intersection and sHEVs powertrain models along with the corresponding approximation and convexification. The decentralized model predictive control problem is formulated in Section 3. Numerical results and analyses are provided in Section 4. Finally, concluding remarks are given in Section 5.

This section provides an introduction to the autonomous intersection crossing control approach via the decentralized model predictive control (DMPC) technique with a prediction horizon $N_{p}\!\in\!\mathbb{N}_{>0}$. Hence, it is necessary to formulate the intersection control problem in a discretized form with a sampling distance interval $\Delta s\!\in\!\mathbb{R}_{>0}$. Meanwhile, let us define that $L\!+\!S\!=\!\alpha\Delta s$, $L\!=\!\alpha_{1}\Delta s,\,S\!=\!\alpha_{2}\Delta s$. The discretized dynamics (19) of CAV $i\!\in\!\mathcal{N}$ in the space domain is given by:

The decentralized system relies on the information exchange between the intersection coordinator and the CAVs. For an arbitrary CAV $i$, a unique identity is assigned to record its crossing order and direction when it enters the CZ. The identity is defined as $(i,d_{i},\mathcal{I}_{i}(k))$, where $i\!\in\!\mathcal{N}$ is the crossing order of vehicle $i$ to enter the MZ, $d_{i}$ is an index denoted as the traveling direction (north, south, east and west), and $\mathcal{I}_{i}(k)$ is the information set generated by the coordinator at step $k$ by collecting the past optimal state sequences of vehicle $i$:

where $\mathbf{x}_{i}(k)\!=\![x_{i}(k|k),x_{i}(k\!+\!1|k),\ldots,x_{i}(k\!+\!N_{p})]^{\top}$ is the optimal state sequence computed by the local MPC on CAV $i$ at step $k$ within the prediction horizon $N_{p}$.

Then, based on the identity of each CAV and the information exchange rule shown below, these information sets $\mathcal{I}_{i}(k)$ will be sent to specific CAVs which require them to support their DMPC algorithm. Taking an arbitrary CAV $i$ as an example, the information exchange rule is shown as follows:

1. 1.

   For $\mathcal{C}_{i}\notin\emptyset$, CAV $i$ requires $\mathcal{I}_{h}(k)$ of CAV $h\!\in\!\mathcal{C}_{i}$ ($h\!<\!i$), which is immediately ahead of CAV $i$ in the same lane.

2. 2.

   For $\mathcal{L}_{i}\!\notin\!\emptyset$, CAV $i$ requires $\mathcal{I}_{q}(k)$ of CAV $q\!\in\!\mathcal{L}_{i}$ ($q\!<\!i$), which is the last element (CAV index) in set $\mathcal{L}_{i}$.

3. 3.

   For $\mathcal{O}_{i}\!\notin\!\emptyset$, CAV $i$ requires $\mathcal{I}_{p}(k)$ of CAV $p\!\in\!\mathcal{O}_{i}$ ($p\!<\!i$), which is the last element (CAV index) in set $\mathcal{O}_{i}$.

For the decentralized MPC framework, each CAV can achieve the collision avoidance constraints (5), (6), (9) based on the information set $\mathcal{I}_{i}(k)$ assigned from the coordinator subject to the information exchange rule. To achieve the rear-end collision avoidance constraints, an arbitrary CAV $i$ requires the information set of the CAV $h$ which is directly ahead of CAV $i$, such that ${h}\!\in\!\mathcal{C}_{i}$. This constraint is reformulated as follows:

where $j\!\in\!\mathbb{N}_{[0,N_{p}-1]}$, and $t_{h}(k\!+\!j\!+\!1|k)$ is the past information of CAV $h$ saved in the information set $\mathcal{I}_{h}(\bar{k})$, where $\bar{k}\!>\!k$ is the corresponding distance step of CAV $h$ when the $i$th CAV is at step $k$, that is $t_{h}(\bar{k}\Delta s)\!=\!t_{i}(k\Delta s)$.

The reformulation of the lateral collision avoidance constraint (6) between CAV $i$ and CAV $q\in\mathcal{L}_{i}$ in the decentralized framework is given as below:

where $k+j+1=\alpha_{1}$ with $\alpha_{1}\!\in\!\mathbb{N}_{[k,\,k+N_{p}]}$. The exit time of the ${q}$th CAV of MZ, $\hat{t}_{q}^{\mathcal{L}}(L+S)$, is estimated by:

where $l\!\in\!\mathbb{N}_{[0,N_{p}-1]}$, $\bar{v}_{i}$ is the desired terminal velocity and $\bar{k}$ has $t_{q}(\bar{k}\Delta s)\!=\!t_{i}(k\Delta s)$, which represents the associated step of the ${q}$th CAV. For the first case in (24), since the prediction horizon of the ${q}$th CAV has already covered the terminal distance $L\!+\!S$, $\hat{t}_{q}^{\mathcal{L}}(L\!+\!S)$ can be directly obtained. For the second case in (24), the ${i}$th CAV can not directly derive the terminal time $\hat{t}_{q}^{\mathcal{L}}(L+S)$. Therefore, the exit time of the ${q}$th CAV is approximated by the sum of the present predictive time of the last predictive horizon step of CAV ${q}$, $t_{q}(\bar{k}\!+\!N_{p}|\bar{k})$, and the time such that CAV ${q}$ should spend to finish the rest of the distance with an average speed of its terminal speed of the present horizon and the desired terminal velocity.

In terms of the no collision condition in the MZ between CAV $i$ and CAV $p\!\in\!\mathcal{O}_{i}$, the constraint of (9) is rewritten in a decentralized manner as below:

where $j,l\!\in\!\mathbb{N}_{[0,N_{p}-1]}$, while $\hat{t}_{p}^{\mathcal{O}}(L)$ and $\hat{t}_{p}^{\mathcal{O}}(L+S)$ can be obtained by applying the same method shown in (24).

Therefore, the convex DMPC framework for a single vehicle $i\!\in\!\mathcal{N}$ in space domain is presented as follows:

where $\mathbf{u}_{i}(k)\!=\![u_{i}(k|k),u_{i}(k+1|k),\cdots,u_{i}(k+N_{p}-1|k)]^{\top}$, $x_{i}(k)$ in (26b) is the actual state measured at step $k$, the system dynamics (26c) collects (20), the convex set $\mathbb{X}_{i}$ in (26d) collects (7), (10), (14), (22), (23), and (25), the convex set $\mathbb{U}_{i}$ in (26d) collects (4), (12), (17), and the convex set $\mathbb{X}_{i}\times\mathbb{U}_{i}$ in (26d) collects (1b). The initial condition (26e) provides initial state (entry state at CZ) of each CAV $x_{i}(0)$, which satisfies the Assumptions 2 and 3. The objective function (26a) in this is designed as a multi-objective function, which is described as below:

where $W_{1}$, $W_{2}$, $W_{3}$, $W_{4}$ are the weighting factors. The first three terms aim to minimize the traveling time, and the usage of mechanical braking and the fuel consumption of (11), respectively, and the last term is imposed to achieve the battery charge sustaining (CS) condition over the entire prediction horizon.

This section provides numerical examples to investigate the performance of the proposed convex DMPC strategy for a group of $N(t)\!=\!20$ CAVs at an autonomous intersection crossing. Two convex benchmark vehicle models (i.e. CV (Han et al., 2019) and BEV (Pan et al., 2022b)) are considered and formulated within the same DMPC framework (26) for fair comparisons. The investigation contains twofold, 1) the effectiveness and validity of the proposed convex sHEVs model and the DMPC strategy, and 2) the impact of the fuel and electricity energy economy between sHEVs and the two other vehicle types (i.e., CVs and BEVs) in the context of the electricity and petrol price in the UK over the past 12 months.

The parameters of the intersection in this work are $L\!=\!150$ m and $S\!=\!10$ m with a sampling interval $\Delta s\!=\!2$ m. The prediction horizon length of the DMPC method is defined as $N_{p}\!=\!30$, . The terminal velocity leaving the MZ is set to be identical for all CAVs at $\bar{v}_{i}(L+S)\!=\!10$ m/s for fair and easy comparison in different simulation scenarios. Without loss of generality and credibility of the simulation results, this numerical analysis has applied 10 sets of different initial conditions datasets with an arrival rate of 700 veh/h per lane. Moreover, in each of the datasets, the initial velocity $v_{i}(0)$ and entrance time $t_{i}(0)$ for all CAVs are randomly initialized subject to the constraints imposed in Assumptions 2 and 3. Note that the initial speeds of all CAVs follow the uniform distribution within $[v_{\min},v_{\max}]$ and the entrance times for all CAVs follow Poisson distribution, respectively. Moreover, the direction and type of each CAV are also randomly generated. For simplicity and illustration purposes, the crossing sequence $\mathcal{N}(t)$ in this work is predefined such that all CAVs obey the first-in-first-out policy (i.e., all CAVs enter and leave the MZ in the same order they arrive at the CZ). The proposed convex DMPC strategy is solved by YALMIP and MOSEK (Löfberg, 2004) in MATLAB 2021b on a personal computer with an M1 chip and 8 GB of RAM.

In the first instance, the optimal traveled distance trajectories of the 20 CAVs (i.e., 7 CVs, 6 sHEVs, and 7 BEVs) obtained by solving convex DMPC (26) at an average traveled time of approximately 12s are presented in Fig. 4. It can be observed that all vehicles follow the cooperatively assigned crossing order, as they leave the CZ in the same order as they enter the CZ. Moreover, none of the trajectories shown with the same color intersect. For two consecutive vehicles traveling in perpendicular directions, the latter one enters the MZ only after the first one leaves the MZ. This result shows that the rear-end and lateral collision avoidance are satisfied for all CAVs, which verifies the effectiveness and validity of the convex sHEVs model and DMPC strategy.

To compare the performance between sHEVs and the other two benchmark vehicle models (i.e., CVs and BEVs) with various electricity and petrol fuel prices, three autonomous intersection problems with only one vehicle type in each case are formulated under the same proposed DMPC framework (26). Fig. 5 presents the trade-offs of the three autonomous intersection optimization problems for a series of combinations of weighting factors in (26a) between the average energy price and traveled time of all 20 CAVs under ten simulation trials with randomly generated initial conditions. Note that petrol fuel and electric energy usages are converted to energy prices in this study based on the datasets of Oct. 2021 and Oct. 2022 obtained from (TRADING ECONOMICS, 2022; Department for Business, Energy & Industrial Strategy, 2022). As shown in Fig. 5, the average price of CVs is always the highest among the rest cases, which indicates the importance of traffic electrification as it can significantly reduce the average operating cost. In terms of sHEVs and BEVs, the gap between each average cost is reduced as the average travel time decreases. Taking the optimal results with the average travel time at 11.5 s as an example, the traveling cost of sHEVs already reaches the same level as BEVs in Oct. 2022, while a 53% average price gap can be found in the results in Oct. 2021. These results show the advantage of sHEVs as the electricity price increases, especially when the optimization emphasizes more on the minimization of the travel time to achieve efficient transportation.

To further investigate the findings in terms of the reduced average cost between sHEVs and BEVs in Fig. 5, a bar chart with detailed average cost from 2021 Q4 to Oct. 2022 at an average time cost of approximately $12$ s is presented in Fig. 6. The top figure in Fig. 6 shows the price of petrol fuel increased by 14% while the electricity price witnessed a sharp rise of 52% compared to the 2021 Q4 electricity market prices. The corresponding consequences for the vehicle traveling cost are shown at the bottom in Fig. 6, where the gap of the average price between sHEVs and BEVs on operation dropped from 83% to 30% from 2021 Q4 to Oct. 2022. Although both CVs and sHEVs consume petrol fuel for operation primarily, the hybrid powertrain can provide higher energy efficiency. Overall, these findings indicate that the sharply increasing electricity price over the last year is weakening the benefit of BEVs, meanwhile highlighting the importance and benefits of sHEVs.

This paper addressed the signal-free intersection crossing problem for connected and automated vehicles (CAVs) with a series hybrid electric powertrain. The dynamic model of sHEVs is convexified in the space domain. The optimization problems are formulated within a convex decentralized model predictive control (DMPC) framework to ensure a rapid search and unique optimal solution. Numerical examples first validate the effectiveness of the proposed methods with convex relaxation and approximation. The performance of the proposed approach is evaluated against the optimal results yielded by solving the same decentralized intersection problems but with two convex benchmark models of conventional vehicles (CVs) and battery electric vehicles (BEVs). By utilizing the petrol price and electricity charging prices from 2021 Q4 to Oct 2022, it can be found that although the last year witnessed a sharply increasing electric price, CVs are still the most expensive to travel with. Moreover, the average energy price to travel with sHEVs approaches or even sometimes reaches the same level as with BEVs, especially under an efficient transportation strategy (less average traveling time consumption), which highlights the importance and the benefits of sHEVs in the current increasing electricity price market situation in the UK.

Future work will focus on a more comprehensive analysis of the performance of various HEV types under the turbulent electricity market, including the parallel HEVs, and plug-in electric vehicles for autonomous intersection crossing problem. Robust control strategies will also be developed to reduce and tackle the potential uncertainties in mathematics models, communications, and measurements.