Coordinating Truck Platooning by Clustering Pairwise Fuel-Optimal Plans

Platooning is a common element in intelligent transportation systems. It refers to a group of vehicles forming a road train without any physical coupling between the vehicles. A short inter-vehicle distance is maintained by automatic control and vehicle-to-vehicle communication. Platooning is beneficial in various ways. Due to the small inter-vehicle gaps the total road throughput can be increased [1]. It can also help facilitate the driverless operation of the trailing vehicles. In this paper, our main motivation to study platooning is the potential to reduce the fuel consumption of the follower vehicles in the platoon. Similar to what racing cyclists exploit, the follower vehicles experience a reduction in air drag which translates into reduced fuel consumption [2].

Using platooning to reduce fuel consumption leads to a challenging coordination problem. This differs from a classical setting where platoons are primarily a measure to increase road throughput and where a majority of the vehicles can platoon. Consider two trucks that travel between the same two regions but from different locations within the regions and at approximately the same time. Then the trucks can adjust their speeds slightly at the beginning of their journeys, form a platoon at the start of the common part of their route and thus save fuel during most of their journeys. The catch is that this might involve one of the trucks having to drive faster before the two merge, which increases air-drag and consequently fuel consumption during that phase. One truck might also slow down to let the other truck catch up but then travel at an increased speed later on to arrive at its destination on time. If many trucks are involved, it is not straightforward to compute a plan for all trucks that would be globally fuel-optimal. This is the problem we address in this paper.

Variations of this problem have been considered in literature. In [3] the authors formulate a mixed integer linear programming problem, without considering the speed dependency of fuel consumption, and prove that the problem is NP-hard. In [4] the authors consider a simple catch-up coordination scheme and evaluate it on real fleet data. In [5] local controllers for coordinating the formation of platoons are proposed. In [6] the authors use data-mining to identify economic platoons based on different criteria. They allow that trucks wait for other trucks to form the platoon. In [7] the problem is formulated as an optimization problem. The proposed method is, however, not applicable to large numbers of trucks due to the combinatorial complexity.

Clustering is present in a variety of different contexts. A large body of research focuses on clustering methods as an analysis tool to structure, understand, and classify large data sets [8, 9]. Examples include the clustering of graphs in the scope of community detection [10, 11, 12]. Our clustering algorithm is inspired by partitioning around medoids [13]. A closely related area to our subproblem of choosing coordination leaders is leader election, where a group of agents has to jointly determine a leader [14, 15]. When we interpret pairwise fuel savings as preferences for trucks being coordination leaders, we can see our algorithm as local leader election. In one variant of the proposed clustering algorithm we consider that leader and follower share their benefit from platooning according to a fixed ratio. This setting is related to coalitional game theory [16], which is a widely used framework in communication theory.

We derive a powerful, efficient, and scalable method to coordinate platooning of a large number of trucks in a fuel-optimal way. The main contribution is that the coordination problem is formulated as a clustering problem based on pairwise fuel-optimal speed profiles. We derive pairwise fuel-optimal plans based on a first-order fuel model and propose a clustering algorithm. We discuss the convergence of the clustering algorithm and demonstrate the entire method using Monte Carlo simulations.

The paper is organized as follows. We start in Section II by modeling a single truck as a hybrid system, where the discrete jumps model transitions between road segments. In Section III we develop a fuel-optimal plan for two trucks, where the leader keeps a constant speed and the the follower adapts its speed to merge with the leader. In Section IV we build on these pairwise plans to construct a similarity graph and formulate the problem to select good leaders and match followers to each of them. In Section V we propose a clustering algorithm to find approximate solutions to the problem and analyze its convergence. In Section VI the method is demonstrated by Monte Carlo simulations.

In this section we model a single truck as a hybrid system, define platooning, and introduce a fuel consumption model.

We consider a highway network in which a set of geographic locations are connected by one-way road segments. We model the road network as a weighted, directed graph $\mathcal{G}=(\mathcal{N},\mathcal{E},W)$, where the nodes $\mathcal{N}$ correspond to geographic locations. An edge $(i,j)\in\mathcal{E}\subset\mathcal{N}\times\mathcal{N}$ models a road segment from the location of node $i$ to the location of node $j$. The weight $W(e)>0$ of edge $e\in\mathcal{E}$ models the length of the road segment. The state of a truck $(x,\ell)$ comprises the edge $\ell\in\mathcal{E}$ that corresponds to the truck’s current road segment and the distance $x\in[0,W(\ell)]$ from the beginning of that road segment.

As long as the system state $(x,\ell)$ is in the flow set, $x$ evolves continuously according to (1). When the state reaches the jump set, i.e., when the truck reaches the end of the segment, the state is reset according to the jump map (2). Figure 1 illustrates the trajectory of such a system. For theoretical considerations of such models see [17].

A truck has a transport assignment to travel through the road network from a start node $n^{\mathrm{S}}\in\mathcal{N}$ at time $t^{\mathrm{S}}$ to a destination node $n^{\mathrm{D}}\in\mathcal{N}$ where it must arrive at time $t^{\mathrm{D}}$. We assume that there is a unique shortest path $\bar{p}=\bar{p}[1],\bar{p}[2],\dots,\bar{p}[P]$, $\bar{p}[i]\in\mathcal{E}$ for $i\in\{1,\dots,P\}$ from $n^{\mathrm{S}}$ to $n^{\mathrm{D}}$ with respect to $W$ and we let the truck travel along that path: $\ell^{+}(\bar{p}[i])=\bar{p}[i+1]$. With this notation we can define the trajectory of a truck (compare to Figure 1).

In the following we introduce a system composed of $K$ trucks. We introduce for each truck $k\in\{1,\dots,K\}$ its state $(x_{k},\ell_{k})$. Truck $k$ starts at node $n_{k}^{\mathrm{S}}$ at time $t_{k}^{\mathrm{S}}$ and must arrive at node $n_{k}^{\mathrm{D}}$ at time $t_{k}^{\mathrm{D}}$. Its shortest path is denoted $\bar{p}_{k}$ and has length $P_{k}$. Its speed is denoted $v_{k}$. We have the start position $(x_{k}^{\mathrm{S}},\ell_{k}^{\mathrm{S}})=(0,\bar{p}_{k}[1])$ and the destination position $(x_{k}^{\mathrm{D}},\ell_{k}^{\mathrm{D}})=(W(\bar{p}_{k}[P_{k}]),\bar{p}_{k}[P_{k}])$.

In this section we derive a pairwise fuel-optimal speed plan. We consider a pair of trucks: a coordination leader (CL) and a coordination follower (CF). These concepts differ from the concepts of a platoon leader and platoon follower that were introduced in Section II. The CL keeps a constant speed while the CF selects at the beginning of its journey a speed that allows it to merge into a platoon with the CL. Then the two platoon until they split up, followed by that the CF selects a speed so that it arrives at its pre-specified arrival time at its destination. Figure 2 illustrates these three phases. We first derive the optimal speed during the first and the last phase neglecting that trucks can only platoon on common road segments. We add this constraint in the second part of the section. In Section IV, we build on such pairwise plans and obtain an overall plan for all $K$ trucks. First, we define the default speed profile, i.e., a constant speed.

The following proposition gives the optimal rendezvous speed for a CF when there are no restrictions as to where the CL and the CF can meet.

The proof of Proposition 1 can be found in the Appendix.

The optimal speed after splitting up for the CF to meet its deadline is also the appropriate $v_{\mathrm{s}}^{*}$ depending on whether the CF has to speed up or slow down in order to arrive at the specified time.

In the remainder of this section, we discuss how Proposition 1 can be used to compute an optimal speed profile when the CL and the CF travel on different but intersecting paths. Figure 2 illustrates the discussion.

To this end, we define the distance $d_{p}$ between two states $(x_{1},\ell_{1})$, $(x_{2},\ell_{2})$ with respect to a path $p$.

Consider a CL with index $0$ and a CF with index $1$. Two trucks can platoon only on the road segments corresponding to common edges of their paths. Because their paths are shortest paths it can be shown that the shared edges between two paths form a path as well (Lemma 1 in [7]), i.e., two paths meet and split up at most once. Trucks $0$,$1$ start at $(x_{0}^{\mathrm{S}},\ell_{0}^{\mathrm{S}})$, $(x_{1}^{\mathrm{S}},\ell_{1}^{\mathrm{S}})$ at time $t_{0}^{\mathrm{S}}$, $t_{1}^{\mathrm{S}}$ and arrive at $(x_{0}^{\mathrm{D}},\ell_{0}^{\mathrm{D}})$, $(x_{1}^{\mathrm{D}},\ell_{1}^{\mathrm{D}})$ at time $t_{0}^{\mathrm{D}}$, $t_{1}^{\mathrm{D}}$, respectively. We denote the position at which the CL and the CF start platooning at time $t^{\mathrm{M}}$ as $(x^{\mathrm{M}},\ell^{\mathrm{M}})$ and where they split at time $t^{\mathrm{Sp}}$ as $(x^{\mathrm{Sp}},\ell^{\mathrm{Sp}})$. These meeting points have to lie on the trajectory of the CL with constant speed $v_{0}$:

When platooning with the CL the planned trajectory of the CF consists of three phases: from start to the meeting point with speed $v^{\mathrm{S}}$, from meeting point to the split point platooning as platoon follower of $0$ with speed $v_{0}$, and from the split point to the destination with speed $v^{\mathrm{Sp}}$. We define $d^{\mathrm{S}}=d_{\bar{p}_{1}}\big{(}(x_{1}^{\mathrm{S}},\ell_{1}^{\mathrm{S}}),(x^{\mathrm{M}},\ell^{\mathrm{M}})\big{)}$ and $d^{\mathrm{Sp}}=d_{\bar{p}_{1}}\big{(}(x^{\mathrm{Sp}},\ell^{\mathrm{Sp}}),(x_{1}^{\mathrm{D}},\ell_{1}^{\mathrm{D}})\big{)}$. We have the relations

We define the virtual position difference at the start/end of the CF’s trajectory as

which is equivalent to $\Delta d$ in Proposition 1. If $\Delta d^{\mathrm{S}}>0$ then $v^{\mathrm{S}}>v_{0}$, if $\Delta d^{\mathrm{S}}<0$ then $v^{\mathrm{S}}<v_{0}$, if $\Delta d^{\mathrm{Sp}}>0$ then $v^{\mathrm{Sp}}>v_{0}$, and if $\Delta d^{\mathrm{Sp}}<0$ then $v^{\mathrm{Sp}}<v_{0}$. Then, we can compute according to (18) the appropriate $v^{*}$ for the first and the last phase.

This derivation has ignored so far that the first possible point to merge is when the CL’s and the CF’s paths meet. If $v^{*}$ leads to a distance from $(x_{1}^{\mathrm{S}},\ell_{1}^{\mathrm{S}})$ to the merge point that is too small, then the CL selects a speed that lets the CL and CF merge at the position where the two paths meet, denoted here $(0,\ell^{F})$. This speed is $v^{\mathrm{S}}=d_{\bar{p}_{1}}\big{(}(x_{1}^{\mathrm{S}},\ell_{1}^{\mathrm{S}}),(\ell^{F},0)\big{)}/(t^{\mathrm{M}}-t_{1}^{\mathrm{S}})$. The corresponding case might occur at split up, so that $v_{1}^{\mathrm{Sp}}=d_{\bar{p}_{1}}\big{(}(W(\ell^{L}),\ell^{L}),(x_{1}^{\mathrm{D}},\ell_{1}^{\mathrm{D}})\big{)}/(t_{1}^{\mathrm{D}}-t^{\mathrm{Sp}})$, where $(W(\ell^{L}),\ell^{L})$ is the position where the CL’s and the CF’s paths split up.

The first check to test if platooning is possible and beneficial is, whether the calculated merge point lies before the split point or not, i.e., whether $d^{\mathrm{S}}+d^{\mathrm{Sp}}<d_{\bar{p}_{1}}\big{(}(x_{1}^{\mathrm{S}},\ell_{1}^{\mathrm{S}}),(x_{1}^{\mathrm{D}},\ell_{1}^{\mathrm{D}})\big{)}.$ If this condition is fulfilled, we can calculate the fuel cost for the CF with the speed profile that is adapted for platooning with the CL as follows

We summarize the results of the Section in definition of the adapted speed profile.

In this section we combine default speed profiles and adapted speed profiles from Section III in order to minimize the total fuel consumption $\sum\limits_{k=1}^{K}f_{\mathrm{c},k}$. We select a number of CLs that stick to their default speed profile. All other trucks are assigned to their best CL if there is one. The best CL for a CF is the one that yields the largest fuel saving for the CF. Since the speed profile of the CL is not affected by the CF, we can assign several CFs to a CL independently. Note that the method presented in this section can accommodate default and adapted speed profiles different from the ones presented in Section III. Like this factors such as traffic, different fuel models, soft constraints on the arrival time, etc. can be considered.

Based on the results from Section III, we calculate the potential fuel savings gained from platooning for all pairs of trucks, where one takes the role of the CL and the other the role of the CF. We collect the results in a weighted directed graph, which we call the coordination graph. Recall that the total fuel consumption of a truck $i$ with the default speed profile is denoted $\bar{f}_{\mathrm{c},k}$. We denote the fuel consumption of a CF $i$ with CL $j$ as $f_{\mathrm{p},(i,j)}$. If $i$ and $j$ platooning with $i$ being the CF of $j$ is not possible, then $f_{\mathrm{p},(i,j)}=\bar{f}_{\mathrm{c},k}$.

We introduce the set of in-neighbors of a node $n$ as $\mathcal{N}_{\mathrm{i}}(n)=\{i\in\mathcal{N}_{\mathrm{c}}:\exists(i,n)\in\mathcal{E}_{\mathrm{c}}\}$ and the set of out-neighbors of a node $n$ as $\mathcal{N}_{\mathrm{o}}(n)=\{i\in\mathcal{N}_{\mathrm{c}}:\exists(n,i)\in\mathcal{E}_{\mathrm{c}}\}$. We define that the maximum over an empty set is zero, i.e., $\max\limits_{i\in\emptyset}(\cdot)=0$.

Next, we formulate the problem of finding a fuel optimal set of CLs $\mathcal{N}_{\mathrm{l}}$.

If $(i,j)\in\mathcal{E}_{\mathrm{c}}$ with $i\in\mathcal{N}_{\mathrm{c}}\setminus\mathcal{N}_{\mathrm{l}}$ and $j=\arg\max\limits_{j\in\mathcal{N}_{\mathrm{o}}(i)\cap\mathcal{N}_{\mathrm{l}}}(W_{\mathrm{c}}(i,j))$, we say that $i$ is the CF of $j$ and $j$ is the CL of $i$. If $j$ has no out-neighbor in $\mathcal{N}_{\mathrm{l}}$, then $\max_{j\in(\mathcal{N}_{\mathrm{o}}(i)\cap\mathcal{N}_{\mathrm{l}})}(W_{\mathrm{c}}(i,j))=\max_{j\in\emptyset}(W_{\mathrm{c}}(i,j))=0$.

This problem is similar to k-medoids clustering [13]. Medoids clustering means selecting a, typically fixed, number of cluster centers from a set of data points. The remaining data points are assigned to the closest cluster center. The objective is to find cluster centers in such a way that the sum of the distances of all nodes to the closest cluster center is minimal. In our problem we can identify the CLs with the medoids.

Albeit similar to k-medoids clustering, there are some significant differences. We maximize similarity within the cluster as opposed to minimizing the distance. Similarity from node $i$ to node $j$ is here how much fuel is saved if node $i$ selects node $j$ as a CL. This implies that the similarity between two nodes is not symmetrical, non-negative, and can be zero. The number of clusters is not fixed but part of the optimization and not every node has to be assigned to a cluster. It is easy to see, that a solution with zero or $K$ clusters is not optimal if $\mathcal{E}_{\mathrm{c}}\neq\emptyset$. This is different from an application where the sum of the distances to the cluster centers is minimized. If every node is a cluster center, this trivially corresponds to the smallest possible objective value. Thus, it is not trivial to find (approximate) solutions to the problem.

In this section, we describe four variants of an approximation algorithm (Algorithm 1) to Problem 1 inspired by partitioning around medoids (PAM) [13]. Algorithm 1 works according to the following scheme. At the beginning $\mathcal{N}_{\mathrm{l}}$ is empty, i.e., there are no CLs. In each round a node $n\in\mathcal{N}_{\mathrm{c}}$ is selected for which it is beneficial to be added to $\mathcal{N}_{\mathrm{l}}$ or removed from $\mathcal{N}_{\mathrm{l}}$, and $\mathcal{N}_{\mathrm{l}}$ is updated accordingly. This is repeated until no further improvement is possible. We introduce a function $\Delta u(n,\mathcal{N}_{\mathrm{l}})$ that measures how much is gained from switching whether $n$ belongs to $\mathcal{N}_{\mathrm{l}}$ or not.

We consider two methods to select $n$ and two choices of $\Delta u(n,\mathcal{N}_{\mathrm{l}})$. The first method for selecting $n$ is to select it in a greedy manner according to $n=\arg\max\limits_{\bar{n}\in\mathcal{N}_{\mathrm{c}}}\Delta u(\bar{n},\mathcal{N}_{\mathrm{l}})$. This corresponds to what is done in the “build phase” of PAM. The second method is to choose $n$ randomly with equal probability from the set $\{\bar{n}\in\mathcal{N}_{\mathrm{c}}:\Delta u(\bar{n},\mathcal{N}_{\mathrm{l}})>0\}$. The two different choices for $\Delta u(n,\mathcal{N}_{\mathrm{l}})$ are elaborated in the following Sections V-A and V-B.

In this section we present a simulation study of the platoon coordination method developed in this paper. The road network that was used in the simulations was generated randomly by sampling 100 location points uniformly in a square of side length 800. All combinations of these locations were sorted by their Euclidean distance. Then, starting from the combination with the shortest distance, the combinations of locations were connected by two road segments (one in each direction) with length according to the Euclidean distance, if there had not already been a path between the two locations that was at most $1.5$ times longer than the Euclidean distance. The road network is shown in Figure 4.

We evaluated Algorithm 1 in four variants (greedy/random node selection and total/pairwise gain) with Monte Carlo simulations. In the figures and the following discussion we use “total”/“pairwise” to refer to $\Delta u$ according to (12)/(15), and “greedy”/“random” to refer to selecting the node with largest $\Delta u$ or a random node in each iteration. We considered a nominal speed of $80$ according to which arrival times were set, i.e., the constant speed of a CL $j$ is $v_{j}=80$. For the fuel model we considered that $F^{0}=1$, $F^{1}=1/80$, and $F_{p}^{0}=.9F^{0}$, $F_{p}^{1}=.9F^{1}$. We get that (18) evaluates to $80\pm 35.777=(44.223,115.777)$. For pairwise gain we terminated Algorithm 1 additionally once a specific $\mathcal{N}_{\mathrm{l}}$ reoccurred in order to avoid ending up in an infinite loop as described in Example 1.