1 Introduction

Vehicular networking, based on wireless communications between vehicles and with other infrastructures, enables a variety of new vehicular safety applications. One of the most challenging applications is Cooperative Vehicle Safety Systems (CVSSs) [1]. In CVSSs, each node (vehicle) periodically delivers their own state information (e.g., vehicle position, heading, and acceleration) and event (e.g., slamming changing lane) to the nodes in their neighborhood over a shared channel. Neighboring nodes receiving this information analysis these received data and predict possible potential collisions [2]. The standards for CVSSs communication is IEEE 802.11p and IEEE 1609 standards [3], which is defined under a DSRC framework [4]. Based on these standards, CVSSs can detect potential threats and provide the vehicle collision warning in real time. CVSSs play a critical role in collision avoidance, and it has been showed that CVSSs could reduce over 75 % of a nation’s crashes [5].

The performance of vehicle tracking, i.e., vehicle tracking accuracy, is the basis for CVSSs [2]. Vehicle tracking accuracy is the error between the real position of a sending vehicle and the last known position retrieved from a received message by its neighboring nodes [6]. The performance of vehicle tracking, however, is significantly affected by the network. In CVSSs, the wireless channel resource in the network is limited. Large number of vehicles in a scenario with high vehicle density will compete for access to the limited channel resource to periodically broadcast vehicle state information. The channel will be saturated even for medium vehicle density [7]. It is shown in [8] that (with 100 neighboring nodes delivering 10 packets per second, each of size 500 bytes) the generated channel load will exceed the channel’s ideal capacity (3 Mbps). In CSMA, a high channel load will result in a significant amount of packets collisions. As a consequence, the state information of the neighboring nodes cannot be up-to-date, which adversely affects the vehicles tracking accuracy and consequently, brings potential collisions among vehicles.

Controlling transmission power can be used to limit the channel load while providing high tracking accuracy for vehicles in CVSSs. In vehicular networking, it is more important to gain higher tracking accuracy for closer neighboring vehicles [9]. Transmission power could be decreased to reduce the load of periodic state messages on the channel and hence successfully broadcast tracking information for closer neighboring vehicles. Transmission power is one of the most important adjustable parameters in vehicular networking to maintain the reliable network-level and application-level performance under different traffic and network conditions [10].

In the literature, several solutions [2, 9–12] have been proposed to tackle channel congestion and vehicle tracking problems by controlling the transmission power or packets transmission rate. All these solutions, however, rely on channel status to determine the optimal transmission power or transmission rate. In these solutions, channel status is detected through the number of message exchange or collision rate on the channel. The collisions or the number of message exchange, however, are hard to detect in broadcast environment with no acknowledgement mechanism in vehicular networking. Meanwhile, actions to control the transmission power are undertaken only after channel congestion status has been detected. Thus, there need some time to recover from the congestion state. These approaches expose the CVSSs to the risk of low tracking accuracy during an interval of channel congestion, which is not suitable to real-time vehicle tracking application in CVSSs.

In CVSSs, the tracking accuracy is mainly depended on a network performance metric: the information dissemination rate (IDR). IDR measures how many packets from a node are successfully received by all neighboring nodes per unit time in the network [2]. The channel load of the network, however, is largely depended on the dynamic vehicle density. Thus, it is necessary to use the real-time vehicle density information to control the transmission power in order to limit the channel load while at the same time achieving the high IDR for each node. However, in the power control problem, increasing the power of one node may result in the increase of the interference for other neighboring nodes, degrading their network performance. That is, the increase of one node’s power is at the cost of performance degradation of other nodes and usually causes the network state to deviate from the ideal value [13]. Therefore, it is imperative to design a dynamic feedback control scheme to determine the real-time power strategies in order to mitigate the adverse effects of dynamic vehicle density and interference on the performance of tracking accuracy.

Against this background, a dynamic feedback power control scheme (DFPCS) is proposed in order to deal with the effect of continuous change of vehicle density and the interference on vehicle tracking in CVSSs. This scheme is based on a proportional-integral-derivative control (PID control) [14] theoretic approach. The PID control is a closed-loop control approach that adjusts the process parameters in a feedback manner in order to make the performance of a process reach the ideal value. The PID control can cope with the interference and uncertainties and has strong ability to guarantee the robustness of a system to continuous dynamics and interference. In the proposed dynamic feedback control scheme, a predictive model is proposed to obtain the ideal IDR and optimal transmission power according to the current vehicle density. Based on the ideal IDR and optimal transmission power, a feedback control model based on a PID controller is designed. This feedback control model evaluates the error between the ideal IDR from the predictive model and the current real IDR from the real information dissemination process to produce the real-time power strategies. These real-time power strategies will be finally implemented in the real network and hence guide the real information dissemination process to the ideal IDR state. Simulations are carried out using traffic simulator VISSIM and network simulator NS3, and the results confirm that the proposed dynamic feedback power control scheme is robust to variations of vehicle density and interference, which can remarkably improve the performance of the vehicle tracking in CVSSs.

The contribution of the paper is proposing a dynamic feedback power control scheme to reduce the channel load while improving the performance of vehicle tracking in CVSSs under the effect of the dynamic change of vehicle density and interference factors. In this proposed scheme, a dynamic IDR model that captures the high dynamics of vehicle density is proposed. Based on this model, a predictive model is introduced to obtain the ideal IDR in corresponding to the current vehicle density. This model predicts the ideal IDR in a dynamic and receding-horizon fashion manner. The obtained ideal IDR will provide a benchmark for the feedback power control. Then, a feedback control model is designed to produce the real transmission power strategies to guide real network state to evolve towards the ideal IDR state. Distinct from all the previous studies, the proposed scheme operates in a closed-loop feedback fashion and hence, can dynamically guide the network under control in such a way that the high tracking accuracy for each node could be achieved under the conditions of varying traffic situation and the effect of interference.

In the next section, we review related work. The vehicle tracking problem and PID theory are described in Sect. 3. Based on PID, a dynamic feedback power control scheme is proposed in Sect. 4. Section 5 provides a dynamic information dissemination Model. In Sect. 6, we develop a predictive model. Section 7 presents a feedback power control model. Section 8 gives the evaluation of the performance of the proposed dynamic feedback power control scheme. The conclusions are given in Sect. 9.

2 Related Work

There have been many studies on congestion control for vehicle tracking in CVSSs. These solutions mainly focus on either transmission rate or transmission power control to limit the channel load and improve the tracking performance.

One such solution relies on transmission rate control to reduce the channel load while improving the tracking accuracy. The US Vehicle safety communication consortium (VSCC) proposed a Beaconing strategy [15], in which vehicles send state information every 100 ms to their neighboring nodes, in order to allow their neighboring nodes to tracking the sending nodes. Rezaei et al. [16] proposed an adaptive transmission rate scheme for CVSSs to reduce the channel load. This scheme adjusts the transmission rate according to a threshold crossing policy defined for tracking error violation. When the real tracking error exceeds the threshold value, the vehicle will increase its transmission rate. On the contrary, when the real tracking error is lower than the threshold value, it will increase the transmission rate. However, this scheme assumes that there is no packets loss or time delay in the communication channel, which does not hold in actual communication process. Rezaei et al. [5] proposed a scheme of dynamic transmission rate adjustment with repetition to reduce the communication load while improving vehicle tracking accuracy. In [6], a distributed transmission rate control algorithm is proposed. The algorithm is dynamically adapted to tracking errors of neighboring nodes and packets collision ratio in the channel. The transmission rate is adjusted based on tracking accuracy and network status, which can reduce the channel load and attain the acceptable tracking accuracy. Bansal et al. [17] proposed a transmission rate control strategy to adjust each vehicle’s transmission rate according to the channel status and vehicles’ speed. The strategy reduces the transmission rate for a high channel load and highly dynamic vehicles, which reduces the channel load and provides the high tracking accuracy. However, when in a scenario with heavy traffic, the transmission rate control method will has limited effect in reducing the channel load.

Another solution relies on transmission power control to limit the channel load. Fallah and Huang [2] analyzed the sizes of transmission rate and transmission range that affect the network status and vehicles tracking performance, and then designed a feedback scheme for transmission power control based on the channel busy ratio. The proposed transmission power control scheme can improve tracking accuracy under different network conditions. Fallah et al. [9] proposed a combined rate and power control scheme to control network congestion caused by periodic data traffic. The approach uses the observations of channel busy time to derive the optimal transmission power to reduce the load generated by periodic messages. The work in [11] showed that when the channel occupancy is maintained between (0.4, 0.8), a good performance of information receiving rate could be achieved. A feedback scheme for transmission power control based on the channel occupancy was then designed to improve the vehicle tracking performance. Torrent-Moreno et al. [12] proposed a distributed and fair transmission power control scheme called a D-FPAV based on the well-known max–min fair principle. The method restricts the load on the shared channel to a predefined MaxBeaconingLoad (MBL) and thus, reserves bandwidth for emergency messages. The above schemes mainly rely on channel occupancy or channel collision probability to decide how a transmission power action should be undertaken. But the actions to reduce the load are taken only after congestion has been happened. This exposes CVSSs to the risk of high tracking errors during the time in which the channel is recovered from the congested condition. Moreover, the interference generated from power control that affects the network performance is not considered. In vehicular networking, channel load is largely depended on the vehicle density. Therefore, it is necessary to design a closed-loop feedback power control strategy to mitigate the adverse influence of rapid change of vehicle density and interference factors in order to maintain accurate tracking accuracy for each node.

Distinct from all the previous congestion control schemes, we present a dynamic feedback control scheme for transmission power adaption based on PID to improve the tracking accuracy for each node. The PID-based feedback control scheme aims to adjust the transmission power in a dynamic and closed-loop manner to reduce the channel load while at the same time improving the tracking accuracy for all nodes under the conditions of dynamic vehicle density and interference factors.

3 Preliminaries

This section gives a brief description of the vehicle tracking problem in CVSSs and the PID control theory.

3.1 Vehicle Tracking Problem

The vehicle tracking function blocks inside each vehicle are shown in Fig. 1. For n ≥ 2 nodes, each node i, \(i \in \{ 1,2, \ldots ,n\}\), has a plant and a bank of neighbor estimators. The plant describes the state of node i. The neighbor estimators operate simple kinematics models for estimating the state of the neighbors. Let \(\tilde{x}_{ji} (t)\) be the receiver j’s estimation of sender i’s position and \(\tilde{v}_{ji} (t)\) be the receiver j’s estimation of sender i’s speed. The kinematics model is formulated as a simple discrete equation, which is switched as following two modes [5]:

  1. 1.

    If no state information of node i is received at time t at receiver j, use the estimated state of node i at time t − 1 to estimate the state of node i at time t, i.e., [5],

    $$\begin{aligned} \tilde{x}_{ji} (t) = \tilde{x}_{ji} (t - 1) + \tilde{v}_{ji} (t - 1)\Delta t \hfill \\ \tilde{v}_{ji} (t) = \tilde{v}_{ji} (t - 1) \hfill \\ \end{aligned}$$
    (1)

    where \(\Delta t\) is the time interval in-between message transmissions.

  2. 2.

    Else if the state information of node i is received at t, use it to reset estimation state of node i, i.e., [5],

    $$\begin{aligned} \tilde{x}_{ji} (t) = x_{i} (t) \hfill \\ \tilde{v}_{ji} (t) = v_{i} (t) \hfill \\ \end{aligned}$$
    (2)

    where \(x_{i} (t)\) is the position of node i and \(v_{i} (t)\) is the speed of node i. Thus, the position error between the actual position and the estimated position of sender i at receiver j is calculated as follows [5]:

    $$\varepsilon_{ji} (t) = x_{i} (t) - \tilde{x}_{ji} (t)$$
    (3)
Fig. 1
figure 1

The function blocks in vehicle tracking

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The objective of our power control is to achieve the minimum vehicle tracking error ε ji (t) under the change of vehicle density.

3.2 PID Control Theory

The idea behind the PID control theory is to design a closed-loop feedback controller that produces the control strategies to guide the controlled process to an ideal state [14].

The basic function blocks of a PID control system are shown in Fig. 2, in which we can see that a PID control system mainly consists of three components: a set point, a PID controller, and a process.

Fig. 2
figure 2

The function blocks in a PID control system

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Set point A set point denotes an ideal performance of a system. Here, the set point R in (t) represents the target objective of a system.

PID Controller A PID controller is a closed-loop feedback controller that use a control law u(t) to reach the target performance. The PID controller is composed of three components, namely the proportional component (P), the integral component (I) and the derivative component (D). P accounts for the current error, I for the past errors, and D for the changing speed of the errors. By adjusting these three terms, the PID controller can produce the real control law u(t) for a target performance of the process.

Process The process is the controlled dynamic system.

As shown in the Fig. 2, Y out (t) is the real performance of a process and R in (t) is the value of the set point. The difference between them is denoted by e(t) as follows [14]:

$$e(t) = R_{in} (t) - Y_{out} (t)$$
(4)

In a PID control system, based on the error e(t) between a real process state and an ideal set point, the controller produces the control law or control actions in order to make the output of the actual plant converge to the output of the set point. The PID control is an effective method to achieve the target performance of a system in the face of interference and uncertainties in the process dynamics.

CVSSs are the typical dynamical and uncertain systems due to the rapid change of vehicle density and interference. In order to achieve the expected tracking accuracy under the change of vehicle density and the effect of interference, in next section, a dynamic feedback power control scheme based on PID control is proposed. This dynamic feedback power control scheme can guarantee a high tracking performance for each node in the presence of rapid change of vehicle density and interference.

4 Dynamic Feedback Power Control Scheme

Based on PID control, a dynamic feedback power control scheme for CVSSs is proposed in Fig. 3. The scheme consists of four key components: the traffic situation, the predictive model, the feedback power control model and the real information dissemination process.

Fig. 3
figure 3

The proposed dynamic feedback power control scheme

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4.1 Traffic Situation

The purpose of the traffic situation module is to describe the real dynamic traffic conditions. We use the traffic simulator VISSIM as the representation of the actual traffic states. As have been mentioned, traffic conditions heavily affect the network performance and consequently, directly affect the tracking process. The significant effect of traffic situation on communication process and tracking process requires a new design of power control scheme that acts depending on the characteristics of the dynamic traffic situation, in order to control channel load and improve the tracking accuracy. Based on the traffic situation module, the traffic states are collected and evaluated to obtain the current vehicle density. The obtained vehicle density is then fed into the following predictive model and feedback power control model for further evaluation, in order to produce the real-time power control strategies.

4.2 The Predictive Model

The purpose of the predictive model is to produce the ideal network performance. The objective of CVSSs is to pursue high tracking accuracy for each node. The tracking accuracy, however, is highly depended on the performance of IDR in IEEE 802.11p. Therefore, we regard the IDR as the ideal network state. Based on the current vehicle density from the actual traffic situation,the predictive model produces the target or ideal IDR in corresponding to the current vehicle density. The obtained ideal IDR will serve as a benchmark for the following feedback power control. Meanwhile, the predictive model seeks the transmission power control strategies to achieve such ideal IDR. Due to the fact that the vehicle density is highly dynamic, only a short-time prediction for network state is feasible in such a dynamic and uncertain traffic situation. Therefore, the predictive model evaluates and predicts the state of IDR in a short-term manner, which can minimize the prediction errors eventually.

4.3 The Feedback Power Control Model

In CVSSs, the inevitable interference in the power control problem may cause the performance of IDR to deviate from the ideal state, leading to the degradation of the vehicle tracking performance. The objective of the feedback power control model is to alleviate the effect of the interference and uncertain feature of the network on the performance of IDR. The feedback power control model is designed based on a PID controller, which dynamically adjusts transmission power for each node to mitigate the effects of interference on IDR of the underlying network. The feedback power control model produces the real-time power adjustment strategies according to the difference between the ideal IDR from the predictive model and the current real IDR from the real information dissemination process, in order to make the response of the real IDR the same as that of the predictive model. That is, the purpose of feedback power control model is to make the error between the ideal IDR and real IDR converge to zero.

4.4 Real Information Dissemination Process

The real information dissemination process describes the actual network state considering the effect of dynamic density and interference factor. The actual network state characters the real information dissemination rate in CVSSs. In the paper, we use the network simulator NS3 to represent the real communication process and network state. The produced power strategies from the feedback power control model will be finally implemented to guide the actual network to evolve towards the ideal IDR, so that accurate vehicle tracking can be achieved under various vehicle densities and interference factors.

The proposed PID-based scheme is in contrast to traditional methods [2, 6, 9, 11] that only use the network status, such as channel busy ratio or collision ratio, to design adjustment strategies for network parameters. These traditional methods do not take into account the highly dynamic and time-varying characteristics of vehicle density. The proposed scheme integrates the dynamic traffic situation and the real network state in a unified framework, which can produce the real-time power control strategies and thus, alleviate the effect of dynamic vehicle density and the interference factors on the performance of vehicle tracking.

Before discussing the main components (predictive model and feedback power control model) of the proposed scheme, we first present a dynamic information dissemination model. This model describes the evolvement of IDR under the effect of dynamic vehicle density.

5 The Dynamic Information Dissemination Model

In CVSSs, a reliable and accurate tracking mainly depends on a performance metric: the information dissemination rate [2]. The IDR describe the tracking information dissemination ability of the network. This metric is depended on the probability of a successful packet reception, which is affected by the vehicle density. In this section, we first present the probability model of a successful packet reception. Then, we give the IDR model with the nature of dynamic vehicle density.

5.1 The Probability Model of a Successful Packet Reception

The probability of a successful packet reception is affected by the channel contention and the hidden nodes interference. These two major factors, however, are all affected by the vehicle density. In this section, we first present a model that characterizes the effect of channel contention on successful reception of packets under the change of vehicle density. Then, we give a model that depicts the effect of hidden nodes on the successful reception of packets by taking into account the vehicle density.

5.1.1 Effect of Channel Contention on Successful Reception of Packets

In the IDR model, we assume that vehicles send state packets according to a Poisson distribution with packets arrival rate R i . During the broadcasting process of packets, a node transmits a packet in a randomly selected time slot. Let T VS be the average transmission time of a packet, the probability that node i transmits a state packet in T VS is given by \(P_{i}^{a}\), i.e., [20].,

$$P_{i}^{a} = 1 - e^{{ - R_{i} T_{VS} }}$$
(5)

in which the T VS will be derived based on a Markov chain in the following section.

In CVSSs, the underlying network utilizes the carrier sense multiple access/collision avoidance (CSMA/CA) protocol to regulate the channel access [18]. CSMA/CA uses the back-off mechanism to wait for a random back-off time before access to the channel for transmitting a packet. The back-off process is modeled as a Markov chain model whose states are back-off counter values [19].

In the Markov chain, when a node has a state packet for transmission, it will wait for an interval of AIFS before it can broadcast it, where AIFS = AIFSN·µ is the arbitration interframe space for the transmission of state packets with AIFSN = 6 and the time slot µ = 13 vs. Let p denote the busy channel probability. If the channel is sensed busy by a given node i during the interval of AIFS, node i will select a contention window W i from \([0, \, 1, \, 2, \ldots ,W_{min} - 1]\) as a back-off counter and start a back-off process, where W min is the minimum contention window size specified in IEEE 802.11p. During the back-off process, if the channel is idle, the back-off mechanism will decrements the back-off counter with probability (1 − p); otherwise, it will froze the back-off counter for the period of a packet transmission until the channel is sensed idle again. Once the back-off counter arrives at the zero state, the vehicle will broadcast its packet. The Markov chain model for the back-off process is shown in Fig. 4 [20].

Fig. 4
figure 4

Markov chain model for the back-off process

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Let \(S = \{ 0, \, 1, \, 2, \ldots ,W_{min} - 1\}\) be the states of the Markov chain model. Based on the above back-off process in the Markov chain, the one-step transition probability matrix is defined as Q = [q mn ] l×l , where q mn is the transition probability from state m (m ∈ S) to state n (n ∈ S) and l = W i . In the transition probability matrix Q = [q mn ] l×l , q mn is defined as follows:

$$q_{mm} = \left\{ {\begin{array}{*{20}l} {p,} \hfill & {if\;m = n} \hfill \\ {1 - p,} \hfill & {if\;n = m + 1\;or\;m = n + 1} \hfill \\ {{p \mathord{\left/ {\vphantom {p {W_{\hbox{min} } ,}}} \right. \kern-0pt} {W_{\hbox{min} } ,}}} \hfill & {if\;m = 0\;or\;n = 0} \hfill \\ \end{array} } \right.$$
(6)

The first condition in Eq. (6) denotes that the back-off process transfers from state {m} to state {m} with probability p.

The second condition in Eq. (6) denotes that the back-off process transfers from the state {m} to state {m − 1} with probability (1 − p).

The third condition in Eq. (6) denotes that the transition probability from the state {0} to another state is p/W min . This transition probability means that the transmission of a new packet starts by randomly choosing a new contention window W i (W i  ∈ S) at the zero state with a probability of p/W min .

We define the probabilities of the steady states in Markov chain as a W min -dimensional vector of \(\pi = [b_{0} ,b_{1} ,b_{2} , \ldots ,b_{{W_{min} - 1}} ]\), where b i denotes the probability that the back-off counter reaches the steady state i. Based on properties of Markov model [19], the W i -dimensional vector satisfies the following equations:

$$\left\{ \begin{array} {l}\pi Q = \pi \hfill \\ \sum\limits_{k = 0}^{{W_{\hbox{min} } - 1}} {b_{k} } = 1 \\ \end{array} \right.$$
(7)

where Q is state transition probability matrix defined in Eq. (6).

By solving the above equations, the steady state probability b 0 can be obtained as given by Eq. (8) [20].

$$b_{0} = \frac{2(1 - p)}{{2 - 3p + pW_{\hbox{min} } }}$$
(8)

A node can send a packet only when the current state of the back-off process reaches to state b 0 with probability of (1 − p). Moreover, the vehicle also has a packet for transmission. The probability that node i can send a packet is given by [20]:

$$\tau = \frac{{2(1 - p)^{2} }}{{2 - 3p + pW_{\hbox{min} } }}(1 - e^{{ - R_{i} T_{VS} }} )$$
(9)

The channel is busy if at least one node within the communication range of the sender transmits a packet. However, the vehicle density in the road section is not constant, which makes the number of nodes within the communication range variable. Thus, the time-varying characteristic of vehicle density has to be considered in the channel contention model. Given the vehicle density ρ(t) of a road segment at time t, the probability \(p(t)\) that the channel is busy at time t is given by follows:

$$p(t) = 1 - (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}}$$
(10)

where \(\tau\) is the probability of sending a packet by each node, \(r_{i} (t)\) is the communication range of node i, and \(\rho (t) \cdot r_{i} (t)\) is the number of nodes in the channel contention area.

After some algebraic manipulations, Eq. (10) can be expressed as follows:

$$\tau = 1 - (1 - p(t))^{{\frac{1}{{^{{\rho (t) \cdot r_{i} (t)-1}} }}}}$$
(11)

The Eq. (11) is a continuous and monotone increasing function in the range of \(p(t) \in \left( {0, \, 1} \right)\). It is also easy to see that equation defined by Eq. (9) is a continuous function in the range of p ∈ (0, 1). Thus, the uniqueness of the solution of Eqs. (9) and (11) is proven. The solution to Eqs. (9) and (11) can be solved.

All nodes located in the communication range of sender i contend with sender i to obtain access to the channel for transmission. For a successful reception by a receiver, one of the conditions that has to be satisfied is none of the neighboring nodes located in the communication range of node i transmitting when node i is sending. That is, a successful transmission occurring at a slot time is given that exactly one node transmits on the channel. Let \(\rho (t)\) denote the vehicle density and r i (t) be the communication range of node i at time t. For each \(i \in \left\{ {1, \ldots ,n_{i} \left( t \right)} \right\},\;{\text{where}}\;n_{i} \left( t \right) = \rho \left( t \right) \cdot r_{i} \left( t \right)\) is the number nodes located in the communication range at time t, by considering the dynamics of vehicle density, the probability that none of the neighbors send packets when a packet is being transmitted by node i at time t is:

$$P_{i}^{C} (t) = (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}}$$
(12)

where \(\tau\) is the probability of sending a packet by each node located in the communication range of node i, and \(\rho (t) \cdot r_{i} (t)\) denotes the number of nodes located within the communication range of node i.

5.1.2 Effect of Hidden Nodes on Successful Reception of Packets

Apart from the effect of CSMA contention on the successful reception of packets, hidden nodes also destroy the successful reception of packets at a receiver, which adversely affects the vehicle tracking performance. We now describe how a hidden node collides with a sender at receivers when a sender is sending [2].

As shown in Fig. 5, assuming that I is a sender and K is a receiver, the nodes that are hidden to I is M and N, that is M and N are hidden nodes. The transmission from any hidden node M or N collides with the transmission of I given that node I is sending. Let T t denote the transmission time of a packet and T s denote the duration of a timeslot. Let L = T t /T s denote the number of slots that a packet transmission takes. For successful reception by a receiver, it is necessary that none of the hidden nodes transmit during a period of L time slots before or after I sends its packets [2]. That is, packets collisions will happen if at least one of the hidden nodes transmits packets during 2L duration. From the above analysis, we can see that the number of hidden nodes affect the successful reception probability at a receiver. The number of hidden nodes, however, is also not constant due to the rapid change of vehicle density. The dynamics of vehicle density have to be considered in the collision probability of hidden nodes. Given the vehicle density ρ(t) at time step t, for a receiver j, the probability \(P_{j}^{H} (t)\) that none of the hidden nodes at receiver j send packets when node i is transmitting is:

$$P_{j}^{H} (t) = (1 - \tau )^{{\rho (t) \cdot d_{ij} (t) \cdot 2L}}$$
(13)

where d ij (t) denotes the distance between node i and receiving node j at time t, \(\tau\) is the probability of sending a packet by each hidden node within the hidden nodes area of node i, and L denotes the number of slots that a packet transmission takes.

Fig. 5
figure 5

Contention area and hidden nodes area

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To calculate the probability of successful reception by a receiver, it is imperative that no nodes within the sender’s communication range transmit packets in the same time slot in which the sender is transmitting. At the same time, no hidden nodes are transmitting in an interval of one packet transmission before or after the sender starts its transmission. For each \(j \in \left\{ {1, \ldots ,n_{i} \left( t \right) - 1} \right\}\), where \(n_{i} \left( t \right) = \rho \left( t \right) \cdot r_{i} \left( t \right)\) is the number of nodes located in the communication range at time step t, by taking into account the dynamic nature of vehicle density, putting all the above conditions together, the probability \(P_{j}^{succ} (t)\) that a receiver j successfully receive the packet from sender i at time t is defined as follows:

$$\begin{aligned} P_{j}^{succ} (t) & = P_{i}^{C} (t) \cdot P_{j}^{H} (t) \\ & = (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}} \cdot (1 - \tau )^{{\rho (t) \cdot d_{ij} (t) \cdot 2L}} \\ \end{aligned}$$
(14)

5.2 Information Dissemination Rate

In CVSSs, the tracking accuracy mainly depends on the amount of successfully received state information and number of nodes that received the information. These two elements are captured by a measure of IDR. IDR measures how many packets from a node are successfully received by all neighboring nodes per unit time. IDR includes the above two elements: the number of nodes located in the communication range of a sender and the successful reception rate of packets by each neighboring node, into a measurable parameter [2]. The rate of successfully reception of packets is dependent on the transmission rate of a sender and the probability of successful reception by receivers. The number of neighboring nodes, however, is variable because of dynamic vehicle density. Thus, the dynamic characteristic of vehicle density has to be considered in IDR model. From the description above, the IDR model of sender i is defined as follows:

$$IDR_{i} (t) = \sum\limits_{j = 1}^{{\rho (t) \cdot r_{i} (t) - 1}} {R_{i} (t) \cdot P_{j}^{succ} (t)}$$
(15)

where ρ(t) is the vehicle density, \(R_{i} (t)\) the transmission rate of sender i and \(P_{j}^{succ} (t)\) is probability of successful reception of receiver j at time t.

The successfully transmission rate of sender i is defined as [2]:

$$R_{i} (t) = \frac{\tau }{{T_{VS} }}$$
(16)

where \(\tau\) is the probability of sending a packet by each node, \(T_{VS} = P_{s} T_{s} + P_{o} T_{o} + P_{c} T_{c}\) is the average size of a slot, \(P_{s} = \rho (t) \cdot r_{i} (t) \cdot P_{i}^{s}\) is the probability of all nodes within communication range successfully transmit packets and \(P_{i}^{s} = \tau \cdot (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}}\) is the probability that node i successfully transmits a packet, \(P_{o} = (1 - \tau )^{{\rho (t) \cdot r_{i} (t)}}\) is the probability of the channel being idle, \(P_{c} = 1 - P_{s} - P_{o}\) is the probability of packets collision, T s is the duration of the channel is busy for transmitting a packet, T o is the duration of the idle period, and T c is the average time that the channel is busy during a collision.

For each node \(i \in \left\{ {1, \ldots ,n_{i} \left( t \right)} \right\}\), where \(n_{i} \left( t \right) = \rho \left( t \right) \cdot r_{i} \left( t \right)\) is the number of neighboring nodes of node i at time t, combing Eqs. (14), (15) and (16), the IDR model that characters the dynamic characteristics of vehicle density is defined as follows:

$$IDR_{i} (t) = \sum\limits_{j = 1}^{{\rho (t) \cdot r_{i} (t) - 1}} {\frac{{\tau \cdot (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}} \cdot (1 - \tau )^{{\rho (t) \cdot d_{ij} (t) \cdot 2L}} }}{{T_{VS} }}}$$
(17)

6 Prediction Model

In the paper, we focus on the maximization of IDR performance for each node, in order to improve the tracking accuracy for all nodes in CVSSs. In this section, we give a predictive model to determine transmission power actions such that the best performance of IDR can be achieved according to the current vehicle density. The predictive model is applied to achieve the ideal IDR and moreover to find the transmission power actions to gain such IDR simultaneously. The predictive model predicts the ideal IDR and the corresponding transmission power in a dynamic and receding-horizon fashion. At the start of each predictive horizon, the vehicle density is fixed using the latest traffic situation. Let t be the start of a predictive horizon. At each time step t, for each \(i \in \left\{ {1, \ldots ,n_{i} \left( t \right)} \right\}\) with \(n_{i} \left( t \right) = \rho \left( t \right) \cdot r_{i} \left( t \right)\), the predictive model is expressed using the following non-linear optimal problem subject to constrains:

$$\begin{aligned} \hbox{max} J_{i} (t) & = \sum\limits_{j = 1}^{{\rho (t) \cdot r_{i} (t) - 1}} {\frac{{\tau \cdot (1 - \tau )^{{\rho (t) \cdot r_{i} (t) - 1}} \cdot (1 - \tau )^{{\rho (t) \cdot d_{ij} (t) \cdot 2L}} }}{{T_{VS} }}} \\ & \quad s.t.\;r_{\hbox{min} } \le r_{i} (t) \le r_{\hbox{max} } \\ \end{aligned}$$
(18)

where \(r_{\hbox{min} }\) is the minimum transmission range and \(r_{\hbox{max} }\) is the maximum transmission range. The limits, \(r_{\hbox{min} }\) and \(r_{\hbox{max} }\), come from the safety requirement in CVSSs. The minimum transmission range is around 50 m and the maximum transmission range is around 500 m [9].

We now explain how the predictive model can be applied for transmission power control to maximize IDR performance based on the dynamic vehicle density. Let t be the start of a rolling horizon, at each time step t, each node first measures the current vehicle density ρ(t) of a road segment. Next, the predictive model solves the optimization problem (18) to determine the current optimal transmission power actions for all nodes that maximize the performance criterion \(J_{i} (t)\) subject to constrain on communication range. Finally, the horizon will be moved one step forward once it finishes current period, and the optimization procedure over the new horizon is repeated by measuring the new vehicle density ρ(t + 1) at time step t + 1.

Once the optimal communication range is obtained using Eq. (18), we should map it to actual power value. In this paper, we use the Two-ray ground reflection propagation model, which is a propagation model for the modeling of propagation loss and propagation delay in vehicular networking. The Two-ray ground reflection propagation model is given as follows [21]:

$$P_{j}^{t} (t) = \frac{{P_{i}^{t} (t)G_{t} G_{r} h_{t}^{2} h_{r}^{2} }}{{d^{4} L_{c} }}$$
(19)

where \(P_{j}^{t} (t)\) is the received power of a receiver j, \(P_{i}^{t} (t)\) is the transmitted power of sender i, \(G_{t}\) is the transmission gain, \(G_{r}\) is the reception gain, \(h_{t}\) and \(h_{r}\) are the heights of the transmit and receive antennas respectively, \(d\) is the relative distance between a sender i and a receiver j, and \(L_{c}\) denotes the loss coefficient.

In this model, given the transmitted power level \(P_{i}^{t} (t)\) of node i, the reliable communication range between two vehicles can be determined if the received power equals the threshold level. The threshold level can guarantee a maximum acceptable bit error rate [22]. Thus, based on Eq. (19), we can map the obtained transmission range to actual transmission power value. Therefore, using the predictive model, the generated ideal IDR and the corresponding transmission power can be determined and then, can be used to as a benchmark when design the feedback transmission power control strategies.

7 The Feedback Power Control Model

The inevitable interference generated by the power control problem will cause the real network state to deviate from the expected value. In order to stabilize the real IDR state at the ideal IDR value, and to enhance the robustness of CVSSs in the face of dynamical vehicle density and interference, in this section, a feedback power control model is designed using PID control [14]. The key idea of this feedback power control model is to tune the transmission power of each node such that the real IDR state converges to the ideal value. The feedback power control model, essentially, is a closed-loop controller that employs feedback to determine how well the target IDR has been achieved.

The classical PID includes three components: the proportional component, the integral component and derivative component. Since the derivative component is too sensitive to noise [23], the feedback power control model only exploits the PI (proportional-Integral) control. Figure 3 shows the feedback principle of the power control model. The basic idea is as follows. Let t 0 be the start of the receding-horizon for solving the predictive model and T be the sampling period of the real network state with time step \(t_{s} = kT \left( {k = 1, \, 2, \ldots ,n} \right)\). At time step t s , we use the IDR value \(I_{i}^{in} (t_{s} )\) produced from predictive model as the set point of the feedback power control model. Then, we use the IDR \(I_{i}^{out} (t_{s} )\) from the real information dissemination as the output of the feedback model, feed it back, and compare it with \(I_{i}^{in} (t_{s} )\). Based on the error between \(I_{i}^{in} (t_{s} )\) and \(I_{i}^{out} (t_{s} )\), the PI controller calculates a new transmission power for the next feedback. For each node \(i \in \left\{ {1, \ldots ,n_{i} \left( t \right)} \right\}\), the feedback controller for power control is defined as follows:

$$P_{i} (t_{s} ) = P_{i} (t_{s} - 1) + K_{i}^{P} e_{i} (t_{s} ) + K_{i}^{I} \int_{{t_{0} }}^{{t_{s} }} {e_{i} (\chi )d(\chi )}$$
(20)

where \(P_{i} (t_{s} )\) is the transmission power of node i at time step t s , \(K_{i}^{P}\) is the proportional gain, \(K_{i}^{I}\) is the integral gain, \(e_{i} (t_{s} ) = I_{i}^{in} (t_{s} ) - I_{i}^{out} (t_{s} )\) with \(I_{i}^{in} (t_{s} )\) denoting the ideal IDR of node i generated by the predictive model and \(I_{i}^{out} (t_{s} )\) denoting the actual IDR of node i from the real network at time step t s , and \(P_{i} (t_{s} - 1)\) is the transmission power of node i at time step t s  − 1. Because of the highly dynamic characteristics of the vehicle density, the parameters \(K_{i}^{P}\) and \(K_{i}^{I}\) are not constant. In order to achieve the rapid response capability of this power control strategy, \(K_{i}^{P}\) and \(K_{i}^{I}\) should be tuned adaptively in this feedback power control model. To determine the parameters \(K_{i}^{P}\) and \(K_{i}^{I}\), an objective function is defined as follows:

$$J_{i} (K_{i} ) = \frac{1}{2}\int_{{t_{0} }}^{{t_{s} }} {e_{i}^{2} (\eta )} d(\eta )$$
(21)

where \(K_{i} = [K_{i}^{P} ,K_{i}^{I} ]\).

The objective of the feedback power control model is to make the error between the ideal IDR and real IDR converge to zero. Thus, we can determine the parameters \(K_{i}^{P}\) and \(K_{i}^{I}\) by minimizing the objective function (21). Using the gradient method, the gradient of \(J_{i}\) with respect to \(K_{i}\) is given as follows:

$$\frac{{\partial J_{i} }}{{\partial K_{i} }} = \int_{{t_{0} }}^{{t_{s} }} {e_{i} (\eta )} \frac{{\partial e_{i} (\eta )}}{{\partial K_{i} }}d(\eta )$$
(22)

We can adjust the parameters \(K_{i}\) along the negative gradient direction of \(J_{i} (K_{i} )\), i.e.,

$$\Delta K_{i} = - \lambda_{i} \frac{{\partial J_{i} }}{{\partial K_{i} }} = - \lambda_{i} \int_{{t_{0} }}^{{t_{s} }} {e_{i} (\eta )} \frac{{\partial e_{i} (\eta )}}{{\partial K_{i} }}d(\eta )$$
(23)

where \(\lambda_{i} > 0\) is the step size, \(\Delta K_{i} = K_{i} - K_{i0}\) with \(K_{i0}\) denoting the initial value of the parameters \(K_{i}\). Thus, the tuned parameters \(K_{i}\) is given as follows:

$$K_{i} = - \lambda_{i} \int_{{t_{0} }}^{{t_{s} }} {e_{i} (\eta )} \frac{{\partial e_{i} (\eta )}}{{\partial K_{i} }}d(\eta ) + K_{i0}$$
(24)

In order to obtain the adaptive law of the parameters \(K_{i}\), take the derivative of \(K_{i}\) with respective to t for Eq. (24), i.e.,

$$\frac{{dK_{i} }}{dt} = - \lambda_{i} e_{i} \frac{{\partial e_{i} }}{{\partial K_{i} }}$$
(25)

Equation (25) is the adaptive law of the tuned parameters \(K_{i}\). We can use a discretized method to solve Eq. (25). By solving the Eq. (25), the dynamic controller parameters \(K_{i}\) can be found which will be fed into the feedback controller in Eq. (20). Thus, the real-time power control strategies are produced and then, implemented in the actual information dissemination process and hence, guide the IDR state of the actual network towards the ideal state.

8 Performance Evaluation

This section first introduces simulation settings, and then presents simulation results.

8.1 Simulation Settings

To test the performance of our proposed power control scheme, we have conducted traffic/network simulation experiments for CVSSs using VISSIM [24] and NS3 [25] simulators for road traffic and network simulations, respectively. The main simulation scenario in VISSIM is a straight 1-km of a 4-lane highway. In each simulation, the speed of vehicles follow a uniform distribution between v min and v max with mean µ = (v min  + v max )/2 and variance σ 2 = (v min  − v max )2/2. We set v min  = 20 m/s and v max  = 30 m/s, which is typical for highways [26]. Table 1 lists the simulation parameters used in the experiments.

Table 1 Values of parameters used in the simulation
Full size table

During the simulation, we sample at 10 Hz to collect the vehicle state information, such as vehicle position, speed and heading. At each 100 ms time step, every node generates a packet and broadcasts it to its neighbors. Upon receiving the vehicle state information from the shared channel, each vehicle resets its estimation of vehicles located in its communication range. In this simulation, receding-horizon would be moved 10 s forward when it finishes the optimization during the current horizon. During the next horizon of 10 s, another power control process will be started.

8.2 Simulation Results

The vehicle density has been selected to cover scenarios between congested traffic and free-flow traffic on a four-lane highway. According to [2], the vehicles’ speed in a scenario of free-flow traffic is 32.9 m/s and the vehicles’ speed in a scenario of congested traffic is 6.1 m/s. The time-headway between vehicles includes the response time of drivers and the braking time. The response time of drivers is set to 0.8 s and the maximum braking deceleration is 8 m/s2 [27]. Under the scenario of free-flow traffic, the maximum baking distance is S fb  = (32.9 m/s)2/2 × 8 m/s2 ≈ 67.7 m; Under the scenario of congested traffic, the maximum baking distance is S cb  = (6.1 m/s)2/2 × 8 m/s2 ≈ 2.25 m; The critical inter-vehicle spacing in a scenario of free-flow traffic is S f  = 32.9 m × 0.8 s + 67.7 m = 93.92 m and the critical inter-vehicle spacing in a scenario of congested traffic is S c  = 0.8 s × 6.1 m/s +2.25 m = 7.13 m. Assuming 4 m of vehicle length, the vehicle density in a four-lane congested highway is ρ c  = 4 lanes × 1/(7.13 m + 4/2 m) ≈ 0.4 vehicles/m and the vehicle density in a four-lane free flow road is ρ f  = 4 lanes × 1/(93.92 m + 4/2 m) ≈ 0.04 vehicles/m. In this section, we verify the performance of the proposed DFPCS under the vehicle density range from 0.04 to 0.4 vehicles/m.

To validate the proposed information dissemination model in Eq. (17), we compare the computed IDR from the theoretical model with the obtained IDR from the simulation. The values of the parameters used to obtain the results, for both the analytical model and the simulation runs, are given in Table 1. The experiments have been repeated by varying the vehicle density from 0.1 to 0.4 vehicles/m. In the simulation, all nodes use the same transmission power and adjust it in synchronization with other nodes. The results are averaged over 10 simulation runs. Figure 6 plots the average IDR obtained by each node with different transmission power using the theoretical model and simulation. As can be seen, the theoretical model matches well with the simulation results and hence, the information dissemination model is highly accurate. This result proves the suitability of proposed analytical model for various scales of CVSSs with different vehicle densities. Moreover, from Fig. 6, we can see that there exists a best performance of IDR for different vehicle density. We regard this best IDR state as the ideal state. Hence, this ideal state will serve as a benchmark for the feedback power control.

Fig. 6
figure 6

IDR comparisons between analytical model and simulation

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To verify the convergence performance of the real-time response to the dynamic changes of vehicle density, we simulate a scenario where the vehicle density changes. We assume that at t = 10 s, the vehicle density changes from a constant 0.1 to 0.2 vehicles/m, at t = 20 s, the vehicle density changes from 0.2  to 0.4 vehicles/m and at t = 30 s, the vehicle density returns to 0.1 vehicles/m. Figure 7 shows how the DFPCS reacts when the vehicle density changes. As can been seen, DFPCS converges to the optimal IDR during each stage when the vehicle density changes. Figure 8 shows the convergence rate of the proposed DFPCS when the vehicle density changes. From Fig. 8, we can see that DFPCS converges to the ideal value fast during each stage when the vehicle density changes. After only about 3 iterations, the network state converges to the optimal IDR, which fully demonstrates that the proposed DFPCS has a good real-time responding characteristic.

Fig. 7
figure 7

Evolution of the IDR under DFPCS

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Fig. 8
figure 8

The convergence of DFPCS during the change of vehicle density

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Figure 9 shows how the transmission power varies when the vehicle density changes using the proposed DFPCS. It is clear that, as the vehicle density increases, the transmission power decreases. This means that, to achieve the optimal IDR for high vehicle densities, vehicles have to reduce their transmission powers and provide better IDR performance to nearer vehicles. Meanwhile, when the vehicle density is low, the feedback power control strategy increases the power and leaves the limited channel resource to faraway neighbors in order to provide better IDR for farther nodes, which in turn track farther vehicles in CVSSs.

Fig. 9
figure 9

Evolution of the transmission power under DFPCS

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To test the capacity and effectiveness of the proposed DFPCS, we compare tracking errors of DFPCS with those of Beaconing strategy and adaptive transmission rate control strategy (denoted as ATRCS). Beaconing strategy is a solution proposed by vehicle safety communication consortium (VSCC) [15]. In Beaconing strategy, vehicles send state information every 100 ms to their neighboring nodes. ATRCS is a well-known transmission control strategy that adapts the packets transmission rate based on the tracking errors to improve the performance of vehicle tracking [6]. During each simulation run, we collect statistics from neighboring vehicles within the communication range set by Beaconing strategy, ATRCS and DFPCS, respectively, and calculate the tracking errors of these three strategies. In this simulation, we use the measure of average tracking errors as the main performance metric for CVSSs simulations.

These comparisons are shown in Fig. 10 and Table 2, respectively. From Fig. 10, we can see that during each stage of change of the vehicle density, the proposed DFPCS performs well and the tracking errors produced by DFPCS are lower than Beaconing strategy and ATRCS. The tracking errors using the Beaconing strategy are heavily increased when the vehicle density is high. In Beaconing strategy, the transmission power is restricted to a fixed value (25 dBm). Thus, in the scenario with high vehicle density, more nodes will lie within the communication range and consequently, these nodes would simultaneously try to access the channel for broadcasting packets all the time, which results in high packet collisions. As a consequence, Beaconing strategy suffers higher tracking errors when there is a high vehicle density. In ATRCS, the packets transmission rate is dynamically adjusted based on the estimated tracking error and channel busy ratio. When the channel busy ratio is high, ATRCS will decrease the packet generate rate to reduce the channel collision ratio. The tracking accuracy, however, is directly dependent on the amount of packets transmitted to receivers and thus, the tracking accuracy is directly dependent on the transmission rate of state packets. Once the transmission rate is decreased, not enough packets is available for neighboring nodes to track the sending nodes in real time, which in turn causes large tracking errors. In the case where the vehicle density is high, the DFPCS decreases the transmission power to reduce the opportunity to access the channel all the time for nodes, which reduces the collision loss ratio of packets and hence, improves the tracking accuracy for nearer nodes. This is the ideal behavior since the nearby neighboring nodes are more dangerous for the sending nodes and therefore, it is more important to provide higher tracking accuracy for closer neighboring vehicles in CVSSs. Table 2 shows the tracking errors produced by Beaconing strategy, ATRCS and DFPCS when the vehicle density is set to 0.1vehicle/m, 0.2vehicle/m, 0.3vehicle/m and 0.4vehicle/m, respectively. We can see that our proposed DFPCS generates higher tracking accuracy compared with these traditional schemes. Although for lower vehicle density, the Beaconing strategy and ATRCS work better than DFPCS, the channel is not used to its potential. When facing low vehicle density, the proposed DFPCS increases the transmission power to provide better tracking performance to farther nodes.

Fig. 10
figure 10

Tracking errors versus vehicle density

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Table 2 The comparison of tracking errors among three strategies
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Therefore, our DFPCS is adaptive to the dynamic change of vehicle density. When there is a high vehicle density, DFPCS can reduce the channel load and provide higher tracking accuracy for nearer nodes. When there is a low vehicle density, DFPCS can make full utilization of the limited channel resource and hence, enable nodes at far distances to track each other. These above testing results demonstrate that the DFPCS is robust to varying traffic conditions.

9 Conclusion and Future Work

In this paper, a dynamic feedback control scheme for transmission power adaption in CVSSs is proposed in order to improve the tracking accuracy under the continuous change of vehicle density and the effect of interference factors. We first introduce a dynamic IDR model by considering the nature of time-varying vehicle density. The dynamic IDR model quantifies the ability of the network to disseminate vehicle tracking information under the impact of varying vehicle density. Based on the IDR model, we propose a predictive model to generate the ideal IDR state to achieve the optimal tracking accuracy in corresponding to the current vehicle density. Then, a feedback power control model is designed to produce the real power control strategies to make the response of the real information dissemination process become the same as the ideal IDR state. Distinct from previous power control schemes, the proposed scheme adjusts the transmission power in a closed-loop and feedback manner. The simulation results reveal that this proposed dynamic feedback power control scheme can achieve the robust tracking accuracy under various traffic situations and the effect of interference.

Our proposed scheme can be extended in several ways to determine the other network parameters, such as transmission rate and contention window, to improve the tracking accuracy. Moreover, the tracking performance is also affected by the physical dynamics of vehicle movement. Therefore, transmission power and transmission rate should be jointly controlled to achieve the optimum performance of the vehicle tracking under the scenarios with varying vehicle density and high dynamics of vehicle movement.