In this section, we formulate the AUV-aid data gathering problem with the objectives of minimizing the AUV travel distance, minimizing the total energy consumption and maximizing the network lifetime.
3.1. Motivation and Overview
Leveraging the sub-sea MI communication model in [
27], as shown in
Figure 2, the MI channel capacity can exceed 5 ×
bps. Moreover, we have built up a sub-sea MI testbed as shown in
Figure 3. A data transmission based on Quadrature Phase Shift Keying (QPSK) modulation scheme with symbol rate equal to 100 kHz is demonstrated in
Figure 4. Limited by the size of the water tank, we were only able to separate the transmitter and receiver at the distance of 0.7 m. The testing results show that the data transmission is 100% successful. Thus, leveraging the advantages of reliable and high speed MI communication, long distance acoustic communication and the mobility of AUVs, UWSNs can dramatically reduce energy cost on data gathering with high reliability.
Consider an AUV and many sensors form a underwater wireless sensor network. Each sensor is anchored to the target area for ocean monitoring/sensing applications. Both magnetic-induction (MI) and acoustic communication modules are equipped to every sensor for short distance high speed communication and long distance low speed communication, respectively. We assume that the perceived data are periodically gathered and delivered to the designated surface station. During each period, the AUV travels from the surface station, dives into the water and visits a subset of the sensors for data collection, brings the collected data back to the surface station, and prepares for the next round data collection. Since the MI communication range is quite short, the AUV should visit the sensor position exactly for MI communication. These sensors that are not visited by the AUV transmit their sensing data to chosen sensors via acoustical channels.
Suppose there are sensors, and the i-th sensor is anchored at the coordinate . Let denote the acoustic sub-channel set. The transmission range on the m acoustic sub channel is , and the interference range of this channel is . The neighbor sensor set for each sensor can be obtained according to their coordinates and communication range. For simplicity, in this paper, we assume the AUV’s speed is constant at V, and the MI communication rate is constant at .
3.2. Problem Formulation
As mentioned above, we have three objectives for this problem, and the first objective is to minimize the AUV traveling distance:
where variable
is the distance between sensor
i and sensor
j and
is a binary variable that indicate the AUV travel path. If the AVU travel from sensor
i to sensor
j,
. Otherwise,
.
The second objective is to minimize the total energy consumption:
where
is the data amount from sensor
i to sensor
j.
and
denote the energy consumption of sensor
i on transmitting and receiving through acoustic channels, respectively.
is the neighbour sensor set of sensor
i.
The final objective is to maximize the network lifetime. We define the network lifetime as the sensing round that the first sensor runs out of energy. Therefore, the objective of maximizing the network lifetime can be seen as maximizing the lifetime that is the minimum lifetime in the network:
where
is the energy consumption on this round predefined sensing task,
is the residual energy of sensor
i and
is the energy consumption on transmitting its own data through MI.
To achieve the above objectives, we have the following constraints:
Notations used in above equations are listed in
Table 2. Equations (
5) and (6) are acoustic-channel interference constraints. Because of the interference in acoustical communication, each sensor can only transmit to or receive from another sensor through a specific acoustic channel at a time (Equation (
5)). We use a binary variable
to indicate the sub-channel state. If node
i transmits data to node
j on sub-channel
m at time
t,
. Otherwise,
. In Equation (6),
is the sensor set within the
jth sensor’s interference range on channel
m. If sensor
j is within the interference range of sensor
k on sub-channel
m, and sensor
k is transmitting data to its neighbor
l via
m sub-channel, then sensor
j’s neighbor
i will fail to transmit to sensor
j through sub-channel
m at this time because of signal interference (Equation (6)).
Equation (8) to (14) are data flow constraints. We define
and
to represent the sensor sets that will be visited by AUV and will not be visited by AUV, respectively. Then there is no data flow to
sensors (Equations (8) and (9)) and no data flow from
sensors (Equations (10) and (11)). Moreover, the data flow from an
sensor equals the data it sensed (Equation (
12)), and the outgoing data of each sensor is the sum of the incoming data and its sensing data (Equation (13)). Variable
in Equation (
12) is the sensing rate of sensor
i. In addition, the data flow on each sub-channel should not exceed the link capacity (Equation (14)). Variable
in Equation (14) is the channel capacity between sensor
i and sensor
j on sub channel
m.
Equation (15) means that, during the data gathering process, each node should not spend more energy than its remained energy. Variable
in Equation (16) is the buffer size of the
ith sensor, and Equation (16) constrain the data amount stored on an
sensor should not exceed the buffer size of this sensor. A binary variable
is used to indicate the sensor type. If sensor
i is an
sensor, then
. Otherwise,
. Equations (18) and (19) are the travel tour constraints that each chosen sensor must visit exactly once. In addition, the AUV stop time on each sensor for data gathering is:
Since there is a polynomial-time reduction from the traditional traveling salesman problem to the above problem by setting and to zero, the above problem is an non-deterministic polynomial-time hard (NP-hard) problem. We implement the problem using Matlab+YALMP+GUROBI and found that, for a small scenario with no more than eight nodes, it takes more than an hour to find the optimal solution. Considering that this problem is NP-hard, the execution time would grow exponentially, and we propose a heuristic algorithm to solve this problem in the following section.