The radio model for a sensor node’s energy consumption assumes that all of the nodes maintain a minimum level of successful communication. There are different energy calculations between transmitting and receiving data. Generally, a radio model defines the energy consumption of a transmitter-receiver as E_elec J/bit, and the energy consumption of a transmitter amplifier by a signal to noise ratio as E_amp J/bit. Therefore, the following equation is required for sending a k (bit) message to a node with a distance of d (meters) [4]: (1) E Tx ( k , d ) = k   E elec + kE amp d 2

Equation (1), E_Tx, is the transmitting energy of a radio beam: (2) E Rx ( k , d ) = kE elec

In Equation (2), E_R, is the receiving energy of a radio beam. We set up a distance d to R for multi-hop based clustering as seen in the following energy equations: (3) E Rx ( k , R ) = kE elec = l (4) E Tx ( k , R ) = k   E elec + kE amp R 2 = l + μ R 2

The clustering energy consumption of sensor networks is divided into two categories: intra energy generated in a local cluster and inter energy generated when a cluster-head sends the aggregated data to a sink. In each category, the total energy consumption of the networks, E_cluster, can be measured by adding the cluster-heads energy, E_ch, to the member nodes energy, E_member as follows: (5) E Tx ( k , R ) = k   E elec + kE amp R 2 = l + μ R 2where ‘m’ is the optimal number of cluster-heads and N is the number of sensor nodes in the networks.

For calculation, we assume the sensor networks are organized as follows: the size of the network is A × A. Sensor nodes are equally distributed over the networks. For simplicity, a cluster-head is located in middle of a local cluster as shown in Figure 1(a). We assume that the distance of the sensor nodes located farthest from the cluster-head is ‘a’ and the radius of a local cluster which is the distance between the cluster-head and a sink, ‘s’. The communication radius of a sensor node is R, (r ≤ R), R is the possible radius of communication within its neighbor nodes and r is the communication radius of sensor nodes [11,12]. Based on these assumptions, we find the energy and size change of a local cluster based on a change in the number of cluster-heads.

The total number of cluster-heads in a network is defined as m, therefore the area of the sensor networks is the same as the total area of local clusters which can be calculated as a²π × m = A². From this, we can figure out the radius of a local cluster, ‘a’, as: (6) a = A m π

The number of nodes in a local cluster can be expressed as N/m. As shown in Figure 1(b), a local cluster can be divided into a number of n rings within a radius of nodes, R, for multi-hop communication. The n^th ring represents the distance at the farthest nodes, ‘a’. Additionally, it represents hop counts between the cluster-head and the farthest nodes. So, nodes’ hop counts in the n^th ring can be described as a/R. We know the average number of sensor nodes with n^th hop counts as compared to the area of a local cluster within the area, which is to subtract the area of the (n−1)^th ring from the area of the n^th ring [14]. Therefore, n^th_{avg_node}, the average number of nodes with n^th hop counts is: (7) n avg_node th = N [ π a 2 − π ( nR ) 2 ] π a 2 m

The average number of nodes with (n−1)^th hop counts, (n−1)^th_{avg_node} is: (8) ( n − 1 ) avg_node th = N { π ( nR ) 2 − π [ ( n − 1 ) R ] 2 } π a 2 m

When nodes with n^th hop counts send data packets to nodes with (n−1)^th hop counts, the number of relay packets can be determined by dividing Equation (7) into Equation (8). With n^th hop counts in a local cluster, the average number of relay packets, k_{n_Intra}, is: (9) k n_Intra = N [ π a 2 − π ( nR ) 2 ] / π a 2 m N { π ( nR ) 2 − π [ ( n − 1 ) R ] 2 } / π a 2 m = a 2 − n 2 R 2 R 2 ( 2 n − 1 )

In an intra-cluster, a node’s energy consumption is divided into two types: transmission energy of own sensing data and relay energy of neighbor’s sensing data. Therefore, the average energy consumption of nodes with n^th hop counts, E_{mem_intra}, is to add packet transmission energy, E_Trans to own packet transmission energy, E_Relay, of packet relay energy. This equation can be represented as: (10) E mem intra = E Relay × k n Intra + E Trans                                 = [ E T x ( k , R ) + E R x ( k , R ) ] × k n_Intra + E T x ( k , R )                                 = [ ( l + μ R 2 ) + l ] k n Intra + ( l + μ R 2 )                                 = ( 2 l + μ R 2 ) a 2 − n 2 R 2 R 2 ( 2 n − 1 ) + ( l + μ R 2 )

The average number of packets which a cluster-head receives from member nodes is N/m. The energy consumption of cluster heads for aggregate packets is E_agg. The received packets are sent to a sink node by cluster heads during inter-cluster transmission. At this time, the transmission energy of a cluster head occurs when sending packets to the sink, E_Trans. The average energy consumption of a cluster head, E_ch, can be represented as: (11) E ch = E Agg × N m + E Tran = E Agg × N m + ( l + μ R 2 )

Generally, a sink node is located close to the sensor network. The distance between a cluster-head and a sink is from one hop R to n^th node’s distance added to one hop R, R ≤ s ≤ a A + R. Therefore, we can assume that the average distance between a cluster-head and a sink, s, is: (12) s = ( 2 A + 2 R ) 2

As the number of relay packets is also increased in proportion to s, in inter-clusters, the average number of relay packets, k_{n_Inter}, is: (13) k n_Inter = sm R × 1 m = s R

In an inter-cluster, as the CHs can only send the packet to a sink node, we just find the relay packet energy. To achieve this, during inter-cluster transmission, the average energy consumption between a cluster-head and a sink, E_{mem_inter}, is as follows, according to Equation (13): (14) E mem_inter = E Relay × k n_Inter                                       = ( 2 l + μ R 2 ) × k n Inter = s ( 2 l + μ R 2 ) R

Based on the clustering energy equations, we can find the average energy consumption of member nodes, E_member, by using the following added equations: (15) E member = E mem Intra + E mem Inter                               = ( 2 l + μ R 2 ) a 2 − n 2 R 2 R 2 ( 2 n − 1 ) + ( l + μ R 2 ) + ( 2 l + μ R 2 ) s R

For network configuration, we assume the following network topology, as described in Table 1.

We set up the size of the networks to be 100 m × 100 m, with a possible node communication radius, R, set at 5 meters. To prevent an isolated node, the number of network nodes is 400. The sensor node’s initial energy is 1 J (Joule) and the data packets of a node are 525 bytes between a cluster-head and member node, and a sink and a cluster-head. As described previously, a sink node is located outside of the sensor networks with the distance between a sink and the networks defined as R. For constant energy, we set up the transmission/receiving energy, E_elec, to be 50 nJ/bit and the amplifier energy, E_amp, to be 10 pJ/bit/m². The aggregation energy per a packet, E_f or E_agg, is 0.21 mJ/bit [4]. After cluster configuration, we assume that the process of transmitting sensing data to a sink is one round. Thus, member nodes send their own sensing data packet to cluster-heads, cluster-heads aggregate and process them, and then the cluster-heads transmit the processed data to a sink over a data path. In these processes, we omit the messages or packets for clustering as negligible.

In an intra-cluster scenario, if there is a single cluster-head, as illustrated in Figure 2(A), the number of relay packets of member nodes is more than 2,400 per round, based on multi-hop communication. As the number of cluster-heads increases, Figure 2(A) shows that the number of relay packets is decreased.

The reason for this is that the average distance between a cluster-head and member nodes is decreased. In the case of cluster-heads being more than 27% of the networks, a local cluster does not generate the relay packets, as the distance between a cluster-head and member nodes is a < R. On the other hand, the packets which a cluster-head collects from member nodes are fixed, regardless of changing the number of cluster-heads, because the total aggregated data from nodes is always 400, as seen in Figure 2(B). Therefore, the average energy consumption of a node only affects intra-cluster energy consumption, as seen in Figure 2(C). As shown in Figure 2(C,D), the energy consumption of a node is in proportion to the intra-cluster energy consumption.

In an inter-cluster, with an increase of cluster-heads, there is an increase of relay packets between a cluster-head and a sink. As illustrated in Figure 3(A), if sensor networks have only one cluster-head, there are 15 relay packets in the inter-cluster. When cluster-heads are more than 30% of the sensor networks they send more than 1,800 relay packets to a sink. An increase in the number of relay packets affects inter-cluster energy consumption, as seen in Figure 3(B). This is the reason most of the energy consumption in an inter-cluster is only affected by the relay packets which cluster-heads transmit to a sink. Even if the inter-cluster energy is increased, the average inter-cluster energy consumption is fixed after 12 cluster-heads, 3% among sensor nodes, as shown in Figure 3(C), because increased cluster-heads distribute the energy loads evenly in the networks.

In Figures 2(D) and 3(B), the amount of intra-cluster energy is inversely proportional to the amount of inter-cluster energy. Therefore, it is important to obtain a balance between the two. Using the two energy consumptions, we can find the total energy consumption of sensor networks, as shown in Figure 4(A). In this figure, in the case of a single cluster-head, the total energy consumption of the network is 1.05 J. This value is very large compared to the initial sensor energy. If the number of cluster-heads is 24, or 6%, the total energy consumption is minimal at 0.32 J. If the number of cluster heads number is more than 24, the energy consumption is gradually increased. Figure 4(A) shows that the ratio of cluster-heads should be between 4% and 8% in order to consume the minimum amount of energy. Considering the average energy consumption per round, we can find the number of rounds of a node. As seen in Figure 4(B), in the case of only one cluster-head, a node can be alive below 400 rounds. In the case of 24 cluster-heads however, a node can be alive for more than 1,200 rounds. So, to establish 4% to 8% of cluster-heads among sensor nodes is the most energy efficient method to set up for the optimal number of cluster-heads.