Innovative Fitness Functions for Robust Energy Management in WSNs

Wireless Sensor Networks (WSNs) are widely recognized as a crucial enabling technology for Internet of Things applications. One of the primary challenges in designing WSNs is to ensure efficient energy management, which involves minimizing and uniformly distributing energy consumption among the wireless sensors to extend the network lifetime. In this paper, we propose a set of mathematical tools in the form of fitness functions that can be used to measure, compare, and control how the network manages energy consumption among wireless sensors. Furthermore, we present an Optimized Energy Balanced and Distributed Clustering (OEBDC) protocol for WSNs, which utilizes these fitness functions to manage energy resources more efficiently by promoting uniform energy consumption, minimizing communication overhead, and extending network lifetime. The proposed tools can be integrated with other WSN protocols to manage energy resources according to specific requirements that suited different applications. We have evaluated the performance of the proposed protocol against well-established routing protocols for WSNs and found that OEBDC achieves a notable advantage in extending network lifetime compared to other protocols, while also demonstrating robust control in managing energy resources.


Overview and Background
Energy management in wireless sensor networks is crucial for maximizing network lifetime, optimizing resource utilization, reducing costs, minimizing environmental impact, ensuring reliable data collection, and enhancing network scalability.It can summarize the benefits of energy management in WSNs as follows [1][2][3][4][5]: Prolonged Network Lifetime: Energy management techniques aim to extend the network's operational lifetime by conserving energy resources.Since many WSNs are deployed in remote or inaccessible areas, it is often impractical to replace or recharge their batteries frequently.By efficiently managing energy consumption, the network can operate for an extended period without requiring maintenance.
Scalability and Deployment Flexibility: Energy management enables the deployment of larger sensor networks or increases the number of deployed nodes within a given energy budget.By optimizing energy usage, the network can accommodate a greater number of nodes, leading to increased coverage, data collection, and monitoring capabilities.
Cost Reduction: Energy-efficient protocols and algorithms can help minimize the energy consumption of individual sensor nodes, reducing the overall operational costs associated with battery replacement or maintenance.Additionally, by maximizing the network lifetime, energy management techniques can delay or eliminate the need for costly network infrastructure upgrades.
Reliable and Timely Data Collection: Energy management techniques ensure that sensor nodes remain operational for an extended period, allowing continuous and uninterrupted data collection.By conserving energy and avoiding premature node failures, the network can provide reliable and timely information, which is crucial for various applications, such as environmental monitoring, surveillance, and industrial automation.
Network Quality of Service (QoS): Effective energy management enhances the overall performance and QoS of the wireless sensor network.By reducing energy consumption, the network can allocate resources more efficiently, increase throughput, reduce latency, and improve packet delivery rates.This, in turn, enhances the reliability and responsiveness of the network, enabling better application-level performance.
To manage energy resources efficiently, the sensor nodes need to organize themselves into clusters that are managed by cluster heads.The job of the sensor nodes is to collect information, aggregate it and send it to their cluster heads.The cluster heads collect information from the sensor nodes and send data directly to the base station for further processing.The communication between cluster heads and the base station may occur on a single-hop or multi-hop communication.Typically, hundreds to thousands of sensors are distributed randomly or regularly into the area under the study.Hence, the sensor nodes should be cost-effective and efficient in terms of energy consumption to maximize their lifetime.The reliability of WSNs increases as the number of deployed wireless sensors increases.The main job of sensor nodes is to organize in a manner such as collecting information, aggregating, and sending it back to the base station for further processing.
WSNs are different from the conventional wireless network since their components, i.e., sensor nodes, are limited in terms of processing power and energy resources.Conventional wireless network protocols do not typically apply to WSNs due to several limitations.Firstly, sensors are deployed randomly into a field with no infrastructure.Hence, it is the responsibility of these sensors to localize themselves.Secondly, the limited energy resources of sensors impose limitations on the amount of processing that these sensors can perform.Thirdly, WSNs should continuously re-organize their sensors due to changes in a network configuration that may result from either external, environmental, or internal factors such as sensor nodes dying [2].
The key idea behind the extension of WSNs lifetime is to equally distribute energy consumption among wireless sensors and minimize communication overhead.This requires careful energy management throughout network layers.According to [1], the percentage amount of energy consumption across wireless communication modules is; sensing (2%), processing (5%), transmission (32%), reception (30%), idle (30%), and stand by (1%) as shown in Fig. 1.The most energy consumption is in transmission, reception, and idle modules with a total percentage of 92%.Thereby, there is a need for new methods that efficiently manage energy consumption in transmission, reception, and idle modules [1,3].

Major Contribution
The main contribution of this work is divided into two parts.First, we mathematically formulate and propose a set of fitness functions that can be integrated with protocols-based WSNs for guiding them toward achieving efficient management of energy consumption.Energy management is achieved in terms of balancing energy consumption among wireless sensors, minimizing communication overhead, maximizing protocol-energy efficiency, and maximizing average network lifetime.In addition, the proposed fitness functions can be used for evaluating and comparing protocols' performance.
Second, we propose Optimized Energy Balanced and Distributed Clustering (OEBDC) protocol that minimizes and uniformly distributes energy consumption among wireless sensors.In the design process of the OEBDC protocol, three factors are considered crucial in saving energy consumption in transmission, reception, and idle modules.The three factors are the percentage ratio of cluster heads, the size of the cluster, and how many consecutive time trails a sensor node has served as a cluster head.The OEBDC protocol achieves the goal of minimizing and uniformly distributing energy consumption among wireless sensors by optimizing the three factors that we define as hyper-parameters.The suggested optimization algorithm utilizes hyper-parameters as constraint variables and fitness functions as objective functions.
Mainly, the OEBDC protocol has two phases: advertisement and transmission.The purpose of the advertisement phase is to select an optimal number of cluster heads, which in turn make a decision either to stay as a cluster head or to initiate a re-clustering process.The purpose of the transmission phase is to transmit the sensed information from wireless sensors to the destination with a minimum amount of energy consumption.The performance of the OEBDC protocol is simulated and tested with different proposed fitness functions.In addition, we compared the performance of the OEBDC protocol with similar standard WSNs based protocols.Simulation results have shown that the OEBDC protocol has achieved two improvements in comparison to other protocols: extending network lifetime by 42% and balancing energy consumption among wireless sensors.

Article Organization
The rest of this paper is organized as follows.Section 2 explores protocols based on WSNs.In Sect.3, we introduce the OEBDC protocol.The sensor node's radio model and evaluation metrics (fitness functions) are presented in Sects.4 and 5 respectively.Section 6 discusses the results and finally, conclusion points are given in Sect.7.

Related Work
In this section, we present a summary of related research efforts in the field of WSNs.There are numerous protocols proposed that target different WSNs aspects such as energy efficiency, load balancing, coverage, connectivity, scalability, data fusion, medium delay, security, and robustness [4][5][6][7][8][9][10][11].These aspects are listed in descending order concerning their importance in WSNs applications.A comprehensive explanation of these aspects can be found in [12].Routing protocols can be classified as flat and hierarchical routing.In flat routing, all sensor nodes have the same functionality and perform similar tasks, and data transmission is based on a single-hop transmission toward the base station.Whereas in hierarchical routing, sensor nodes are grouped into clusters based on predetermined rules [2].In hierarchical routing, nodes perform different tasks based on their functionality as either a sensor node or a cluster head [13].Since this paper is concerned with energy efficiency, load balancing, single-hop transmission, and hierarchical protocol structure, we will only review the main protocols that target these ends.As a result, we excluded other protocols that do not satisfy these characteristics.
In [14], a pioneering protocol based on hierarchical clustering is proposed called Low Energy Adaptive Clustering Hierarchy (LEACH).The purpose of LEACH is to save energy consumption by rotating the role of the cluster head randomly among sensor nodes.Despite the many advantages that LEACH offered, it has the disadvantage of inefficient management of energy consumption.To overcome random cluster head selection in LEACH, a protocol called Hybrid Energy-Efficient Distributed clustering (HEED) is proposed [15].In HEED, a cluster head is selected based on two parameters, the node residual's energy and energy communication cost for a given cluster head.However, the limitation of HEED is that it could generate more cluster heads than needed which results in more consumed network energy.HEED was further improved by the so-called: Distributed Weight-based Energy-efficient Hierarchical Clustering protocol (DWEHC) [16].DWEHC has aimed at balancing cluster size and minimizing communication cost in the intra-cluster using the sensor node's location information.Unfortunately, DWEHC consumes a considerable amount of energy in overhead communication during cluster formation.
Soon after LEACH emergence, numerous energy routing protocols are proposed to mitigate the LEACH downside called LEACH variants or successors [12].We will only review LEACH variant protocols that are closely related to our approach and meet the criteria described earlier in this section.LEACH-Centralized (LEACH-C) [17] used a base station for clustering purposes that minimizes consumed energy by eliminating overhead communications at the cost of equipping each sensor node with a Global Positioning System (GPS).GPS could result in more energy consumption per sensor and increased cost.LEACH-Deterministic Cluster Head Selection (LEACH-DCHS) [18] has modified LEACH by adding the residual sensor node energy to the threshold equation to improve the process of cluster head selection.The main downside is the considerable number of overheads required that affect network energy.More Energy Efficient-LEACH (MELEACH) [19] followed a procedure that minimizes communications distance among sensor nodes at the expense of increasing communication overhead.Advanced-LEACH (ALEACH) [20] restricts the selection of cluster heads based on two factors: current state probability and general probability.Although ALEACH achieved good energy consumption but is not suitable for large-scale networks since it uses direct communication between cluster heads and the base station.
Quadrant Cluster-based LEACH (Q-LEACH) [21] combined the characteristic of directional routing protocol [22] and LEACH.Q-LEACH divided the network area into four clusters.The cluster heads of the four-quadrant communicate with each other to define the best route from source to destination.The drawback of Q-LEACH is the considerable amount of intra-cluster communications that affected the network lifetime.Ma, Ning, et al. [23] proposed an energy-aware routing protocol using the Voronoi theory, a modern computer graphics theory.The Voronoi graph theory has been used to improve the process of cluster establishment by finding the number of sensor nodes in a cluster.The proposed protocol was compared with LEACH and LEACH-C.The results show the ability of the proposed protocol to increase the remaining network energy by 54.9% and 29.7% for LEACH and LEACH-C respectively.However, the suggested protocol requires the calculation of distance for each sensor node [24].Suggested using Particle Swarm Optimization (SPO) to introduce energy balanced clustering protocol called EBC-SPO.Furthermore, multiple constraints such as average intra-cluster distance, residual energy, and average cluster size were used as fitness functions for selecting and distributing the CHs uniformly.The performance of the proposed protocol is compared with the LEACH and other protocols where the EBC-SPO shows its ability to outperform other protocols.Once again, the suggested protocol requires the locations of sensor nodes to be known in advance.
Zhidong, et al. [25] propose a new WSN clustering routing protocol based on using a hierarchal clustering algorithm such as the AGNES algorithm.To evaluate its performance, the proposed protocol was compared with well-known energyaware routing protocols such as LEACH.Although the suggested protocol increased the network lifetime by approximately 40% compared with LEACH, it needs the locations of nodes and base station [26].Suggested a new approach to select the optimal cluster head that leads to enhancing the lifetime of WSN.The authors used Multiple Attribute Decision-making (MADM) methods to find the ranking of multiple attributes that identify the best CHs selection process.The performance of the suggested approach is compared with other protocols such as LEACH and LEACH-C where the results show the ability of the proposed approach to outperform LEACH and LEACH-C.However, this protocol requires the location of nodes and gateway are known in advance.
Behera et al. in [27] introduce a modified version of LEACH called R-LEACH.R-LEACH aims to improve the performance of LEACH through the process of selecting cluster heads (CH) in each round.In R-LEACH, the selection of CH considers the residual energy of non-cluster nodes in each round, and the node with the highest residual energy is chosen as the CH.To evaluate the performance of R-LEACH, the authors compare it with other similar protocols using metrics such as first node dies, and last node dies.While R-LEACH demonstrates a 66% improvement in network lifetime compared to LEACH, it is important to note that the cluster size is determined based on a predefined threshold.
Finally [28], has summarized three network parameters that affect energy consummation in WSNs.First, the optimal number of cluster heads for a given network setup.Second, the optimal cluster size for each cluster head.Third, the duration of a sensor node plays the role of a cluster head.In other words, we do not want to select different cluster heads for each new round of communication, and neither hold the cluster head role for a long time.These factors play a crucial role in saving network energy and balancing energy consumption among sensor nodes.Based on these factors, we propose a protocol that uses a surrogate optimization method for finding the optimal parameters for a given network setup.The protocol is called Optimized Energy Balanced and Distributed Clustering.

Optimized Energy-Balanced and Distributed Clustering (OEBDC) Protocol
Minimized energy consumption in WSNs is achieved by the careful design of protocols that minimize overhead communications and balance the energy consumption among sensor nodes.Extending network lifetime has a direct connection to the network energy while balancing energy consumption among sensor nodes ensures that all network sectors are active before network breakdown.In hierarchal protocols, many factors affect the lifetime of the sensor nodes.First, the percentage number of cluster heads ( P ) that affects non-uniform energy consumption.A small percentage causes a heavy load on the cluster heads that leads to quick consumption of their energy while many cluster heads cause too much communication overhead.Second, the optimal Cluster Size ( CS ) that a cluster head manages.The amount of energy consumed by a cluster head greatly depends on its cluster size.For example, the unequal cluster size among different clusters causes unbalanced energy consumption among cluster heads.The stored energy of a cluster head with a large cluster size consumed more energy compared to a cluster head with a small cluster size.That means unbalanced cluster size causes unequal energy consumption among sensor nodes that eventually lose the sectors of network coverage as time runs.
Third, how long does a cluster head keep its role?The number of consecutive rounds that sensor nodes serve as cluster heads can greatly affect network lifetime.For example, a small number means more communication overhead during cluster formation while a large number causes draining energy of cluster heads more quickly causing unbalancing energy consumption.As a result, optimizing all three factors mentioned above minimizes overhead communications and saves network energy.In summary, there are optimal values for the factors mentioned above that play important roles in extending network lifetime.
For a given network, the optimal percentage number of cluster heads is P .The cluster size is described as CS i , i = 1, 2, ..., K.Where K representing the number of clusters, which is also equal to the number of cluster heads.
The number of consecutive rounds a sensor node served as a cluster head is N r = 1, 2, ..., J , where J an integer number represents the largest number allowed for N r .Optimizing the parameters P, CS i and N r against network lifetime should mini- mize overhead communications and balance energy consumption, thereby, extending network lifetime.Table 1 shows the definition of the symbol used throughout the paper.After defining these parameters, the next step is to define a protocol structure that implements these definitions.
There are four phases of the OEBDC protocol: nomination, advertisement, association, and data transmission.The purpose of the nomination stage is for the selection of cluster heads.In the first round, the selection of cluster head is a random process since initially, all sensor nodes have equal storage energy while the rest of the network lifetime follows a different procedure as described in detail later in this 76 Page 8 of 28 section.Each sensor in the network generates a random number with a uniform distribution between 0 and 1.If the generated number is less than P , the sensor node becomes a cluster head.Following this procedure ensured that the total number of cluster heads K = N × P falls within the allowed range.
Having assigned all cluster heads in the network, they broadcast messages to the network to form clusters.In the advertisement phase, all sensor nodes should keep their receiver active to receive the advertisement messages.The communication protocol for the advertisement phase is Carrier Sense Multiple Access (CSMA).The next phase is the association phase in which each sensor node informs the nearest cluster head to be a member of its cluster-based on the maximum received power.
In the association phase, sensor nodes can use the RSSI (Received Signal Strength Indicator) for distance estimation, which helps to select the nearest cluster head for association based on the received signal levels from the cluster heads.

Protocol efficiency (t)
The number of active sensor nodes as time runs arg ( (t)) The argument of the function (t) ⌊x⌋ Rounding a number x to the nearest integer no more than x After forming clusters, sensor nodes collect information and send it to their cluster head.Each cluster head aggregates the collected information and sends it to the Base Station (BS).This phase is called the data transmission phase and the communication protocol used is called Time Division Multiple Access (TDMA).In the next time trial (nomination phase) the selection process of the cluster head is different from the first trial since the energy distributed among sensor nodes is unbalanced.
Each cluster head CH i , i = 1, 2, ..., P × N = K should satisfy two criteria to keep its role as cluster head, otherwise, it nominates the next cluster head.First, the number of rounds ( N r ) that a cluster served as a cluster head is less than a predeter- mined integer number J .Second, it should check its cluster size CS i if it is within an allowed range CS min ≤ CS i ≤ CS max , i = 1, 2, ..., K .Where CS min and CS max are the minimum and maximum allowed cluster sizes defined as follows The term ((1 − P)∕ P) = ((N − K)∕ K) represents the optimal cluster size for a given K .r is a parameter that controls the size of CS min , and CS max .Its values are limited to 0 ≤ r ≤ 1 .To clarify the concept, assume we have a network with N = 100 and P = 5% hence, the cluster heads population is K = N × P = 5 and the optimal cluster size is ((N − K)∕ K) = 19 .For a given cluster head if N r > J or the cluster size is not within the range CS min ≤ CS i ≤ CS max , i = 1, 2, .., 5 then the clus- ter head ranks the sensor nodes belonging to its cluster in descending order based on residual energy.The sensor node that has the maximum residual energy is nominated to be the next cluster head.Algorithm (1) summarizes the steps of the OEBDC protocol.
From the previous discussion, it is clear that three parameters save energy consumption in the OEBDC protocol which are P, r and N r .These parameters need to be optimized to minimize energy consumption in the OEBDC protocol.It can be considered as a constraint that minimizes the target function (ex.Network lifetime) for a given network setup.There are different optimization techniques used to solve different problems.In our optimization problem, we need to define the objective function that we want to optimize and then define the constraint(s) or variable(s) that is used to minimize the objective function.
To optimize the system variables, a parametric optimization method can be used.In this method, one input of the system is varied while the other variables are kept constant and the system behavior is noted, i.e., minimizing the objective function.Parametric optimization methods may not be efficient in time-consuming simulation problems and they might only improve system performance partially.Since our objective function is not easy to evaluate i.e. we used a computer simulation to generate it, we used a method called Simulation-Based Optimization (SBO) [29].
Once the mathematical formulation of the system is identified, the computerbased simulation can provide the key variables used to optimize the system performance.In SBO, the optimum system parameters that minimize the objective function are estimated iteratively.In each iteration, the system moves closer to the optimal values with minimum computation cost and time.
Figure 2 shows the process of SBO for WSNs.The basic idea of SBO is to integrate the WSNs numerical simulator with an optimization method.The WSN simulator is created with network fixed parameters, then the network hyper-parameters needed to optimize ( P, r, N r ) are varied.The numerical optimization methods iter- atively track the objective function, for example, network lifetime.The new constraints parameters values are fed again to the simulator and the process is iteratively repeated until the best-valued variables that minimized the objective function are found.The simulator model used for WSNs as well as the different objective functions for WSNs is explained in detail in Sects.4 and 5.
The missing part in the algorithm defined in Fig. 2 is the type of optimization method that fits our problem.There are different methods proposed for SBO such as stochastic approximation, global response surface, cross-entropy, genetic algorithm, and surrogate [29,30].There is no superior optimization method that fits all problems.Each method has characteristics that are more suited for some applications.Since our network simulator is time-consuming, we apply a method of optimization approach that works for a time-consuming simulation called surrogate optimization [30].
Surrogate optimization is used to find optimal system variable values for expensive or time-consuming objective functions for a given multi-dimensional constraint variable.Mainly surrogate optimization consists of two phases that are iteratively repeated which are: (1) Build a Surrogate model: In this phase, random points are generated within multi-dimensional constraint variables.These points are used to estimate the surrogate model of the objective function by interpolating them using the radial basis function.(2) Search for Minimum: In this phase, thousands of random points are generated within constraint variables bound for searching for a global minimum of the objective function.These points are evaluated against a merit function.The best point that minimized the objective function is selected as a candidate point or called an adaptive point.The adaptive point is used again in the surrogate model and the search phase is repeated until it reaches the stopping criteria.

Modeling Energy Consumption of a Wireless Sensor
Figure 3 shows the basic energy model of a wireless sensor node that helps in studying the performance of different energy-aware protocols based WSNs.As shown, the node transceiver consists of two parts: transmitter and receiver.The transmitter electronics are responsible for the construction of a signal before transmission from the antenna.The amount of energy consumed at the transmitter is E ele = 50nJ∕bit .The amount of energy dissipated in the amplifiers is amp = 100pJ∕bit∕m 2 which is required to achieve acceptable signal energy to noise energy ratio.On the receiver side, the amount of energy dissipated in receiving a single bit is E ele = 50nJ∕bit .The wireless energy model presented in [14] is used to simulate the performance of the OEBDC protocol.It can summarize the energy radio model as follows.There are two packets in WSNs communications.Data packets are used to carry information and signal packets for managing data communications.
The amount of energy consumed in transmitting a data packet of length k 1 (bit∕packet) is.
, where E radio = E elec × data rate.The OEBDC protocol workflow given in Algorithm 1 is simulated using the radio energy models (2)(3)(4)(5)(6)(7)(8).To implement the simulation, we used the object-oriented capabilities available in the MATLAB R2022a package.In the next section, we defined several evaluation metrics that we used to guide the OEBDC protocol for managing energy resources efficiently.

Evaluation Metrics
To quantify and manage energy consumption in the WSNs as well as compare protocols based on WSNs, different assessment metrics (fitness functions) are used.Fitness functions are also used to guide the optimization algorithm toward achieving different requirements such as extending network lifetime, minimizing communication overhead, balancing energy consumption, maximizing protocol-energy efficiency, and maximizing average network lifetime.In this section, we discuss popular assessment metrics.In addition, we propose and formulate new metrics called the effective network lifetime and network lifetime slop to quantify the network lifetime.

Network Constellation
A two-dimensional plot that shows the physical locations of sensor nodes and their current states: live or die.

Network's Lifetime
Display the number of sensor nodes alive as a function of time.This metric helps compare the performance of energy-aware protocols in WSNs.

FND, HNA, and LND
FND stands for First Node Dies in the network as time runs while HNA stands for Half Node Alive and LND is the Last Node Dies.These metrics are important in showing which protocol extended network lifetime as much as possible.These metrics are estimated from the network lifetime.The mathematical definitions for FND, HNA and LND are given in ( 9), (10), and ( 11) respectively.where (t) represents the number of sensor nodes alive as time runs (network's life- time), arg( (t)) is the argument operator for the function (t) i.e., for a given (t) return the time step t , and ⌊x⌋ is an operator that rounds a number x to the nearest integer no more than x.

Protocol Efficiency
In WSNs, there are two packets used in managing WSNs data transfer; data and signal packets.Overhead communications are necessary for organizing sensor nodes' communications; however, they consume energy as well.In [28], we proposed this metric for measuring the amount of energy usage in data communication compared to communication overhead.
Let the normalized energy ( E no ) dissipated in overheads transmission be: the total number of sensor nodes in the network is N .The total energy used in over- head communications is E o and the individual initial sensor node energy is E i .Fol- lowing the same approach above the normalized energy dissipated in data communication ( E nd ) is defined as: where E d is the total energy dissipated in data communications.Using ( 12) and ( 13), the overall normalized energy dissipated in the network is E t is the overall energy consumed in the network.Based on the previous discus- sion, the protocol efficiency ( P ) is defined as P is used to force the OEBDC protocol to minimize the amount of energy used in overhead communications compared to data communications.

Effective Network Lifetime (ENL)
We propose this metric to measure the performance of an energy-aware protocol in terms of the accumulation number of sensor nodes alive throughout running time.In other words, ENL represents the average network lifetime.We have numerically computed the area under the curve of network lifetime and then averaged the results by dividing by N .ENL indicates on average how fast the network loses its coverage sectors as time progresses.It can be defined as follows: ENL Defined in (16) helps in quantifying and comparing the performance of dif- ferent protocols regardless of their relevant network lifetime behavior.The idea of ENL is inspired by the concept of equivalent noise bandwidth from communication theory.Figure 4 clarifies the concept of ENL where the actual network lifetime is plotted in green color and the effective network lifetime in pink color.
According to ( 16) the area under the curves of the ENL and actual network lifetime are equal.The ENL converts the actual network lifetime into an ideal network lifetime and provides a common standard for quantitative measures of the network performance over different protocols-based WSNs.ENL is used to guide the OEBDC protocol to maximize the average network lifetime.

Slope of Network Lifetime ( SNL)
Based on the definition of ENL, we propose a mathematical tool called SNL that measures how energy is consumed uniformly among sensor nodes by measuring the slope between FND and LND as follows The reason for choosing −(2∕ ) tan −1 (⋅) in the definition of SNL is to limit its values to [0, 1] For an ideal network performance SNL = 1 and FND = HNA = ENL = LND(Fig.4).As SNL → 1 the network distributes energy consumption more uniformly among sensor nodes.In other words, SNL is a tool that helps in guiding the optimization algorithm to force the OEBDC protocol to balance the energy consumption among wireless sensors.Besides, it is used for comparing different protocols in terms of a quantitative measure of how uniform energy consumption is distributed among wireless sensors.In the next section, we evaluate the proposed OEBDC protocol using different fitness functions FND, HNA, ENL, p , SNL, and LND.Table 2 shows a definition summary of fitness functions used in the optimization process.It also highlighted the main objective of each fitness function, in which each fitness function is designed to achieve a specific goal.
In the next section, we evaluate fitness functions and compare the proposed protocol with standard protocols that have similar architecture to the OEBDC protocol to highlight the effectiveness of the OEBDC protocol.

Results and Discussions
In this section, the performance of OEBDC is extensively studied by simulating WSNs using the radio model discussed in Sect. 2. The simulated WSN has dimensions of 50 × 50 meters and the base station coordinate is located at 25 × 100 meters.To make the simulation feasible, each sensor node is considered as an object class that has different characteristics such as sensor node identification number, sensor node status (alive/dead), and other parameters defined in Table 3.Typical WSNs operate in the 2.4 GHz band using 5 MHz channels and can transmit up to 250 Kbps [31].When the simulation starts, each sensor node is loaded with 0.5 J of energy.The data packet length is 2000 bits while the signal packet length is 64 bits.As time runs, the network energy resources are used for transmitting information to the base station.The energy stored in sensor nodes is gradually consumed and the simulation stops when all sensor nodes in the network die.
The discussion of results is divided into two parts.The first part is focused on the assessment of the performance of the OEBDC protocol against fitness functions while the second part is focused on the performance of OEBDC versus LEACH and its variance.

Performance of OEBDC Protocol with Different Fitness Functions
To evaluate the performance of OEBDC protocol, we used the surrogate optimization method to find the optimal network parameters values P , N r and r against dif- ferent network fitness functions defined in Sect. 5 such as FND, HNA, LND, ENL, SNL, and p .In addition, we explore the capability of the proposed fitness functions in the management of energy consumption in WSNs.
Table 4 shows the result of the simulation of WSNs for the network parameters (constraints) P , N r and r against different fitness functions FND, HNA, LND, ENL, SNL, and p for different network sizes N = 50, 100, 150 and 200.Table 4 illustrates how the network parameters and the network size ( N ) affected different fitness functions.As shown and for a given network size (N), the constraints variables are changed against different fitness functions.For example, when N = 100 ,  The reason that N r takes large values compared to other cases is the fitness function p forced the optimizer solver to maximize N r values to reduce overhead communications as much as possible.The same strategy applies when optimizing the network against HNA.In this case, the variables r and N r takes large values compared to other cases to reduce overhead communications that result in maximizing the fitness function HNA.Another example is the fitness function SNL forces the optimizer solver to minimize N r to the lowest value that is 1 to ensure balancing of energy consumption among all sensors in the network at the expense of increasing overhead communications.In addition, the constraint variables values in some cases interact in a complex way to achieve the fitness function goal.The same conclusion can be extended to other cases.More information can be achieved by generating plots that visualize and compare information generated in the Table 4. Figures 5,6,7,8 show network lifetime versus time steps for different fitness functions and network populations ( N ).These figures are generated from the data presented in Table 4.
As shown, optimizing network performance for different fitness functions has a direct effect on how the network distributes energy consumption among different sensors.It was noted the LND fitness function shows an extension of network lifetime.It also shows losing coverage area on other network sectors earlier compared to other fitness functions.
For example, the fitness functions FND, HNA, and ENL show more stable performance regarding maintaining average network lifetime.In other words, they keep the activity of different network coverage areas as long as possible before the network breakdown.Figures 5,6,7,8 show the ENL achieves the best performance in terms of average network lifetime compared to FND and HNA.In This behavior indicates that optimizing the network against SNL results in distributing energy consumption uniformly among sensor nodes.In summary, the fitness functions under discussion provide more flexibility in protocols-based WSNs in terms of the following cases.Use the energy resources for data communication compared to overhead communications as in the case of p .Maximizing average network lifetime as in the cases of FND, HNA, and ENL.Extending the network lifetime as much as possible as in the case of LND.Distribute energy consumption uniformly among all sensor nodes as in the case of SNL.In addition, the proposed fitness functions can be used for comparing the performance of protocols based WSNs.These fitness functions treat different wireless sensors in the network fairly.In other words, the proposed fitness functions do not support quality of service in terms of energy management based on sensors locations.For example, in some applications some sectors in the network may need more energy management compared to other sectors.One proposal to target this issue is to introduce additional protocol parameters that can manage network energy based on the sensor location.
Figure 9 shows different fitness functions (FND, HNA, LND and ENL) plotted against the total number of sensor nodes for a given network coverage area.As shown for a given network dimension, there is an optimal number of nodes needed to deploy in order to maximize HNA, LND and ENL which is about 200 sensors in this setup.Deploying more sensor nodes means more energy in the network, however a tradeoff between performance and cost should be considered carefully.One future direction of this study is to investigate the optimal number of sensor nodes that need to be deployed for a given network budget.

Evaluation of OEBDC Protocol Versus LEACH and its Variants
In this section, we compare the performance of the OEBDC protocol against LEACH and its variants such as ALEACH, and LEACH-DCHS.The reason we considered these protocols among others is their structure is similar to OEBDC protocols in terms of distributed clustering management and incorporating the node residual energy in the cluster head selection.
Figure 10 shows a comparison between the proposed protocol and three related energy-aware routing protocols, which are LEACH, ALEACH, and LEACH-DCHS.The comparison was carried out using different fitness functions based on OEBDC protocols.We excluded FND and HNA from the comparison since ENL provided similar and better behavior in terms of average energy consumption.In Fig. 10 the constraint parameters of OEBDC are optimized against ENL, LND, and SNL for N = 100 , and the values for P, r and N r are taken from the Table 4.As shown, the proposed protocol has flexible and notable performance among different metric functions.
In addition, OEBDC has three features that enable the network to manage energy resources more efficiently.First, it distributes energy uniformly among sensor nodes.Second, it minimizes overhead communications as much as possible.These two factors guarantee that all network coverage sectors are active until the network breakdown.This behavior is clearly shown in Fig. 10 in which the network breakdown curve shows a fast fall in a short time from full 100 active sensor nodes to 0 as in the Fig. 10 Active nodes versus time steps for different protocols when N = 100 case of the OEBDC-ENL.Third, the proposed protocol is extending network lifetime by 42% compared to others.
The performance of OEBDC is also quantified and compared with other protocols in terms of other network assessment tools such as FND, HNA, LND, ENL, and p .Since ENL, FND, HNA, and LND are closely related parameters, they are compared in Fig. 11 while SNL and p comparison is shown in Table 5.Table 6 shows percentage improvements of the OEBDC protocols compared to other protocols under comparison.The comparison is done over all evaluation metrics (fitness functions).The results show the OEBDC protocols excel all protocols under comparison among  all network assessment metrics.It achieved notable improvement results in terms of LND and SNL by 44% and 160% respectively.It can improve the performance of the OEBDC protocol in terms of energy consumption by modifying the protocol structures to support multi-hop communication as well as providing energy management based on the sensor nodes locations.In summary, the strong parts of the proposed protocol are the careful design of its structure as well as using optimized parameters that result in minimizing communication overhead, distributing energy consumption uniformly among sensor nodes, and extending network lifetime.
To statistically validate the results of the OEBDC protocol against other protocols under study, we run two-sample t-test for different fitness functions under study.The purpose of the two-sample t-test is to compare the performance of two protocols in two hypothesis cases.The null hypothesis is that the results of the two protocols are not statistically significant against the alternative hypothesis.All the p-values shown in Table 7 reject the null hypothesis at the default 5% significance level which indicates that the results of the OEBDC protocol are statistically significance.
To that end, a comparison between the OEBDC protocol versus LEACH and its variants in terms of strength, weakness, and performance is shown in Table 8.

Conclusion
In this work, we proposed a protocol called OEBDC for applications-based WSNs.
has been shown to minimize and balance the energy consumption among wireless sensors and extend network lifetime.In addition, we proposed new metrics used in the assessment of protocols-based WSNs as well as in optimizing their performance.The performance improvement is done in terms of minimizing overhead communications, balancing energy consumption among sensor nodes, and extending the average network lifetime.In addition, we have compared the performance of the OEBDC protocol with standard protocols based WSNs.Results showed the proposed protocol provided notable results in extending network lifetime compared to others over different evaluation metrics.The future direction for this work is to use a multi-objective optimizing algorithm over different fitness functions to maximize the average network lifetime.

Fig. 1
Fig. 1 Energy consumption of various modules in WSNs

Fig. 2
Fig. 2 Process of simulation-based optimization for WSNs

Fig. 3
Fig. 3 Radio energy model for a sensor node

Fig. 4
Fig. 4 Ideal (effective) network lifetime versus actual network lifetime Maximize the time trails until first node die Half Nodes Alive ( HNA) HNA = arg ( (t) = ⌊N∕2⌋) Maximize the time trails until 50% node alive Last Node Dies ( LND) LND = arg ( (t) = 0) Maximize the time trails until last node die Effective Network Lifetime (ENL) Maximize the time trails until on average half node alive Slope of Network Lifetime (SNL) SNL = − 2 tan −1 (LND)− (FND) LND−FND Minimize the time trials that the network takes from full to zero coverage the values of constraints variables P , r and N r against p are 2%, 0.1, and 45 respectively.

Fig.
Fig. Active node versus time steps for different target functions when N = 50 for OEBDC protocol

Fig. 6 Fig. 7
Fig. 6 Active node versus time steps for different target functions when N = 100 for OEBDC protocol

Fig. 8 Fig. 9
Fig. 8 Active node versus time steps for different target functions when N = 200 for OEBDC protocol

Fig. 11
Fig. 11 of different protocols in term of ENL, FND, HND and LND when N = 100

Table 1
Symbol definition i , i = 1, 2, .., N Number of sensor nodes for the ith cluster (cluster size) N Total number of sensor nodes in the network K Number of cluster heads N r = 1, 2, ..., J The number of consecutive rounds a sensor node served as a cluster head CH i , i = 1, 2, ..., P × N

Table 2
Definition of the fitness functions

Table 5
Comparison of different protocols versus p and SNL for N = 100Bold is to highlight the performance of the suggested protocol