Fault resilience in network of energy harvesters

Energy harvesters (EH) that scavenge energy from ambient environment are gaining popularity and are used for powering low demand devices on account of their low power outputs. Enhancement of the power is achieved through an array or network of identical EH. The focus of this study is on investigating how the network topology affects the harvesting efficiency per EH, using complex network theory. The studies are presented with respect to vibration induced EH, specifically, the commonly used network of coupled pendulums oscillating in a magnetic field, with the pendulum supports being subjected to vibrations. Questions on the EH efficiency are investigated with respect to the number of EH in the network, its topology and the effects of faults which lead to loss of regularity. Additionally, the effects of parametric random variabilities in the individual EH are investigated with respect to the harvesting efficiency. This study shows that EH efficiency is best for regular networks, can be enhanced by increasing connectivity but up to a limit and is resilient against few local faults. The performance drops with larger number of faults or due to parametric uncertainties. The findings of this study are expected to be of use in design and maintenance of EH networks.


Introduction
Energy harvesters (EH) are small miniature devices that are designed to scavenge energy from ambient environmental conditions, that would otherwise be lost.These are increasingly gaining popularity on account of their ability to provide a secondary clean source of power that complements the energy source from the traditional grid and have found use in a host of applications that require reusable batteries, ranging from powering sensors, condition monitoring, remote sensing and powering low power LED devices.By dint of their objective and design, the power output from these EH are low.Consequently, there is significant research attention on attempts to scale up the power output of these devices, primarily, by connecting an array of identical EH in a network.However, studies in these directions have essentially focussed on the physics of coupled EH and to the best of the author's knowledge, no study has been attempted to investigate the network topology on the harvesting efficiency of EH networks.This study aims to address this gap.
Though the aim of this study is to investigate the effects of the network topology on the harvesting efficiency of a network of EH, the usefulness of the findings can be demonstrated meaningfully when the physics of the EH are appropriately modelled into the analysis.With this in view, this study undertakes this investigation with respect to a special class of EH-vibration induced energy harvesting devices.Vibration induced energy harvesters are devices that scavenge energy from ambient vibrations to usable power [1,2].Early designs of vibration induced energy harvesters operated on the principle of attaining 1:1 resonance with the excitations.However, the operability of these EH were limited to a very narrow band in the frequency spectrum of the excitation.To address the narrow band power generation, two electromagnetic harvesters with magnetic and mechanical couplings experiencing harmonic support excitations was utilized to generate power across a wide frequency spectrum [3].Several attempts to enhance the operation spectrum of EH were made by building non-identical EH that ensured the entire spectrum of the excitation frequency was available for energy harvesting but this proved to be not very efficient.Increase in the efficiency has been achieved by designing nonlinear harvesters that operate at a broader spectrum [4][5][6]; this includes harvesting through electromagnetic induction as well [2,7,8].Researchers have examined power generation through electromagnetic induction from an array of cantilever beams [2] with varying lengths.Since non-identical beams generate less power compared to identical beams, this limitation is compensated by increasing the number of beams.The power harvested can be scaled up by building a network of EH, such as a chain or array of coupled EH [7,9,10].Coupled EH not only leads to an increase in the power harvested but also enhances the operational bandwidth, especially if the EH behave nonlinearly.There has also been studies on an array of pendulums as electromagnetic broadband energy harvester [8].The authors have carried out studies both numerically and experimentally, where multiple energy harvesters are coupled through springs to increase the power harvested.In recent research, the base-excited double pendulum has emerged as an innovative method for generating electricity on a small scale.This system effectively harnesses energy across both linear and non-linear operational states [11].
Researchers have explored the dynamics of an array of energy harvesters subjected to base excitation, particularly studying how grounding springs and variations in pendulum lengths influence power generation [7].The study demonstrated that there exists a critical limit for the number of harvesters beyond which the harvested power cannot be further increased.The emphasis of this work is on devising strategies to enhance harvested power beyond the saturated value demonstrated in [7] without changing the number of harvesters involved.The effects of the network topology on the efficiency of energy harvesting for a network of vibration induced energy harvesters, specifically, rigid pendulums with attached magnets such that the varying magnetic field induces voltage in electrical coils placed strategically is investigated.The oscillations in the pendulum are induced through the supports which are subjected to ambient excitations.While the physics of EH from such devices are well understood and have been discussed extensively in the literature [7,8,12,13], the focus of this study is on investigating the effects of the network topology on the yield efficiency per harvester.Specifically, the performance of N coupled EH arranged in a regular topology vis-a-vis complex networked topology is investigated.Questions related to increasing the connectedness of the network nodes are addressed as well as the scalability in terms of the network size is investigated.As the probability of faults increase as the network size increases, the harvesting efficiency in the presence of random faults in the network are investigated.Finally, the effects of parametric random uncertainties in the network EH is studied.The remaining paper is structured as follows: section 2 presents the mathematical model for the EH being considered in this study.Section 3 presents in details the numerical results and the corresponding analysis that has been carried out.The paper ends with discussions presented in section 4.

Mathematical model
The governing equations for each EH, when coupled as a network of N EH, is given in the general form as where, θ i is the m-dimensional state vector associated with each EH, θi = f i (θ i ) represents the dynamics for the uncoupled EH, h i (•) is the pairwise coupling function and k i is the strength of the coupling.For identical EH, In network parlance, each EH is a node and the network coupling topology is modelled through the adjacency matrix A, whose elements A ij take values of 1 or 0, depending on whether a coupling exists between nodes i and j.The voltage induced in each EH in the network is v i = ψ( θi ) and the power harvested is P i = g(v 2 i ), where ψ(•) and g(•) are functions of θi .For this study, the nodes are assumed to be identical electromagnetic pendulum EH [8], shown in figure 1.In this figure, a regular network with each node having four connections is shown.Two nodes i and j are highlighted in red and a zoomed-in view of the EH is also shown in figure 1.The governing equations of each EH are given by θi = Here, M i is mass of the pendulum bob, assumed to be of magnetic material, l i is the length, k is the stiffness of the coupling between nodes i and j, a is the axial distance from the support where the couplings are attached, c is damping and θ i is the angular displacement from the vertically down equilibrium position (corresponding to θ i = 0) for the ith pendulum.Vibrations are induced into each pendula in the network on account of support accelerations ẍg where x g = X g sin ωt with X g being the magnitude of excitation and ω is the frequency of excitation; see figure 1. Electrical coils are assumed to be arranged along the arc of the pendulum motion such that the motion of the magnetic bob induces voltage by electromagnetic induction.The length of the coil wire is L, B is the magnetic flux density and c e is the electrical damping.Parallel resistors R l are connected across the coil.The magnetic field interactions between the two pendulum bobs are neglected.The voltage induced in the i th EH is v i = BLl i θi and the corresponding power is Assuming the electrical coils to be all connected in parallel, the total power is The support acceleration is taken to be harmonic with frequency ω.Equation ( 2) can be expressed in the non-dimensionless form in terms of these dimensionless parameters: Here ω n = √ g/l is the linear natural frequency of each pendulum.Using the dimensionless parameters, equation ( 2) can be written as, Note that the assumption of low amplitude oscillations are not invoked and hence the governing equations retain their nonlinearities through the trigonometric functions in f(•).Additionally, the coupling function h(•) is also nonlinear.
For the numerical simulations, the coupled non-dimensional governing equations are numerically integrated using 4th order Runge-Kutta with a time step of 0.01 for time duration of T = 10 × 10 5 non-dimensional time units.The numerical values of the parameters considered are β = 0.1 and f = 0.05 is the non-dimensional amplitude for the support acceleration.The value of N is taken to be 100.All the simulations were performed in double precision using C++ programs, that were parallelized using OpenMP.The initial conditions of each node were assumed to be uniformly distributed between [0,1] radians or [0, 57.3 • ].The first 4 × 10 5 non-dimensional time units were disregarded from the analysis.

Results
The defects are classified as (a) faults in connections between the EH, which in turn, affect the network topology and (b) random variabilities in the parameters of the individual EH, leading to parametric uncertainty.The effects of these on the harvesting efficiency of the network is discussed in this section.

Harvesting efficiency of a regular network
A symmetrical network comprising of N nodes in a ring formation is considered.Each EH is assumed to be coupled only to the neighbouring node on each side.A ring configuration is considered to eliminate boundary effects.The number of connections of each node is its degree m i and is 2 in this case.Since the network is symmetric, the average degree m = ( The maximum power harvested from each Maximum normalised power (saturated value) for lightly damped system is higher than that of heavily damped system; Inset: ring topology, m = 2.
node, denoted by P max , occurs when the excitation is equal to the damped natural frequency and is proportional to the corresponding maximum angular velocity θi .As the network parameters are changed, the natural frequencies also change.However, to ensure fair comparison, Ω is suitably adjusted for each network configuration such that the P max used for comparisons is indeed the maximum.As P t scales linearly with N, to compare the harvesting efficiency for the different scenarios investigated, the per node harvesting power, expressed as ζ = P t /N, is used as the order parameter, which is a measure of the collective dynamical characteristic of a complex network.Figure 2(a) shows the variation of ζ with Ω for an uncoupled EH and a ring network with N = 100.The values of damping ratios are taken to be γ f = 0.01, γ e = 0.009 and γ f = 0.03, γ e = 0.01.These values are considered to study the effect of damping ratios on ζ.This figure reveals that for both the uncoupled and coupled cases, the parameter ζ tends to be higher for lower damping ratios γ f and γ e .This trend occurs because reduced damping ratios imply lesser resistance opposing the system's motion.Consequently, lower damping ratios lead to higher oscillation amplitudes compared to systems with higher damping, thereby resulting in the higher values of ζ.Also, for N = 100, when the damping ratios is lower, the range of Ω for which energy is harvested is greater than that of high values of damping ratio (see Purple and green line in figure 2(a)).
As the power harvested by a single EH is minimal, multiple EH are coupled to increase ζ. Figure 2(b) shows the variation of ζ with N for m = 2 for both the values of damping ratios.This indicates that (i) it is more efficient to have a network of N coupled EH than N uncoupled ones, (ii) the power yield increases with N but (iii) beyond a critical N, there is no further increase in ζ even though (iv) the total power increases linearly with N. (v) For low damping ratios, the saturated value of ζ is greater compared to higher damping ratios.From figure 2(b), it is shown that the value of ζ saturates after N = 50.By selecting N = 100, it is ensured that the numerical results presented maintain consistency and do not vary with N. Unless specified otherwise, subsequent numerical simulations employ values of 0.03 for γ f and 0.01 for γ m .
Increasing the nodal degree by connecting each node to larger number of nodes and by preserving the regularity of the network topology further increases ζ as shown in figure 3(a).For m = 30, ζ is almost one order higher than for the uncoupled EH.However, further increasing the density of the couplings leads to a drop in ζ, drops below the ζ levels for m = 2 when m > 80 and at m = 99, ζ is equal to the uncoupled case (m = 0).This is expected as m = 99 is the all-in-all coupled case and ζ becoming equal to the uncoupled case implies that perfect synchronization in the network is achieved.This is verified from the time histories of an order parameter, defined as H(τ ) = ( ∑ i N θ i (τ ))/N and shown in figure 3(b) for the last 10 5 time steps to ensure the system is in steady state.Unlike for m = 4, the time history for H(τ ) for m = 99 follows a pure sinusoid implying that the oscillations in all the nodes are in perfect phase.In this synchronized state, the inter node coupling does not contribute to the dynamics and hence ζ becomes equal to the uncoupled case.An important result from figure 3(a) is that there is an optimal m where ζ is maximum.

Effect of faulty links and network topology on ζ
As the number of links in a network increases, the probability of some of the links becoming faulty or losing functionality also increases.A faulty link in a regular network destroys the network symmetry.An analysis of the network performance as the regularity of the network topology is disturbed can be carried out by using  the percentage of faulty links, p, as a metric for loss of regularity.The selection of faulty links is implemented numerically as follows: (1) Calculate the total number of connections ℓ in the network by summing all the non-zero elements in A and dividing by 2 (since the graph is undirected).(2) Estimate ℓ p = p% × ℓ which is the number of faulty connections in the network.Therefore ℓ p denotes the number of non-zero elements of A that should be made 0. (3) As the network in this study is undirected, A is a symmetric matrix.Therefore, consider the non-zero elements of A above the main diagonal and store the index of each element (i, j) in a new array S. Let the size of this array be q.(4) Next, generate a uniformly distributed random integer z between 0 and q − 1.The adjacency matrix A is updated by considering the element in S corresponding to z as 0. The element (j, i) is also made 0 to ensure symmetry.Repeat steps 3 and 4 till ℓ p links are removed.Figure 4(a) presents a schematic representation of a symmetric network with N = 14 but with three faulty links, which are marked in blue dotted lines.The absence of these links leads to a loss of symmetry.
From a mechanistic point of view, fewer links in the network makes the system less stiff, which in turn, alters its intrinsic properties such as the natural frequencies of the system.This implies that the energy harvesting capabilities of the network of EH with faults get altered.Figure 4(b) shows the variation of ζ with p, for a network with N = 100 and for three cases of m.As p increases, the total number of links in the system decreases as a result of which the system becomes dynamically less stiff.Interestingly, ζ is observed to become identical to the uncoupled EH beyond a threshold value of p for a given network indicating that the faulty links, distributed randomly in the network, eliminates the effect of the network coupling.Clearly, the threshold number of faulty links at which this occurs is lower for network topologies with lower m.The adjacency matrix in figure 4 represents nodes connected solely to their immediate nearest neighbors.To explore the impact of non-adjacent couplings, a network with m = 2, establishing connections with the second and third nearest neighbors is introduced.Figures 5(a) and (b) compares the variation of ζ with Ω for p = 2% and for p = 10% respectively, for the cases of m = 2-when the connections are with the nearest neighbour (blue line), for the case when the couplings are with the second nearest neighbor (red line) and for  the case when the couplings are with the third nearest neighbor (green line).Interestingly, these figures depict that there is a minimal distinction observed in the relationship between ζ and Ω.This is because, the change of connectivity from nearest to second and third nearest neighbor does not alter the natural frequency of the system significantly.The absence of a substantial alteration in the natural frequency of the system despite the change in connectivity can be attributed to the structural consistency within the Laplacian matrix, given by D − A where D represents the diagonal matrix containing degrees on its diagonal and A denotes the adjacency matrix.This lack of significant change occurs because the modification in connectivity only affects the positions of the sub-diagonal elements (diagonals above and below the principal diagonal) within the matrix structure.Due to this, the behaviour of the system to external excitation remains the same as observed from figures 5(a) and (b).
Loss of regularity in the network can occur by indiscriminate random linking of the individual EH as well.This is specially true when repairing a faulty coupling by linking with a node without preserving the topological symmetry.Here, the total number of links in the network do not change but the number of links for each node now forms a distribution, referred to as the degree distribution.
Figure 6 shows two such network topologies and the corresponding degree distribution of the nodes.Here, P netw is the probability of rewiring.The network in figure 6(a) has a low P netw = 0.02 implying most nodes have equal degrees with only a few nodes having either more or fewer links as can be seen in the degree distribution in figure 6(a).Such networks are referred to as small-world networks [14].On the other hand, figure 6(b) shows the network topology when P netw = 0.5; the corresponding degree distribution in figure 6(b) shows greater variability.These networks are classified as random networks [15].The nodes in figure 6 are color coded according to their degree distribution.Figure 7(b) is demarcated into three distinct regions.Regime 1 comprises of networks where P netw < 0.003 representing topology with less than 0.3% of rewired links.This implies that in a network with N = 100 and m = 2, there are less than 1 rewired link out of a total of 100 links.Such a topology, for all intents and purposes, is regular and there is no change in ζ.Regime 2 is characterized by a slight drop in ζ and can be defined to be the regime where 0.003 < P netw ⩽ 0.2.The number of rewirings in this topology are larger but the network still retains the characteristics of a regular network.This regime is identified as small-world.When P netw > 0.02, the number of random rewirings is large, the regularity of network topology is lost yielding a random network and leading to sharp fall in ζ.These results lead to four important conclusions.First, the power yield ζ is highest for networks with regular topology, the yield becoming progressively lower as the regularity of the network is destroyed.Secondly, networks have an in-built resilience to withstand small local defects that disturbs its regular topology.As the extent of irregularity becomes more, this resilience breaks down.Thirdly, the network resilience is more when m is larger as even though the drop in ζ can be significant, the yield can still be higher for a network with the same P netw but with lower m.Finally, beyond a value of P netw , ζ converges to the value of the single uncoupled EH irrespective of m.This is observed for a threshold P netw when there still exists a large number of links in the network but these couplings appear to have no effect on the network dynamics.

Effect of parametric uncertainty on ζ
The investigations so far considered all EH to be identical.However, in reality, manufacturing imperfections exist despite the best tolerance levels, implying that the EH in the network follow a distribution, inducing parametric uncertainty.These imperfections, known as mistuning, lead to disturbing the regularity of the networked system even in a topology that is perfectly regular.Since the mistuning is assumed to be on account of manufacturing tolerances, they can be assumed to be random and is referred to as random mistuning [16].To investigate their effects on ζ, the pendulum lengths are assumed to follow a Gaussian distribution about its nominal design value with standard deviation σ, such that 0 < σ ⩽ 0.10.Higher values of σ indicate greater randomness in the properties of the individual EH, and therefore greater deviation from a regular networked system, even though its topology is regular.Figure 8(a) shows the variation of ζ for different values of σ for m = 2, 4 and 6.It is observed that ζ decreases with an increase of σ for all three networks.It should also be noted that ζ for the network m = 2 falls below the power yield for the single uncoupled EH for σ > 0.05.These results clearly illustrate that loss of regularity is an important parameter for decrease in the performance of a networked system.Observations from figure 8(a) indicate that an increase in σ leads to a substantial decrease in ζ.In a system with interconnected non-identical oscillators, variations in pendulum oscillation occur due to the dependence of oscillation on the length of the pendulum.Consequently, at a specific excitation frequency, some pendulums experience greater oscillations compared to others, resulting in differences in their amplitude of oscillation within the system.This results in the decrease of ζ as shown in figure 8(b) for σ = 0, 0.04, 0.08.Similar observations for two coupled electromagnetic pendulum have been reported in [8].
Next, the performance of networked system is investigated, where the EH are assumed to be randomly mistuned following a distribution along with irregularity in the network topology on account of defects in the couplings.Here, case 1 considers a network of mistuned EH with the fraction of faulty links being p. Case 2 considers a network of mistuned EH with the probability of rewirings being defined in terms of P netw .In case 1, the number of total links decrease with an increase in p while in case 2, the total links in the network remain unchanged but the degree of the nodes follow a distribution.Figure 8   with p for three networks with m = 2, 4 and 6, when σ = 0.05.At this value of σ for a regular network with m = 2, ζ was observed to be equal to the uncoupled EH.As in a network with identical EH, an increase of p was observed to lead to a decrease in ζ.However, even for p = 0, ζ is observed to be at least half of what was observed in a network of identical EH.For higher values of p, ζ is observed to be significantly lower than the yield from an uncoupled EH.This indicates that random mistuning contributes significantly to the loss of regularity in the system and is detrimental to the network performance.
Figure 9(a) shows the variation of ζ with P netw for the same parameters of the networks considered in figure 8(b).Here, the number of links in the network remain the same for all values of P netw .As in figure 7(b), the power yield is classified into three distinct regimes.In Regime 1, where the number of rewirings are few, there is minimal drop in ζ for all the three networks indicating the network resilience to small defects in topology.Regime 2 characterised as small-world network shows a sharper decrease in ζ as P netw increases.In Regime 3, where the network topology is random, the drop in ζ is significant.Comparing with the yield values with those reported in figure 7(b) leads to two important observations: (a) the power yield is significantly lower in all regimes when the EH are randomly mistuned and (b) for all three networks (m = 2, 4, 6), ζ falls lower than the yield of the uncoupled nominal EH.In fact, for m = 2, ζ is lower than that of the uncoupled nominal EH for all P netw .These results therefore indicate that parametric uncertainty plays a greater role in disturbing the network symmetry and having a deletrious effect on the performance of the network.
Figure 9(b) summarizes the variation of the total power P t = Nζ harvested by a network of N coupled EH, for the various cases investigated.For uncoupled EH; P t increases linearly with N see blue dashed line.The power yield is significantly enhanced by introducing coupling in a regular topology; see black dashed line (m = 2) and red dashed line (m = 6).However, parametric uncertainty decreases the harvesting efficiency as can be seen from the green and the magenta curves.In fact, for m = 2, parametric uncertainty with σ = 0.05 leads to lower power yield than a set of N uncoupled EH.However, increasing m still leads to enhancement of power harvesting in comparison to uncoupled EH.

Conclusion
Complex networks [15], known for their intricate interdependencies among components, are prone to disruptions and failures.Recent research, such as that by Kurths et al [17], illustrate how cutting connections between nodes compromises the stability of power grids, emphasizing the crucial role of individual connections in overall network robustness.Additionally, investigations into ageing effects on biological neuron networks [18] reveal the gradual loss of node functionality during ageing transitions.Furthermore, the application of complex network theory in assessing power grid resilience, as seen in Indian contexts [19], showcases the adaptability of this theory across diverse domains.Using the methodology adopted in the literature to study the robustness of complex networks, our study aims to comprehensively analyze the performance efficiency and robustness of vibration-induced EH networks, demonstrating its engineering applicability within this context.A network of vibration induced EH holds significance as it serves as a strategic framework for amplifying harvested power.This approach has been instrumental in achieving substantial scalability of energy extraction [7,9,10].A comprehensive analysis of the performance efficiency and the robustness of a network of energy harvesters have been carried out numerically.The harvesting efficiency per harvester has been used as the metric for analyzing the performance of the network.The efficiency have been investigated with respect to the topology of the network in terms of its regularity of the number of connections with each EH, the total number of EH in the network and the connectedness of each EH.Subsequently, the effects of local defects in the network have been studied.Specifically, investigations were carried out with respect to three types of defects: (a) topological defects, which introduce deviations from its regular topology, (b) functional defects, when some connections in the network lose their functionality over time and (c) parametric uncertainties that arise due to manufacturing variations, leading to uncertainties in the EH parameters.The key findings emerging from this study are as follows: (i) the harvesting efficiency per harvester is best for networks with regular topology vis-a-vis complex networks, (ii) the efficiency is enhanced as the number of EH in the network increases but no further enhancement is possible beyond a certain threshold number of harvesters in the network, (iii) the enhancement in the efficiency saturates at 2.5 times the efficiency of uncoupled harvesters, (iv) there exists an optimal connectedness of the network with respect to the harvesting efficiency, (v) random parametric uncertainties in the EH contributes to loss of regularity in the networked system and leads to a loss in the network performance, (vi) the performance of the network is resilient against few defects as their effects remain localised without affecting the performance of the majority of the harvesters, but (vii) the power yield drops drastically if the parametric uncertainty or the number of defects increase beyond 2%-5%.These findings are expected to be of use in designing networks of similar EH and for designing maintenance schedules.While this study primarily analyzes the robustness of EH networks, the findings suggest the broader applicability of this analysis to diverse complex networks.More interesting questions regarding the estimation of number of EH required to reach the saturation value needs to be explored as further extensions of the present study.

Figure 1 .
Figure 1.Schematic representation of a regular network of EH.Each node in the network has four connections (m = 4).Two nodes marked as i and j are highlighted in red, in the network, and a zoomed-in view of the EH is shown, with the accompanying labels explained in the text below.

Figure 3 .
Figure 3. (a) Variation of ζ with m.There is an optimum value of m for which ζ is maximum (b) Variation of H(τ ) with non-dimensional time steps τ for m = 4 and m = 99.For m = 99, H(τ ) shows a sinusoid variation, implying the EH are in perfect synchrony.

Figure 4 .
Figure 4. (a) Schematic representation of a network with faulty links (marked in blue dotted lines) with N = 14 and m = 4.(b) Variation of ζ with p for N = 100.As p increases, initially there is a decrease in ζ and then it attains a constant value of that of uncoupled EH.

Figure 5 .
Figure 5. Variation of ζ with Ω with nearest, second nearest and third nearest neighbor connectivity for (a) p = 2% (b) p = 10%.Transitioning from nearest to second and third nearest neighbor connectivity shows no discernible change in system behavior.

Figure 6 .
Figure 6.(a) Small-world network topology : Pnetw = 0.02 and the corresponding degree distribution.(b) Random network with Pnetw = 0.5 and the corresponding degree distribution.Nodes are color coded according to their degree distribution.

Figure 7 (
a) depicts a schematic representation of a network with N = 14, where the initial symmetry is disrupted due to the random linking of edges.The green-colored edges in the original network are rewrired (see the network in the right panel) contributing to the loss of the network symmetry.The variation of ζ with P netw for three networks with m = 2, 4 and 6 are shown in figure 7(b).It is seen that for the same values of P netw , ζ is higher as m increases and this is consistent with the results presented in figure 4(b).

Figure 7 .
Figure 7. (a) Schematic representation of deviation from a regular network with N = 14 and m = 4 (links marked in green represent rewired edges).(b) Variation of ζ with Pnetw for N = 100.Drop in ζ is maximum when Pnetw > 0.02.
(c) shows the variation of ζ

Figure 9 .
Figure 9. (a) Plot of ζ as a function of Pnetw; mistuned case with N = 100.(b) Pt − N plots with and without mistuning.Increasing m significantly increases power yield, makes the EH network robust to parametric uncertainties.