Table 1 Notation for SolarTC _k ( i ) and FTTC _k ( i )

E _residual (i)

Amount of residual energy at node n _i

E_threshold(i)

Energy threshold for changing mode, (P_sys(i)/P_solar(i))C

m _i

1 if sensor node n _i is in fault-tolerant mode, otherwise 0

s _i

1 if sensor node n _i has decided its operating power, otherwise 0

r _i ,p _i

Current transmission range and power of sensor node n _i ($p_i=r_i^{\alpha }$), respectively

$r_i^{\text{max}}$,$p_i^{\text{max}}$

Maximal transmission range and power of node n _i , respectively

N_all(i)

{$n_j\mid \text{dist}(n_i,n_j)\le r_i^{\text{max}}$}

FTN_all(i)

{n _j ∈ N_all(i) ∣ n _j is in fault-tolerant mode}

FTN_covered(i)

{n _j ∈ FTN_all(i) ∣ dist(n_i,n_j) ≤ r_i} ∪ {n _j ∈ FTN_all(i)∣($r_i\le \text{dist}(n_i,n_j)\le r_i^{\text{max}})\wedge$ (n _i is k-connected to n _j in the localized view of the topology at n _i )}

$p_i^{\text{LMST}}$

Transmission power of node n _i needed to maintain the LMST topology (1-connectivity)

$p_i^{\text{min}}$

Transmission power of node n _i needed to reach the closest k fault-tolerant neighbors (FT-neighbors), or at least to maintain the LMST topology (1-connectivity)

$p_i^{\text{FTN}}\left(n\right)$

Transmission power of node n _i needed to reach the closest n FT-neighbors (including the sink),$p_i^{\text{FTN}}\left(0\right)=p_i^{\text{LMST}}$

Δp _i

Incremental power needed to reach the nearest node in FTN_all(i)-FTN_covered(i)