Table 3 List of abbreviations

EESS

Electrical energy supply system

RF

Regression function

I ⊂ N₀

With \( i,{i}_0,{i}^{\prime },{i}_0^{\prime}\in I \) and \( i,{i}_0^{\prime}\le {i}_0,{i}^{\prime}\le {i}_0^{\prime } \)

T ⊂ R

With t, t^′, ∆t, ∆t_R, t_A, t_E, τ ∈ T and ∆t = t^′ − t

N ⊂ R³

With \( {x}_i,{x}_i^{\prime },{x}_j,{x}_{n_0}\in N \), pairwise coprime

E^D(t, x_i)

Energy demand function; short form \( {E}_i^N(t) \)

\( {E}_{\mathrm{max}}^D\left({x}_i\right) \)

Maximum energy demand; short form \( {E}_{\max, i}^N(t) \)

\( {f}_i^D(t) \)

Time-dependent behavior of user i

E^S(t, x_i)

Energy supply function; short form \( {E}_i^B(t) \)

\( {E}_{j_0}^D\frac{(t)}{E_{j_0}^S}(t) \)

Bundled energy demand/supply of j₀  microcells

\( {E}_{\mathrm{Nat}.}^D\frac{(t)}{E_{\mathrm{Nat}.}^S}(t) \)

Energy demand/supply function of a national macrocell

u

Energy density of the electromagnetic field

S

Poynting vector

J

Current density

E

Electric field strength

H

Magnetic field strength

ρ

Electrical charge density

\( \frac{c_0}{c} \)

Speed of light in vacuum/in a medium

\( {E}_i^{ST} \)

Stationary supply component of the ith microcell

c_i ∈ R

Stationary state constant of the ith microcell

\( {E}_{j_0}^{ST} \)

Stationary supply constant of a macrocell

P^S(t, x_i)

Output power of the ith source; short form \( {P}_i^B(t) \)

\( {P}_i^{D_{\mathrm{max}}} \)

Maximum load (power consumption) of the ith microcell

\( {P}_i^{S_{\mathrm{max}}} \)

Maximum supply (power generation) of the ith microcell

\( {\dot{P}}^S\left(t,{x}_i\right) \)

Power dynamics of the ith microcell; short form \( {\dot{P}}_i^B(t) \)

P(j₀)

Total power from j₀ microcells

P(k₀)

Total power from k₀ current sources

\( {\wp}_{\mathrm{macrocell}}^N \)

Maximum load of a macrocell, analogous to power generation

\( {\wp}_{\mathrm{Nat}.}^N \)

Maximum load of a national macro cell, analogous to power generation

S₁, …, S₅

Structure variables, components of S^∗

S^∗ ∈ R⁵

Technological structure vector

\( \varDelta {t}_{r_i} \)

Time shift in the ith microcell due to the relativity principle

\( \varDelta {t}_{s_i} \)

Time shift in the ith microcell considering real sources

\( \varDelta {t}_{j_0} \)

Total time delay within a macrocell of j₀ microcells

J ⊂ N₀

With j, j₀ ∈ J and j ≤ j₀

K ⊂ N₀

With k, k₀ ∈ K and k ≤k₀

Ν ⊂ N₀

With n, n₀ ∈ N and n ≤ n₀

(N, d)

Metric space on the set N with metric d

σ

Electrical conductivity

Ω_u ⊂ R⁴

Technological solution space with u ∈ Ω_u

\( {\varOmega}_{u_0},{\varOmega}_{u_I},{\varOmega}_{u_{II}} \)

Subsets of the technological solution space

LE

Unit of length

r _s

Substantial risk factor

r₁, r₂

Sub-risk factors

μ

Failure factor

p _i

Failure likelihood of the ith microcell

\( {P}_{i_0} \)

Failure likelihood of a macrocell with i₀  users

S^∗∗ ∈ R⁶

Extension of the structure vector S^∗  with the substantial risk factor

S ₆

Structure variable for the substantial risk factor

v _tv

Availability

v _{tv, B}

Sustainability boundary; sustainable availability boundary

\( {v}_{tv}\bullet {E}_{\mathrm{Nat}.}^D(t) \)

National sustainability

\( {v}_{tv,n}\bullet {E}_n^D(t) \)

Regional sustainability in a macrocell

v

State vector with sustainability component

Ω ∈ R⁵

Sustainable technological solution set with

\( {E}_{T_{\mathrm{Ref}.}}^N \)

Annual energy demand in a reference year

\( {\lambda}_i^{-} \)

ith supply factor

\( {\lambda}_{\mathrm{min}}^{-} \)

National supply factor

h(t)

Distribution of annual energy demand with \( {\int}_{t_0}^{t_0+365}h(t) dt=1 \)

g(x) ∈ [0; 1]

Weights between \( {\lambda}_{min}^{-} \) and r_s

\( {E}_{T_{\mathrm{Ref}.}}^D \)

Annual reference energy arbitrary initial value \( {\int}_{t_0}^{t_0+365}{E}^D(t) dt \)