Contents
A.1 Formula sheet
Circuits
U = Z·I S = U·I* = P + jQ ZL = jLω ZC = 1/(jCω) ω = 2πf
Three-phase: U = √3·V P = √3·U·I·cos φ QC = P(tan φ₁ − tan φ₂) C = QC/(ωU²)
Three-phase: U = √3·V P = √3·U·I·cos φ QC = P(tan φ₁ − tan φ₂) C = QC/(ωU²)
Networks
Zb = Ub²/Sb Ib = Sb/(√3Ub) ΔU/U ≈ (RP + XQ)/U² ΔP = R(P² + Q²)/U²
Transformer: Zsc = uscU²/Sn Rsc = PscU²/Sn² Xsc = √(Zsc² − Rsc²)
Load flow: Si* = Vi*ΣkYikVk Gauss-Seidel: Vi = [(Pi − jQi)/Vi* − Σk≠iYikVk]/Yii
Transformer: Zsc = uscU²/Sn Rsc = PscU²/Sn² Xsc = √(Zsc² − Rsc²)
Load flow: Si* = Vi*ΣkYikVk Gauss-Seidel: Vi = [(Pi − jQi)/Vi* − Σk≠iYikVk]/Yii
Short circuit (IEC 60909-0)
I"k3 = cUn/(√3|Z₁|) I"k2 = cUn/|Z₁ + Z₂| I"k1 = √3cUn/|Z₁ + Z₂ + Z₀|
I"kE2E = √3cUn|Z₂|/|Z₁Z₂ + Z₁Z₀ + Z₂Z₀| ip = κ√2 I"k κ = 1.02 + 0.98e−3R/X S"k = √3UnI"k
I"kE2E = √3cUn|Z₂|/|Z₁Z₂ + Z₁Z₀ + Z₂Z₀| ip = κ√2 I"k κ = 1.02 + 0.98e−3R/X S"k = √3UnI"k
Solar
δ = 23.45 sin(360(284 + n)/365) ωs = arccos(−tan φ tan δ) Kt = H/H₀
Hβ = H[(1 − Hd/H)Rb + (Hd/H)(1 + cos β)/2 + ρ(1 − cos β)/2]
P = Pp(G/1000)(1 + γ(Tc − 25)) Tc = Ta + (NOCT − 20)G/800 E = Pp·Hβ·PR
Hβ = H[(1 − Hd/H)Rb + (Hd/H)(1 + cos β)/2 + ρ(1 − cos β)/2]
P = Pp(G/1000)(1 + γ(Tc − 25)) Tc = Ta + (NOCT − 20)G/800 E = Pp·Hβ·PR
Stand-alone PV (NRJXpert method)
Qd = Ed/(U·ηconv·0.98·0.9) C = Qd·autonomy(h)/24 Np,batt = ⌈C/(Cbatt·DoD)⌉
PWM: Np = ⌈Qd/(0.9·H·Imp)⌉ MPPT: Pp = QdU/(H·0.9·fT·0.96) charge controller and inverter: × 1.25
PWM: Np = ⌈Qd/(0.9·H·Imp)⌉ MPPT: Pp = QdU/(H·0.9·fT·0.96) charge controller and inverter: × 1.25
Wiring
Ib = 1.25·I In ≥ Ib Iz0(S)·kθ·kg ≥ In kθ = √((90 − θa)/60)
ΔU = b(ρL/S cos φ + λL sin φ)I (b = 2 DC and single-phase, 1 three-phase on V) ρCu = 0.0225 ρAl = 0.036 Ω·mm²/m λ = 0.08 mΩ/m
ΔU = b(ρL/S cos φ + λL sin φ)I (b = 2 DC and single-phase, 1 three-phase on V) ρCu = 0.0225 ρAl = 0.036 Ω·mm²/m λ = 0.08 mΩ/m
Wind
P = ½ρAv³ f(v) = (k/c)(v/c)k−1e−(v/c)k v̄ = cΓ(1 + 1/k) ½ρc³Γ(1 + 3/k)
v(H) = v(Hmeas)ln(H/z₀)/ln(Hmeas/z₀) E = 8 760∫P(v)f(v)dv(1 − losses) CF = E/(8 760Pn)
v(H) = v(Hmeas)ln(H/z₀)/ln(Hmeas/z₀) E = 8 760∫P(v)f(v)dv(1 − losses) CF = E/(8 760Pn)
Solar water pumping
TDH = HG + PC Eel = 2.725·V·TDH/(ηpηmηconv) Pc = Eel/(Ens·ηc)
Qmax = πV/(2ΔT) PH = Qmax·TDH/367 Pc = PH/(ηpηmηconvηc)·1 000/G (V in m³/d, Q in m³/h, PH in kW)
Qmax = πV/(2ΔT) PH = Qmax·TDH/367 Pc = PH/(ηpηmηconvηc)·1 000/G (V in m³/d, Q in m³/h, PH in kW)
Economics
CRF = a(1 + a)N/((1 + a)N − 1) NPV = ΣFn/(1 + a)n LCOE = Cdisc/ΣEn/(1 + a)n
real rate = (1 + a)/(1 + i) − 1 minimum subsidy = max(0, −NPV/I₀)
real rate = (1 + a)/(1 + i) − 1 minimum subsidy = max(0, −NPV/I₀)
Smart grids (levels 6 and 7)
SAIDI = Σ(Ni·ti)/NT SAIFI = ΣNi/NT CVRf ≈ 2Z + I RoCoF ≈ ΔP·f₀/(2H·ΣSn)
NTL = Einjected − Emetered − Etechnical precision = TP/(TP + FP) recall = TP/(TP + FN) frame success = 1 − (1 − pⁿ)^(r+1)
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NTL = Einjected − Emetered − Etechnical precision = TP/(TP + FP) recall = TP/(TP + FN) frame success = 1 − (1 − pⁿ)^(r+1)
A.2 Wiring tables (copper, XLPE insulation, installation method C, 30 °C)
| Cross-section (mm²) | 1.5 | 2.5 | 4 | 6 | 10 | 16 | 25 | 35 | 50 | 70 | 95 | 120 | 150 | 185 | 240 | 300 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Iz0, 2 loaded conductors (A) | 24 | 33 | 45 | 58 | 80 | 107 | 138 | 171 | 209 | 269 | 328 | 382 | 441 | 506 | 599 | 693 |
| Iz0, 3 loaded conductors (A) | 22 | 30 | 40 | 52 | 71 | 96 | 119 | 147 | 179 | 229 | 278 | 322 | 371 | 424 | 500 | 576 |
Source: IEC 60364-5-52, table B.52.5 (values used by NRJXpert). Aluminium: about 78 % of these values, from 16 mm².
| Ambient temperature (°C) | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
|---|---|---|---|---|---|---|---|---|
| kθ (XLPE insulation, 90 °C) | 1.04 | 1.00 | 0.96 | 0.91 | 0.87 | 0.82 | 0.76 | 0.71 |
| Bunched circuits | 1 | 2 | 3 | 4 | 6 | 9 |
|---|---|---|---|---|---|---|
| Grouping factor kg (indicative) | 1.00 | 0.80 | 0.70 | 0.65 | 0.57 | 0.50 |
Standard ratings (A): 2, 4, 6, 10, 16, 20, 25, 32, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1 000, 1 250, 1 600.
↑ Back to topA.3 Short circuit: factor c and coefficient κ
| Nominal voltage | cmax | cmin |
|---|---|---|
| ≤ 1 kV (tolerance +6 %) | 1.05 | 0.95 |
| ≤ 1 kV (tolerance +10 %) | 1.10 | 0.95 |
| > 1 kV | 1.10 | 1.00 |
| R/X | 0 | 0.1 | 0.2 | 0.3 | 0.5 | 0.7 | 1.0 |
|---|---|---|---|---|---|---|---|
| κ | 2.00 | 1.75 | 1.56 | 1.42 | 1.24 | 1.14 | 1.07 |
A.4 Orders of magnitude
| Quantity | Typical value |
|---|---|
| Irradiation in Senegal | 4.3 (December) to 6.7 (April) kWh/m²/d on the horizontal plane in Dakar; ≈ 2 000 kWh/m²/yr |
| Grid-connected PV yield | 1 500 to 1 650 kWh/kWp/yr (PR 0.75 to 0.82) |
| Coastal wind at 10 m | 4 to 6 m/s on average (northern coast), lower inland |
| Solar pumping | rural demand 20 to 50 l/inh/d; pump × motor efficiency 30 to 50 %; tank of 1 to 3 days of consumption |
| Lifetimes | modules 25 years; lead-acid batteries 3 to 7 years, LFP 10 to 12 years; inverters 8 to 12 years; charge controllers 10 to 15 years; cables 25 years |
| Efficiencies | inverter 92 to 97 %; MPPT 96 to 98 %; lead-acid battery 80 to 85 %, lithium 92 to 95 % |
| Genset consumption | 0.25 l/kWh at full load, 0.35 to 0.45 l/kWh at low load |
| Discount rate | 6 to 12 %; inflation 2 to 4 % |
| Non-technical losses | from 1 to more than 15 % of the energy supplied depending on the network; smart meter + installation ≈ 50 to 100 kF |
A.5 Calculators
↑ Back to topA.6 Bibliography and standards
- H. Saadat, Power System Analysis, McGraw-Hill — load flow, short circuit, symmetrical components.
- J. J. Grainger, W. D. Stevenson, Power System Analysis, McGraw-Hill.
- W. H. Kersting, Distribution System Modeling and Analysis, CRC Press — backward/forward sweep, unbalanced networks.
- M. E. Baran, F. F. Wu, “Network reconfiguration in distribution systems for loss reduction and load balancing”, IEEE Trans. Power Delivery, 1989 — IEEE 33-bus network.
- J. A. Duffie, W. A. Beckman, Solar Engineering of Thermal Processes, Wiley — solar geometry, tilted plane.
- J. F. Manwell, J. G. McGowan, A. L. Rogers, Wind Energy Explained, Wiley — Weibull, power curves.
- IEC 60909-0:2016 (short-circuit currents); IEC 60364 / NF C 15-100 (LV installations); UTE C15-712-1 (PV installations); IEC 61400-12-1 (wind turbine power curves); IEEE 1547-2018 (DER interconnection); IEEE 2030.7 / 2030.8 (microgrids); IEC 62056 (DLMS/COSEM); OpenADR 2.0b; IEC 62443 and IEC 62351 (cybersecurity).
Author
November 2026