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Renewable Energy

Clean power maths: solar yield, wind turbines, capacity factor, batteries and LCOE

Solar PV Energy Yield

NEWIntermediate
E=A×r×H×PRE = A \times r \times H \times PR

Annual energy from a solar array's area, efficiency and sunlight.

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Wind Turbine Power

NEWAdvanced
P=12ρAv3CpP = \tfrac{1}{2}\rho A v^3 C_p

Power a wind turbine extracts, rising with the cube of wind speed.

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Betz Limit

NEWAdvanced
Cp,max=16270.593C_{p,max} = \frac{16}{27} \approx 0.593

Maximum fraction of wind energy any turbine can capture.

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Capacity Factor

NEWIntermediate
CF=EactualPrated×tCF = \frac{E_{actual}}{P_{rated} \times t}

Ratio of actual output to theoretical maximum.

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Levelised Cost of Energy

NEWAdvanced
LCOE=C+Mt(1+r)tEt(1+r)tLCOE = \frac{C + \sum \frac{M_t}{(1+r)^t}}{\sum \frac{E_t}{(1+r)^t}}

Lifetime cost per unit of energy produced.

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Battery Energy Capacity

NEWBasic
E=V×QE = V \times Q

Energy stored from voltage and charge capacity.

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Battery Runtime

NEWIntermediate
t=E×DoDPt = \frac{E \times DoD}{P}

How long a battery powers a load given depth of discharge.

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Solar Payback Period

NEWBasic
T=CST = \frac{C}{S}

Years to recover a solar install from annual savings.

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Optimum Panel Tilt (rule of thumb)

NEWIntermediate
βϕ\beta \approx \phi

Fixed panels roughly tilted to the site latitude capture the most annual energy.

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Inverter Efficiency

NEWBasic
η=PacPdc\eta = \frac{P_{ac}}{P_{dc}}

Fraction of DC power converted to usable AC.

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Hydroelectric Power

NEWIntermediate
P=ρgQHηP = \rho g Q H \eta

Power from falling water: flow rate times head.

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CO₂ Emissions Avoided

NEWBasic
m=E×fm = E \times f

Carbon saved by generating clean energy instead of fossil power.

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Solar Power on a Surface

NEWBasic
P=G×A×cosθP = G \times A \times \cos\theta

Sunlight power hitting a tilted surface.

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Panel Output After Degradation

NEWIntermediate
Pn=P0(1d)nP_n = P_0 (1 - d)^n

Panel power after n years of gradual degradation.

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Rotor Swept Area

NEWBasic
A=πr2A = \pi r^2

Area swept by turbine blades, which sets its power potential.

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