Renewable Energy

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

This Renewable Energy formula collection brings together the core maths behind solar, wind, hydro and batteries, from power output and capacity factor to levelised cost of energy and emissions avoided. It’s designed for students, exam revision and anyone working on real projects, helping you estimate performance, compare technologies and make smarter clean-energy decisions with confidence.

Solar PV Energy Yield

NEWIntermediate

E = A \times r \times H \times PR

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

Wind Turbine Power

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

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

Betz Limit

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

Maximum fraction of wind energy any turbine can capture.

Capacity Factor

NEWIntermediate

CF = \frac{E_{actual}}{P_{rated} \times t}

Ratio of actual output to theoretical maximum.

Levelised Cost of Energy

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

Lifetime cost per unit of energy produced.

Battery Energy Capacity

E = V \times Q

Energy stored from voltage and charge capacity.

Battery Runtime

NEWIntermediate

t = \frac{E \times DoD}{P}

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

Solar Payback Period

T = \frac{C}{S}

Years to recover a solar install from annual savings.

Optimum Panel Tilt (rule of thumb)

NEWIntermediate

\beta \approx \phi

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

Inverter Efficiency

\eta = \frac{P_{ac}}{P_{dc}}

Fraction of DC power converted to usable AC.

Hydroelectric Power

NEWIntermediate

P = \rho g Q H \eta

Power from falling water: flow rate times head.

CO₂ Emissions Avoided

m = E \times f

Carbon saved by generating clean energy instead of fossil power.

Solar Power on a Surface

P = G \times A \times \cos\theta

Sunlight power hitting a tilted surface.

Panel Output After Degradation

NEWIntermediate

P_n = P_0 (1 - d)^n

Panel power after n years of gradual degradation.

Rotor Swept Area

A = \pi r^2

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