# Theory

Before diving into the theory and developing the basic equation for hydropower power generation, take a look at these basic physics concepts. If there are any that are not clear to you, click them and read about them in the Glossary:

Where does hydropower come from?

It comes from solar energy, as follows:

• The sun evaporates lakes and oceans and lifts the moisture into the atmosphere.
• It also creates the weather, which moves the moisture-laden air around so that some precipitates as rain and snow into the hydro catchment.
• The snow melt and rain then flows down the water courses of the catchment to a reservoir or head pond.
• Here the water has Potential Energy relative to the tailwater level of the hydro station.
• The role of the hydro station is to convert this potential energy to electrical energy.

Potential Energy

Assume we have one cubic metre of water at a height of one metre above the tailrace level of a hydro station.

It has a mass of 1000 litres * 1 kg/litre = 1000 kg

If the gravitational acceleration, g, at this location is 9.81 m/s², this mass has a weight of 9810 Newtons.

Weight is a force. When the volume of water is lowered to the tailrace, this force is displaced by the height of 1 m and the work recovered is:

W = F* s = 9810*1 = 9810 Joules.

We say that the potential energy of the water has been converted to Work.

Expanding this logic, when a volume of water V is lowered a distance H to the tailrace, the potential energy recovered as work is:

W = 9810 * V * H Joules.

Now:

Power(Watts) = Work(Joules) / Time(seconds) …

… so if we convert this much potential energy to work in each second, we produce
P = 9810*V/t*H Watts

The volume flowing per second is the flow discharge, Q, so the equation becomes:

Power in watts = 9810*Q*H

This is the theoretical power before taking into account Losses.

Losses

When the potential energy of flowing water is converted to electrical energy there are energy losses along the way.

Water flowing from the intake to the powerhouse experiences friction losses as the water flows through trash rack, changes in the shape of the water passages, and turbulence created by the gate slots, bends and bifurcations. All along its passage through the water conveyance system, the water is dragging against the walls of the water conveyances. This creases friction loss. These losses reduce the hydraulic head difference across the turbine, and reduce the amount of power generated (see Gross Head and Net Head in the Glossary)

There are also various losses within the turbine. These are accounted for in the efficiency of the turbine (see Efficiency in the Glossary)

If the rotational speed of the turbine is suitable for economical design of a generator, the generator is directly coupled to the turbine. In the case of low head turbines, however, the turbine speed is often too low for an economical generator design and a gear box must be used to increase the shaft speed. This “speed increaser” gearbox has internal friction losses that further reduce the power supplied to the generator.

Generators also have internal losses due to electrical losses, magnetic losses and windage. These are accounted for in the efficiency of the generator.

The power leaving the generator flows through buswork and High Voltage Switchgear en route to the generator transformer. There are electrical losses in both these components, but there is an even bigger loss caused by the tapping off power to operate the generator exciter (the system that turns the rotating poles of the generator into electro-magnets) and the station service system (the electrical system that runs the power station, providing lights, heating and cooling, and powering the other numerous pieces of mechanical equipment and electrical equipment (see Glossary).

When the power reaches the generator transformer, it is converted from generator voltage (typically 10 to 14 kV)  to the transmission line voltage (often just called the Line Voltage. There are electrical losses in the transformer, and in the electrical transmission system.

After this long discussion on losses, you will be surprised if there is any useful power yet. In fact when you add them all up, the losses may account for only 15% to 25% of the available potential energy, which makes hydro very efficient relative to other power sources.

Hydropower Formula

The water power available to a turbine is as follows:

Pw = 𝛄*Q*Hnet

where Pw is the power in kilowatts (kW); 𝛄 is the specific weight of water in Newtons/m³, Q is the flow in m³/s and Hnet is the net head in metresw.

No turbine can convert all this power to mechanical power on the turbine shaft. The percentage of power that is converted is called the turbine efficiency, ζt, so the turbine shaft power is as follows:

Pt = 𝛄*Q*Hnet*ζt

Turbine efficiency varies with the type of turbine and the head and flow at which it is operating, but peak efficiency can approach 90%

Similarly, the generator connected to the turbine shaft cannot convert all the mechanical shaft power into electrical energy. The percentage of power that is converted is called the generator efficiency, ζg. Generator efficiencies typically approach 98%. The electrical power produced by a turbine/generator is therefore as follows:

P = 𝛄*Q*Hnet*ζt*ζg

This is the BASIC HYDROPOWER FORMULA.

As explained in the section on Losses, some of the generator power is used to run the generator and the power station, and there are further electrical losses all the way to the electrical customer.