# Hydropower plant in equilibrium

I am trying to set up an equation relating the mechanical energy in a hydropower plant to its electrical output. What I picture is the relation between the energy input from the moving water, to the rotational energy of the turbine to the electric output of the generator.

I know I can calculate the power P directly using the formula P = d*h*r*g*k, where d is the density of water, h is the height, r is the flow rate, g is the gravity constant and k is an efficiency coefficient.

However, I´d like to set up an equilibrium equation where I can vary the flow of water and see the effect on the angular speed of the turbine and furthermore the electric power output from the generator.

I guess I could do this by calculating the kinetic energy of the water and translate this into rotational energy of the turbine (using 0.5*I*ω^2 where I is the moment of inertia and ω the angular velocity ?). This energy would in turn be converted into electrical energy less some friction and magnetic resistance.

I do however not exactly see how I should insert the time dimension in order to obtain an equation of a system in equilibrium (i.e. where the water flow is constant and the output including friction and resistance is constant).

Did I explain this right and can anyone help me with my problem?

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If this an AC generator synched to a transmission system, the angular speed of the generator will not be allowed to vary as that would change the frequency of the generated voltage. In this case, generator speed is fixed but generator torque (and consequently mechanical input power) will change if you vary the flow rate.

Even if there is no grid to synch with, most AC machines are run at constant speed to get a fixed electrical output frequency.

If it is an isolated DC system, then speed might be allowed to vary, but I have never seen such a system.

• davenn
anorlunda
Staff Emeritus
Dr. D is correct. If the plant is connected to the grid, speed will not vary. The equilibrium equation is therefore (mechanical power - losses) = electrical power.

Losses are typically low. If we assume losses to be zero, then a generator is simply a perfect converter of mechanical power to electric power. You never need to consider speed or torque, or voltage or current, or anything other then energy conservation as long as your interest is confined to equilibrium.

Thank you for your replies! I see your point -that is perhaps the reason that hydropower plants often have several turbines/generators in parallel: if the water flow is low they will only use one turbine at near constant speed, while is the flow is larger they might add another turbine, also at constant speed?

There is still one thing that is not clear to me: If you have a constant water flow hitting the turbine, how can you calculate the angular speed? I guess you could use the potential energy of the water over a certain time period and translate this into rotational energy as in the formula I mentioned above: 0.5*I*ω^2 where I is the moment of inertia and ω the angular velocity. If I know the moment of inertia, I could solve for the ω.
But if I think of the water flow as a constant force F hitting the turbine blades, is there a way to use Newton´s 2nd to find the angular speed? The acceleration cannot be infinite, hence there must be some resistance in the system that creates an equilibrium: constant water flow and constant angular speed?

russ_watters
Mentor
No, angular speed of the generator is pretty much completely independent of the energy captured by it, since it spins at constant speed and doesn't absorb the energy as kinetic energy in its rotation. So the speed is something you'd just read off the nameplate or datasheet of the turbine.

But if you think of the turbine only and ignore the generator, then the angular speed of the turbine must be dependent on the water flow? I mean if you reduce the water flow by 50%, the force from the water on the turbine blades must be reduced and hence the angular speed of the turbine must decrease?
Right?

anorlunda
Staff Emeritus
Wrong. The turbine and generator are rigidly bolted together.

russ_watters
Mentor
But if you think of the turbine only and ignore the generator, then the angular speed of the turbine must be dependent on the water flow? I mean if you reduce the water flow by 50%, the force from the water on the turbine blades must be reduced and hence the angular speed of the turbine must decrease?
Right?
No. Just like with linear motion, force relates to acceleration, not speed. There is no direct relationship between force and speed. And in the case of the generator, the net force is always zero anyway: so there is no acceleration except when you first turn it on.

russ_watters
Mentor
You're welcome. Maybe it'll help if I give the next step in the logic:

When you first turn-on the turbine, there is an unbalanced force (torque), causing an acceleration. The turbine spins-up to a speed where the opposing torque of the generator equals the torque on the turbine and the acceleration stops.

and the next step...
For a non-grid-tied dc generator, a change in the flow rate will produce an unbalanced torque, and a change in the speed. The new speed will be a new equilibrium between the torques.

anorlunda
Staff Emeritus
No. Just like with linear motion, force relates to acceleration, not speed. There is no direct relationship between force and speed. And in the case of the generator, the net force is always zero anyway: so there is no acceleration except when you first turn it on.
That is a hard concept for laymen IMHO because in everyday life the most common use of force is to overcome friction. To make your car go faster, you must step on the gas. There are no friction-free examples of "a particle in motion remains in motion" in everyday life.

In physics, such as when discussing Newton's laws, we most often neglect friction, or lump it in with other forces. Friction is nuisance in physics because there is no law of friction, just an empirical approximation of molecular forces.

Before the circuit breaker closes to connect the generator to the grid, the speed of the turbine-generator settles where the force provided by the water balances friction at that speed. To spin faster, you need more water flow. As soon as the breaker closes, friction becomes negligible and completely new balances apply.

Even when the generator feeds a load while not connected to the grid, power to the load las little to do with speed. A speed governor control system becomes dominant.

Sorry to flog this to death, but I think that glomming over the differences in these frictive and non-frictive cases causes many misunderstandings between science and lay people.