PHES model (pumped hydro energy storage)

[Meneer]

TL;DR

Need help to correctly calculate the head loss and flow rate of a small PHES, so the turbine output power can be calculated.

Hello!

I am trying to model a small pumped hydro energy storage system, with two closed reservoirs (upper and lower).
However, I am assuming here that I only know the volume of both reservoirs, the penstock diameter and length, the height difference between both reservoirs, and the turbine efficiency. The fluid used is water.

This picture hopefully clarifies what I mean:

To determine the power output of the turbine I use P=ρ g Q η (H - Hlosses).
Therefore I would need to calculate both the flow rate Q and Hlosses
To calculate the flow rate Q , I was planning on using the Bernoulli equation between point 1 and 2 on the image, so I can assume P₁ ,P₂ , v₁ and v₂ are 0.
This would simplify the Bernoulli equation from P₁/ρ + (V₁²)/2g + h₁ = P₂/ρ + (V₂²)/2g + h₂ + H_losses to 0 + 0 + H = 0 + 0 + 0 + H_losses.

With the Darcy Weisbach equation (assuming no minor losses for now) I would then put this in terms of H = (f) (L/D) (v²/2g) .
with unknowns v and f, the Moody diagram and an iterative process can be used to determine both these factors.
The the flow rate is calculated by Q=V*A

However, I am not sure if this is a correct way of doing this, since I really need the flow velocity at point 3, as here the flow goes into the turbine. Also, from how I understand it now, if I do it like this the Head losses will be as large as the gross head which does not make sense.
Can someone please shed some light on this and help me in the right direction? Thank you!

 

[anorlunda]

The question is more clear than the intent. Are you trying to model an existing pumped hydro plant, or are you designing a new one?

You don't mention the gates. Gates before the turbine regulate the flow from zero to max. Max flow is a property of the plant design. If you are modeling an existing plant, that should be given, not calculated.

With losses, the same thing applies. For an existing plant, those numbers should be given, not calculated. For design purposes, you need to know every twist and turn in the penstock, and the roughness of the surfaces.

Losses in the turbine are calculated by the turbine manufacturer (likely by FEA analysis, confirmed by tests) and guaranteed. They also change substantially with lifetime as the surface of the blades get pitted. Nobody other than the manufacturer is expected to calculate them.

When I help utilities model power plants for purposes of power grid performance, or for purposes of operator training, all the models are done in units of per-unit. That is much simpler to handle than real units. The per unit base quantities (rated power, flow, etc.) are supplied by the designers, or determined by tests.

 

[Meneer]

Dear Anorlunda, thank you for your reply!

My intent is to design a model for a small-scale PHES plant which can be used to determine the potential power output, when only the reservoir sizes and the topography are known. So this is not about existing pumped hydro plants, but new ones.

I understand that the flow rate is dependent on the gates, but for this case I would like to calculate the maximum flow rate to be used by the turbine, depending on the head and diameter of the pipe. velocity is only obtained by gravitational force, and the gate is assumed to be completely open in this case. If you know of any way to model the flow rate based on how much the valve is opened that would be even better, but for now assuming the valve is either completely closed or opened is sufficient.

I hope I clarified my intent, if not please let me know and I will elaborate!