# PHES model (pumped hydro energy storage)

• Meneer
In summary, you need to know the Bernoulli equation between points 1 and 2 on the diagram, and the Darcy Weisbach equation to determine the flow rate.
Meneer
TL;DR Summary
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 + Hlosses to 0 + 0 + H = 0 + 0 + 0 + Hlosses.

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!

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.

russ_watters and berkeman
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!

## 1. What is a PHES model?

A PHES model, or pumped hydro energy storage model, is a type of energy storage system that uses water to store and release energy. It involves pumping water from a lower reservoir to a higher reservoir when excess energy is available, and then releasing the water back down through turbines to generate electricity when energy is needed.

## 2. How does a PHES model work?

A PHES model works by using pumps to transport water from a lower reservoir to a higher reservoir, where it is stored as potential energy. When energy is needed, the water is released back down through turbines, which generate electricity as it flows back to the lower reservoir. This process can be repeated as needed to store and release energy.

## 3. What are the benefits of using a PHES model?

There are several benefits to using a PHES model for energy storage. It is a reliable and proven technology, with a long lifespan and low maintenance requirements. It also has a high efficiency, with the ability to store and release large amounts of energy quickly. Additionally, PHES models can help to balance the grid by providing a steady and flexible source of energy.

## 4. Are there any drawbacks to using a PHES model?

While PHES models have many benefits, there are some drawbacks to consider. They require large amounts of land and water resources, which can be challenging to find in certain locations. They also have high upfront costs for construction and installation. Additionally, the environmental impact of building a PHES model, such as disrupting ecosystems and altering water flow, should be carefully considered.

## 5. How is a PHES model different from other types of energy storage?

There are several types of energy storage systems, including batteries, compressed air, and thermal storage. A PHES model is unique in that it uses water as the storage medium. This allows for large-scale storage and long-term energy storage, making it a valuable tool for balancing the grid and integrating renewable energy sources. It also has a longer lifespan compared to other storage technologies, making it a more sustainable option in the long run.

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