Battery/Generator Runtime Estimation

Hello! Welcome to the final lesson of our course on electricity.

In our previous session, we tackled the critical task of sizing a generator, focusing on its ability to handle both continuous running power and the significant surge power from starting motors. We learned that selecting a generator is a matter of ensuring its capacity (in kVA) exceeds the system's peak demand.

Today, we answer the final practical question: once we know a power source can run our devices, for how long can it do so? This lesson will equip you to estimate the runtime of a device on a battery or generator based on its power consumption and the energy source's capacity.

This final piece of the puzzle brings together concepts of power, energy, and efficiency, providing a complete framework for analyzing and planning for off-grid or backup power needs.

From Power to Energy: The Importance of the Watt-hour

Throughout this course, we've worked extensively with power, measured in Watts (W). Power is the rate at which energy is used or produced. However, to calculate runtime, we need to know the total amount of energy a source can store. This is measured in Watt-hours (Wh).

• Power (W): How fast energy is flowing. (Analogous to speed in km/h).
• Energy (Wh): The total amount of energy available. (Analogous to total distance in km).

A device drawing 100 watts for 2 hours consumes $100\text{W}\times 2\text{h}=200\text{Wh}$ of energy.

Batteries are often rated in Amp-hours (Ah), which measures electric charge. To convert this to energy, you must also know the battery's nominal voltage. Your background in econometrics, where models often require careful unit handling to be meaningful, makes the importance of this distinction clear.

The fundamental relationship is:
$\text{Energy (Wh)}=\text{Charge (Ah)}\times \text{Voltage (V)}$

Because it directly represents energy, the Watt-hour (or kilowatt-hour, kWh) is the most reliable unit for comparing energy sources and calculating runtime.

The Basic Runtime Calculation

In an ideal world, the calculation for runtime is simple division:

$\text{Runtime (hours)}=\frac{\text{Battery Energy Capacity (Wh)}}{\text{Load Power Consumption (W)}}$

For example, a fully charged 1200 Wh battery powering a constant 60 W load would ideally last:
$\text{Runtime}=\frac{1200\text{Wh}}{60\text{W}}=20\text{hours}$

However, real-world performance is always lower than this ideal figure due to several factors. Let's incorporate them to build a more accurate model.

Refining the Model: Real-World Inefficiencies

Just as economic models must account for transaction costs and externalities, our runtime calculation must account for real-world losses.

1. Usable Capacity and Depth of Discharge (DoD)

You cannot—and often should not—drain a battery to 0%. The maximum percentage you can safely use is called the Depth of Discharge (DoD). It varies significantly by battery chemistry:

• Lead-Acid: Typically have a DoD of 50% to preserve their lifespan. Draining them further causes rapid degradation.
• Lithium Iron Phosphate (LiFePO4): Can often be discharged to 80-100% with thousands of cycles, making them far more energy-dense in practice.

Therefore, the actual available energy is Total Wh × DoD %.

2. System Efficiency Losses

Energy is lost as heat during use. There are two main sources of loss to consider when running AC appliances from a DC battery:

• Battery Efficiency: The process of discharging a battery is not 100% efficient. Some energy is lost internally. Lithium batteries are very efficient (around 95%), while lead-acid batteries are less so (around 85%).
• Inverter Efficiency: An inverter converts the battery's DC power to AC power for your appliances. This process itself consumes power. A typical inverter is about 90% efficient, meaning for every 100 W your appliance needs, the inverter must draw about $100/0.9=111$ W from the battery.

The following video provides an excellent walkthrough of how to incorporate these factors.

Our more robust formula now looks like this:

$\text{Runtime (h)}=\frac{\text{Total Wh}\times \text{DoD \%}\times \text{Battery Eff. \%}}{\left(\frac{\text{Appliance Power (W)}}{\text{Inverter Eff. \%}}\right)}$

Additionally, all-in-one "power stations" have their own internal 'phantom' loads (screens, processors, etc.) that reduce the advertised Wh capacity to a lower usable capacity. As shown in the "Easily Calculate Power Station Run-times!" video (03:57-05:33), this usable capacity is often around 85-90% of the advertised value even before considering the main inverter losses for the load itself.

Handling Variable Loads

Many appliances, like refrigerators or well pumps, cycle on and off. They don't draw their full power continuously. To handle this, we must calculate their average power consumption.

For example, a refrigerator might run its 150 W compressor for 20 minutes every hour (a 33% duty cycle).

• Average Power = $150\text{W}\times 0.33=50\text{W}$

The most accurate way to determine average power for a device is to measure its total energy consumption over a long period (e.g., 24 hours) using a simple plug-in watt meter. If a fridge consumes 1200 Wh in 24 hours, its average power is $1200\text{Wh}/24\text{h}=50\text{W}$.

A Comprehensive Example

Let's put it all together. You want to run a small off-grid setup with a 12V, 200Ah LiFePO4 battery.
The loads are:

1. Lights: 30 W, running continuously.
2. Small Fridge: 120 W when running, with a 25% duty cycle (measured over 24h).

Assumed efficiencies:

• Battery DoD: 90%
• Battery Efficiency: 95%
• Inverter Efficiency: 90%

1. Calculate Total Energy Capacity:
$\text{Total Energy}=200\text{Ah}\times 12\text{V}=2400\text{Wh}$

2. Calculate Usable Energy at the Battery Terminals:
$\text{Usable Energy}=2400\text{Wh}\times 0.90(\text{DoD})\times 0.95(\text{Batt. Eff.})=2052\text{Wh}$

3. Calculate Average Load Power Drawn from the Inverter:

• Lights Power = 30 W
• Fridge Average Power = $120\text{W}\times 0.25=30\text{W}$
• Total Average Load = $30\text{W}+30\text{W}=60\text{W}$

4. Calculate Effective Power Draw from the Battery:
The inverter's inefficiency increases the load on the battery.
$\text{Effective Power Draw}=\frac{60\text{W}}{\text{0.90 (Inv. Eff.)}}\approx 66.7\text{W}$

5. Calculate Final Estimated Runtime:
$\text{Runtime}=\frac{\text{Usable Energy (Wh)}}{\text{Effective Power Draw (W)}}=\frac{2052\text{Wh}}{66.7\text{W}}\approx 30.8\text{hours}$

Notice how the ideal calculation ($2400\text{Wh}/60\text{W}=40\text{hours}$) is almost 10 hours longer. Accounting for real-world factors is essential for accurate planning.

Test your understanding!

You have a 24V, 100Ah lead-acid battery bank. You want to power a 150W computer setup.
Calculate the estimated runtime using the following assumptions:

• Lead-Acid DoD: 50%
• Battery Efficiency: 85%
• Inverter Efficiency: 90%

Show answer

1. Calculate Total Energy Capacity:
$\text{Total Energy}=100\text{Ah}\times 24\text{V}=2400\text{Wh}$

2. Calculate Usable Energy:
$\text{Usable Energy}=2400\text{Wh}\times 0.50(\text{DoD})\times 0.85(\text{Batt. Eff.})=1020\text{Wh}$

3. Calculate Effective Power Draw from the Battery:
$\text{Effective Power Draw}=\frac{150\text{W}}{\text{0.90 (Inv. Eff.)}}\approx 166.7\text{W}$

4. Calculate Final Estimated Runtime:
$\text{Runtime}=\frac{1020\text{Wh}}{166.7\text{W}}\approx 6.1\text{hours}$

Conclusion and Course Summary

Today you have learned how to build a robust model to estimate the runtime of an energy source, moving from a simple ideal formula to one that incorporates the practical constraints of efficiency and usable capacity.

Key Takeaways:

• Energy (Wh), not power (W), determines runtime. The core relationship is Energy (Wh) = Charge (Ah) × Voltage (V).
• The basic runtime formula is Runtime = Energy / Power.
• A realistic estimate must adjust for Depth of Discharge (DoD), battery efficiency, and inverter efficiency.
• For non-continuous loads, you must use the average power consumption to calculate runtime.

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Course Wrap-up

Congratulations on completing this course! You have journeyed from the foundational principles of electricity to a deep, practical understanding of power systems.

We began with the fundamentals of DC circuits, applying Ohm's and Kirchhoff's laws. We then delved into the underlying physics, exploring the nature of electric and magnetic fields, the mechanism of electromagnetic induction, and their ultimate unification in Maxwell's Equations.

With that theoretical foundation, we moved to the applied world of AC circuits, analyzing impedance and power factor. This enabled us to understand the operation of generators and motors, calculating EMF, torque, and efficiency. Finally, we put everything together in this last module on Applied Power Systems, learning to size generators for complex loads and, today, to calculate their operational runtime.

You have achieved your goal of understanding mains power, generators, and the core equations and units, all grounded in the physics of electromagnetism. You are now well-equipped to quantitatively analyze and reason about a wide array of electrical systems. I hope this knowledge serves you well in your future projects and intellectual pursuits.