15. Battery and Backup System Calculations

Principle

Sizing a battery backup system—for a UPS, emergency lighting, or renewable storage—involves determining the required battery capacity (typically in ampere-hours, Ah) to supply a given load for the desired duration. Key factors include:

  • Load Profile: The power or current demand during the backup period, which can be constant or varying.
  • Autonomy Time: The required duration of battery support (e.g., 15 minutes for UPS hold-up, 8 hours for telecom backup).
  • Depth of Discharge (DoD): Batteries are rarely discharged to 100% to prolong life; designs typically limit DoD to 50–80%.
  • Temperature and Aging: Battery capacity decreases in cold conditions and over time. A margin (often ~125% of the calculated capacity) is included to account for capacity loss at end-of-life.

Basic Battery Capacity Calculation

One common approach is to calculate the required energy in kilowatt-hours (kWh) and convert it to Ah. For example, if the load is given in kW:

Battery capacity (kWh) = (Load in kW / Efficiency) × Duration (h)

Alternatively, if the load is specified in amperes:

Required Ah = (Load current (A) × Duration (h)) / DoD

Efficiency (η), which accounts for battery and inverter losses, might be around 0.9 for inverter-based systems.

Example – UPS Sizing

Consider a UPS supporting 5 kW of critical load for 15 minutes (0.25 h) with an overall efficiency of 0.9. Using a 240 V DC battery system:

  1. Calculate the DC power needed:
    PDC = 5 kW / 0.9 ≈ 5.56 kW
  2. Energy required over 0.25 h:
    E = 5.56 kW × 0.25 h ≈ 1.39 kWh
  3. At 240 V, the required capacity is:
    Ah = 1.39 kWh / 240 V ≈ 5.8 Ah
  4. To limit discharge to 60% DoD:
    Required Ah = 5.8 / 0.6 ≈ 9.7 Ah

In this example, you would select a battery rated for at least ~10 Ah at the 15-minute discharge rate. Note that battery capacities are usually specified at a particular discharge rate (such as C10 or C20), and high-rate discharges result in lower effective capacity per Peukert’s law.

Step-by-Step Example – Battery Bank for Solar Storage

Suppose you need to supply a remote telemetry station drawing 1 A at 48 V (i.e., 48 W) for 3 days (72 h) of autonomy with no sun. To preserve battery life, the design limits discharge to 50% (DoD = 50%):

  1. Calculate the required energy:
    E = 48 W × 72 h = 3456 Wh = 3.456 kWh
  2. For a 48 V system, 100% usage would require:
    Ah = 3456 Wh / 48 V ≈ 72 Ah
  3. Considering a 50% DoD, the battery bank must provide:
    Required Ah = 72 / 0.5 = 144 Ah
  4. Accounting for temperature effects and aging (e.g., multiplying by 1.2), you might specify a ~180 Ah 48 V battery bank.

If using 12 V batteries, four in series (to obtain 48 V) of ~180 Ah each would be required.

Backup Generator and UPS Interplay

Battery sizing for a UPS is designed to provide hold-up power until a backup generator starts. Typically, the UPS must supply power for 10 seconds (start-up time) plus an additional 10–30 seconds for the transfer of load. For longer outages, a generator is used rather than relying on extensive battery capacity.

Battery Charger Calculation

The charger must supply both the load and the current required to recharge the battery within the desired time. For example, if a battery bank is 200 Ah and you want to recharge from 50% to full in 8 hours:

  1. Required recharge capacity = 100 Ah in 8 h → approximately 12.5 A.
  2. If a concurrent load draws 10 A during recharge, the charger must supply:
    12.5 A + 10 A ≈ 24 A
  3. Rounding for inefficiency, a ~25 A charger would be specified.

Niche Applications

  • High-Rate Discharge vs. Deep-Cycle Batteries: UPS systems use batteries rated for high discharge rates, while solar or telecom applications typically use deep-cycle batteries.
  • Lithium-Ion Batteries: These allow deeper discharges (up to 80–90%) and generally have higher efficiency (~95%), but calculations follow a similar method.
  • Capacitor-Based Backups: For very short ride-through times (seconds), supercapacitors can be sized using energy formulas (e.g., 0.5 C V²).
  • Battery Short-Circuit Current: In some designs, worst-case short-circuit currents are calculated to ensure DC breakers or fuses can handle the surge.
  • Temperature Compensation: In cold climates, battery capacity drops significantly; sizing may double capacity to ensure sufficient runtime.

Industry Relevance

Battery backup systems are critical in UPS for IT, emergency lighting, telecom base stations, renewable energy storage, and electric vehicles. Proper sizing ensures that batteries provide the required backup duration without being overstressed, which prolongs battery life and maintains system reliability. In many applications, standards specify minimum backup times—for example, emergency lighting systems must operate for at least 1 hour, and fire alarm systems often require 24 hours of standby power.

Standards

  • IEEE 485: Provides guidelines for sizing lead-acid batteries in stationary applications, including correction factors for temperature and aging.
  • NFPA 111: Standard on Stored Electrical Energy Emergency and Standby Power Systems outlines requirements for battery backup in critical applications.
  • EN 50171: Specifies requirements for central battery systems in emergency lighting, including backup duration and recharge times.
  • Telecom and EV Standards: Various standards (e.g., ITU recommendations, IEC 62660 for EV batteries) guide battery sizing for specific applications.

Software Tools

  • Battery Sizing Programs: Many vendors (e.g., Enersys, East Penn) offer online tools where you input load profiles, ambient conditions, and reserve margins to select the appropriate battery model and quantity.
  • Excel Calculators: Numerous spreadsheets implement IEEE 485 methods by breaking the load profile into time intervals and applying discharge curves.
  • UPS Runtime Estimators: Many UPS manufacturers provide charts or online tools that calculate runtime based on the UPS model and connected load.
  • Simulation Tools: Software such as Homer or PVsyst simulates battery usage over multiple days, incorporating stochastic variations in renewable input and load.
  • Battery Management System (BMS) Data: Modern BMS systems track state-of-charge and discharge profiles, which can help in sizing expansions or verifying that existing systems meet new backup requirements.

Conclusion

Accurate battery and backup system calculations ensure that critical loads receive power during outages without oversizing expensive battery banks. By accounting for load profiles, autonomy time, DoD, efficiency, and aging, designers can create reliable, cost-effective backup solutions that comply with industry standards.