# Battery Sizing

## Introduction

Figure 1. Stationary batteries on a rack (courtesy of Power Battery)

This article looks at the sizing of batteries for stationary applications (i.e. they don't move). Batteries are used in many applications such as AC and DC uninterruptible power supply (UPS) systems, solar power systems, telecommunications, emergency lighting, etc. Whatever the application, batteries are seen as a mature, proven technology for storing electrical energy. In addition to storage, batteries are also used as a means for providing voltage support for weak power systems (e.g. at the end of small, long transmission lines).

### Why do the calculation?

Sizing a stationary battery is important to ensure that the loads being supplied or the power system being supported are adequately catered for by the battery for the period of time (i.e. autonomy) for which it is designed. Improper battery sizing can lead to poor autonomy times, permanent damage to battery cells from over-discharge, low load voltages, etc.

### When to do the calculation?

The calculation can typically be started when the following information is known:

• Battery loads that need to be supported
• Nominal battery voltage
• Autonomy time(s)

## Calculation Methodology

The calculation is based on a mixture of normal industry practice and technical standards IEEE Std 485 (1997, R2003) "Recommended Practice for Sizing Lead-Acid Batteries for Stationary Applications" and IEEE Std 1115 (2000, R2005) "Recommended Practice for Sizing Nickel-Cadmium Batteries for Stationary Applications". The calculation is based on the ampere-hour method for sizing battery capacity (rather than sizing by positive plates).

The focus of this calculation is on standard lead-acid or nickel-cadmium (NiCd) batteries, so please consult specific supplier information for other types of batteries (e.g. lithium-ion, nickel-metal hydride, etc). Note also that the design of the battery charger is beyond the scope of this calculation.

There are five main steps in this calculation:

1) Collect the loads that the battery needs to support
2) Construct a load profile and calculate the design energy (VAh)
3) Select the battery type and determine the characteristics of the cell
4) Select the number of battery cells to be connected in series
5) Calculate the required Ampere-hour (Ah) capacity of the battery

### Step 1: Collect the battery loads

The first step is to determine the loads that the battery will be supporting. This is largely specific to the application of the battery, for example an AC UPS System or a Solar Power System.

### Step 2: Construct the Load Profile

Refer to the Load Profile Calculation for details on how to construct a load profile and calculate the design energy, $E_{d} \,$, in VAh.

The autonomy time is often specified by the Client (i.e. in their standards). Alternatively, IEEE 446, "IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and Commercial Applications" has some guidance (particularly Table 3-2) for autonomy times. Note that IEEE 485 and IEEE 1115 refer to the load profile as the "duty cycle".

### Step 3: Select Battery Type

The next step is to select the battery type (e.g. sealed lead-acid, nickel-cadmium, etc). The selection process is not covered in detail here, but the following factors should be taken into account (as suggested by IEEE):

• Physical characteristics, e.g. dimensions, weight, container material, intercell connections, terminals
• application design life and expected life of cell
• Frequency and depth of discharge
• Ambient temperature
• Charging characteristics
• Maintenance requirements
• Ventilation requirements
• Cell orientation requirements (sealed lead-acid and NiCd)
• Seismic factors (shock and vibration)

Next, find the characteristics of the battery cells, typically from supplier data sheets. The characteristics that should be collected include:

• Battery cell capacities (Ah)
• Cell temperature
• Electrolyte density at full charge (for lead-acid batteries)
• Cell float voltage
• Cell end-of-discharge voltage (EODV).

Battery manufacturers will often quote battery Ah capacities based on a number of different EODVs. For lead-acid batteries, the selection of an EODV is largely based on an EODV that prevents damage of the cell through over-discharge (from over-expansion of the cell plates). Typically, 1.75V to 1.8V per cell is used when discharging over longer than 1 hour. For short discharge durations (i.e. <15 minutes), lower EODVs of around 1.67V per cell may be used without damaging the cell.

Nickel-Cadmium (NiCd) don't suffer from damaged cells due to over-discharge. Typical EODVs for Ni-Cd batteries are 1.0V to 1.14V per cell.

### Step 4: Number of Cells in Series

The most common number of cells for a specific voltage rating is shown below:

12V 6 9-10
24V 12 18-20
48V 24 36-40
125V 60 92-100
250V 120 184-200

However, the number of cells in a battery can also be calculated to more accurately match the tolerances of the load. The number of battery cells required to be connected in series must fall between the two following limits:

(1) $N_{max} = \frac{V_{dc} (1+V_{l,max})}{V_{c}} \,$

(2) $N_{min} = \frac{V_{dc} (1-V_{l,min})}{V_{eod}} \,$

where $N_{max} \,$ is the maximum number of battery cells

$N_{min} \,$ is the minimum number of battery cells
$V_{dc} \,$ is the nominal battery voltage (Vdc)
$V_{l,max} \,$ is the maximum load voltage tolerance (%)
$V_{l,min} \,$ is the minimum load voltage tolerance (%)
$V_{c} \,$ is the cell charging voltage (Vdc)
$V_{eod} \,$ is the cell end of discharge voltage (Vdc)

The limits are based on the minimum and maximum voltage tolerances of the load. As a maximum, the battery at float voltage (or boost voltage if applicable) needs to be within the maximum voltage range of the load. Likewise as a minimum, the battery at its end of discharge voltage must be within the minimum voltage range of the load. The cell charging voltage depends on the type of charge cycle that is being used, e.g. float, boost, equalising, etc, and the maximum value should be chosen.

Select the number of cells in between these two limits (more or less arbitrary, though somewhere in the middle of the min/max values would be most appropriate).

### Step 5: Determine Battery Capacity

The minimum battery capacity required to accommodate the design load over the specified autonomy time can be calculated as follows:

$C_{min} = \frac{E_{d} (k_{a} \times k_{t} \times k_{c})}{V_{dc} \times k_{dod}} \,$

where $C_{min} \,$ is the minimum battery capacity (Ah)

$E_{d} \,$ is the design energy over the autonomy time (VAh)
$V_{dc} \,$ is the nominal battery voltage (Vdc)
$k_{a} \,$ is a battery ageing factor (%)
$k_{t} \,$ is a temperature correction factor (%)
$k_{c} \,$ is a capacity rating factor (%)
$k_{dod} \,$ is the maximum depth of discharge (%)
Table 1. Temperature correction factors for vented lead-acid cells (from IEEE 485)

Select a battery Ah capacity that exceeds the minimum capacity calculated above. The battery discharge rate (C rating) should also be specified, approximately the duration of discharge (e.g. for 8 hours of discharge, use the C8 rate). The selected battery specification is therefore the Ah capacity and the discharge rate (e.g. 500Ah C10).

An explanation of the different factors:

• Ageing factor captures the decrease in battery performance due to age.
The performance of a lead-acid battery is relatively stable but drops markedly at latter stages of life. The "knee point" of its life vs performance curve is approximately when the battery can deliver 80% of its rated capacity. After this point, the battery has reached the end of its useful life and should be replaced. Therefore, to ensure that battery can meet capacity throughout its useful life, an ageing factor of 1.25 should be applied (i.e. 1 / 0.8). There are some exceptions, check with the manufacturer.
For Ni-Cd batteries, the principles are similar to lead-acid cells. Please consult the battery manufacturer for suitable ageing factors, but generally, applying a factor of 1.25 is standard. For applications with high temperatures and/or frequent deep discharges, a higher factor of 1.43 may be used. For more shallower discharges, a lower factor of 1.11 can be used.
• Temperature correction factor is an allowance to capture the ambient installation temperature. The capacity for battery cells are typicall quoted for a standard operating temperature of 25C and where this differs with the installation temperature, a correction factor must be applied. IEEE 485 gives guidance for vented lead-acid cells (see figure right), however for sealed lead-acid and Ni-Cd cells, please consult manufacturer recommendations. Note that high temperatures lower battery life irrespective of capacity and the correction factor is for capacity sizing only, i.e. you CANNOT increase battery life by increasing capacity.
• Capacity rating factor accounts for voltage depressions during battery discharge. Lead-acid batteries experience a voltage dip during the early stages of discharge followed by some recovery. Ni-Cds may have lower voltages on discharge due to prolonged float charging (constant voltage). Both of these effects should be accounted for by the capacity rating factor - please see the manufacturer's recommendations. For Ni-Cd cells, IEEE 1115 Annex C suggests that for float charging applications, Kt = rated capacity in Ah / discharge current in Amps (for specified discharge time and EODV).

## Worked Example

Figure 2. Load profile for this example

### Step 1 and 2: Collect Battery Loads and Construct Load Profile

The loads and load profile from the simple example in the Energy Load Profile Calculation will be used (see the figure right). The design energy demand calculated for this system is Ed = 3,242.8 VAh.

### Step 3: Select Battery Type

Vented lead acid batteries have been selected for this example.

### Step 4: Number of Cells in Series

Suppose that the nominal battery voltage is Vdc = 120Vdc, the cell charging voltage is Vc = 2.25Vdc/cell, the end-of-discharge voltage is Veod = 1.8Vdc/cell, and the minimum and maximum load voltage tolerances are Vl,min = 10% and Vl,max = 20% respectively.

The maximum number of cells in series is:

$N_{max} = \frac{V_{dc} (1+V_{l,max})}{V_{c}} \,$
$= \frac{120 \times (1 + 0.2)}{2.25} = 64 \,$ cells

The minimum number of cells in series is:

$N_{min} = \frac{V_{dc} (1-V_{l,min})}{V_{eod}} \,$
$= \frac{120 \times (1 - 0.1)}{1.8} = 60 \,$ cells

The selected number of cells in series is 62 cells.

### Step 5: Determine Battery Capacity

Given a depth of discharge kdod = 80%, battery ageing factor ka = 25%, temperature correction factor for vented cells at 30 deg C of kt = 0.956 and a capacity rating factor of kc = 10%, the minimum battery capacity is:

$C_{min} = \frac{E_{d} \times k_{a} \times k_{c} \times k_{t}}{V_{dc} \times k_{dod}} \,$
$= \frac{3,242.8 \times 1.25 \times 1.1 \times 0.956}{120 \times 0.8} = 44.4 \,$ Ah

## Computer Software

Some battery manufacturers (such as Alcad) also provide software programs to size batteries using basic input data such as load profiles, autonomies, etc. The software will size the batteries and will often also provide details regarding different battery rack (or enclosure) dimensions.

## What Next?

Using the results of the battery sizing calculation, the approximate dimensions of the batteries can be estimated based on typical vendor information. This will assist in determining the size, number and dimensions of the battery racks or cabinets required, which can then be used as input into the equipment / room layouts. Preliminary budget pricing can also be estimated based on the calculation results.