# 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)

## IEEE Definitions

IEEE Std. 485-1997 provides some definitions related to the battery sizing terminology:

battery duty cycle: The loads a battery is expected to supply for specified time periods.

cell size: The rated capacity of a lead-acid cell or the number of positive plates in a cell.

equalizing charge: A prolonged charge, at a rate higher than the normal float voltage, to correct any inequalities of voltage and specific gravity that may have developed between the cells during service.

full float operation: Operation of a dc system with the battery, battery charger, and load all connected in parallel and with the battery charger supplying the normal dc load plus any charging current required by the battery. (The battery will deliver current only when the load exceeds the charger output.)

period: An interval of time in the battery duty cycle during which the load is assumed to be constant for purposes of cell sizing calculations.

rated capacity (lead-acid): The capacity assigned to a cell by its manufacturer for a given discharge rate, at a specified electrolyte temperature and specific gravity, to a given end-of-discharge voltage.

valve-regulated lead-acid (VRLA) cell: A lead-acid cell that is sealed with the exception of a valve that opens to the atmosphere when the internal gas pressure in the cell exceeds atmospheric pressure by a preselected amount. VRLA cells provide a means for recombination of internally generated oxygen and the suppression of hydrogen gas evolution to limit water consumption.

vented battery: A battery in which the products of electrolysis and evaporation are allowed to escape freely to the atmosphere. These batteries are commonly referred to as “flooded.”

## Battery Characteristics and Types

### Battery Components

Battery technology has not changed much in the last 100 years. The standard construction method involves flooding lead plates in sulfuric acid. The chemical reaction between the positively charged lead plate and the negatively charged acid allows the battery to store and “give” electricity. The thickness of the lead plate is closely related to the lifespan of the battery because of a factor called “Positive Grid Corrosion”. The positive lead plate gradually wears away over time. Thicker plates are used in deep cycle batteries. This usually translates to a longer battery life. Although plate thickness is not the only factor related to longer lifespan, it is the most critical variable.

### Battery Lifespan

Most of the loss incurred in charging and discharging batteries is due to internal resistance, which is eventually wasted as heat. Efficiency ratios are relatively high considering that most lead acid batteries are 85 to 95 percent efficient at storing the energy they receive. Deep cycle batteries used in renewable energy applications are designed to provide many years of reliable performance with proper care and maintenance. Proper maintenance and usage play a major role in battery lifespan. Toiling over your battery bank daily with complex gadgets and a gallon of distilled water, however, is not necessary. The most common causes of premature battery failure include loss of electrolyte due to heat or overcharging, undercharging, excessive vibration, freezing or extremely high temperatures, and using tap water among other factors

### Battery Charging Stages

There are three basic stages in charging a battery: bulk, absorption, and float. These terms signify different voltage and current variables involved in each stage of charging.

• Bulk Charge: In the first stage of the process, current is sent to the batteries at the maximum safe rate, batteries will accept it until voltage is brought up to nearly 80-90 percent full charge level. There are limits on the amount of current the battery and/or wiring can take.
• Absorption Charge: In the second stage, voltage peaks and stabilizes and current begins to taper off as internal resistance rises. The charge controller puts out maximum voltage at this stage.
• Float Charge: This can also be referred to as trickle charging or a maintenance charge. In this stage, voltage is reduced to lower levels in order to reduce gassing and prolong battery life. The main purpose of this stage is basically to maintain the battery’s charge in a controlled manner. In Pulse Width Modulation (PWM) the charger sends small, short charging cycles or “pulses” when it senses small drops in voltage.

#### Depth of Discharge (DOD)

The Depth of Discharge (DOD) is used to describe how deeply the battery is discharged. If the battery is 100% fully charged, it means the DOD of this battery is 0%. If the battery has delivered 30% of its energy, here are 70% energy reserved, the DOD of this battery is 30%. And if a battery is 100% empty, the DOD of this battery is 100%. DOD always can be treated as how much energy that the battery delivered.

### Determining battery state of charge

There are a few ways to determine the state of charge on a battery, each with their own level of accuracy. As there is no direct method to measure a battery’s state of charge, there are numerous ways to go about it. One way to gauge a battery is by measuring its static voltage and comparing it to a standardized chart. This is the least accurate method, but it only involves an inexpensive digital meter. Another method of gauging the battery involves measuring the density or specific gravity of the sulfuric acid electrolyte. This is the most accurate test, yet it is only applicable to the flooded types. This method involves measuring the cell’s electrolyte density with a battery hydrometer. Electrolyte density is lower when the battery is discharged and higher as the cells are charged. The battery’s chemical reactions affect the density of the electrolyte at a constant rate that is predictable enough to get a good indication of the cell’s state of charge. Using an amp-hour meter one can also obtain an accurate indication of the battery’s state of charge. Amp-hour meters keep track of all power moving in and out of the battery by time, and the state of charge is determined by comparing flow rates.

### Amp-Hour rating & Capacity

All deep cycle batteries are classified and rated in amp-hours. Amp-hours is the term used to describe a standardized rate of discharge measuring current relative to time. It is calculated by multiplying amps and hours. The generally accepted rating time period for most manufacturers is 20 hours. This means that the battery will provide the rated amperage for about 20 hours until it is down to 10.5 volts or completely dead. Some battery manufacturers will use 100 hours as the standard to make them look better, yet it can be useful in long-term backup calculations.

### Renewable Applications

There are three main types of batteries that are commonly used in renewable energy systems, each with their own advantages and disadvantages. Flooded or “wet” batteries are the most cost efficient and the most widely used batteries in photovoltaic applications. They require regular maintenance and need to be used in a vented location, and are extremely well suited for renewable energy applications. Sealed batteries come in two varieties, the gel cell and Absorbed Glass Mat (AGM) type. The gel cell uses a silica additive in its electrolyte solution that causes it to stiffen or gel, eliminating some of the issues with venting and spillage. The Absorbed Glass Mat construction method suspends the electrolyte in close proximity with the plate’s active material. These batteries are sealed, requiring virtually no maintenance. They are more suitable for remote applications where regular maintenance is difficult, or enclosed locations where venting is an issue.

• a) Flooded Lead Acid (FLA)

Flooded lead acid batteries are the most commonly used batteries, and have the longest track record in solar electric systems. They usually have the longest life and the lowest cost per amp-hour of any of the other choices. The downside is that they do require regular maintenance in the form of watering, equalizing charges and keeping the terminals clean. These cells are often referred to as “wet” cells, and they come in two varieties: the serviceable, and the maintenance-free type (which means they are designed to die as soon as the warranty runs out). The serviceable wet cells come with removable caps, and are the smarter choice, as they allow you to check their status with a hydrometer.

• b) Gelled Electrolyte Sealed Lead Acid (GEL)

Gel sealed batteries use silica to stiffen or “gel” the electrolyte solution, greatly reducing the gasses, and volatility of the cell. Since all matter expands and contracts with heat, batteries are not truly sealed, but are "valve regulated". This means that a tiny valve maintains slight positive pressure. AGM batteries are slowly phasing out gel technology, but there still are many applications for the gel cells. The recharge voltage for charging Gel cells are usually lower than the other styles of lead acid batteries, and should be charged at a slower rate. When they are charged too fast, gas pockets will form on the plates and force the gelled electrolyte away from the plate, decreasing the capacity until the gas finds its way to the top of the battery and recombines with the electrolyte.

• c) Sealed Absorbed Glass Mat (AGM)

Absorbed Glass Mat (AGM) is a class of valve-regulated lead acid battery (VLRA) in which the electrolyte is held in glass mats as opposed to freely flooding the plates. This is achieved by weaving very thin glass fibers into a mat to increase surface area enough to hold sufficient electrolyte for the lifetime of the cell. The advantages to using the AGM batteries are many, yet these batteries are typically twice the cost of their flooded-cell counterpart. On the plus side, these cells can hold roughly 1.5 times the amp hour capacity of a similar size flooded battery due to their higher power density. Another factor that improves their efficiency is the higher lead purity used in AGM cells. Because of their sandwich construction, each plate no longer has to support its own weight. Their low internal resistance allows them to be charged and discharged much faster than other types. AGM cells function well in colder temperatures and are highly resistant to vibration. There are many advantages to using the AGM cells over their flooded counterpart that are beyond the scope of this article.

#### Maintenance & Monitoring

Proper maintenance and monitoring will greatly extend the life of your batteries. Flooded batteries need to be checked regularly to make sure electrolyte levels are full. The chemical reaction releases gases, as water molecules are split into hydrogen and oxygen. This, in turn, consumes water and creates the need to replace it regularly. Only distilled water should ever be used in batteries, and you should never add any kind of acid solution. The connections from battery to battery and to the charging and load circuits should always be kept clean and free of corrosion. Corrosion is created upon charging, when a slight acid mist forms as the electrolyte bubbles. Corrosion buildup will create a good deal of electrical resistance, eventually contributing to a shortened battery life and malfunctions. A good way to keep up on the terminals is to regularly clean them with a baking soda solution

#### Future Trends

Companies world-wide are quickly adjusting to the increased global market for solar systems by developing batteries that are better suited for photovoltaic systems. At some distant point in the future, it is likely that lead-acid batteries will become extinct, as newer technologies in lithium ion and Nickel metal hydride continue to evolve. Because lead-acid batteries are under the hood of virtually every car, advancements in lead-acid technology, however are still being made. New developments in lead-acid technology usually originate in the auto industry. Efficiency ratings are constantly going up, as new sensors and improved materials are helping batteries achieve longer lifespan.

## 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:

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

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 (%)
$k_{e} \,$ is a system efficiency (%)
$k_{dod} \,$ 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).
• System efficiency is an allowance for losses in the battery (e.g. coloumbic efficiency) and power electronics (e.g. inverter and/or charger).

## 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.

### Android App

If you have an android phone then we suggest using our app "Battery Sizing Tool".

## 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.