# AC UPS Sizing

## Contents

## Introduction

This calculation deals with the sizing of an AC uninterruptible power supply (UPS) system (i.e. rectifier, battery bank and inverter). In this calculation, it is assumed that the AC UPS is a double conversion type with a basic system topology as shown in Figure 1.

An external maintenance bypass switch and galvanic isolation transformers are other common additions to the basic topology, but these have been omitted from the system as they are irrelevant for the sizing calculation.

### Why do the calculation?

An AC UPS system is used to support critical / sensitive AC loads. It is typically a battery-backed system which will continue to operate for a specified amount of time (called the autonomy) after a main power supply interruption. AC UPS systems are also used as stable power supplies that provide a reasonably constant voltage and frequency output, independent of voltage input. This is particularly useful for sensitive electrical equipment on main power supplies that are prone to voltage / frequency fluctuations or instability.

The AC UPS sizing calculation determines the ratings for the main AC UPS system components: 1) rectifier, 2) battery banks and 3) inverter.

In some cases, the manufacturer will independently size the system and it is only necessary to construct the AC UPS load schedule and load profile. However the calculation results will also help determine the indicative dimensions of the equipment (e.g. size of battery banks) for preliminary layout purposes.

### When to do the calculation?

The AC UPS sizing calculation can be done when the following prerequisite information is known:

- UPS loads that need to be supported
- Input / Output AC voltage
- Autonomy time(s)
- Battery type

## Calculation Methodology

The calculation procedure has four main steps:

- 1) Determine and collect the prospective AC UPS loads
- 2) Construct a load profile and determine the UPS design load (VA) and design energy (VAh)
- 3) Calculate the size of the stationary battery (number of cells in series and Ah capacity)
- 4) Determine the size of the inverter, rectifier/ charger and static switch

### Step 1: Collect the AC UPS Loads

The first step is to determine the type and quantity of loads that the AC UPS system will be expected to support. For industrial facilities, this will typically be critical instrumentation and control loads such as the DCS and ESD processor and marshalling hardware, critical workstations and HMI's, telecommunications equipment and sensitive electronics. The necessary load data should be available from the instrumentation and control engineers.

For commercial facilities, UPS loads will mainly be server, data / network and telecommunications hardware.

### Step 2: Load Profile, Design Load and Design Energy

Refer to the Load Profile Calculation for details on how to construct a load profile, calculate the design load ([math]S_{d} \,[/math]) and design energy ([math]E_{d} \,[/math]). The "Autonomy method" for constructing load profiles is typically used for AC UPS systems.

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. Sometimes a single autonomy time is used for the entire AC UPS load, which obviously makes the construction of the load profile easier to compute.

### Step 3: Battery Sizing

Refer to the Battery Sizing Calculation for details on how to size the battery for the AC UPS system. The following sections provide additional information specific to battery sizing for AC UPS applications.

#### Nominal Battery (or DC Link) Voltage

The nominal battery / DC link voltage is often selected by the AC UPS manufacturer. However, if required to be selected, the following factors need to be considered:

- DC output voltage range of the rectifier – the rectifier must be able to output the specified DC link voltage
- DC input voltage range of the inverter – the DC link voltage must be within the input voltage tolerances of the inverter. Note that the battery end of discharge voltage should be within these tolerances.
- Number of battery cells required in series – this will affect the overall dimensions and size of the battery rack. If physical space is a constraint, then less batteries in series would be preferable.
- Total DC link current (at full load) – this will affect the sizing of the DC cables and inter-cell battery links. Obviously the smaller the better.

In general, the DC link voltage is usually selected to be close to the nominal output voltage.

#### Number of Cells in Series

The number of battery cells required to be connected in series must be between the two following limits:

(1) [math] N_{max} = \frac{V_{dc} (1+V_{i,max})}{V_{f}} \, [/math]

(2) [math] N_{min} = \frac{V_{dc} (1-V_{i,min})}{V_{eod}} \, [/math]

where [math]N_{max}[/math] is the maximum number of battery cells

- [math]N_{min}[/math] is the minimum number of battery cells
- [math]V_{dc}[/math] is the nominal battery / DC link voltage (Vdc)
- [math]V_{i,max}[/math] is the inverter maximum input voltage tolerance (%)
- [math]V_{i,min}[/math] is the inverter minimum input voltage tolerance (%)
- [math]V_{f}[/math] is the nominal cell float (or boost) voltage (Vdc)
- [math]V_{eod}[/math] is the cell end of discharge voltage (Vdc)

The limits are based on the input voltage tolerance of the inverter. As a maximum, the battery at float voltage (or boost if applicable) needs to be within the maximum input voltage range of the inverter. Likewise as a minimum, the battery at its end of discharge voltage must be within the minimum input voltage range of the inverter.

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 appropriate).

### Step 4: UPS Sizing

#### Overall UPS Sizing

Most of the time, all you need to provide is the overall UPS kVA rating and the UPS vendor will do the rest. Given the design load ([math]E_{d} \,[/math] in VA or kVA) calculated in Step 2, select an overall UPS rating that exceeds the design load. Vendors typically have standard UPS ratings, so it is possible to simply select the first standard rating that exceeds the design load. For example, if the design load 12kVA, then the next size unit (e.g. 15kVA UPS) would be selected.

#### Rectifier / Charger Sizing

The rectifier / charger should be sized to supply the inverter at full load and also charge the batteries (at the maximum charge current). The design DC load current can be calculated by:

- [math] I_{L,dc} = \frac{S}{V_{dc}} \, [/math]

where [math]I_{L,dc}[/math] is the design DC load current (full load) (A)

- [math]S[/math] is the selected UPS rating (kVA)
- [math]V_{dc}[/math] is the nominal battery / DC link voltage (Vdc)

The maximum battery charging current can be computed as follows:

- [math] I_{c} = \frac{C k_{l}}{t_{c}} \, [/math]

where [math]I_{c}[/math] is the maximum DC charge current (A)

- [math]C[/math] is the selected battery capacity (Ah)
- [math]k_{l}[/math] is the battery recharge efficiency / loss factor (typically 1.1) (pu)
- [math]t_{c}[/math] is the minimum battery recharge time (hours)

The total minimum DC rectifier / charger current is therefore:

- [math] I_{dc} = I_{L,dc} + I_{c} \, [/math]

Select the next standard rectifier / charger rating that exceeds the total minimum DC current above.

#### Inverter Sizing

The inverter must be rated to continuously supply the UPS loads. Therefore, the inverter can be sized using the design AC load current (based on the selected UPS kVA rating).

For a three-phase UPS:

- [math] I_{L} = \frac{S}{\sqrt{3} V_{o}} \, [/math]

For a single-phase UPS:

- [math] I_{L} = \frac{S}{V_{o}} \, [/math]

where [math]I_{L}[/math] is the design AC load current (full load) (A)

- [math]S[/math] is the selected UPS rating (kVA)
- [math]V_{o}[/math] is the nominal output voltage (line-to-line voltage for a three phase UPS) (Vac)

Select the next standard inverter rating that exceeds the design AC load current.

#### Static Switch Sizing

Like the inverter, the static switch must be rated to continuously supply the UPS loads. Therefore, the static switch can be sized using the design AC load current (as above for the inverter sizing).

## Worked Example

### Step 1 and 2: Collect the AC UPS Loads and Construct Load Profile

For this example, we shall use the same loads and load profile detailed in the Energy Load Profile Calculation example. The load profile is shown in the figure right and the following quantities were calculated:

- Design load [math]S_{d}[/math] = 768 VA
- Design energy demand [math]E_{d}[/math] = 3,216 VAh

### Step 3: Battery Sizing

For this example, we shall use the same battery sizes calculated in the Battery Sizing Calculation worked example. The selected number of cells in series is 62 cells and the minimum battery capacity is 44.4 Ah. A battery capacity of 50 Ah is selected.

### Step 4: UPS Sizing

#### Overall Sizing

Given the design load of 768 VA, then a 1 kVA UPS would be appropriate.

#### Rectifier Sizing

Given a nominal dc link voltage of 120Vdc, the design DC load current is:

- [math] I_{L,dc} = \frac{S}{V_{dc}} \, [/math]
- [math] = \frac{1,000}{120} = 8.33 \, [/math] A

Suppose the minimum battery recharge time is 2 hours and a recharge efficiency factor of 1.1 is used. The maximum battery charging current is:

- [math] I_{c} = \frac{C k_{l}}{t_{c}} \, [/math]
- [math] = \frac{50 \times 1.1}{2} = 27.5 \, [/math] A

Therefore the total minimum DC rectifier / charger current is:

- [math] I_{dc} = I_{L,dc} + I_{c} \, [/math]
- [math] = 8.33 + 27.5 = 35.83 \, [/math] A

A DC rectifier rating of 40A is selected.

#### Inverter and Static Switch Sizing

Suppose the nominal output voltage is 240Vac. The design AC load current is:

- [math] I_{L} = \frac{S}{\sqrt{3} V_{o}} \, [/math]
- [math] = \frac{1,000}{\sqrt{3} \times 240} = 2.406 \, [/math] A

An inverter and static switch rating of 5A is selected.

## Computer Software

Preliminary sizing is normally done manually. Notwithstanding this, many AC UPS manufacturers provide sizing tools as part of their service package (for example, see the APC online UPS selector tool).

## What next?

Using the results of the UPS sizing calculation, the approximate dimensions of the batteries and UPS cabinet can be estimated based on typical vendor information. This will assist in developing the equipment / room layouts. Preliminary budget pricing can also be estimated based on the calculation results.