**Over the years, engineers have developed many definitions for measuring power with distorted voltages and currents. However, none of these definitions successfully characterize all of the distinguishing power components necessary for billing purposes, nor do they explain how to compensate for components that are not useful. To deal with this problem, the Institute of Electrical and Electronics Engineers (IEEE) organized a task force, chaired by Dr. Alex Emmanuel of Worcester Polytechnic Institute. The task force developed Standard 1459-2000, the “IEEE Trial Use Standard for Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions.” This standard is the subject of this month's column.**

In the power business, nothing is more basic and important than measuring delivered energy for billing purposes. In addition to active (or real) delivered power, industry personnel often measure reactive power and/or total apparent power (definitions to follow). This allows them to factor in the additional system capacity required to supply a load's total power requirements.

Conventional induction disk watt-hour meters provide accurate real-power measurements. Engineers can use the same meters to measure reactive power by shifting the voltage 90 deg. Recently, engineers have developed electronic meters that accurately measure voltages and currents and calculate different power and energy quantities of interest. They also provide additional capabilities, including flexible demand periods, time-of-use rate calculations, and power quality measurements.

All of this works great for the assumed conditions of balanced systems and sinusoidal voltages and currents. The definitions and formulas used today for active, reactive, and apparent power have been used since the 1940s. When these definitions were developed, loads were dominated by motors, lighting, and other linear loads. This allowed the calculations to be simplified by taking advantage of the nondistorted voltages and currents and relatively balanced systems.

But what happens when harmonic components distort voltages and currents? This is an important question, considering the increasing applications of electronic equipment in all customer categories. Power electronic devices such as computer power supplies, adjustable-speed motor drives, phase-controlled rectifiers, thyristor-controlled loads, fluorescent lighting with electronic ballasts (including compact fluorescents) are everywhere. These nonlinear, harmonic-producing loads result in distorted voltages and currents that change the characteristics of the power delivered and the system capacity required to supply it.

### Sinusoidal Conditions

Let's begin by reviewing the definitions for balanced and sinusoidal conditions. In these cases, we can use single-phase measurements for single-phase and 3-phase systems.

The instantaneous power (*p*) is defined as the instantaneous voltage (*v*) multiplied by the instantaneous current (*i*). Now divide this power into two components — active and reactive.

**Active power (P)**, measured in watts, is the integral of the instantaneous power signal over some integral number of cycles:

Where: T=1/*f* = the period of one cycle in seconds, *k* is an integer number, and τ is the time at which the measurement starts.

This is the power component that utilities normally charge for and the component measured by a watt-hour meter. It results from the component of the current that is inphase with the voltage. In the case of a sinusoidal system, the active power is given by:

Where: *V* is the rms voltage magnitude, *I* is the rms current magnitude, and *θ* is the angle between the voltage and the current.

**Reactive power (Q)**, measured in vars, is the power associated with the component of the current that is out-of-phase (in quadrature) with the voltage. This type of energy oscillates between the source and inductances, capacitances, and moving masses that pertain to electromechanical systems. The average value of this rate of flow is zero, and the net transfer of energy to the load is zero. It is obtained in a similar manner to active power, resulting in the following definition:

Note that reactive power is positive if the load is inductive and negative if the load is capacitive.

**Apparent power (S)** is the product of the rms voltage and current. It's measured in volt-amperes (VA):

Apparent power is important because it represents the total capacity that must be available to supply power to the load — even though only a portion of this is useful power. Some utilities use a demand charge based on the total apparent power to account for this system capacity requirement.

**Power factor** is defined as the ratio of active power (*P*) to total apparent power (*S*). This ratio represents the portion of the delivered power that is useful for performing work:

In sinusoidal situations, power factor is equal to the cosine of the angle between the voltage and current (cosθ).

Active, reactive, and apparent power form the power triangle shown in **Fig. 1**, on page 8.

### Distorted Voltages and Currents

What happens when the voltage and current are not pure sinusoidal waveforms? In this case, it's useful to separate the voltage or current signals into two distinctive components — the power system fundamental frequency components *v _{1}* and

*i*, and the remaining terms

_{1}*v*and

_{h}*i*, which include all of the distortion components (both integer and noninteger harmonics). Consequently, the rms values of these components are related as follows:

_{h}These definitions provide the basis for the traditional measurements of total harmonic distortion (THD) in voltages and currents:

With harmonic distortion included, the power triangle that defines the power factor and the individual power components doesn't work. Some new definitions are needed. Rather than reviewing the whole history of definitions for power components that include harmonic distortion, we'll focus on the definitions developed for Standard 1459-2000 and the basis for this set of definitions.

The philosophy behind Standard 1459 involves defining a number of different components that could be useful in defining the responsibility and the costs associated with supplying power. These components can then be combined as necessary for particular situations. Here are the important components:

**Active power (P)** is divided into a fundamental component and a harmonic component. This is important because, typically, only the fundamental component benefits the load.

For example, motors produce no useful work from the power at harmonic frequency — they only cause additional heating in the motors. In most practical situations, the harmonic component of the active power is quite small.

Where:

Electronic meters can separate these two components if utility personnel want to base electricity charges on only the fundamental active power, rather than the total active power. (Currently, most electronic meters would not make this distinction and would try to accurately calculate the total active power). Conventional induction disk watt-hour meters can't make the distinction between the fundamental active power and the total active power. However, the responses of conventional watt-hour meters provide some attenuation at harmonic frequencies, which makes the meters focus primarily on the fundamental component of the active power.

It's possible to divide **reactive power (Q)** into two components in a similar manner. However, it's more useful to use a different breakdown of the nonactive power components because of distortion power components. The main purpose for defining a component called reactive power is to develop procedures and equipment for controlling the voltages and losses in a system at the fundamental frequency. Only fundamental frequency reactive power is important for these objectives.

For instance, you can size a capacitor bank to correct for fundamental frequency reactive power, but this capacitor bank would not compensate (and could magnify) nonfundamental frequency components.

The definition for total **apparent power (S)** remains the same — the product of the rms voltage and rms current (which now both include harmonic components). However, it's a good idea to separate the apparent power into fundamental and nonfundamental frequency apparent power. The fundamental frequency apparent power (*S*1) and its components (*P*1 and *Q*1) define the rate of flow of electromagnetic field energy associated with 50Hz and 60Hz voltages and currents.

These components can be used for power system design and evaluation in traditional manners.

The total apparent power will then consist of a number of additional components when the harmonic effects are taken into account. The expansion of rms voltages and currents into fundamental and harmonic terms is used to resolve the apparent power into its different components as follows:

Where: *S _{N}* is the nonfundamental frequency apparent power:

### Distortion Power Components

The breakdown of the total apparent power helps determine three different distortion power components that make up the nonfundamental frequency apparent power (*S _{N}*).

**Current distortion power (vars).** The harmonic distortion in the current interacting with the fundamental frequency component of the voltage causes this component. It's usually the largest of the distortion power components. To develop penalties for harmonic current injection into the power system, you might use this component as the basis for these calculations.

**Voltage distortion power (vars)**. This component is caused by the harmonic distortion in the voltage interacting with the fundamental frequency current. It's usually not as significant because voltage distortion tends to be smaller than current distortion. Also, because many people consider voltage quality to be the responsibility of the utilities, it may not be wise to make this the basis of charges for nonfundamental frequency apparent power.

**Harmonic apparent power (VA).** This is usually the smallest of the nonfundamental frequency apparent power components. It is defined as follows:

Sometimes it may be useful to divide harmonic apparent power into two separate components, one of which we have already defined above (*P _{H}*).

The other term, *D _{H}*, can be called harmonic distortion power (vars).

**Power factor**. The definitions for power factor do not change. Power factor is a measure of how efficiently the active power is being supplied. However, only the fundamental frequency power factor is useful for designing traditional power factor correction solutions that help improve voltage profiles and losses due to fundamental frequency reactive power flows on the system. This is sometimes referred to as displacement power factor. We will use the symbol *PF*_{1} for this quantity. Here are the power factor definitions:

There are simplified calculations for true power factor based on realistic distortion levels in voltages and currents, but we will not address those in this article.

**Table 1**, on page 12, summarizes the different definitions presented and groups them according to their use and whether they are associated with fundamental frequency or harmonic components. These are the definitions proposed by meter (and other monitoring equipment) manufacturers.

Standard 1459 provides a simple example that illustrates these calculations in a practical situation. The example is based on supplying a thyristor-controlled load, such as a dimmer switch controlling a light. The voltage and current waveforms at the metering point are shown in **Fig. 2**.

The important quantities needed to calculate the different power components are as follows:

The fundamental frequency apparent power (*S*_{1}) for this example is 1229.7VA. **Table 2**, on page 12, summarizes the different power components in percentages of this base value. The true power factor is 0.58 and the displacement power factor is 0.696. Note that an electronic power supply load might have a similar true power factor but could have a displacement power factor close to unity. Also note that the nonfundamental frequency apparent power (684VA) is a significant component of the total apparent power in this example. This will be true in any case where current distortion is significant. To correct for this component, you would have to use some type of harmonic control rather than traditional power factor correction.

### Conclusion

You may have noticed that we did not address unbalanced conditions in this article. This is because unbalanced conditions make situations even more complicated. You may want to investigate this further by obtaining the standard from the IEEE (www.ieee.org).

The definitions proposed in Standard 1459-2000 provide a blueprint for meter manufacturers and other monitoring equipment manufacturers to implement accurate power measurements in environments with significant distortion using a standardized approach. The standard breaks the power measurements into components that can characterize three things: the useful real power delivered to the load, the reactive power that can be compensated with conventional power factor correction, and the power components that require other methods of compensation (such as active filters). If we are going to charge for the system capacity required to supply these harmonic components, we must agree on standard definitions to measure them.

*Mark McGranaghan directs power quality projects and product development at Electrotek Concepts in Knoxville, Tenn. You can reach him at* mark@electrotek.com.

*Erich Gunther is responsible for technology development at Electrotek Concepts in Knoxville, Tenn. He is also the chief architect for the Dranetz-BMI Signature System ^{™}. You can reach him at* erich@electrotek.com.

### Acknowledgment

*The authors wish to thank Alex Emmanuel for his significant contributions to the article*.