Harmonic studies play an important role in characterizing and understanding the extent of harmonic problems.

Harmonic studies are often performed when:

1. Finding a solution to an existing harmonic problem

2. Installing large capacitor banks on utility distribution systems or industrial power systems


3. Installing large nonlinear devices or loads

4. Designing a harmonic filter

5. Converting a power factor capacitor bank to a harmonic filter.

Harmonic studies provide a means to evaluate various possible solutions and their effectiveness under a wide range of conditions before implementing a final solution. In this article, methods for carrying out harmonic studies are presented.

Harmonic Study Procedure:


The ideal procedure for performing a power systems harmonics study can be summarized as follows:

1. Determine the objectives of the study. This is important to keep the investigation on track. For example, the objective might be to identify what is causing an existing problem and solve it. Another might be to determine if a new plant expansion containing equipment such as adjustable-speed drives and capacitors is likely to have problems.

2. If the system is complex, make a pre-measurement computer simulation based on the best information available. Measurements are expensive in terms of labor, equipment, and possible disruption to plant operations. It will generally be economical to have a good idea what to look for and where to look before beginning the measurements.

3. Make measurements of the existing harmonic conditions, characterizing sources of harmonic currents and system bus voltage distortion.


4. Calibrate the computer model using the measurements.

5. Study the new circuit condition or existing problem.

6. Develop solutions (filter, etc.) and investigate possible adverse system interactions. Also, check the sensitivity of the results to important variables.

7. After the installation of proposed solutions, perform monitoring to verily the correct operation of the system. Admittedly, it is not always possible to perform each of these steps ideally.


The most often omitted steps are one, or both, measurement steps due to the cost of engineering time, travel, and equipment charges. An experienced analyst may be able to solve a problem without measurements, but it is strongly recommended that the initial measurements be made if at all possible because there are many unpleasant surprises lurking in the shadows of harmonics analysis.

Developing a System Model for Harmonic Study:

There are two fundamental issues that need to be considered in developing a system model for harmonic simulation studies. The first issue is the extent of the system model to be included in the simulation. Secondly, one must decide whether the model should be represented as a single- phase equivalent or a full three-phase model. As an example of model extent, suppose a utility plans to install a large capacitor bank on a distribution feeder and would like to evaluate the frequency response associated with the bank. Representing the entire distribution system is usually not practical because it would be time-consuming to develop the model and it would strain computational resources to run simulations.

One approach would be to start developing a model one or two buses back from the bus of interest and include everything in between. Another approach would be to start with a small simple circuit that accurately represents the phenomena and add more of the system details to determine the impact on the solution result.

At the point when adding more system details does not change the analysis results, the physical system is sufficiently represented by the simulation model. In modelling distribution systems for harmonic studies, it is usually sufficient to represent the upstream transmission system with a short circuit equivalent at the high-voltage side of the substation transformer.


The leakage impedance of the transformer dominates the short-circuit equivalent and effectively isolates the transmission and distribution for many studies. However, if there is a capacitor bank near the high-voltage side of the transformer, part of the transmission system must be modelled to include the capacitor bank.

The combination of the transformer and the capacitor bank may behave as a filter for some frequency as seen from the low-voltage side of the transformer. Distribution system components downstream from the substation transformer (or at the low-voltage side) such as feeder lines, capacitor banks, key service transformers, and end-user capacitor banks must be represented. Since the feeder capacitor banks dominate the system capacitance, it is usually acceptable to neglect capacitance from overhead feeder lines. However, if there is a significant amount of UD cable, cable capacitance should be represented, especially if the study is concerned with higher-order harmonics.

The analyst must then decide if the model should be represented as a complete three-phase model or a single-phase equivalent. A single-phase equivalent model is generally simpler and less complicated to develop compared to a three-phase model. However, it is often inadequate to analyze unbalanced phenomena or systems with numerous single-phase loads. Fortunately, there is a rule that permits the simplified positive sequence modelling for many three- phase industrial loads.

Determining the response of the system to positive-sequence harmonics is straightforward since both utility and industrial power engineers are accustomed to doing such modelling in their load flow and voltage drop analyses. The rule may be simply stated- When there is a delta winding in a transformer anywhere in series with the harmonic source and the power system, only the positive-sequence circuit need be represented to determine the system response. It is impossible for zero- sequence harmonics to be present; they are blocked.

Figure 7.3 illustrates this principle, showing what models apply to different parts of the system. Both the positive- and negative-sequence networks are generally assumed to have the same response to harmonics. Sometimes measurements will show triplen harmonics in the upstream from a delta winding. One normally assumes these harmonics are zero sequence. They may be, depending on what other sources are in system.

However, they can also be due to unbalanced harmonic sources, one example of which would be an arc furnace. Only the triplens that are in phase are zero sequence and are blocked by the delta winding. Therefore, it is common to include triplen harmonics when performing analysis using a positive-sequence model.

The symmetrical component technique fails to yield an advantage when analyzing four-wire utility distribution feeders with numerous single-phase loads. Both the positive- and zero sequence networks come into play. It is generally impractical to consider analyzing the system, manually, and most computer programs capable of accurately modelling these systems simply set up the coupled three-phase equations and solve them directly.

Fortunately, some computer tools now make it almost as easy to develop a three-phase model as to make a single-phase equivalent. It takes no more time to solve the complete three-phase model than to solve the sequence networks because they would have to be coupled also. Not only does the symmetrical component technique fail to yield an advantage in this case, but analysts often make errors and inadvertently violate the assumptions of the method.

It is not generally recommended that harmonic analysis of unbalanced circuits be done using symmetrical components. It should be attempted only by those who are absolutely certain of their understanding of the method and its assumptions.

Modelling Harmonic Sources:

Most harmonic flow analysis on power systems is performed using steady-state, linear circuit solution techniques. Harmonic sources, which are nonlinear elements, are generally considered to be injection sources into the linear network models. They can be represented as current injection sources or voltage sources. For most harmonic flow studies, it is suitable to treat harmonic sources as simple sources of harmonic currents.

This is illustrated in Fig. 7.4 where an electronic power converter is replaced with a current source in the equivalent circuit. The voltage distortion at the service bus is generally relatively low, less than 5 percent. Therefore, the current distortion for many nonlinear devices is relatively constant and independent of distortion in the supply system.

Values of injected current should be determined by measurement. In the absence of measurements and published data, it is common to assume that the harmonic content is inversely proportional to the harmonic number. That is, the fifth-harmonic current is one-fifth, or 20 percent, of the fundamental, etc. This is derived from the Fourier series for a square wave, which is at the foundation of many nonlinear devices.

However, it does not apply very well to the newer technology PWM drives and switch-mode power supplies, which have a much higher harmonic content. When the system is near resonance, a simple current source model will give an excessively high prediction of voltage distortion. The model tries to inject a constant current into a high impedance, which is not a valid representation of reality.

The harmonic current will not remain constant at a high voltage distortion. Often, this is inconsequential because the most important thing is to know that the system cannot be successfully operated in resonance, which is readily observable from the simple model. Once the resonance is eliminated by, for example, adding a filter, the model will give a realistic answer.

For the cases where a more accurate answer is required during resonant conditions, a more sophisticated model must be used. For many power system devices, a Thevenin or Norton equivalent is adequate.

The additional impedance moderates the response of the parallel resonant circuit. A Thevenin equivalent is obtained in a straightforward manner for many nonlinear loads. For example, an arc furnace is well represented by a square-wave voltage of peak magnitude approximately 50 percent of the nominal ac system voltage. The series impedance is simply the short-circuit impedance of the furnace transformer and leads (the lead impedance is the larger of the two).

Unfortunately, it is difficult to determine clear-cut equivalent impedances for many nonlinear devices. In these cases, a detailed simulation of the internals of the harmonic-producing load is necessary. This can be done with computer programs that iterate on the solution or through detailed time- domain analysis.

Fortunately, it is seldom essential to obtain such great accuracy during resonant conditions and analysis do not often have to take these measures. However, modelling arcing devices with a Thevenin model is recommended regardless of need.

Computer Tools for Harmonics Analysis:

The characteristics of such programs and the heritage of some popular analysis tools are described here.

First, it should be noted that one circuit appears frequently in simple industrial systems that does lend itself to manual calculations (Fig. 7.6). It is basically a one-bus circuit with one capacitor.

Two things may be done relatively easily:

1. Determine the resonant frequency. If the resonant frequency is near a potentially damaging harmonic, either the capacitor must be changed or a filter designed.

2. Determine an estimate of the voltage distortion due to the current.

Given that the resonant frequency is not near a significant harmonic and that projected voltage distortion is low, the application will probably operate successfully. Unfortunately, not all practical cases can be represented with such a simple circuit. In fact, adding just one more bus with a capacitor to the simple circuit in Fig. 7.6 makes the problem a real challenge to even the most skilled analysts. However, a computer can perform the chore in milliseconds. To use the computer tools commonly available, the analyst must describe the circuit configuration, loads, and the sources to the program.

Data that must be collected include:

i. Line and transformer impedances

ii. Transformer connections

iii. Capacitor values and locations (critical)

iv. Harmonic spectra for nonlinear loads

v. Power source voltages.

These values are entered into the program, which automatically adjusts impedances for frequency and computes the harmonic flow throughout the system.

Capabilities for Harmonics Analysis Programs:

Acceptable computer software for harmonics analysis of power systems should have the following characteristics:

1. It should be capable of handling large networks of at least several hundred nodes!

2. It should be capable of handling multiphase models of arbitrary structure. Not all circuits, particularly those on utility distribution feeders, are amenable to accurate solution by balanced, positive sequence models.

3. It should also be capable of modelling systems with positive sequence models. When there can be no zero-sequence harmonics, there’s no need to build a full three-phase model.

4. It should be able to perform a frequency scan at small intervals of frequency (e.g., 10 Hz) to develop the system frequency-response characteristics necessary to identify resonances.

5. It should be able to perform simultaneous solution of numerous harmonic sources to estimate the actual current and voltage distortion.

6. It should have built-in models of common harmonic sources.

7. It should allow both current source and voltage source models of harmonic sources.

8. It should be able to automatically adjust phase angles of the sources based on the fundamental frequency phase angles.

9. It should be able to model any transformer connection.

10. It should be able to display the results in a meaningful and user-friendly manner.

Harmonic Analysis by Computer Historical Perspective:

The most common type of computer analysis of power systems performed today is some form of power flow calculation. Most power engineers have some experience with this class of tool. Other common computer tools include short-circuit programs and, at least for transmission systems, dynamics (transient stability) programs. Harmonics and electromagnetic transients tools have traditionally been in the domain of specialists due to the modelling complexities.

While power flow tools are familiar, their formulation is generally unsuitable for harmonics analysis. Of the tools in common usage, the circuit model in short-circuit programs is closer to what is needed for harmonic flow analysis in networks. In fact, prior to the advent of special power systems harmonic analysis tools, many analysis would use short-circuit programs to compute harmonic distortion, manually adjusting the impedances for frequency.

This is an interesting learning experience for the student, but not one that the practitioner will want to repeat often. Of course, one could also perform the analysis in the time domain using electromagnetic transients programs, but this generally is more time-consuming and is excessive for most problems. Today, most power system harmonics analysis is performed in the sinusoidal steady state using computer programs specially developed for the purpose.

It is encouraging to see many vendors of power system analysis software providing some harmonics analysis capabilities in their packages, although the main application in the package may be a power flow program. It is useful to see how this has evolved. Unlike power flow algorithms, few of the developers have written technical papers documenting their efforts. Therefore, it is difficult to trace the history of harmonics analysis in power systems through the literature.