This book covers instantaneous power theory as well as the importance of design of shunt, series, and combined shunt-series power active filters and hybrid passive-active power filters

* Illustrates pioneering applications of the p-q theory to power conditioning, which highlights distinct differences from conventional theories

* Explores p-q-r theory to give a new method of analyzing the different powers in a three-phase circuit

* Provides exercises at the end of many chapters that are unique to the second edition

Hirofumi Akagi is a Professor in the department of electrical and electronic engineering at the Tokyo Institute of Technology. His research interest includes power conversion systems and its applications to industry, transportation, and utility. He has authored and coauthored some 120 IEEE Transactions papers and two invited papers published in Proceedings of the IEEE in 2001 and 2005. He was elected as an IEEE Fellow in 1996, a Distinguished Lecturer of the IEEE Power Electronics and Industry Applications Societies for 1998-1999. He received six IEEE Transactions prize paper awards, and 15 IEEE Industry Applications Society Committee prize paper awards. He is the recipient of the 2001 IEEE Power Electronics William E. Newell Award, the 2004 IEEE Industry Applications Society Outstanding Achievement Award, the 2008 IEEE Richard H. Kaufmann Technical Field Award, and the 2012 IEEE Power & Energy Society Nari Hingorani Custom Power Award. Dr. Akagi served as the President of the IEEE Power Electronics Society for 2007-2008. Since January 2015, he has been serving as the IEEE Division II Director.

Edson Hirokazu Watanabe is a Professor at COPPE/Federal University of Rio de Janeiro, where he teaches Power Electronics. His main fields of interests are converters analysis, modeling and design, active filters and FACTS technologies. Dr. Watanabe has more than 50 journal papers and more than 200 conference papers. He is a member of the IEE-Japan, The Brazilian Society for Automatic Control, The Brazilian Power Electronics Society, CIGRE and Power Engineering, Industry Applications and Power Electronics Societies of IEEE. In 2005, he was admitted to the National Order of Scientific Merit, Brazil. In 2013, he received the IEEE Power & Energy Society Nari Hingorani FACTS Award and became member of the National (Brazil) Academy of Engineering and in 2015 he was elected a member of the Brazilian Academy of Sciences.

Mauricio Aredes received the B.Sc. degree from UFF - Fluminense Federal University, Rio de Janeiro State in 1984, the M.Sc. degree in Electrical Engineering from UFRJ - Federal University of Rio de Janeiro in 1991, and the Dr.-Ing. degree (summa cum laude) from Technische Universität Berlin in 1996. In 1997, he became an Associate Professor at the Federal University of Rio de Janeiro, where he teaches Power Electronics. His main research area includes HVDC and FACTS systems, active filters, Custom Power, Renewable Energy Systems, and Power Quality Issues.

# CHAPTER *1*

*INTRODUCTION*

**THE INSTANTANEOUS ACTIVE AND REACTIVE POWER** theory, or the so-called "*p-q* theory," was introduced by Akagi, Kanazawa, and Nabae in 1983. Since then, it has been extended by the authors of this book, as well as other research scientists. This book deals with the theory in a complete form for the first time, including comparisons with other sets of instantaneous power definitions. The usefulness of the *p-q* theory is confirmed in the following chapters dealing with applications in controllers of compensators that are generically classified here as active power line conditioners.

The term "power conditioning" used in this book has much broader meaning than the term "harmonic filtering." In other words, the power conditioning is not confined to harmonic filtering, but contains harmonic damping, harmonic isolation, harmonic termination, reactive-power control for power factor correction, power flow control, and voltage regulation, load balancing, voltage-flicker reduction, and/or their combinations. Active power line conditioners are based on leading edge power electronics technology that includes power conversion circuits, power semiconductor devices, analog/digital signal processing, voltage/current sensors, and control theory.

Concepts and evolution of electric power theory are briefly described later. Then, the need for a consistent set of power definitions is emphasized to deal with electric systems under nonsinusoidal conditions. Problems with harmonic pollution in alternating current systems (ac systems) are classified, including a list of the principal harmonic-producing loads. Basic principles of harmonic compensation are introduced. Finally, this chapter describes the fundaments of power flow control. All these topics are the subjects of scope and will be discussed deeply in the following chapters of the book.

## 1.1 Concepts and Evolution of Electric Power Theory

One of main points in the development of alternating current (ac) transmission and distribution power systems at the end of the nineteenth century was based on sinusoidal voltage at constant-frequency generation. Sinusoidal voltage with constant frequency has made easier the design of transformers and transmission lines, including very long distance lines. If the voltage were not sinusoidal, complications would appear in the design of transformers, machines, and transmission lines. These complications would not allow, certainly, such a development as the generalized "electrification of the human society." Today, there are very few communities in the world without ac power systems with "constant" voltage and frequency.

With the emergence of sinusoidal voltage sources, the electric power network could be made more efficient if the load current were in phase with the source voltage. Therefore, the concept of reactive power was defined to represent the quantity of electric power due to the load current that is not in phase with the source voltage. The average of this reactive power during one period of the line frequency is zero. In other words, this power does not contribute to energy transfer from the source to the load. At the same time, the concepts of apparent power and power factor were created. Apparent power gives the idea of how much power can be delivered or consumed if the voltage and current are sinusoidal and perfectly in phase. The power factor gives a relation between the average power actually delivered or consumed in a circuit and the apparent power at the same point. Naturally, the higher the power factor, the better the circuit utilization. As a consequence, the power factor is more efficient not only electrically but also economically. Therefore, electric power utilities have specified lower limits for the power factor. Loads operated at low power factor pay an extra charge for not using the circuit efficiently.

For a long time, one of the main concerns related to electric equipment was power factor correction, which could be done by using capacitor banks or, in some cases, reactors. For all situations, the load acted as a linear circuit drawing a sinusoidal current from a sinusoidal voltage source. Hence, the conventional power theory based on active-, reactive-, and apparent-power definitions was sufficient for design and analysis of power systems. Nevertheless, some papers were published in the 1920s, showing that the conventional concept of reactive and apparent power loses its usefulness in nonsinusoidal cases [1,2]. Then, two important approaches to power definitions under nonsinusoidal conditions were introduced by Budeanu [3,4] in 1927 and Fryze [5] in 1932. Fryze defined power in the time domain, whereas Budeanu did it in the frequency domain. At that time, nonlinear loads were negligible, and little attention was paid to this matter for a long time.

Since power electronics was introduced in the late 1960s, nonlinear loads that consume nonsinusoidal current have increased significantly. In some cases, they represent a very high percentage of the total loads. Today, it is common to find a house without linear loads such as conventional incandescent lamps. In most cases, these lamps have been replaced by electronically controlled fluorescent lamps. In industrial applications, an induction motor that can be considered as a linear load in a steady state is now equipped with a rectifier and inverter for the purpose of achieving adjustable speed control. The induction motor together with its drive is no longer a linear load. Unfortunately, the previous power definitions under nonsinusoidal currents were dubious, thus leading to misinterpretations in some cases. Chapter 2 presents a review of some theories dealing with nonsinusoidal conditions.

As pointed out earlier, the problems related to nonlinear loads have significantly increased with the proliferation of power electronics equipment. The modern equipment behaves as a nonlinear load drawing a significant amount of harmonic current from the power network. Hence, power systems in some cases have to be analyzed under nonsinusoidal conditions. This makes it imperative to establish a consistent set of power definitions that are also valid during transients and under nonsinusoidal conditions.

The power theories presented by Budeanu [3,4] and Fryze [5] had basic concerns related to the calculation of average power or root-mean-square values (rms values) of voltage and current. The development of power electronics technology has brought new boundary conditions to the power theories. Exactly speaking, the new conditions have not emerged from the research of power electronics engineers. They have resulted from the proliferation of power converters using power semiconductor devices such as diodes, thyristors, insulated-gate bipolar transistors (IGBTs), gate-turn-off thyristors, and so on. Although these power converters have a quick response in controlling their voltages or currents, they may draw reactive power as well as harmonic current from power networks. This has made it clear that conventional power theories based on average or rms values of voltages and currents are not applicable to the analysis and design of power converters and power networks. This problem has become more serious and clear during comprehensive analysis and design of active filters intended for reactive-power compensation as well as harmonic compensation.

From the end of the 1960s to the beginning of the 1970s, Erlicki and Emanuel-Eigeles [6], Sasaki and Machida [7], and Fukao et al. [8] published their pioneer papers presenting what can be considered as a basic principle of controlled reactive-power compensation. For instance, Erlicki and Emanuel-Eigeles [6] presented some basic ideas like "compensation of distortive power is unknown to date.." They also determined that "a non-linear resistor behaves like a reactive-power generator while having no energy-storing elements" and presented the very first approach to active power-factor control. Fukao et al. [8] stated that "by connecting a non-active-power source in parallel with the load, and by controlling it in such a way as to supply reactive power to the load, the power network will only supply active power to the load. Therefore, ideal power transmission would be possible."

Gyugyi and Pelly [9] presented the idea that reactive power could be compensated by a naturally commutated cycloconverter without energy storage elements. This idea was explained from a physical point of view. However, no specific mathematical proof was presented. In 1976, Harashima et al. [10] presented, probably for the first time, the term "instantaneous reactive power" for a single-phase circuit. That same year, Gyugyi and Strycula [11] used the term "active ac power filters" for the first time. A few years later, in 1981, Takahashi et al. published two papers [12,13] giving a hint of the emergence of the instantaneous power theory or "*p-q* theory." In fact, the formulation they reached can be considered a subset of the *p-q* theory that forms the main scope of this book. However, the physical meaning of the variables introduced to the subset was not explained by them.

The *p-q* theory in its first version was published in the Japanese language in 1982 [14] in a local conference, and later in *Transactions of the Institute of Electrical Engineers of Japan*...