# Milestone-Proposal:Hirofumi Akagi

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Docket #:2023-10

*This proposal has been submitted for review.*

**To the proposer’s knowledge, is this achievement subject to litigation?**
No

**Is the achievement you are proposing more than 25 years old?**
Yes

**Is the achievement you are proposing within IEEE’s designated fields as defined by IEEE Bylaw I-104.11, namely: Engineering, Computer Sciences and Information Technology, Physical Sciences, Biological and Medical Sciences, Mathematics, Technical Communications, Education, Management, and Law and Policy. **
Yes

**Did the achievement provide a meaningful benefit for humanity?**
Yes

**Was it of at least regional importance?**
Yes

**Has an IEEE Organizational Unit agreed to pay for the milestone plaque(s)?**
Yes

**Has the IEEE Section(s) in which the plaque(s) will be located agreed to arrange the dedication ceremony?**
Yes

**Has the IEEE Section in which the milestone is located agreed to take responsibility for the plaque after it is dedicated?**
Yes

**Has the owner of the site agreed to have it designated as an IEEE Milestone?**
Yes

**Year or range of years in which the achievement occurred:**

1983-1986

**Title of the proposed milestone:**

The p-q Theory of Three-Phase Instantaneous Real and Imaginary Powers

**Plaque citation summarizing the achievement and its significance:**

In 1983, the Tokyo Institute of Technology and the Nagaoka University of Technology established the p-q theory for three-phase circuits, giving a mathematical proof to physical meanings of each-phase instantaneous active and reactive powers. It was followed by experimental verification of the validity and effectiveness. Since the 1990s, this theory has been applied to design and control of three-phase grid-tied converters.

**200-250 word abstract describing the significance of the technical achievement being proposed, the person(s) involved, historical context, humanitarian and social impact, as well as any possible controversies the advocate might need to review.**

**IEEE technical societies and technical councils within whose fields of interest the Milestone proposal resides.**

**In what IEEE section(s) does it reside?**

Tokyo Section

### IEEE Organizational Unit(s) which have agreed to sponsor the Milestone:

**IEEE Organizational Unit(s) paying for milestone plaque(s):**

**Unit:** Tokyo Section

**Senior Officer Name:** Haruko Kawahigashi

**Unit:** Tokyo Section

**Senior Officer Name:** Hiroshi Suzuki

**IEEE Organizational Unit(s) arranging the dedication ceremony:**

**Unit:** Tokyo Section

**Senior Officer Name:** Haruko Kawahigashi

**Unit:** Tokyo Section

**Senior Officer Name:** Hiroshi Suzuki

**IEEE section(s) monitoring the plaque(s):**

**IEEE Section:** Tokyo Section

**IEEE Section Chair name:** Kiyoharu Aizawa

### Milestone proposer(s):

**Proposer name:** Hiorfumi Akagi

**Proposer email:** *Proposer's email masked to public*

**Proposer name:** Hideaki Fujita

**Proposer email:** *Proposer's email masked to public*

**Proposer name:** Makoto Hagiwara

**Proposer email:** *Proposer's email masked to public*

**Please note:** your email address and contact information will be masked on the website for privacy reasons. Only IEEE History Center Staff will be able to view the email address.

**Street address(es) and GPS coordinates in decimal form of the intended milestone plaque site(s):**

2-12-1, O-okayama, Meguro, Tokyo, 152-8550, Japan, GPS; 35.60680380, 139.68476719 and 1603-1, Kamitomioka, Nagaoka, Niigata, 940-2188, Japan GPS; 37.42446849, 138.77701238

**Describe briefly the intended site(s) of the milestone plaque(s). The intended site(s) must have a direct connection with the achievement (e.g. where developed, invented, tested, demonstrated, installed, or operated, etc.). A museum where a device or example of the technology is displayed, or the university where the inventor studied, are not, in themselves, sufficient connection for a milestone plaque**.

**Please give the address(es) of the plaque site(s) (GPS coordinates if you have them). Also please give the details of the mounting, i.e. on the outside of the building, in the ground floor entrance hall, on a plinth on the grounds, etc. If visitors to the plaque site will need to go through security, or make an appointment, please give the contact information visitors will need. **
The museum in the Tokyo Institute of Technology and the techno-museum in the Nagaoka University of Technology are the two intended sites of the milestone plaque. The two universities made a significant contribution to the emergence of the theory in terms of generation and establishment. Therefore, the two sites are the ultimate choice for the plaque to be displayed.

**Are the original buildings extant?**

Yes, the original buildings are extant, in which the theory was generated and established.

**Details of the plaque mounting:**

The plaque is to be displayed, together with a copy of the world’s-first paper on the theory presented at the international power electronics conference (IPEC-Tokyo) sponsored by the IEE of Japan (IEEJ) in March 1983 and two copies of the ensuing experimental papers published in the IEEE Transactions on Industry Applications in May 1984 and May 1986, along with a few panels for a concise explanation of the theory in English and Japanese. The intended sites are on the first basement floor of the museum in the Tokyo Institute of Technology, and on the first floor of the techno-museum in the Nagaoka University of Technology.

**How is the site protected/secured, and in what ways is it accessible to the public?**

Each of the two museums provides a security control system with camaras and other security equipment. Each exhibition place is publicly accessible without appointment during normal business hours except Saturday, Sunday, and holidays.

**Who is the present owner of the site(s)?**

The Tokyo Institute of Technology, established in 1881, and the Nagaoka University of Technology (formerly, the Technological University of Nagaoka), established in 1976, are both “public” universities. The present owner is President, Dr. Kazuya Masu for the Tokyo Institute of Technology, and President, Dr. Shigeharu Kamado for the Nagaoka University of Technology.

**What is the historical significance of the work (its technological, scientific, or social importance)? If personal names are included in citation, include justification here. (see section 6 of Milestone Guidelines)**

Extended Summary

In December 1957, General Electric Company invented silicon-controlled rectifiers capable of controlling their turn-on timing, thus resulting in initiating the era of power electronics. Later, silicon-controlled rectifiers were internationally renamed “thyristors.” In the middle of the 1970s, bipolar junction transistors (BJTs) capable of switching on and off were put on the market. Then, BJTs were replacing thyristors gradually. This replacement allowed power electronics experts to apply pulse-width modulation (PWM) to single-phase and three-phase voltage-source converters without any auxiliary circuit for commutation. This epoch-making advance made it possible to actively control three-phase instantaneous currents at the ac side of the PWM converter. At the same time, it encouraged power electronics experts to do comprehensive research on “instantaneous reactive power” and/or “instantaneous reactive current.” However, no one had succeeded in establishing any convincing theory before 1983.

Hirofumi Akagi received his M. S. and Ph. D. degrees in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in March 1976 and in March 1979, respectively. Immediately, he joined as an assistant professor the department of electrical engineering at the Nagaoka University of Technology (formerly, the Technological University of Nagaoka), Nagaoka, Japan. Since January 2000, he has been a professor, currently a professor emeritus, with the Tokyo Institute of Technology.

In November 1981, Akagi defined and formulated a new pair of instantaneous real and imaginary powers, p and q, for each of three-phase circuits, lumping three single-phase circuits together. Three months later, he gave a strict proof based on mathematical expressions (See equations (10) to (15) in the supporting file named "Akagi-Figures-Equation.pdf") to a clear explanation of the physical meanings of q as well as the other new pair of instantaneous active and reactive powers. He presented the world’s-first paper on the theory (named the “p-q theory” later) at an international conference in March 1983 [1]. The p-q theory in the time domain was characterized by using only the information of the present voltages and currents. This definition and formulation made it applicable to any waveform without any restriction, including non-periodic and transient-state waveforms. The conference paper was followed by two experimental papers published in the IEEE Transactions on Industry Applications in May 1984 [2] and May 1986 [3].

The p-q theory is summarized as follows: A three-phase three-wire circuit had no neutral wire between the source and the load, in which the sum of the three-phase instantaneous currents ia, ib, and ic was always zero, that is, ia + ib + ic = 0. (See Fig. 1 in the supporting file named "Akagi-Figures-Equation.pdf") This equation enabled to reduce the number of the independent currents from three to two. Akagi applied a so-called “α-β transformation” to the three-phase voltages and currents. This transformation made it possible to convert the three-phase three-wire circuit to a two-phase three-wire circuit. Lumping the two single-phase circuits together, he reconsidered the already-existing instantaneous real power p as a “scalar product” of a two-dimensional instantaneous voltage space vector and a two-dimensional instantaneous current space vector. By contrast with it, Akagi defined and formulated the new instantaneous imaginary power q as a “vector product” of the voltage and current space vectors. (See equation (9).) The most important and significant peace to the formulation was that the determinant of the two-dimensional matrix linking the two-dimensional power space to the two-dimensional current space was always a positive number except for the unrealistic conditions of eα = eβ = 0. This implied that the inverse matrix existed, allowing bidirectional mapping between the current space and the power space. Next, Akagi defined and formulated instantaneous active and reactive powers for each of the α-phase and the β-phase. Thus, the following two items came into existence among the four instantaneous active and reactive powers in the two-phase three-wire circuit:

1) One arithmetic sum of the instantaneous active power in the α-phase and that in the β-phase was always equal to the three-phase instantaneous real power p. This meant that the two instantaneous active powers were transferred from the source to the load.

2) The other arithmetic sum of the instantaneous reactive power in the α-phase and that in the β-phase was always zero. This meant that the two instantaneous reactive powers circulated between the α-phase and the β-phase from a point of view of power flow.

To avoid confusion and misunderstanding in the usage of technical terminology, Akagi distinguished one pair of instantaneous real and imaginary powers for each of three-phase circuits from the other pair of instantaneous active and reactive powers for each phase in a three-phase circuit. He gave a strict mathematical proof (See equations (10) to (15) in the supporting file named "Akagi-Figures-Equations.pdf") to the above items 1) and 2). At present, the power electronics community refers to it as the p-q theory. The naming came from the formulation of the pair of p and q. (See equation (9) in the supporting file named "Akagi-Figures-Equations.pdf")

As soon as Akagi established the p-q theory, he applied it to a three-phase instantaneous imaginary-power compensator consisting of a three-phase voltage-source pulse-width modulation (PWM) converter with a small dc capacitor, intending to verify the validity of the p-q theory. Akagi and his team verified experimentally that this compensator had the capability of eliminating all the instantaneous reactive powers related to q from a three-phase full-bridge thyristor converter operating even in transient-sate conditions. (See Figs. 2 to 4 in the supporting file named "Akagi-Figures-Equations.pdf") The compensator using six bipolar junction transistors (BJTs) and six free-wheeling diodes had no bulky capacitor connected at the dc side because the instantaneous active powers related to p was excluded from the target of compensation. The compensating characteristics had been impossible to obtain through the application of any traditional reactive-power theory based on single-phase circuits. He had the world’s-first experimental paper published in the IEEE Transactions on Industry Applications in May 1984 [2]. Next, he applied the p-q theory to a three-phase pure active filter for power conditioning. He had the world’s-second experimental paper published in the same IEEE Transactions in May 1986 [3].

The p-q theory has made a theoretical and practical contribution to three-phase grid-tied power conversion systems in system design and control strategy. Applications of the p-q theory to the grid-tied systems include pure/hybrid active filters for power conditioning [5]-[6], static var generators (SVG) installed on the premises of a railway substation in the Japanese bullet train system [7], wind-turbine/photovoltaic systems for actively using renewable energy resources, uninterruptible power supplies (UPS), battery energy storage systems (BESS), static synchronous compensators (STATCOM) for 50Hz or 60-Hz transmission systems, modular multilevel converters for high-voltage direct-current (HVDC) transmission systems, and so on. According to Google Scholar, the conference paper presented in 1983 [1], the two IEEE Transactions papers published in 1984 [2] and 1986 [3], and the world’s-first monograph on the p-q theory, entitled “Instantaneous Power Theory and Applications to Power Conditioning” published in 2007 (first edition) and 2017 (second edition) [4] reached 1650, 4877, 1023, and 3580 citations, respectively. These individual citations are still counting, although the three papers were presented or published nearly 40 years ago.

Research Background up to 1981

“Active power consumes electric power whereas reactive power does not.” Generally, it is true and correct on a time average but it may not on a time instant from an instantaneous point of view, irrespective of single-phase or three-phase circuit application. A course named “electric circuit theory,” or something similar, is mandatory for undergraduate students who are majoring in electrical engineering at universities and technical colleges around the world. Professors or instructors teach the concept of active power and reactive power, in which the waveforms of voltage and current in single-phase circuits are assumed to be sinusoidal under steady-state conditions. These assumptions make it easy to define, formulate, and understand active power and reactive power for any single-phase circuit. The definition, formulation, and understanding are then expanded directly into three-phase circuits considered as a set of three single-phase circuits.

A continuous advance in semiconductor technology made it possible to put bipolar junction transistors (BJTs) intended for applications to power electronics on the market in the middle of the 1970s. The emergence of BJTs encouraged power electronics experts to do comprehensive research on “instantaneous reactive power” and/or “instantaneous reactive current.” The authors of [8], who were experts of power electronics, introduced a technical term “instantaneous reactive power” to a single-phase circuit, defining and formulating it on a time average with a time window of half a cycle. However, many power electronics experts had a negative opinion towards considering the reactive power on a time average to be “instantaneous.” In addition, the formulation on a time average made the resultant reactive power variant or inaccurate because the reactive power depended strongly on the time window, especially when the waveform of current was distorted, non-periodic, and/or randomly changing. In other words, this situation automatically rendered it inaccurate because the use of information of past voltage and current resulted in being no longer “instantaneous.” Power electronics experts were struggling to solve the above-mentioned issues and/or conflicts.

The authors of [9], who were experts of power engineering, challenged to solve the above-mentioned issues for three-phase circuits, using a three-phase vector diagram of voltage and current. They had a short (two-page-long) paper published in Proceedings of the IEEE in August 1978. It presented two different computing methods for three-phase instantaneous active and reactive powers based on measurements at one instant only. The authors of [9] intended to apply the two methods to protection, load-behavior modeling, state estimation, and reactive-power compensation. They focused on making numerical comparisons in positive-sequence, negative-sequence, and zero-sequence reactive powers between the two methods and an existing method. However, they made neither discussion on any physical meaning of their instantaneous reactive power for each of three-phase circuits, nor description of any instantaneous reactive power for each phase in a three-phase circuit.

In the late 1970s, some power electronics experts recognized and understood that a significant difference in the entity of reactive power existed between single-phase circuits and three-phase circuits [10]. Let a set of three-phase sinusoidal and balanced voltages be ea, eb, and ec in steady-state conditions. Assume that the three-phase currents iCa, iCb, and iCc were flowing into a traditional three-phase reactive-power compensator with a phase difference of ±90 degrees between the voltage and the current in each phase. (See supporting materials.) The above situation made the following equation come into existence: (See the supporting file named "Akagi-Figures-Equations.pdf") Equation (1) implied that the three-phase reactive-power compensator did not need any energy storage device or component as long as any amount of purely leading or lagging reactive current was flowing in, or out of, the compensator.

From the middle of the 1970s to the 1980s, one of the most attractive research targets for power electronics experts, especially in academia, was a so-called “field-oriented control” of three-phase induction and synchronous motors. Before attention was paid to the research targets, research scientists and engineers in the field of power systems and electric machines had applied the d-q transformation and instantaneous space vectors to their research, analysis, and/or design. An advance in research on the field-oriented control made power electronics experts familiar with both d-q transformation and instantaneous space vectors.

Fig. 1. Three-phase three-wire circuit, including the instantaneous voltages and currents. (See the supporting file named "Akagi-Figures-Equations.pdf")

Fig. 1 shows a three-phase three-wire circuit without neutral wire between the source and the load. Takahashi presented an experimental paper on a so-called “power distortion compensator” consisting of a three-phase 24-pulse thyristor converter with asymmetrical gate control at the IEEE Industry Applications Society Annual Meeting in October 1980 [11]. At that time, the two authors, Takahashi and Nabae, were Akagi’s colleague and boss in the power electronics research group at the Nagaoka University of Technology, respectively. Akagi supported experimental work in this research, so that his name was included in the acknowledgement of the conference paper [11].

Assuming that the three-phase instantaneous voltages ea, eb, and ec were sinusoidal and balanced in Fig. 1, Takahashi and Nabae applied the α-β transformation in (2) and (3) to the three-phase instantaneous voltages and currents, respectively. (See the supporting file named "Akagi-Figures-Equations.pdf") Then, they applied the d-q transformation to the two-phase three-wire circuit, where ω is the line angular frequency. Next, they defined and formulated a pair of the instantaneous active current ip and the instantaneous reactive current iq for the two-phase circuit, as shown in (4). (See the supporting file named "Akagi-Figures-Equations.pdf") They gave a plausible explanation that the two-phase instantaneous active current corresponded to an instantaneous active power. They made no description of the two-phase instantaneous reactive current except for the following simple one: The dc component of iq corresponded to a reactive power. Moreover, they made no description of any instantaneous reactive current for each phase in the two-phase three-wire circuit. Hence, they were unable to catch the entity of the instantaneous reactive current or power for three-phase circuits, although they might have come close to the entity or might have got in contact with its “tail.”

Akagi’s Research at the Tokyo Institute of Technology from 1974 to 1979

Hirofumi Akagi received his M. S. and Ph. D. degrees in electrical engineering from the Tokyo Institute of Technology in 1976 and 1979, respectively, after receiving his B. S. degree in electrical engineering from the Nagoya Institute of Technology in 1974. As a graduate student, he devoted himself to doing theoretical and experimental research on line-commutated cycloconverters using thyristors for five years in total. The cycloconverter acts as an ac-to-ac direct frequency changer, in which the output or motor frequency is limited to be much lower than the input or line frequency from the principles of operation. A three-phase-input and three-phase-output line-commutated cycloconverter using many thyristors has no energy storage device such as inductors and capacitors inside. Although it is similar in circuit configuration and operational principle to a three-phase full-bridge thyristor rectifier [12], the control angle of the thyristors in the cycloconverter is changing moment by moment. The cycloconverter has the following features and issues:

1) The input or line current contains complicated harmonic and sideband currents, where the term “complicated” means that the harmonic and sideband frequencies are related to both input and output frequencies, as well as to circuit configurations.

2) Any line-commutated cycloconverter operates with a “lagging” power factor at the input or line side even if it drives a three-phase synchronous motor operating with a “leading” power factor. In February 1979, a senior professor, who participated as an examiner in Akagi’s final defense examination, asked the following question: “Can you give any general explanation to the reason why the cycloconverter always operates with a lagging power factor at the input or line side, irrespective of operating with any (leading, lagging, or unity) power factor at the output or motor side?” Akagi answered “It comes from line commutation of thyristors.” However, he was unable to give any further explanation. As soon as he established the p-q theory in 1983, he was able to answer the question completely. Akagi has greatly appreciated the professor for giving him the homework assignment because it acted as a trigger for the emergence of the p-q theory.

3) In the late 1970s, a practical three-phase-input and three-phase-output line-commutated cycloconverter for a high-power, low-speed motor drive consisted of 72 thyristors and three three-phase delta/delta/star-winding transformers, thus resulting in being a pulse number of 12. Akagi did the following thought experiment: As the pulse number was getting higher and higher from 12 to 24, 48, 96, -----, the three-phase voltages and currents at the output or motor side were coming closer and closer to three-phase sinusoidal and balanced waveforms, assuming that the three-phase voltages at the input or line side were sinusoidal and balanced. However, the three-phase currents at the input side did not get purely sinusoidal because the waveform of current in each phase still contained sideband currents at frequencies of fin ± 6nfout for n = 1, 2, 3, -----, where fin is the input frequency, and fout is the output frequency. This unexpected but curious insight gained from the thought experiment coincided with that from theoretical analysis: No change occurred in the sideband currents at frequencies of fin ± 6nfout even if the pulse number was increased from 12 to 96. However, technical papers available at that time made no description of the reason why no change occurred in the sideband currents although the other high-order harmonic and sideband currents disappeared or got negligibly small. Akagi continued to think of how to give a lucid explanation to the unexpected insight gained from the thought experiment and the theoretical analysis. Hence, he gave himself the homework assignment. As a result, it acted as the other trigger for the emergence of the p-q theory.

Looking back now, Akagi is convinced that the cycloconverter is one of the most interesting and effective research targets for understanding reactive power as well as harmonic and sideband currents. However, he was not conscious of it at that time because he was just a graduate student. His research experience at the Tokyo Institute of Technology had impacted on the emergence of the p-q theory in 1983 at the Nagaoka University of Technology.

Akagi’s Research at the Nagaoka University of Technology from 1979 to 1981

Akagi joined the department of electrical engineering at the Nagaoka University of Technology (formerly, the Technological University of Nagaoka) as an assistant professor in April 1979. As soon as Akagi joined, he started to do intensive research on power conversion systems and their applications including a three-phase three-level neutral-point-clamed pulse-width modulation (PWM) inverter and field-oriented control of three-phase induction and synchronous motors. First of all, he tried to understand instantaneous space vectors and coordinates- transformation techniques such as the d-q transformation for analysis and control of the three-phase motors, not only through carefully reading and understanding technical papers and monographs but also through carrying out experiments.

One of the most surprising happenstances through Akagi’s lifetime research experiences was that he was able to combine his research experience at the Tokyo Institute of Technology from April 1974 to March 1979 with that at the Nagaoka University of Technology from April 1979 to 1981. This happenstance, as well as the good fortune, made Akagi initiate and establish the p-q theory. In other words, the p-q theory would not have emerged from the Nagaoka University of Technology unless Akagi’s theoretical and experimental research on line-commutated cycloconverters at the Tokyo Institute of Technology as a graduate student. In addition, the p-q theory would not have emerged if Akagi had worked for the Tokyo Institute of Technology as a junior faculty member or a post-doctoral fellow immediately he received his Ph. D. degree from the Tokyo Institute of Technology.

The p-q Theory in Three-Phase Three-Wire Circuits [1]

Akagi applied the α-β transformation in (2) and (3) to the three-phase three-wire circuit shown in Fig.1. This made it possible to convert the three-phase circuit into a two-phase (α-phase and β-phase) three-wire circuit, assuming the following equations. (See the supporting file named "Akagi-Figures-Equations.pdf") The three-phase instantaneous power on the a-b-c stationary reference frames, p is always equal to the two-phase instantaneous power on the α-β stationary reference frames. (See the supporting file named "Akagi-Figures-Equations.pdf") When the two-phase voltages eα and eβ were assumed to be given or known, it was easy to compute the “sole” instantaneous power p as long as the “two” instantaneous currents iα and iβ were given or known. In reverse, it was impossible to compute the “two” instantaneous currents iα and iβ accurately and uniquely even if the “sole” instantaneous power p as well as eα and eβ were known or given. This insight implied that the other unknown instantaneous “power” or something similar was missing or hidden behind the existing instantaneous power p. After all, Akagi formulated it as a “vector product” of the instantaneous voltage and current space vectors by contrast with formulating the instantaneous power p as a “scalar product” of the two space vectors. Thus, the other unknown “power” q was given as follows: (See the supporting file named "Akagi-Figures-Equations.pdf") Either plus or minus polarity resulting from the vector product was acceptable as if to give a traditional leading reactive power either plus or minus sign. Equations (7) and (8), hence, yielded the folllowing key equation: (See the supporting file named "Akagi-Figures-Equations.pdf") Equation (9) led to the following insights into the two-phase three-wire circuit:

1) p and q were based on lumping two independent single-phase circuits together.

2) p and q were complementary to each other with different physical meanings.

Akagi named p and q the instantaneous “real” power and the instantaneous “imaginary” power, respectively, considering “imaginary” to be the antonym of “real.” The physical unit of p was definitely Watt or [W] while that of q was discussed later on because it was not [W]. Since the determinant of the two-dimensional matrix in (9) was equal to eα2 + eβ2 > 0, the inverse matrix existed. (See the supporting file named "Akagi-Figures-Equations.pdf") Equation (10) enabled to divide the α-phase and β-phase instantaneous currents iα and iβ into the two currents related to p, iαp and iβp, and those to q, iαq and iβq as follows: (See the supporting file named "Akagi-Figures-Equations.pdf") The four instantaneous currents in the two-phase three-wire circuit were named as follows: iαp : α-phase instantaneous active current, iαq : α-phase instantaneous reactive current, iβp : β-phase instantaneous active current, and iβq : β-phase instantaneous reactive current.

Let the α-phase and β-phase instantaneous power be pα and pβ, respectively. The active use of all of the above-mentioned instantaneous active and reactive currents allowed the following equation to come out. (See the supporting file named "Akagi-Figures-Equations.pdf") Since the two-phase instantaneous real power p was defined as usual, it was given as a sum of the α-phase instantaneous power pα and the β-phase instantaneous power pβ. (See the supporting file named "Akagi-Figures-Equations.pdf") Equation (13) gained an insight into the entity of the instantaneous active and reactive powers. One sum of the first and second terms at the right-hand side of the bottom of (13) was always equal to the two-phase instantaneous real power p. The other sum of the third and fourth terms was always zero. Thus, the following crucial equations came into existence: (See the supporting file named "Akagi-Figures-Equations.pdf") The four instantaneous powers in the two-phase three-wire circuit were named as follows: (See the supporting file named "Akagi-Figures-Equations.pdf") pαp : α-phase instantaneous active power, pαq : α-phase instantaneous reactive power, pβp : β-phase instantaneous active power, and pβq : β-phase instantaneous reactive power.

All of the instantaneous active and reactive currents and powers in the two-phase three-wire circuit were summarized as follows: (See the supporting file named "Akagi-Figures-Equations.pdf") The mathematical proof from (10) to (15) led to the following new findings:

1) The α-phase instantaneous current iα was divided into the α-phase instantaneous active current and the α-phase instantaneous reactive current uniquely and accurately, using only the information of the present voltages and currents. The β-phase instantaneous current iβ was divided into two currents in the same mathematical process.

2) One arithmetic sum of pαp and pβp was always equal to the instantaneous real power p in (14). This justified naming pαp the α-phase instantaneous active power and pβp the β-phase instantaneous active power.

3) The other arithmetic sum of pαq and pβq was always zero in (15). This justified naming pαq the α-phase instantaneous reactive power and pβq the β-phase instantaneous reactive power. The instantaneous imaginary power q played an essential role in determining both α-phase and β-phase instantaneous reactive powers.

4) It was clear from (11)-(15) that the instantaneous active and reactive powers had the same physical unit as Watt or [W].

Power electronics experts might make the following inquiry, “What is the most important contribution of the p-q theory to electrical engineering?” Each power electronics expert may give different answers. Akagi’s answer would be as follows:

1) The p-q theory defined and formulated the pair of p and q in (9), introducing the concept of “mapping” in mathematics from the two-dimensional instantaneous current space to the two-dimensional instantaneous power space, and vice versa.

2) The p-q theory gave the strict mathematical proof from (10) to (15) to a clear explanation of the physical meanings of q and each-phase instantaneous active and reactive powers.

Moreover, the following inquiry might follow, “Which contribution is more crucial, item 1) or 2)?” Akagi’s answer would be item 2) because it enabled the power electronics community to catch the entity of the three-phase instantaneous imaginary power q and each-phase instantaneous active and reactive powers.

The world’s-first paper on the p-q theory was presented at an international conference in March 1983 [1]. This conference paper defined and formulated a new pair of p and q in (9), including the strict mathematical proof from (10) to (15). Akagi added on to the paper the waveforms of voltage and current by computer simulation to validate the p-q theory.

Experimental Verification of the validity and Effective of the p-q Theory [2]-[3]

Fig. 2. Experimental system configuration that was designed, built, and tested in 1982 [2]. (See the supporting file named "Akagi-Figures-Equations.pdf")

As soon as Akagi established the p-q theory in February 1982, he started to verify the validity and effectiveness of it experimentally. The p-q theory for three-phase circuits was accompanied by a theoretical innovation in dividing each of the three instantaneous currents into instantaneous active and reactive currents uniquely and accurately without any time delay. This innovation came from using only the information of the present voltages and currents, without using any information of past voltages and currents.

Fig. 2 illustrates the experimental system configuration that combined the three-phase 100-V, 2-kW, 50-Hz full-bridge thyristor rectifier and the three-phase 100-V, 0.5-kVA, 50-Hz instantaneous imaginary-power compensator prototype based a three-phase voltage-source pulse-width modulation (PWM) converter using six bipolar junction transistors (BJTs) and six free-wheeling diodes. This compensator had the 5-μF capacitor connected at the dc side. The ratio of the maximum energy stored in the dc capacitor operating at a maximum of 200 V with respect to 0.5 kW was calculated as 0.2 ms. This value was much shorter than those of the dc capacitors used in the other compensators at the time, which is one of the most significant features in the compensator as well as the experimental system configuration.

The goal of this experiment was to verify whether the compensator was able to eliminate all the instantaneous reactive currents related to q from the thyristor rectifier. A small-rated three-phase passive filter consisted of the three delta-connected 0.5-μF capacitors, the per-phase 2.4-mH inductor, and the per-phase leakage inductance inherent in the three-phase 200-V 50-Hz transformer. The aim in installing the passive filter at the ac side of the compensator was to mitigate the high-frequency ripple currents caused by switching action of each BJT, while the aim in installing the 5-μF capacitor at the dc side was, not for energy storage, but for stable switching action of each BJT. The control system consisted of analogue circuits using some operational amplifiers and eight analogue multipliers, along with several analogue sensors of voltage and current, but without any phase-locked-loop (PLL) circuit. Neither analogue low-pass/high-pass filter nor integrator for signal processing was used in the control system. The experimental system illustrated in Fig. 2 was designed, built, and tested in 1982, yielding the experimental results shown in Figs. 3 and 4 later on.

Fig. 3. Experimental waveforms before and after a step change was made in the dc load resistor of the thyristor rectifier, where the control angle of the thyristors was kept zero [2]. (See the supporting file named "Akagi-Figures-Equations.pdf")

Fig. 3 shows experimentally-observed waveforms of voltage and current during a transient state before and after an intentional step change was made in the dc load resistor of the thyristor rectifier. The control angle of the six thyristors was kept zero, so that the thyristor rectifier acted as a three-phase full-bridge diode rectifier. The waveform of the a-phase voltage to the neutral point of the source, ea (top) was in phase with that of the a-phase source current iSa (second top) in the transient state as well as in the steady state. The a-phase load current at the ac side of the rectifier, ia (middle) looked a well-known rectangular waveform with a conducting period of 120 degrees in half a cycle. However, the amplitude of ia started increasing gradually immediately the intentional step change was made. The a-phase compensating current at the ac side of the compensator, iCa (second bottom) was controlled to follow its current reference coming from the control system in a current- feedback manner. The waveform of iSa looked almost sinusoidal but not purely sinusoidal. This reason was explained in detail, in conjunction with experimentally-measured harmonic current spectra in Fig. 4, later on. Four small notches and spikes per cycle appeared on the waveform of iSa because the compensating current was unable to follow its steep current reference during line commutation of one thyristor to another. The details of the notches and spikes were beyond the scope of this experiment.

Fig. 4. Experimental harmonic spectra of iSa (left) and ia (right), where three spectra in each of the two measured results correspond to the fundamental (50 Hz), 5th-harmonic (250 Hz), and 7th-harmonic (350 Hz) currents from the left to the right [2]. (See the supporting file named "Akagi-Figures-Equations.pdf")

Fig. 4 shows experimentally-measured harmonic spectra of iSa and ia in steady-state conditions with an enough passage of time after the intentional step change was made. The waveform of iSa contained much less harmonic current than that of ia. However, iSa was not a purely-sinusoidal waveform. Comparison between the 5th-harmonic and 7th-harmonic spectra of iSa (left) and those of ia (right) in Fig. 4 made it clearer than that between the waveform of iSa (second top) and that of ia (middle) in Fig. 3. The reason was because the three-phase instantaneous imaginary-power compensator devoted itself to eliminating all the three instantaneous reactive currents related to q from the thyristor rectifier. The 5th-harmonic and 7th-harmonic currents remaining in the waveform of iSa and the harmonic spectra of iSa came from the ac component of p. According to the p-q theory, the compensator was unable to eliminate the remaining harmonic currents because it did not have any bulky capacitor connected at the dc side. At that time, no technical paper presented the waveforms of iSa and ia in Fig. 3, the harmonic spectra of iSa and ia in Fig. 4, or something similar. In other words, these experimental results verified that the waveform of iSa was not purely sinusoidal, thus resulting in clarifying and justifying the originality and novelty of the p-q theory.

The world’s-first experimental paper validating the p-q theory was published in the IEEE Transactions on Industry Applications in May 1984 [2]. It presented the experimental system in Fig. 2, the waveforms of voltage and current in Fig. 3, and the harmonic current spectra in Fig. 4. Then, the thyristor rectifier was replaced with a three-phase-input and three-phase-output line-commutated cycloconverter using 36 thyristors with a pulse number of 6, where fin = 50 Hz and fout = 5 Hz. The IEEE Transactions paper included experimental waveforms and harmonic current spectra similar to those in Figs. 3 and 4, but with focus on the most dominant sideband current spectra with frequencies of fin ± 6fout, that is, 20 Hz and 80 Hz.

The world’s-second experimental paper was published in the same IEEE Transactions in May 1986 [3]. The p-q theory was applied to a three-phase active filter for power conditioning. It consisted of four three-phase voltage-source PWM converters using 12 bipolar junction transistor (BJTs) dual (two-in-one) modules and four three-phase 50-Hz transformers. An appropriately-designed dc capacitor was installed between the common positive and negative dc terminals of the four voltage-source PWM converters. The p-q theory made the system design and the control strategy appropriate and/or flexible in response to the objectives of installation of the active filter.

Answers to two “Homework Assignments” on Line-Commutated Cycloconverters

A practical three-phase-input and three-phase-output line-commutated cycloconverter consists of thyristors without energy storage components such as inductors and capacitors. When the cycloconverter is driving a synchronous motor coupled with a mechanical load, increasing the field-winding current makes it possible to control the motor power factor in a range of a lagging to a leading via unity power factor. However, the cycloconverter has always a lagging power factor at the input side. Let pin, qin, pout, and qout be the instantaneous real and imaginary powers at the input and output sides, respectively. It was easy to recognize the existence of an equation of pin = pout because the cycloconverter has no energy storage component inside. However, no one would have recognized the existence of an inequality of qin ≠ qout and even the existence of q before the emergence of the p-q theory in 1983. Akagi gained an insight into the entity of q and the essence of the inequality of qin ≠ qout. His real knowledge and deep insight led to the following conclusion: No direct relation exists in the cycloconverter between the input power factor and the output power factor.

The input current of the cycloconverter includes harmonic and sideband currents at the following frequencies: fin ± 6nfout, (6m ±1) fin, and (6m ±1) fin ± 6nfout for m = 1, 2, 3, ----, and n = 1, 2, 3, -----, where fin is the input frequency, fout is the output frequency, and 6m is the pulse number. Even if the pulse number is getting higher, no change occurs in the sideband currents at frequencies of fin ± 6nfout although the other harmonic and sideband currents at frequencies of (6m ±1) fin and (6m ±1) fin ± 6nfout disappeared or were getting smaller. However, no one would have made a clear explanation of the reason why only the sideband currents at frequencies of fin ± 6nfout are independent of the pulse number. The p-q theory led to the following new findings: 1) The sideband currents inherent in the cycloconverter were independent of pin (= pout), but dependent on qin because qin contained ac components at frequencies of 6nfout with a direct relation to the sideband currents. 2) The ac components stemmed from the operating principles of the cycloconverter, independent of the pulse number. The above new findings gave a lucid answer to the home assignment.

Most importantly, information of the output frequency fout is reflected in qin as ac components at frequencies of 6nfout but not reflected in either qout or pin (= pout). When the pulse number is high enough, the instantaneous real and imaginary powers pin (= pout) and qout have only the individual constant dc components determined by the operating conditions of the cycloconverter. When the “positive” sign is assigned to a “leading” power factor, qin consists of a “negative” dc component and ac components at frequencies of 6nfout. The “negative” sign of qin resulted from line commutation of the thyristors, independent of a motor power factor. Even if the synchronous motor is operating at a “leading” power factor, qin consists of a “negative” dc component and ac components at frequencies of 6nfout whereas qout consists of a “positive” dc component but without ac components.

Questions & Answers on the p-q Theory

Q1: What is a difference(s) in technical terminology between the instantaneous imaginary power and the instantaneous reactive power in three-phase three-wire circuits?

A1: The instantaneous imaginary power was defined and formulated for each of three-phase three-wire circuits, whereas the instantaneous reactive power for each phase in a three-phase three-wire circuit. As a result, a distinct difference existed in physical meaning between the two instantaneous imaginary and reactive powers. This difference pushed Akagi to introduce “imaginary” to the former, and “reactive” to the latter, so as to avoid confusion and misunderstanding in the usage of technical terminology. However, some power electronic experts have often used the two technical terms as a synonym without distinguishing one from the other strictly.

Q2: What made Akagi initiate the p-q theory?

A2: It was both good fortune and happenstance that he was able to combine his research experience at the Tokyo Institute of Technology with that at the Nagaoka University of Technology. However, Akagi’s research experiences in the two different academic organizations had no direct relation even in the field of power electronics.

Q3: Why did not Akagi challenge himself to apply the p-q theory to single-phase circuits?

A3: He was unable to do it even if he wanted to do it. The reason was because it was (is) impossible to define and formulate any instantaneous reactive power for single-phase circuits under the conditions of using only the information of the present voltage and current. However, electrical engineers have been defining and formulating the instantaneous power in single-phase circuits under the above conditions.

Similarity and Difference between the p-q Theory and the d-q Transformation

The formulation of a pair of p and q in (9) looks similar to that of the d-q transformation in (4). In fact, the p-q theory and the d-q transformation would get the same in formulation if the three-phase voltages were assumed to be sinusoidal, balanced, and periodic in steady-state conditions. However, the d-q transformation would be inapplicable to any case except for the above ideal conditions on three-phase voltages. On the other hand, the p-q theory is applicable to any case without any limitation on three-phase voltages. This is a significant and clear difference in formulation and application between the p-q theory and the d-q transformation. Moreover, the following distinct difference exists in physical meaning between the two: The d-q transformation converts two-phase currents on the α-β stationary reference frames to the ensuing two-phase currents on the d-q rotating reference frames, and vice versa. The p-q theory performs mapping from the two-dimensional instantaneous current space to the two-dimensional instantaneous power space, and vice versa, as shown in (9) and (10). In short, mapping in mathematics has a more flexible and broader concept than coordinates transformation in electrical engineering.

Expansion of the p-q Theory into Three-Phase Four-Wire Circuits

Some countries in the world have adopted three-phase four-wire circuits in low-voltage power distribution networks, although Japan uses three-phase three-wire circuits extensively. To provide the international application, Akagi expanded the p-q theory into a three-phase four-wire circuit including a zero-sequence voltage and a zero-sequence current. He applied the α-β-0 transformation to the three-phase voltages and currents on the a-b-c stationary reference frames, and introduced a three-dimensional bidirectional mapping matrix to the three-phase voltages and currents on the α-β-0 stationary reference frames. Hence, Akagi established the expanded p-q theory for three-phase four-wire circuits. It was characterized by separating the zero-sequence instantaneous power formulated as a product of the zero-sequence voltage and current from the other two-phase instantaneous real and imaginary powers [1]-[2].

Contributions to Research and Education

1) The emergence of the p-q theory in 1983 encouraged power electronics experts to do further theoretical research on electric power in the time domain under non-sinusoidal conditions. Alessandro Ferrero, with University of Catania, Italy, and Gabrio Superti-Furga, with University of Basilicata, Italy, tried to develop another instantaneous power theory, partially on a time average [13].

2) Jacques L. Willems, with the University of Gent, Belgium, introduced a technical term “the Akagi-Nabae Power Components” to the title of his IEEE Transactions paper, and expanded the p-q theory into multi-phase systems with more than three phases [14]. It would be extremely rare in the field of electrical engineering that the names of scientists or inventers appeared in the tile of such a prestigious IEEE Transactions paper written by another professor or scientist in a different research field within electrical engineering.

3) At present, many universities teach the p-q theory to graduate students majoring in power electronics as well as power systems.

Scientific Importance of the p-q Theory

The following general distinct difference exists between science and technology: As technology is advancing more and more, technological contributions are getting older and older. As a result, they will be replaced with the corresponding new ones, or they will disappear. On the other hand, scientific “real” contributions will continue to live on for many years. In fact, the p-q theory has been living on for 40 years since 1983. This means that it could be considered as one of the scientific “real” contributions. Moreover, the p-q theory will continue to live on for more than 60 years from now or for more than 100 years after the emergence of the p-q theory in 1983.

Social Importance of the p-q Theory

The emergence of the p-q theory has impacted on professors and scientists in academia as well as research and practical engineers in industry from a theoretical and practical point of view. Since the 1990s, the p-q theory has been applied to system design and control strategy for three-phase grid-tied power conversion systems based on voltage-source pulse-width modulation (PWM) converters or inverters. The three-phase grid-tied systems include pure and hybrid active filters for power conditioning [5]-[6], static var generators (SVG) installed on the premises of a railway substation in the Japanese bullet train system [7], wind-turbine and photovoltaic systems for making active use of renewable energy resources, uninterruptible power supplies (UPS), battery energy storage systems (BESS), static synchronous compensators (STATCOM) for stabilizing 50-Hz or 60-Hz transmission systems, modular multilevel converters for high-voltage direct-current (HVDC) transmission systems, and so on.

**What obstacles (technical, political, geographic) needed to be overcome?**

Terminological Obstacles to Naming p and q, and to Giving an Electrical Unit to q

The formulation of p consisted of two terms in the bottom line of (7). It was clear without any doubt that each term was real “instantaneous power” with a physical unit of Watt or [W] because each term was given by a product of an instantaneous voltage and an instantaneous current in the same phase. The formulation of q also consisted of two terms in the right hand of (8). Although each term looked like “instantaneous power,” it did not represent any real “instantaneous power” from a point of view of both physics and electrical engineering. The reason was because each term was given by another product of an instantaneous voltage in one phase and an instantaneous current in the other phase. Before the emergence of the p-q theory in 1983, no one had defined and formulated either term in (8) as long as Akagi knew, or no one would have made any discussion on it even if someone had defined and formulated it. Hence, he named the new pair of p and q “instantaneous real power” and “instantaneous imaginary power,” respectively, considering “imaginary” to be the antonym of “real.” This made it easy to distinguish one from the other, thus contributing to avoiding confusion and/or misunderstandings in the usage of technical terminology.

The p-q theory defined and formulated another new pair of instantaneous “active” and “reactive” powers in each phase. The reason for this naming was as follows: Both instantaneous active and reactive powers targeted a single-phase circuit such as either the α-phase circuit or the β-phase circuit in a two-phase three-wire circuit. Thus, both powers were the same as traditional active and reactive powers in terms of targeting or starting with, not a three-phase circuit, but a single-phase circuit. Thus, Akagi introduced the well-known dual terms “active” and “reactive” to the two new instantaneous powers, taking into account a tradition in electrical engineering. Finally, the two pairs of core technical terms, one pair of “real” and “imaginary” and the other pair of “active” and “reactive,” were introduced to the two new pairs of powers. Hence, this naming also made it easy to distinguish one pair of instantaneous “real” and “imaginary” powers from the other pair of instantaneous “active” and “reactive” powers.

Some professors and research scientists/engineers in the field of power electronics and power engineering have often recognized and understood as a synonym the instantaneous “imaginary” power for a three-phase circuits and the instantaneous “reactive” power for each phase in the three-phase circuit, without distinguishing one from the other. Akagi supposed that they might be interested in, or make active use of, a numerical value itself of the instantaneous imaginary power in a three-phase circuit from an engineering point of view, although the instantaneous imaginary power was different from the instantaneous reactive power in physical meaning. It was clear from (8) that a physical unit of q was not Watt or [W]. Moreover, an electrical unit of q was neither existing “volt-ampere” or [VA] nor “volt-ampere-reactive” or [var] because the two existing electrical units targeted, or started with, single-phase circuits on a time average. Hence, the authors of [4] introduced to q a new electrical unit, “volt-ampere-imaginary” or [vai]. However, international organizations such as the IEEE and the IEC have not yet accepted it or given any electrical unit to q. It would take many years to overcome this obstacle.

Theoretical Obstacles to Explaining and Proving a Physical Meaning of q

As soon as Akagi defined and formulated the pair of p and q in (9), he decided to solve the following theoretical and challenging issues on q; how to give a clear explanation to the physical meaning of q and how to give a mathematical proof to the explanation. It was in November 1981 that Akagi completed the definition and formulation of the pair of p and q. At that time, he was afraid of being unable to solve the above theoretical issues, even if he spent many years, or forever in the worst case. This reason was so simple that Akagi was unable to answer either “yes” or “no” to the following question: “Whether or not does any solution to the theoretical issues exist?” This question to himself made Akagi decide to do the theoretical research alone because it was too risky for his graduate student to challenge the theoretical issues. If he/she had succeeded in, he/she would have got the best score or grade A+ (the best). If he/she had failed in, he/she would have got the worst score or grade F (fail). However, even If Akagi had failed in, he would have lost nothing except for a long or short time he spent on his theoretical research. Hence, Akagi made up in his mind to “take it easy” because no one knew any solution and even either the existence or the non-existence of a solution. As a result, the situation and his mind made him to do the theoretical research alone, slowly but surely, and discontinuously when he had a free time. At that time, he was so busy in teaching, as well as in supervising his graduate students, that he felt little mental stress and pressure from the discontinuous theoretical research. He had the good fortune in a sense.

Actually, it took three months to solve the theoretical issues from the completion of defining and formulating the pair of p and q in (9) to that of giving the mathematical proof to q from (10) to (15). This proof allowed him to explain the physical meanings of the instantaneous imaginary power q as well as each-phase instantaneous active and reactive powers. Behind Akagi’s successful mathematical proof, he had an inspiration of an innovative but one-step development from (10) to (11) at home in the late evening of one day in February 1982. This inspiration brought the complete mathematical proof to him immediately. Then, he attained the full solution to the theoretical issues on the same day. Professors and research scientists might ask him the following question: “Which were the three months he spent on his mathematical proof, too short or too long?” Akagi’s clear answer was as follows: “Fortunately, the three months were definitely too short against his advance prediction.”

Technical Obstacles to Verifying the Validity of the p-q Theory.

Akagi encountered technical obstacles to experimental validation of the p-q theory. Although bipolar junction transistors (BJTs) and diodes were available from the market around 1980, both had low-voltage and small-current ratings. In addition, the BJT was suffering from a long carrier-storage time, while the diode from a large reverse-recovery current. The world’s-first instantaneous imaginary-power compensator had the ratings of a three-phase 100 V, 0.5 kVA, and 50 Hz. It was based on a three-phase voltage-source pulse-width modulation (PWM) converter using six BJTs and diodes, in which a BJT and a diode were connected in anti-parallel with each other, as shown in Fig. 2.

Akagi and his team made efforts at realizing stable operation of the compensator prototype, putting up with repeating fine adjustments of off-set and drift levels in the analogue multipliers and operational amplifiers used in the control circuit. At that time, it was impossible to apply commercially-available micro-computers and analogue-to-digital (A/D) converters to the control circuit because their processing and/or conversion speeds were too slow to use. They managed to obtain experimental waveforms of voltage and current, as well as experimental harmonic current spectra, from such a prototype as shown in Fig. 2. Finally, they succeeded in attaining the world’s first experimental results as shown in Figs. 3 and 4, thus resulting in verifying the validity of the p-q theory.

Looking back now, the power rating of 0.5 kVA was too low to do experiment even if the imaginary-power compensator was considered to be a downscaled prototype. The reason was because the compensator aimed for medium to high-power applications. Although power semiconductor technology was developing by leaps and bounds, the BJTs and diodes available on the market at that time were the greatest bottleneck to the prototype in terms of low-voltage and small-current ratings, and poor switching characteristics. This bottleneck made it difficult to design and build a compensator prototype with a higher power rating because it needed to provide a high-performance current-feedback system with a broad control bandwidth. The control bandwidth was proportional to the carrier frequency of PWM. The higher the carrier frequency, the broader the control bandwidth. However, it was difficult to satisfy a higher power rating and a higher carrier frequency simultaneously at that time. At present, it is possible to design, build, and test such a compensator with a power rating of 500 kVA or higher, adopting the same power circuit configuration as that in Fig. 2. The reason is that the latest 1.2-kV, 1.2-kA SiC (Silicon Carbide)-MOSFET/SBD dual (two-in-one) modules available from the market have the capability of high-frequency and high-speed switching operation with a much shorter carrier-storage time and a much smaller reverse-recovery current, thus resulting in producing low conducting and switching losses.

**What features set this work apart from similar achievements?**

1) One of the three features of the p-q theory for three-phase circuits was to define and formulate a new pair of three-phase instantaneous real and imaginary powers p and q, respectively, in (9) and (10). This definition and formulation were characterized by lumping three single-phase circuits together, and by using only the information of the present voltages and currents without any time delay [1].

2) Another feature was to give a strict mathematical proof, or the proof by mathematical expressions from (10) to (15), to a clear explanation of the physical meanings of q and the other new pair of instantaneous active and reactive powers [1]. One of the most important insights and/or findings was as follows: One arithmetic sum of the instantaneous active power in the α-phase and that in the β-phase was always equal to the three-phase instantaneous real power p. The other arithmetic sum of the instantaneous reactive power in the α-phase and that in the β-phase was always zero. This insight and new finding allowed Akagi to classify each-phase harmonic and sideband currents into the following three groups from their origin: The first one came only from p, the second one only from q [2], and the third one from both p and q [3].

3) The last feature was to verify experimentally that the world’s-first three-phase instantaneous imaginary-power compensator with a small dc capacitor (ideally without it) was able to eliminate the full, or a part, of harmonic and sideband currents coming only from q or belonging to the above second group [2]. This compensator was followed by a three-phase pure active filter for power conditioning [3]. The p-q theory rendered system design and control strategy appropriate and flexible in response to the objectives of installation of the pure active filter. It was characterized by installing an appropriately-designed capacitor on the dc side, and providing an application-oriented and flexible control strategy for eliminating harmonic and sideband currents coming from both p and q or belonging to the above third group. The world’s-first waveforms and spectra obtained from an experimental system including a three-phase 200-V, 7-kVA, 50-Hz pure active filter verified the validity and effectiveness of both the p-q theory and the pure active filter [3].

**Supporting texts and citations to establish the dates, location, and importance of the achievement:** Minimum of five (5), but as many as needed to support the milestone, such as patents, contemporary newspaper articles, journal articles, or chapters in scholarly books. 'Scholarly' is defined as peer-reviewed, with references, and published. **You must supply the texts or excerpts themselves, not just the references**. At least one of the references must be from a scholarly book or journal article. All supporting materials must be in English, or accompanied by an English translation.

[1] H. Akagi, Y. Kanazawa, and A. Nabae, “Generalized theory of the instantaneous reactive power in three-phase circuits,” Proceedings of IEE of Japan International Power Electronics Conference (IPEC-Tokyo), March 1983, pp. 1375-1386. (1650 citations)

[2] H. Akagi, Y. Kanazawa, and A. Nabae, “Instantaneous reactive power compensators comprising switching devices without energy storage components,” IEEE Transactions on Industry Applications, vol. 20, no. 3, pp. 625-630, May/June, 1984. (4877 citations)

[3] H. Akagi, A. Nabae, and S. Atoh, “Control strategy of active power filters using multiple voltage-source PWM converters,” IEEE Transactions on Industry Applications, vol. 22, no. 3, pp. 460-465, May/June, 1986. (1023 citations)

[4] H. Akagi, E. H. Watanabe, and M. Aredes, “Instantaneous power theory and applications to power conditioning,” IEEE Press, 400 pages, 2007 (first edition), and 472 pages, 2017 (second edition). (3580 citations in total)

[5] H. Akagi, “New trends in active filters for power conditioning,” IEEE Transactions on Industry Applications, vol. 32, no. 6, pp. 1312-1322, Nov./Dec. 1996. (2694 citations)

[6] H. Akagi, “Active harmonic filters (invited paper),” Proceedings of the IEEE, vol. 93, no. 12, pp. 2128-2141, Dec. 2005. (1406 citations)

[7] A. Iizuka, M. Kishida, Y. Mochinaga, T. Uzuki, K. Hirakawa, F. Aoyama, and T. Masuyama, “Self-commutated static var generator at Shintakatsuka substation,” Proceedings of IEEJ International Power Electronics Conference (IPEC-Yokohama), April 1995, pp. 609-614.

[8] F. Harashima, H. Inaba, and K. Tsuboi, “A closed-loop control system for the reduction of reactive power required by electronic converters,” IEEE Transactions on Industrial Electronics and Control Instrumentation, vol. 23, no. 2, pp. 162-166, May 1976.

[9] K. Srinivasan and C. T. Nguyen, “Instantaneous three-phase reactive power for digital Implementation: Definition and determination,” Proceedings of the IEEE, vol. 66, no. 8, pp. 986-987, Aug. 1978.

[10] L. Gyugyi, “Reactive power generation and control by thyristor circuits,” IEEE Transactions on Industry Applications, vol. 15, no. 5, pp. 521-532, Spt./Oct., 1979.

[11] I. Takahashi and A. Nabae, “Universal power distortion compensator of line commutated thyristor converter,” Proceedings of IEEE Industry Applications Society Annual Meeting, Oct. 1980, pp. 858-863.

[12] S. Miyairi, H. Akagi, T. Fukao, and M. Fujita, “Equivalence in harmonics between cycloconverters and bridge converters,” IEEE Transactions on Industry Applications, vol. 15, no. 1, pp. 92-99, Jan/Feb, 1979.

[13] A. Ferrero and G. Superti-Furga, “A new approach to the definition of power components in three- phase systems under non-sinusoidal conditions,” IEEE Transactions on Instrumentation and Measurement, vol. 40, no. 3, pp. 568-577, June 1991.

[14] J. L. Willems, “A new interpretation of the Akagi-Nabae power components for non- sinusoidal three-phase situations,” IEEE Transactions on Instrumentation and Measurement, vol. 41, no. 4, pp. 523-527, Aug. 1992.

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