Notes on p-q theory

Asyraf bin Ahmad Saferi

June 15, 2015

Contents

1 Basis of the p-q theory 11.1 The Clarke Transformation . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Calculation of Voltage and Current Vectors when Zero-Sequence Components are Excluded . . . . . . . . . . . . 3

1.2 Three-Phase Instantaneous Active Power in Terms of Clarke Com-ponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The Instantaneous Powers of the p-q Theory . . . . . . . . . . . 5

2 The p q Theory in Three-phase, Three-Wire Systems 62.1 Comparisons with the Conventional Theory . . . . . . . . . . . . 9

2.1.1 Example #1 - Sinusoidal Voltages and Currents . . . . . 102.1.2 Example #2- Balanced Voltages and Capacitive Loads . . 102.1.3 Example #3 - Sinusoidal Balanced Voltage and Nonlinear

Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 The p q Theory in Three-Phase, Four Wire Systems 153.1 The Zero-Sequence Power in a Three-Phase Sinusoidal Voltage

Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Presence of Negative-Sequence Components . . . . . . . . . . . . 173.3 General Case Including Distortions and Imbalances in the Volt-

ages and in the Currents . . . . . . . . . . . . . . . . . . . . . . . 183.4 Physical Meanings of the Instantaneous Real, Imaginary and

Zero-Sequence Powers . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Avoiding the Clarke Transformation in the p q theory . . . . . 21

1 Basis of the p-q theory

p-q Theory is based on a set of instantaneous power defined in time domain.No restrictions are imposed on the voltage or current waveform and it can beapplied to three-phase systems with or without a neutral wire for three-phasegeneric voltage and current waveforms which makes it valid for for both steadystate and transient state. Following chapters will explain this theory is veryefficient and flexible in designing controllers for power conditioners based onpower electronics devices.

This theory considers the three-phase system as a unit, not a superpositionor sum of three single-phase circuits, compared to other traditional concepts

1

Figure 1: Graphical representation. (a) The abs to transformation(Clarketransformation). (b) Inverse to abc transformation (inverse Clarke transfor-mation).

of power which treat a three-phase system as three single-phase circuits. Thep-q theory first transforms voltage and currents from the abc to 0, an d thendefines instantaneous power in this coordinates.

The p-q Theory uses the 0 transformation, also known as Clarke trans-formation, which consists of a real matrix that transforms three-phase voltagesand currents into 0 stationary reference frames.

1.1 The Clarke Transformation

This transformation maps the three-phase instantaneous voltages in the abcphase, va, vb and vc into the instantaneous voltages on the 0-axes v, v andv0.

Followings are the Clarke Transformation and its inverse transformation ofthree-phase generic voltages: v0v

v

= 23

12

12

12

1 12 120

32

32

vavbvc

(1) vavbvc

= 23

12

1 012 12

32

12 12

32

v0vv

(2)Similarly, three-phase generic instantaneous line currents, ia,ib and ic can

be transformed on the /alpha/beta0 axes by i0ii

= 23

12

12

12

1 12 120

32

32

iaibic

(3)and its inverse transformation is iaib

ic

= 23

12

1 012 12

32

12 12

32

i0ii

(4)2

0 transformation separates the zero-sequence components from the abc-phase components. The and axes make no contribution to zero-sequencecomponents.

For three-phase, three-wire system, no zero-sequence exists. Thus we caneliminate i0 which results in simplification. If three-phase voltages are balancedin a four wire system, no zero-sequence voltage and current components arepresent, hence, v0 can be eliminated.

The Clarke transformation and its inverse transformation when v0 is elimi-nated are:

[vv

]=

2

3

[1 12 120

32

32

] vavbvc

(5)and vavb

vc

= 23

1 0 12 32 12

32

[ vv

](6)

Similar equations hold in the line currents.

[ii

]=

2

3

[1 12 120

32

32

] iaibic

(7)and iaib

ic

= 23

1 0 12 32 12

32

[ ii]

(8)

1.1.1 Calculation of Voltage and Current Vectors when Zero-SequenceComponents are Excluded

If v0 can be neglected, instantaneous voltage vector can be defined from theinstantaneous and -voltage components:

v = v + jv (9)

Similarly, if i0 can be neglected, instantaneous current vector is defined as:

i = i + ji (10)

Above instantaneous vector v and i can be represented on a complex plane,where the real axis is the and imaginary axis is . Note that the vectors definedabove are function of time and should not be misinterpreted as phasors. Thiswill later be used to define a new concept of instantaneous complex apparentpower.

Consider the following sinusoidal balanced phase voltages and line currentsof a three-phase linear circuit.

va(t) =

2V cos(t+ V )

vb(t) =

2V cos(t+ V 2pi3 )vc(t) =

2V cost(t+ V +

2pi3 )

ia(t) =

2Icos(t+ I)

ib(t) =

2Icos(t+ I 2pi3 )ic(t) =

2Icost(t+ I +

2pi3 )

(11)

3

Figure 2: Vector representation of voltages and currents on the referenceframes.

The angles v and i are the voltage and current phases, respectively, withrespect to a given reference.

Set of equation (11) consist of a single symmetrical component in the funda-mental positive sequence which means they are sinusoidal and balanced. Theycan be transformed to the reference frames by using equation (5) and (7).The transformed equation are given by:{

v =

3V cos(t+ V )

v =

3V sin(t+ V )and

{i =

3Icos(t+ I)

i =

3Isin(t+ I)

The voltage e and current vector mathbfi are derived as follows:

e = v + jv e = 3V [cost+ V ) + jsin(t+ V )]e = 3V exp j(t+ V ) (12)

i = i + ji e = 3i[cost+ I) + jsin(t+ I)]e = 3i exp j(t+ I) (13)

Hence, in the case of three-phase balanced sinusoidal system, the voltage andthe current vectors have constant amplitudes and rotate in clockwise direction( sequence) at the angular frequency .

The same result will be found if the instantaneous abc voltages and currentsare taken directly to compose vectors on a complex plane. If we kept the relativeposition of a,b and c axes and the and as suggested in Fig.1, we can confirmthe the result. The a axis must be coincident with the that is the real axisof the complex plane. The axis is the imaginary axis shifted by pi/2 from thereal axis. Thus, the complex voltage vector can be composed by

eabc = va exp j0 + vb exp j2pi/3 + vc expj2pi/3 (14)

Replacing the time function for va,vb, and vc given in 11 and some manipu-lation, the following equation is obtained:

eabc =3

2

2V [cos(t+ V ) + jsin(t+ V )] =

3

2

2V exp j(t+ V ) (15)

It can be noted that the voltage vector eabc in 15 has a constant amplitude,and rotates in clockwise direction at angular frequency , as e in 12. Hence,

4

through comparison, we can write the following relation:

e =

2

3eabc (16)

This vector representation of three-phase instantaneous voltages and currenthas been used increasingly in power electronics field and have found used invector control of ac motor drives, space-vector pulse-width modulation (PWM)and control of power conditioners.

1.2 Three-Phase Instantaneous Active Power in Terms ofClarke Components

The Clarke transformation have the property of being invariant in power. Thisfeature makes it very suitable when the focus is put on the analysis of instan-taneous power in three-phase systems.

While all of the traditional power definition summarized last chapter requirethe system to be in steady state, the three-phase instantaneous active powerhas a clear and universally accepted physical meaning and is also valid duringtransient states.

For a three-phase system with or without a neutral conductor in the steadystate or during transients, the three-phase instantaneous active power p3(t)describes the total instantaneous energy flow per second between two subsys-tems.

Equation for calculating instantaneous active power, p3(t) from instanta-neous phase voltages and line currents:

p3(t) = va(t)ia(t) + vb(t)ib(t) + vc(t)ic(t) p3 = vaia + vbib + vcic (17)where va, vb, and vc are the instantaneous phase voltages and ia, ib, and ic

the instantaneous line currents. For a system without neutral wire, the volt-ages are measured from a commont point of references. (Sometimes called theground or fictitious star point). This reference point can be set arbitrarilyand pe calculated from 17, always results in the same value. For instance, ifthe b phase is chosen as a reference point, the measured phase voltages andthe three-phase instantaneous active power, p3, are calculated as:

p3 = (va vb)ia + (vb vb)ib + (vc vb)ic = vabia + vcbic (18)The three-phase instantaneous active power can be calculated in term s of

the 0 components if 4 and 6 are used to replace the abc variables in 17.

pe = vaia + vbib + vcic p3 = vi + vi + v0i0 (19)Equation ?? shows the property of power invariance as a result o fusing the

Clarke transformation. The p q Theory will exploit this feature.

1.3 The Instantaneous Powers of the p-q Theory

p-q theory is defined in three-phase systems with or without a neutral conductor.Three instantaneous powers - the instantaneous zero-sequence power p0, the

5

Figure 3: Three-phase instantaneous active power

instantaneous real power p, and the instantaneous imaginary power q - aredefined from the instantaneous phase voltages and line currents on the 0axes as p0p

q

= v0 0 00 v v

0 v v

i0ii

(20)In this case, only the instantaneous powers defined on the axes exist.

(Product of v0i0 always zero.) Hence, in three-phase, three- wire systems, theinstantaneous real power p represents the total energy flow per time unit interms of components. p3 = p).

Instantaneous imaginary power q has a nontraditional physical meaning,which will explained in upcoming section.

2 The p q Theory in Three-phase, Three-WireSystems

We can also define the p q Theory for three-phase, three-wire systems usinginstantaneous voltage and current vectors as defined in 9 and 10. The conven-tional concept of the complex power uses a voltage phasor and conjugate of acurrent phasor which make it valid only for system in steady state with a fixedline frequency. Using instantaneous vectors of current and voltage, a new def-inition of instantaneous complex power can be made. Instantaneous complexpower s is defined as the product of the voltage vector e and the conjugate ofthe current vector i, given in the form of complex numbers:

s = e i = (v + jv)(i ji) = (vi + v i) p

+ (vi vi) q

(21)

As can be observed from 20, the instantaneous real power and imaginarypowers defined in 21 are part of it. s has no restrictions on it, since it is definedfor instantaneous voltages and current, so it can be applied for steady state ortransients.

In earlier references ([2]-[5] if I refer the text book), the original definitionof p and q was based on the following equation:

6

[pq

]=

[v vv v

] [ii

](22)

Both of 20 and 22 are applicable to the analysis and design in the same manner.But we decides to adopt 20 because positive value of instantaneous imaginarypower q in 20 corresponds to the product of a positive sequence voltage anda lagging (inductive) positive-sequence current, in agreement with conventionalconcept of reactive power.

For better explanation of physical meaning of the powers defined in p-qTheory, the currents will be set as functions of voltages and the real andimaginary powers p and q. From 20, we can write:[

ii

]=

1

v2 + v2

[v vv v

] [pq

](23)

Expanding the right hand side:[ii

]=

1

v2 + v2

[v vv v

] [p0

]+

1

v2 + v2

[v vv v

] [0q

][ipip

]+

[iqiq

](24)

We can define the current components as below:Instantaneous active current on the axis ip:

ip =v

v2 + v2

(25)

Instantaneous reactive current on the axis iq:

iq =v

v2 + v2

(26)

Instantaneous active current on the beta axis ip:

ip =v

v2 + v2

(27)

Instantaneous reactive current on the beta axis iq:

iq =v

v2 + v2

(28)

The instantaneous power on the and coordinates are defined as p andp respectively, and can be calculated from instantaneous voltages and currentson the axes as below:[

pp

]=

[vivi

]=

[vipvip

]+

[viqviq

](29)

Take note that, for 3-phase 3-wire system, equation 19 is equal to 20. Fromequation 29 and 20 the real power can be given by the sum of p and p .Rewriting the sum by using 29 yield the following equation:

7

p = vip + vip + viq + viq

=v2

v2 + v2

p+v2

v2 + v2

p+vvv2 + v

2

q +vvv2 + v

2

q (30)

Two important points can be taken from above equation.

1. Instantaneous real power p is given by:

vip + vip = pp + pp = p (31)

2. Following relation exits for terms dependent on q

viq + viq = pq + pq = 0 (32)

Above equations suggest the separation of powers in the following types.Instantaneous active power on the axis pp

pp = v ip = v2

v2 + v2

p (33)

Instantaneous reactive power on the axis pq

pq = v iq = vvv2 + v

2

q (34)

Instantaneous active power on the axis pp

pp = v ip =v2

v2 + v2

p (35)

Instantaneous reactive power on the axis pq

pp = v iq = vvv2 + v

2

q (36)

Take note that watt[W] can be used as unit for all powers. The above equationlead to the following important conclusion:

Instantaneous current i is divided into the instantaneous active compo-nent ip and the instantaneous reactive component iq as shown in ??and ??. The same division is made for the currents on the axis.

The sum of the axis instantaneous active power pp given in 33 and the axis instantaneous active power pp given in 35 corresponds to instan-taneous real power p.

The sum of pq and pq is always zero. Hence, they neither make acontribution to the instantaneous nor average energy flow between thesource and the load in a three-phase circuit. This is the reason thatthey were named instantaneous reactive power on the and axes. Theinstantaneous imaginary power q is a quantity that gives the magnitudeof the powers pq and pq.

8

Figure 4: Physical meaning of the instantaneous real and imaginary powers.

As will be shown later, since sum of reactive power pq is always zero, theircompensation does not need any energy storage system.

If we replace the variables with its abc equivalence, the following relationcan be found.

q = vi valphai = 13

[(va vb)ic + (vb vc)ia + (vc va)ib]

=13

(vabic + vbcia + vcaib) (37)

Following important remark can be written:

The imaginary power q is proportional to the quantity of energy that isbeing exchanged between the phases of the system. It does not contributeto the energy transfera between the source and the load at any time.

aThe term energy transfer is used here in a general manner, meaning not only theenergy delivered to the load, but also the energy oscillating between the source and theload

Figure 4 summarizes above explanations about real and imaginary power.Important to note that conventional power theory defined reactive power asa component of the instantaneous(active) power, which has an average valueequal to zero. It is not so for p q Theory. The imaginary power means a sumof products of instantaneous three-phase voltage and current portions that donot contribute to the energy transfer between two subsystems at any time. Theterm instantaneous reactive power in a three-phase system is used as synonymof imaginary power and hence, have the same physical meaning.

2.1 Comparisons with the Conventional Theory

Some examples will be presented to help understand the meaning of p and q.The first one will be on linear circuit with a sinusoidal voltages and currentsand others will be on sinusoidal voltages supplying nonlinear loads.

9

2.1.1 Example #1 - Sinusoidal Voltages and Currents

Suppose a three-phase ideal voltage source supplying power to a three-phasebalanced impedance. The phase voltages and line currents can be expressed as

va(t) =

2V sin(t)

vb(t) =

2V sin(t 2pi3 )vc(t) =

2V sin(t+ 2pi3

and

ia(t) =

2Isin(t+ )

ib(t) =

2Isin(t 2pi3 + )ic(t) =

2Isin(t+ 2pi3 + )

(38)

The transformation of the above voltages and currents are:{v =

3V sin(t)

v =

3V cos(t)and

{i =

3Isin(t)

i =

3Icos(t)(39)

The above two equations make it possible to calculate the real and imaginarypowers: {

p = 3V Icosq = 3V Isin (40)

Both values are constant in this example. The real power p is equal to con-ventional definition of the three-phase active power P3, whereas the imaginarypower q is equal to the conventional three-phase reactive power Q3. This ex-ample shows the correspondences between pq Theory and conventional theoryin the case of having sinusoidal balanced voltages and linear loads.

If the load has inductive characteristics, the imaginary power q has a positivevalue, and if the load is capacitive, negative value, in accordance with the mostcommon definition of reactive power.

2.1.2 Example #2- Balanced Voltages and Capacitive Loads

For a three-phase balanced voltage with a three-phase balanced capacitive load(capacitanceC),the load is balanced under steady state. The real and imaginary powers ob-tained are:

p = 0

q = 3V 2Xc(41)

The term Xc represents reactance of the capacitor. As expected, no power isflowing from the source to the load. Also, the imaginary power is constant andcoincident with the conventional three-phase reactive power.

Let consider a second case, a capacitor with capacitance C is connectedbetween phases a and b. The instantaneous real and imaginary power are givenby: {

p = 3 3V2

XCsin(2t+ pi3

q = 3V 2XC [1 + cos(2t+ pi3(42)

Each power in the above equation has no constant part, and present an oscil-lating part. If we consider from conventional power theory, we would expectonly reactive power (average imaginary power) and no real power at all. How-ever the results are different and should be discussed. The reason why the realpower is not zero is because the capacitor terminal voltage is varying as a si-nusoidal wave and therefore, it is being charged and discharged, justifying theenergy flow given by p. In fact, if it is considered that a turbine is powering

10

Figure 5: Three-Phase voltage source supplying a thyristor rectifier.

the generator, it will have to produce a positive torque when the capacitor ischarging, or a negative torque when it is discharging. This obviously will pro-duce a torque ripple in its shaft. If we consider previous example, having threebalanced capacitors, one capacitor is discharging while the others are charging.In steady-state conditions, there is no total (three-phase) energy flow from thesource to the capacitors. It is clear from this example that under unbalancedload conditions the p q Theory presents some important insights that cannotbe seen with the conventional frequency-domain theory under unbalanced loadconditions.

2.1.3 Example #3 - Sinusoidal Balanced Voltage and Nonlinear Load

Now we assume that a three-phase voltage source is balanced and purely sinu-soidal as in 38, and the load is a thyristor rectifier operating with a firing angleequal to 30o. The commutation angle is assumed to be null and the smoothingreactor at the dc side is large enough to eliminate totally the ripple in the dccurrent. Figure 5 shows the idealized circuit.

Figure 6(a) shows that the rectifier dc output voltage waveform vd and 6(b)shows the a-phase voltage and the ideal current waveforms. This three-phasetypical current waveform contains, beside the fundamental, harmonics on theorder (6n 1;n = 1, 2, 3 . . .). The (6n 1)th harmonics are of the negative-sequence type, whereas the (6n + 1)th harmonics are of the negative-sequencetype (See Fig 2-7 in the textbook). The line currents of a six-pulse thyristorrectifier operating with a firing angle equal to 30o can be represented as:

ia(t) =

2I1sin(t pi6

2I5sin(5t+pi

6

2I7sin(7t pi6 . . . (43)

ib(t) =

2I1sin(t 2pi3 pi

6)

2I5sin(5t+2pi

3 pi

6)

+

2I7sin(7t 2pi3 pi

6

ic(t) =

2I1sin(t+2pi

3 pi

6)

2I5sin(5t 2pi3 pi

6)

+

2I7sin(7t+2pi

3 pi

6) . . .

11

Figure 6: (a) Rectifier dc output voltage vd and (b) a phase voltage and currentwaveforms

Figure 7: Three-Phase voltage source supplying a thyristor rectifier.

12

Figure 7 (a) and (b) show the real power and imaginary power, q, respec-tively. The real power was calculated using the voltages and currents on theac side, and it is the same as it would be calculated by the product of the dcvoltage vd and the dc current. The instantaneous input power and at the acside of the rectifier is equal to the output power at the dc side if there are nolosses in the rectifier. The imaginary power is only defined for ac multiphasecircuits, so it can be only calculated at the ac side.

These two power have constant values and a superposition of oscillating constant?I thoughtits oscillat-ing fromthe figure?

constant?I thoughtits oscillat-ing fromthe figure?

components. It is interesting to separate p and q into two parts:

Real power:p = p+ p (44)

Imaginary Power:q = q + q

The real power p represents the total(three-phase) energy flow per time unityin the circuit. The average value p represents the energy flowing per time unityin one direction only. If p and q are calculated in terms of the abc components,they result in

p = 3V I1cospi6

q = 3V I1sinpi6

(45)

The average values of the real and imaginary power given by the p q Theoryagree with the conventional definition of three-phase active and reactive pow-ers obtained through Budeanu approach. The oscillating part p represents theoscillating energy flow per time unity, which naturally produces zero averagevalue, representing an amount of additional power flow in the system withouteffective contribution to the energy transfer from the source of the load or fromthe load to the source.

Imaginary power q gives the magnitude of the instantaneous reactive powersp() and p(q), or the magnitude of the corresponding powers in teh abc system.As explained before, although the instantaneous reactive powers exist in eachaxis, their sum is zero all the time. The average value of imaginary power qcorresponds to the conventional three-phase reactive power and does not con-tribute to energy transfer. The oscillating component of the imaginary powerq corresponds also to a power that is being exchanged among the three phases,without transferring any energy between source and load. In the present exam-ple, both oscillating real(p and imaginary (q powers are related to the presenceof harmonics exclusively in the load currents.

Figure 8 shows the a-phase current component responsible for producingthe average real power p and the average imaginary power q. For reference, thevoltages in this same phase are plotted together. The currents were calculatedusing (3.23) and (3.25), for p = p, and (3.24) and (3.26), for q = q, followedby to abc transformation. The current i(ap) is perfectly in phase with thevoltage va whereas i(aq is delayed exactly by 90

o, acting like an inductive load,as is well known for a line-commutated thyristor rectifier.

Figure 9 (a), (b) and (c) show the a phase current i(ap responsible forproducing the oscillating power p, the current i(aq that produces the oscillatingpower q, and the sum both, respectively. These currents were calculated using(3.23) and (3.25) for p = p, and (3.24) and (3.26) for q = q, followed by toabc transformation. The sum of the four components, is equal to the originalsquare wave current shown in figure. 3-6 (b).

13

.Figure 8: Voltage in a phase; currents i(ap and i(aq

Figure 9: Currents i(ap, iaq and the sum (i(ap+ iaq).

14

The p q Theory has prominent merit of allowing complete analysis andreal-time calculation of various powers and respective currents involved in athree-phase circuit. Though, it is not the main point. Knowing in real timethe values of undesirable currents in a circuit allow us to eliminate them. Forinstance, if the oscillating powers are undesirable, by compensating the currenti(ap) and i(aq) of the load and their correspondent currents in phases b andc the compensated current drawn from the network would become sinusoidal.this is one of the basic idea of active filtering that will be presented in followingchapter.

3 The p q Theory in Three-Phase, Four WireSystems

The presence of fourth conductor, namely the neutral conductor, is very com-mon in low voltage distribution systems, in addition to the cases of groundedtransmission systems. These systems are classified as three-phase, four wire sys-tems. The simplified transformation and equations used in the previous sectionare not applicable to these cases. This section presents cases involving the threeinstantaneous powers p, q, and p0 of the p q Theory.

The three-phase, four wire systems can include both zero-sequence voltageand current as a generic case. These systems allow all the three-line currentsia, ib and ic to be independent, whereas two of the three line currents are inde-pendent in three-phase, three-wire systems. Therefore, to represent the systemcorrectly, the instantaneous zero-sequence power p0 defined on the 0 referenceframes, p0 has to be introduced as the third instantaneous power in addition tothe instantaneous power p and the instantaneous power q.

They are defined mathematically as: p0pq

= v0 0 00 v v

0 v v

i0ii

(46)We should rewrite the three-phase instantaneous active power in terms of

the 0 components:

p(3) = vaia + vbib + vcic = vi + vi + v0i0 = p+ p0 (47)

This equation shows that the three-phase instantaneous active power p3 isequal to the sum of the real power p and the zero-sequence power p0.

3.1 The Zero-Sequence Power in a Three-Phase SinusoidalVoltage Source

To better understand zero-sequence power, let consider a three-phase sinusoidalvoltage source consists of positive- and zero-sequence voltages at the angularfrequency . Symmetrical components of this voltage source were calculatedbased on their phasors. Although this analysis is only valid for steady-statecondition, it can better facilitate understanding. These symmetrical components

15

were then transformed back to the time domain and rewritten as time functionsas

va =

2V+sin(t+ v+) +

2V0sin(t+ v0) (48)

vb =

2V+sin(t 2pi/3 + v+) +

2V0sin(t+ v0)

vc =

2V+sin(t+ 2pi/3 + v+) +

2V0sin(t+ v0)

It is assumed that the current also has positive- and zero sequence compo-nents, that is

ia =

2I+sin(t+ i+) +

2I0sin(t+ i0) (49)

ib =

2I+sin(t 2pi/3 + i+) +

2I0sin(t+ i0)

ic =

2I+sin(t+ 2pi/3 + i+) +

2I0sin(t+ i0)

Applying Clarke transformation, the following voltages and currents on the0 reference frames are obtained:

v =

3V+sin(t+ v+ (50)

v =

3V+cos(t+ v+

v0 =

6V0sin(t+ v0

and

i =

3I+sin(t+ i+ (51)

i =

3I+cos(t+ i+

i0 =

6I0sin(t+ i0

Since the real and imaginary powers defined in 47 depend only on thepositive-sequence voltage and current, these powers are similar in content andin meaning to those analyzed in previous section. Hence, analysis here will befocused on the zero-sequence power p0 given by

p0 = 3V0I0cos(v0 i0) 3V0I0cos(2t+ v0 + i0) = p0 + p0 (52)

This value has the same characteristics as the instantaneous power in singlephase circuit. It has an average value and an oscillating component at twice theline frequency. The average value p0 represents a unidirectional energy flow. It I dont

knowabout this.Where canI get infoon thisinstan-taneouspowerin singlephase cir-cuit?

I dontknowabout this.Where canI get infoon thisinstan-taneouspowerin singlephase cir-cuit?

has the same characteristics as the conventional (average) active power. Theoscillating component p0 also transfers energy instantaneously. The averagevalue is zero due to its oscillating nature. The analysis shows that, in principle,the average value of the zero-sequence power helps to increase the total energytransfer in the sense can be considered a positive point. However, even for thesimplest case of a zero-sequence component in the voltage and current, the zero-sequence power p0 cannot product constant power p0 alone. Which mean, p0will always consists of p0 plus p0, if cos(v0 i0 6= 0. The elimination of p0 isaccompanied by elimination of p0 together. This is one interesting characteristicof this power, and one of the reasons why it is not welcome in most circuits.

16

In summary, the zero-sequence power p0 exists only if there are zero-sequencevoltage and current It is an instantaneous active power contributing to energyflow, just like a single phase circuit. The average zero-sequence power p0 isalways associated to the oscillating component p0. Therefore, there is no wayto eliminate the oscillating component and keep the average part alone.

3.2 Presence of Negative-Sequence Components

Presence of negative-sequence components may be a serious problem for a three-phase, balanced, positive sequence voltage. Analyzing for the case in which thevoltages are sinusoidal waveforms with a frequency of consisting of positive-,negative, and zero-sequence components as given below:

va =

2V+sin(t+ v+ +Vsin(t+ v) +

2V0sin(t+ v0) (53)

vb =

2V+sin(t 2pi/3 + v+ +Vsin(t+ 2pi/3 + v) +

2V0sin(t+ v0)

vc =

2V+sin(t+ 2pi/3 + v+ +Vsin(t 2pi/3 + v) +

2V0sin(t+ v0)

(54)

and the currents consist of

va =

2V+sin(t+ v+ +Vsin(t+ v) +

2V0sin(t+ v0) (55)

vb =

2V+sin(t 2pi/3 + v+ +Vsin(t+ 2pi/3 + v) +

2V0sin(t+ v0)

vc =

2V+sin(t+ 2pi/3 + v+ +Vsin(t 2pi/3 + v) +

2V0sin(t+ v0)

(56)

Applying the Clarke transformation yields the following voltages and cur-rents:

v =

3V+sin(t+ v+) +

3Vsin(t+ v) (57)

v =

3V+cos(t+ v+) +

3Vcos(t+ v)

v0 =

6V0sin(t+ v0)

and

i =

3I+sin(t+ v+) +

3Isin(t+ v) (58)

i =

3I+cos(t+ v+) +

3Icos(t+ v)

i0 =

6I0sin(t+ v0)

The zero-sequence voltage and current are the same as those in the previouscase; the zero-sequence power is equal to that in the previous case. Nothingchanged in it due to the presence of negative-sequence components, as expected.On the other hand, the real and imaginary powers changed considerably. Thepowers given below are separated in their average and oscillating components:

17

p = 3V+I+cos(v+ i+) + 3VIcos(v i) (59)q = 3V+I+sin(v+ i+) + 3VIsin(v i)p = 3V+Icos(2t+ v+ + i) 3VI+cos(2t+ v + i+)q = 3V+Isin(2t+ v+ + i)33VI+sin(2t+ v + i+)

The following conclusion can be written from the above equations for thereal and imaginary powers:

1. The positive- and negative-sequence components in voltages and currentsmay contribute to the average real and imaginary powers.

2. The instantaneous real and imaginary powers contain oscillating com-ponents due to the cross product of the positive-sequence voltage andthe negative-sequence current, and the negative-sequence voltage and thepositive-sequence current. Hence, even circuits without harmonic compo-nents may have oscillating real or imaginary powers.

3.3 General Case Including Distortions and Imbalances inthe Voltages and in the Currents

General equations relating the instantaneous powers in the p q Theory andthe theory of symmetrical components (also called the Fortescue components)are valid in the steady state. The theory of symmetrical components based onphasors is valid only in the steady state. However, these general equations arefundamental in explaining some important characteristics of the p q Theorythat are valid even during transients. Moreover, the general equations will beuseful in understanding control methods for shunt and series active filters andother active power-line conditioners.

For current analysis, a three-phase, four-wire system considered in the steadystate has generic, but periodic, voltages and currents. They may include thefundamental components as well as harmonic components. Further, each three-phase group of phasors in a given frequency may be unbalanced, which meansthat it may consist of positive-, negative, and zero-sequence components, ac-cording to the symmetrical components theory. Generic periodic voltages andcurrents can be decomposed into Fourier series as

vk(t) =

n=1

2Vknsin(t+ kn) k = (a, b, c) (60)

ik(t) =

n=1

2Iknsin(t+ kn) k = (a, b, c) (61)

where n indicate harmonic order. Equation (60) and (3.3) can be written interms of phasors, including the fundamental (n = 1) and harmonic phasors, asfollows:

Vk =

n=1

Vkn 6 kn =n=1

Vkn k = (a, b, c) (62)

18

Ik =

n=1

Ikn 6 kn =n=1

Ikn k = (a, b, c) (63)

Then, the symmetrical-components transformation is applied to each a b charmonic group of phasors of voltages or currents to determine their positive-,negative-, and zero-sequence components: V0nV+n

Vn

= 13

1 1 11 21 2

VanVbnVcn

(64)The complex number in the transformation matrix is the 120o phase-shift

operator:1 6 120o = expj(2pi/3 (65)

The inverse for 64 is: VanVbnVcn

= 1 1 11 2

1 2

V0nV+nVn

(66)We can derive equivalent function from the phasors given by 66 for time. Hence,rewriting the harmonic voltages in terms of symmetrical components in the timedomain, yield the following expressions for the nth a b c group of harmonicvoltages:

van(t) =

2V0nsin(nt+ 0n) +

2V+nsin(nt+ +n) (67)

+

2Vnsin(nt+ n)

vbn(t) =

2V0nsin(nt+ 0n) +

2V+nsin(nt+ +n (2pi/3))+

2Vnsin(nt+ n + (2pi/3))

vbn(t) =

2V0nsin(nt+ 0n) +

2V+nsin(nt+ +n + (2pi/3))

+

2Vnsin(nt+ n (2pi/3))Similarly for instantaneous current:

Ian(t) =

2I0nsin(nt+ 0n) +

2I+nsin(nt+ +n) (68)

+

2Insin(nt+ n)

Ibn(t) =

2I0nsin(nt+ 0n) +

2I+nsin(nt+ +n (2pi/3))+

2Insin(nt+ n + (2pi/3))

Ibn(t) =

2I0nsin(nt+ 0n) +

2V+nsin(nt+ +n + (2pi/3))

+

2Insin(nt+ n (2pi/3))The above decomposition allows the analysis of an unbalanced three-phase

system as a sum of two balanced three-phase systems plus the zero-sequencecomponent.

If the 0 transformation, as defined in (3.1) and (3.3), is applied, thefollowing expressions for the generic voltage and current transformed in the0-reference frames can be obtained:

19

v =

n=1

3V+nsin(nt+ +n) +

n=1

3Vnsin(nt+ n) (69)

v =

n=1

3V+ncos(nt+ +n) +

n=1

3Vncos(nt+ n)

v0 =

n=1

6V0nsin(t+ 0n)

i =

n=1

3I+nsin(nt+ +n) +

n=1

3Insin(nt+ n) (70)

i =

n=1

3I+ncos(nt+ +n) +

n=1

3Incos(nt+ n)

i0 =

n=1

6I0nsin(t+ 0n)

It is possible to see that the positive- and negative-sequence componentcontribute to the and axis voltages and currents, whereas the 0-axisvoltage and current comprise only zero-sequence components. Further, the realpower p and the imaginary power q, and the zero-sequence power p0 as definedin (3.70), can be calculated by using the generic voltages and currents in termof symmeterical components given by (3.91) and (3.92).

The relationship between the conventional powers and the new powers de-fined in p q Theory is better visualized if the powers p, q, and p0 are separatedin their average values p, q, and p0 and oscillating parts p, q, p0.

The resulting power expressions can be obtained from the book if I am soinclined. write the

equationlater. IfI deemedthe equa-tion to beimportant,maybe?

write theequationlater. IfI deemedthe equa-tion to beimportant,maybe?

3.4 Physical Meanings of the Instantaneous Real, Imagi-nary and Zero-Sequence Powers

Fig 10 summarizes the concepts involved in these powers. The following conclu-sions obtained were similar to the one obtained previously. However this timethey were obtained for generic voltage and current waveforms.

Instantaneous powers that the p q Theory defines in the time domain areindependent of the rms values of voltages and currents. Which includes theconventional frequency-domain concepts of active and reactive power definedfor three-phase sinusoidal balanced systems as a particular case. Therefore, thepq theory is not contradictory but complementary to the conventional theoriesin the frequency domain.

The general equations for the real, imaginary and zero-sequence powers givenin (3.94) to (3.99) are the basis for understanding the energy transfer in a three-phase system. It is possible to understand how all the voltage and current com-ponents achieve energy transfer or induce energy exchange (imaginary power)in a three-phase circuit.

20

Figure 10: Physical meaning of the instantaneous powers defined in the 0reference frame.

Zero-sequence components in the fundamental voltage and currentand/or in the harmonics do not contribute to the real power p or tothe imaginary power q.

The total instantaneous energy flow per time unit, that is, the three-phase instantaneous active power, even in a distorted and unbalancedsystem, is always equal to the sum of the real-power and the zero-sequence power (p3 = p + p0), and may contain average and oscil-lating parts.

The imaginary power q, independent of the presence of harmonic orunbalances, represents the energy quantity that is being exchangedbetween the phases of the system. This means the imaginary powerdoes not contribute to energy transfera between source and the loadat any time.

aThe term energy transfer is used here in a general manner, referring not only tothe energy delivered to the load, but also to the energy oscillation between source andload as well

3.5 Avoiding the Clarke Transformation in the pq theoryThe p q theory in a three-phase, four-wire system defines the real power, theimaginary power, and the zero-sequence power as a function of voltages andcurrents in the 0-reference frames. Some expression relating these powers interms of phase mode, that is, the abc variable, were presented in the previoussections, and the two most useful ones are rewritten below:

p3 = vaia + vbib + vcic = vi + vi + v0i0 = p+ p0 (71)

21

q = vi vi = 13

[(va vb)ic + (vb vc)ia + (vc va)ib] (72)

q =13

(vabic + vbcia + vcaib

Moreover, it might be interesting to determine active (real) and reactive(imaginary) current components directly from the instantaneous abc voltagesand currents. For convenience, the active (real) and reactive (imaginary) currentdecomposition defined in (3.22) is repeated here:

[ii

]=

1

v2 + v2

[v vv v

] [p0

]

real currents

+1

v2 + v2

[v vv v

] [0q

]

imaginary currents

(73)

Applying the appropriate inverse Clarke transformation in (3.102) and tak-ing only the first term in the right side of the expression, determine the abc realcurrents as iapibp

icp

= 23

1 0 12 32 12

32

vi + viv2 + v

2

[vv

](74)

The abc real currents can be calculated from the abc voltages and currents byusing (3.5) and its corresponding matrix transformation for the currents. Thus,the variables can be set in terms of abc variables. After some manipulations,the real currents are found to be

iapibpicp

= (vab vca) ia + (vbc vab) ib + (vca vbc) icv2ab + v

2bc + v

2ca

(vab vca)/3(vbc vab)/3(vca vbc)/3

(75)

Note that line voltages, vab, vbcandvca are used in (3.104). This expression con-firms a fundamental concept introduced in the p q Theory, which establishesthat zero-sequence components do not contribute to the real power. Thus,the abc real currents iap, ibp and icp that correspond to the components in theoriginal currents ia, ib and ic. Moreover, the real currents are not influenced byzero-sequence voltage or current components, neither in voltages nor in currents.The reason is that line voltages never contain zero-sequence components, be-cause vab + vbc + vca = 0. Further, even if zero-sequence currents are consideredin (3.104), it follows that

(vab vca) i0 + (vbc vab i0 + (vca vbc) i0 = 0 (76)which results in zero real current.

If the second term in (3.102) is considered, the abc imaginary currents can bedetermined by using the same procedure as above. After some manipulations,they result in iaqibq

icq

= vbc ia + vca ib + vab icv2ab + v

2bc + v

2ca

vbcvcavab

(77)22

As expected, this equation leads to the following conclusion, abc imaginarycurrents iaq, ibq and icq correspond to the components in the original currentsia, ib, ic which generate only imaginary power. They are not influenced by zero-sequence voltage or current components.

The inverse Clark transformation as given in (3.4) can be decomposed intothe sum of two terms, as follows: iaib

ic

= 13

i0i0i0

+23

1 0 12 32 12

32

[ ii

](78)

On the other hand, (3.102) separates ialpha and ibeta into two components,the real and the imaginary currents. One is dependent only on the real power,and the other is dependent only on the imaginary power. These current compo-nents can be calculated directly from the instantaneous abc voltages and currentsby using (3.104) and (3.106). These equations together with (3.102) and (3.107) allow us to write iaib

ic

= 13

i0i0i0

+ iapibpicp

+ iaqibqicq

(79)

23