Modelling of the Power Factor for AC Linear Circuits under Non-sinusoidal Conditions Horia Andrei#1, Fanica Spinei*2, Costin Cepisca*3, Paul C. Andrei*4, and Nicolae Vasile#5

#Dept. of Electrical Engineering, Valahia University of Targoviste 18-20 Blv. Unirii, Targoviste, Dambovita, 130082, Romania

1 hr_andrei@yahoo.com 5 nvasile@icpe.ro

*Dept. of Electrical Engineering, Politehnica University of Bucharest 313 Splaiul Independentei, Bucharest, 060032, Romania

2 fspinei@elth.pub.ro 3 costin@wing.ro

4 sus_ace@yahoo.com

Abstract—In this paper, a new algorithm to calculate the

power factor of the linear circuits in non-sinusoidal regime by introducing the conditions between the relative values of considered harmonics is presented. The relative values which characterize the voltage and current harmonics have been used. The MATLAB example will verify the efficiency of this method.

I. INTRODUCTION Understanding of the current and voltage harmonics is of

utmost importance both to the beneficiaries who thus can prevent the undesirable effects of non-sinusoidal regime in a given network, and to the possible consumers as for as the corresponding measurement and pricing are concerned. The electroenergetical systems in steady state non-sinusoidal conditions contain voltage and current harmonics within the acceptable limits. These limits can be established by a set of relative values of a voltage and current harmonics which characterize the distribution network and the receptors, are presented in [1], [2], [3]. One of the important factors that established the quality of the electrical energy and the performances of the electroenergetical systems is the power factor. It is very useful to determine the variation of the power factor in correlation with the relative values which characterize the voltage and current harmonics.

This paper presents a new algorithm for a mathematical model of the power factor in linear circuit under non-sinusoidal conditions. An example is provided to show the practical application of the proposed method to the linear circuits.

II. CHARACTERIZATION OF THE NON-SINUSOIDAL REGIME USING THE RELATIVE VALUES OF VOLTAGE AND CURRENT

HARMONICS Let’s consider an electroenergetical system in non-

sinusoidal regime and a linear receptor. The receptor being linear and supplied with a non-sinusoidal voltage, the number n of current harmonics it is the same with the number of the voltage harmonics.

A. Relative Values of Voltage and Current Harmonics To illustrate the relative contributions of the current and

voltage harmonics at the terminals of a linear receptor with kϕ the phase angle of each k harmonic, we can use the

Fourier’s series

∑∑==

+=+=n

kkk

n

kkk tkUtkbUtu

11

1)sin(2)sin(2)( γωμγω (1)

∑∑==

−+=−+=n

kkkk

n

kkkk tkItkaIti

11

1)sin(2)sin(2)( ϕγωεϕγω (2)

where the following relative values which characterize the voltage and current harmonics are introduced: - bk and ak represent the ratio between each k = 1,..,n harmonic of voltage, respectively current, and the rms of the voltage (U) respectively current ( I )

UUb k

k = (3)

II

a kk = (4)

- kμ and kε represent the ratio between each k = 1,..,n harmonic of voltage and, respectively, current, and the fundamental value of voltage (U1) respectively current (I1)

1U

U kk =μ (5)

.1I

Ikk =ε (6)

978-1-4244-5795-3/10/$26.00 ©2010 IEEE 353

The following relations are always true [4], [5]:

11

2

1

2 == ∑∑==

n

kk

n

kk ba (7)

niS

b in

kk

ii ,...,1,

2

1

2

22 =∀==

∑=

μ

μ

μ

μ (8)

niS

a in

kk

ii ,...,1,

2

1

2

22 =∀==

∑=

ε

ε

ε

ε (9)

.111 == με (10)

where ∑=

=n

kkS

1

2μμ , and ∑=

=n

kkS

1

2εε .

B. Power Factor One of the most frequent cases in electrical networks and

equipments [6] - [11], is that the reactance of the receptor is inductive. If we presume for the receptor that the nominal value of voltage and current are 1UU N ≅ respectively 1II N ≅ , and the nominal value of active power is ϕcos11IUPN ≅ , [12], [13], then for each k = 1,..,n harmonics the following relations are true

ϕϕ ktgtg k = (11)

ϕϕ

ϕϕ

ϕϕ coscos

coscos

sincos1222

kk

kNk

Nkk IbIb

k

IbI ==+

= (12)

where ϕcos is the nominal value of the power factor.

The active power and the apparent power dissipated by the receptor supplied with voltage (1) can be expressed

∑∑== +

==n

k

kNn

kkkk tgk

bPIUP1

22

2

1 1coscos

ϕϕϕ (13)

∑∑==

==n

kk

n

kk IUUIS

1

2

1

2 (14)

If we take into consideration the classical definition of the

power factor

SPKP = (15)

we obtain, for a linear and inductive receptor, two analogue relations, [14], [15], expressed of the form

∑= +

=n

k

kP tgk

bK1

22

2

1)(

ϕb (16)

and

∑= +

=n

k

kP Stgk

K1

22

2

)1()(

μϕμµ (17)

where the specific vectors of the relative values of voltage harmonics are [ ]nbbb ,...,, 21=b , respectively

[ ]nμμμ ,...,, 21=µ . In order to provide a relationship between the elements of vectors b , µ , and the elements of the vectors

[ ]naaa ,...,, 21=a , [ ]nεεε ,...,, 21=ε which represent the relative values of current harmonics, the relations (11) and (12) are considered. It is easy to see that

ϕϕ

coscos

1

kk

kk b

IIa == (18)

and

ϕϕ 221

1cos

1tgkb

a

k

k

+= (19)

where k = 1,..,n. We conclude that this condition (19) is verified for any relative values ka and kb of the voltage respectively the current harmonics such that the receptor is linear and inductive.

By (8), (9), and (19) an other condition between the relative values kμ and kε is obtained

)1(

1cos

122 ϕϕμ

εε

μ

tgkSS

k

k

+= (20)

for each k = 1,..,n.

C. Hilbert Space Techniques for Computation the Extreme Points of Power Factor

Let us now define, in stationary regime [16],[17], the following functional +ℜ→ℜnF : in the positive Hilbert space nℜ , representing the power factor at the terminals of the linear receptor

∑=

Δ

+==

n

k

kP tgk

bKF1

22

2

1)(

ϕb (21)

We find the extreme points of these functional by using the

properties of Hilbert space. It is proving [18], [19] that F is

354

continuous and is a class 1−nC function in nℜ . Thus, in stationary regime, the derivative b∂∂F with respect to the relative values of voltage harmonics yields n relations that can be synthetically expressed in the form

nk

tgkb

bbF

n

k

k

k

k,...,1,0

1122

2==

+

=∂∂

∑= ϕ

(22)

On these bases, it emerges that v∂∂F = 0, namely, the

solution obtained also corresponds to an extreme point of the functional. In order to establish the nature of this extreme point, it can be observed that the functional F is a quadratic form, and it is positive defined 0)( ≥bF . Thus, the extreme point of the functional is a minimum, only for the trivial solution [ ]nbbb ,...,, 21=b =0. Here, the trivial case 0)( =bF is of no practical interest, because represents the AC regime of the circuit. From this premise the analytic characterization of the power factor is introduced next in order to find the optimum operating power factor point for real conditions.

D. Mathematical Model of Power Factor However, we have developed a mathematical model in

order to facilitate the analytic characterization of power factor, when b, and n, N∈n , are variables. Using in (21) the basic trigonometrically equations, the power factor can be expressed

∑= +−

=n

k

kP kk

bnK1

222

2

cos)1(cos)(

ϕϕ b, (23)

where )1,0(∈kb (24) and )1,0(cos ∈ϕ (25)

A similar relation through computation of (17), is obtained

∑= −+

=n

k

kP Skk

nK1

222

2

]cos)1([cos)(

μϕμϕµ, (26)

For the relative values ),...,,( 21 nbbb , the mathematical

distribution

,..,nknn

kbk 1,)1(

2 =+

= (27)

can be used, because verifies the conditions (7) and (24), and may be represents as well a practical interest case, [20], [21].

By substituting (27) into (19) is obtained the relative values of current harmonics

222 cos)1(1

)1(2

kknnkak +−+

=ϕ

(28)

In order to impose a good quality of the electric energy in

non-sinusoidal regime, the technical literature uses the limits of the voltage harmonics, [22]-[24]. These limits are identically with the relative values kμ , defined by (5), when the relation 1UU N ≅ is true.

Now, we can compare two set of the kb , PK , and ka values obtained, first, by used the mathematical distribution (27), and relation (23),(28) and, second, by used the maximum admissible limits of relative values kμ imposed by the technical literature, and the relations (8), (19), and (26).

Thus, a mathematical model for the relative values variation (27) has been used to study the behaviour in non-sinusoidal conditions, of a linear inductive receptor, characterized by n and ϕcos .

A MATLAB, [25], [26], systematic implementation is provided to show the practical application of the proposed methods.

III. MATLAB APPLICATION AND RESULTS Let us consider a linear inductive circuit, characterized by

1UU N ≅ , n=1,..,5 number of harmonics, and the nominal values of power factor 5.0cos =ϕ or 8.0cos =ϕ . We note

with superscript ' respectively '' the calculated values in different steps of the mathematical algorithm. MATLAB’s procedure contains:

Step 1) For each value of k a MATLAB sequence is used to calculate the mathematical distribution of '

kb . The power

factor 'PK calculated values for each value of '

kb and for two values of nominal power factor, are presented in Table I, and the characteristic nKP −' is shown in Fig. 1.

The relations (23), and (27) are used. TABLE I

VALUES OF POWER FACTOR FOR A MATHEMATICAL DISTRIBUTION OF RELATIVE VALUES OF VOLTAGE HARMONICS

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 'PK for 5.0cos =ϕ 0.5 0.454 0.426 0.381 0.355

'PK for 8.0cos =ϕ 0.8 0.787 0.745 0.701 0.663

Step 2) for each maximum admissible limits of relative

values kμ , by relation (8) a MATLAB sequence is used to

calculate ''kb . From (26) the ''

PK values are calculated for each

value of ''kb and for two values of nominal power factor.

355

Fig. 1. The characteristic 'PK - n obtained for a mathematical distribution of relative values '

kb

Fig. 2. The characteristic ''PK - n obtained for maximum admissible limits of ''

kb

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The maximum admissible limits for the first 5 voltage harmonics are presented in Table II, the calculated power factor values for each value of nominal power factor are presented in Table III, and the characteristic nKP −'' is shown in Fig. 2.

TABLE II MAXIMUM ADMISSIBLE LIMITS OF VOLTAGE HARMONICS

k 1 2 3 4 5

1/UU kk =μ 1 0.02 0.05 0.01 0.06

TABLE III VALUES OF POWER FACTOR FOR MAXIMUM ADMISSIBLE LIMITS OF

RELATIVE VALUES OF VOLTAGE HARMONICS

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 ''

PK for 5.0cos =ϕ 0.5 0.499 0.499 0.499 0.498

''PK for 8.0cos =ϕ 0.8 0.799 0.799 0.799 0.797

Step 3) For each calculated value of power factor from Table I and III we use a MATLAB sequence to calculate the percentage of their decrease compared to the nominal value of power factor. The relation

(%)100cosϕ

PKp = (29)

is used, and the resulting values are reported in Table IV.

TABLE IV NUMERICAL VALUES OF PERCENTAGE FOR POWER FACTOR

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 'p for 5.0cos =ϕ 100% 90.8% 82.8% 76.2% 71%

'p for 8.0cos =ϕ 100%. 98.3% 93.1% 87.6% 82.9%

''p for 5.0cos =ϕ 100% 99.9% 99.8% 99.8% 99.6%

''p for 5.0cos =ϕ 100% 99.9% 99.8% 79.9% 79.7%

Step 4) The values k, ''

kb , ϕcos , and the relations (19), respectively (28) are used in a MATLAB sequence in order to calculate the relative values of current harmonics '

ka , and ''ka . The resulting values are reported in Table V-VIII.

TABLE V RELATIVE VALUES OF CURRENT HARMONICS FOR A MATHEMATICAL

DISTRIBUTION OF RELATIVE VALUES OF VOLTAGE HARMONICS ( 5.0cos =ϕ )

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 '1a for 5.0cos =ϕ 1 0.577 0.408 0.316 0.258

'2a for 5.0cos =ϕ - 0.453 0.320 0.248 0.202

'3a for 5.0cos =ϕ - - 0.267 0.207 0.169

'4a for 5.0cos =ϕ - - - 0.180 0.147

'5a for 5.0cos =ϕ - - - - 0.132

TABLE VI RELATIVE VALUES OF CURRENT HARMONICS FOR A MATHEMATICAL

DISTRIBUTION OF RELATIVE VALUES OF VOLTAGE HARMONICS ( 8.0cos =ϕ )

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 '1a for 8.0cos =ϕ 1 0.577 0.408 0.316 0.258

'2a for 8.0cos =ϕ - 0.566 0.4 0.310 0.253

'3a for 8.0cos =ϕ - - 0.360 0.279 0.228

'4a for 8.0cos =ϕ - - - 0.250 0.204

'5a for 8.0cos =ϕ - - - - 0.186

TABLE VII RELATIVE VALUES OF CURRENT HARMONICS FOR MAXIMUM ADMISSIBLE

LIMITS OF RELATIVE VALUES OF VOLTAGE HARMONICS ( 5.0cos =ϕ )

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 ''

1a for 5.0cos =ϕ 1 0.999 0.998 0.998 0.998

''2a for 5.0cos =ϕ - 0.010 0.007 0.005 0.004

''3a for 5.0cos =ϕ - - 0.018 0.014 0.011

''4a for 5.0cos =ϕ - - - 0.0028 0.0022

''5a for 5.0cos =ϕ - - - - 0.013

TABLE VIII RELATIVE VALUES OF CURRENT HARMONICS FOR MAXIMUM ADMISSIBLE

LIMITS OF RELATIVE VALUES OF VOLTAGE HARMONICS ( 8.0cos =ϕ )

Harmonics n = 1 n = 2 n = 3 n = 4 n = 5 ''

1a for 8.0cos =ϕ 1 0.999 0.998 0.998 0.998

''2a for 8.0cos =ϕ - 0.013 0.010 0.007 0.006

''3a for 8.0cos =ϕ - - 0.034 0.019 0.015

''4a for 8.0cos =ϕ - - - 0.0039 0.0032

''5a for 8.0cos =ϕ - - - - 0.0192

Consequently, comparing the results obtained by using the

mathematical distribution with the maximum admissible limits of relative values of voltage harmonics, the following remarks can be statues:

• In both cases, the power factor decreases when the number of harmonics increases, which has a practical proved solution in electrical networks;

• The power factor has a proportional dependence with the value of nominal power factor in both cases, which has a practical proved solution in electrical networks ;

• The numerical values of percentage p have an important decrease when using the mathematical distribution of the relative value of voltage harmonics, and these values can be used with efficiency for the limitation of the non-sinusoidal regime effects;

• The relative values of the current harmonics decrease when the number of harmonics increases, and this conclusion is practically proved;

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• In both cases, the relative values of current harmonics have a proportional dependence with the value of nominal power factor, which has a practical proved solution in electrical networks.

IV. CONCLUSIONS In order to determine the network harmonic pollution and

to improve the normative and standards, in this paper a mathematical calculation method has been found. We demonstrate that the power factor depends of the relative values of voltage harmonics and of the number of voltage harmonics.

It is remarkably useful, both in theory and practice, to determine the behavior of the receptor power factor depending on the relative values of the voltage and current harmonics. In this way we localize, by efficient measurements, the sources and the amplitude of non-sinusoidal regime in electricity networks.

The relative values bk , ak , kμ , and kε can be easily measured, and the calculation algorithm implemented for the power factor is not difficult to apply through a mathematical distribution of harmonics which is practically verified.

The authors develop a comprehensive and efficient method to find the quality and the performances of the electrical energy, through the realization of a mathematical model for the power factor.

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