**3. The mathematical models for determining the induced voltages**

The data obtained from measurements on the ground represent an advantage for conceiv‐ ing mathematical models, because the results obtained through mathematical modeling can be managed comparing with the real ones. This fact has lain at the basis of designing the mathematical models, trying to imitate, as realistically as possible, the physical phenom‐ ena that occur in nature.

It should also be mentioned that, at low frequencies, the couplings of the electromagnetic interferences between sources and victims can be separated, through different experi‐ ments, into electric couplings and magnetic couplings, respectively. Mathematical model‐ ing should take into account this observation that leads to achieving, separately, two different models, one for electric and one for magnetic phenomena [8, 9].

**Figure 6.** The variation of the voltages induced through capacitive coupling depending on the length of their parallel

**Figure 7.** The variation of the voltages induced through inductive coupling depending on the length of their parallel

portions.

354 Computational and Numerical Simulations

portions.

But regardless of the type of electromagnetic coupling, the values of induced voltages are dependent on both the geometry of the power lines and the power running on these lines and therefore the mathematical models must include, primarily, the geometric calculation of the supporting pillars of the high voltage overhead power lines with double circuit and the determination of their capacities and, respectively, the mutual inductances between the conductors of the power line with double circuit. The geometric, electric and magnetic parameters represent equation coefficients through which there are determined the voltages induced electrically and magnetically in the conductors of the disconnected circuit of the high voltage power line with double circuit.

### **3.1. Determination of the geometric parameters of the supporting pillars of the high voltage overhead power lines with double circuit**

The calculation of the geometrical parameters of the supporting pillars of the high voltage overhead power lines with double circuit must consider the distances between the conductors of the double circuit, the distances between conductors and their images in the earth and the maximum arrow formed by the conductors of the power line in a standard horizontal opening, as shown in Fig.8, a and b.

**Figure 8.** Explanation of the determination of the geometrical parameters of the high voltage overhead power lines with double circuit. a) The geometrical parameters of the pillar; b) Determination of the average height of the conduc‐ tors from the ground.

The geometric calculation of the supporting pillar of the high voltage overhead power line with double circuit is made using the following algorithm:

**a.** The average distance between the conductors and the return path through the ground is obtained by taking into account the resistivity of the soil, with the expression:

$$D\_{CP} = 550 \sqrt{\frac{\rho}{f}} \tag{1}$$

**d.** The distances between the conductors of the two circuits of the power line with double

( )

1 2 3 2 2 1 21 2 2 2 23

2; 2; 2

2

*d d dd dd*

= = =

*Rr Ss Tt*

*d d hh dd*

== + + -

*d d h dd*

== + -

*d d h dd*

**3.2. The mathematical modeling of the electric (capacitive) coupling**

== + -

2

( )

( ) ( )

In the case of electric (capacitive) coupling, the high voltage overhead power line with double circuit represents a complex set of capacities which are formed due to the differences in potential both between the active circuit phases, because of the different values of the voltage phasers of the three-phases at any moment and between the potentials of the conductors of the active circuit and those of the disconnected circuit and that insulated from the earth. The

(a) (b)

formed between the two circuits; b) The equivalent capacities for a phase of the disconnected circuit.

are calculated by the following expressions:

**•** For partial capacities between phases:

**Figure 9.** The set of capacities formed between the active circuit (RST) and the disconnected one (rst). a) Capacities

The values of the partial capacities between phases and between the phases and the ground

2 2 12 31

Experimental Determinations and Numerical Simulations of the Effects of...

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(4)

357

2

circuit result from the following expressions:

*Sr Rs*

*Ts St*

*Tr Rt*

assembly of capacities which are formed is shown in Fig. 9.

where *ρ* - resistivity of the soil and *f* - the frequency of the power line voltage.

**b.** The average height of the power line conductor from the ground level results from:

$$\hbar\_k = H - a\_1 - \lambda\_{\rm izk} - \frac{2}{3} f\_{\rm max \, k} \tag{2}$$

**c.** The vertical and horizontal distances of the active conductors, for each type of pillar presented in table 1, are determined by the following expressions:

$$\begin{aligned} d\_{RS} &= d\_{rs} = \sqrt{h\_1^2 + \left(d\_2 - d\_1\right)^2} \\ d\_{ST} &= d\_{st} = \sqrt{h\_2^2 + \left(d\_2 - d\_3\right)^2} \\ d\_{RT} &= d\_{rt} = \sqrt{\left(h\_1 + h\_2\right)^2 + \left(d\_3 - d\_1\right)^2} \end{aligned} \tag{3}$$

**d.** The distances between the conductors of the two circuits of the power line with double circuit result from the following expressions:

$$\begin{aligned} d\_{Rs} &= 2d\_1; \quad d\_{Ss} = 2d\_2; \quad d\_{Ti} = 2d\_3\\ d\_{Sr} &= d\_{Rs} = \sqrt{h\_1^2 + \left(2d\_2 - d\_1\right)^2} \\ d\_{Ts} &= d\_{St} = \sqrt{h\_2^2 + \left(2d\_2 - d\_3\right)^2} \\ d\_{Tr} &= d\_{Rt} = \sqrt{\left(h\_1 + h\_2\right)^2 + \left(2d\_3 - d\_1\right)^2} \end{aligned} \tag{4}$$

### **3.2. The mathematical modeling of the electric (capacitive) coupling**

(a) (b)

tors from the ground.

356 Computational and Numerical Simulations

**Figure 8.** Explanation of the determination of the geometrical parameters of the high voltage overhead power lines with double circuit. a) The geometrical parameters of the pillar; b) Determination of the average height of the conduc‐

The geometric calculation of the supporting pillar of the high voltage overhead power line

**a.** The average distance between the conductors and the return path through the ground is obtained by taking into account the resistivity of the soil, with the expression:

r

= (1)

(2)

(3)

<sup>550</sup> *DCP <sup>f</sup>*

**b.** The average height of the power line conductor from the ground level results from:

<sup>3</sup> *<sup>k</sup> izk <sup>k</sup> h Ha f* =-- l

1 max 2

**c.** The vertical and horizontal distances of the active conductors, for each type of pillar

( )

2 2 1 21 2 2 2 23

( )

( ) ( )

2 2 12 31

where *ρ* - resistivity of the soil and *f* - the frequency of the power line voltage.

presented in table 1, are determined by the following expressions:

*d d h dd*

== + -

*d d h dd*

== + -

*d d hh dd*

== + + -

*RS rs*

*ST st*

*RT rt*

with double circuit is made using the following algorithm:

In the case of electric (capacitive) coupling, the high voltage overhead power line with double circuit represents a complex set of capacities which are formed due to the differences in potential both between the active circuit phases, because of the different values of the voltage phasers of the three-phases at any moment and between the potentials of the conductors of the active circuit and those of the disconnected circuit and that insulated from the earth. The assembly of capacities which are formed is shown in Fig. 9.

**Figure 9.** The set of capacities formed between the active circuit (RST) and the disconnected one (rst). a) Capacities formed between the two circuits; b) The equivalent capacities for a phase of the disconnected circuit.

The values of the partial capacities between phases and between the phases and the ground are calculated by the following expressions:

**•** For partial capacities between phases:

$$C\_{ik} = \frac{2\pi\varepsilon\_0 I}{\ln\frac{d\_{ik}}{r\_0}}\tag{5}$$

being charged. But, in this particular case, there are known neither the electric charges of the condensers, nor the voltages at which these condensers are loading. In order to solve this complicated problem we have used analogies between the electric network, which contains only condensers and the electric network containing only resistors. Thus, if the relation for voltage drop, U, between the armatures of a condenser of capacitance C loaded with electric charge Q and voltage drop U, at the hubs of a resistor of resistance R run by currents of intensity

Experimental Determinations and Numerical Simulations of the Effects of...

value R, and value Q is analogous to value I and, as to voltage U, in order to have the same sense in both cases, the sense of current I has to correspond to the sense of electrostatic field E

**Figure 11.** The correspondence between the analogous values of the theory of electrostatics and that of electro-kinet‐

Based on the analogies between the electrostatic and electro-kinetics values, there has been

**Figure 12.** The analogous electro-kinetics scheme of the electric (capacitive) coupling for a phase of the disconnected

In order to determine the voltages induced through the electric (capacitive) coupling there can be applied the method of Kirchhoff's theorems in the case of the analogous electro-kinetics

designed the equivalent electro-kinetics scheme, as shown in Fig. 12.

*<sup>C</sup>* is analogous to

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359

which allows of the establishment of the following correspondences: value <sup>1</sup>

between the condenser armatures, as shown in Fig.11.

I, there results:

*<sup>C</sup>* , respectively:*<sup>U</sup>* <sup>=</sup>*<sup>R</sup>* <sup>⋅</sup> *<sup>I</sup>*,

*<sup>U</sup>* <sup>=</sup> *<sup>Q</sup>*

ics.

circuit.

**•** For partial capacities between the phases and the ground:

$$\mathbf{C}\_{pi} = \frac{2\pi\varepsilon\_0 l}{\ln\left(\frac{2h\_i}{r\_0}\right)}\tag{6}$$

where: *l*- is the length of power line, *dik* - the distances between the phase conductors, according to relations (3) and (4), and *r*0- the radius of the phase conductor.

By transforming the phase capacities connected in delta connection (Fig. 9 a) into the equivalent capacities in Y-connection (Fig. 9 b), the null potential of the two Y-connections are equal with the null potential of the ground and thus the phase capacities are placed in parallel with the phase capacities versus the ground. There results the electrostatic equivalent scheme shown in Fig. 10:

**Figure 10.** The electrical scheme of the capacitive coupling between the two circuits of the power line.

This electrical scheme represents a set of circuits having as passive elements only condensers, and solving of such a problem supposes using Kirchhoff's theorems either for determining the electric charge of the condensers or determining the voltages at which these condensers are being charged. But, in this particular case, there are known neither the electric charges of the condensers, nor the voltages at which these condensers are loading. In order to solve this complicated problem we have used analogies between the electric network, which contains only condensers and the electric network containing only resistors. Thus, if the relation for voltage drop, U, between the armatures of a condenser of capacitance C loaded with electric charge Q and voltage drop U, at the hubs of a resistor of resistance R run by currents of intensity I, there results:

$$
\mathcal{U} = \frac{\mathcal{Q}}{\mathcal{C}}, \text{ respectively:}
\mathcal{U} = \mathbb{R} \cdot I\_{\prime\prime}
$$

0

pe

*ik*

<sup>=</sup> (5)

(6)

2

*<sup>l</sup> <sup>C</sup> <sup>d</sup>*

*ik*

*pi*

**•** For partial capacities between the phases and the ground:

358 Computational and Numerical Simulations

to relations (3) and (4), and *r*0- the radius of the phase conductor.

in Fig. 10:

ln

0

0

pe <sup>=</sup> æ ö ç ÷ ç ÷ è ø

*h r*

2 <sup>2</sup> ln

*<sup>l</sup> <sup>C</sup>*

0

where: *l*- is the length of power line, *dik* - the distances between the phase conductors, according

By transforming the phase capacities connected in delta connection (Fig. 9 a) into the equivalent capacities in Y-connection (Fig. 9 b), the null potential of the two Y-connections are equal with the null potential of the ground and thus the phase capacities are placed in parallel with the phase capacities versus the ground. There results the electrostatic equivalent scheme shown

**Figure 10.** The electrical scheme of the capacitive coupling between the two circuits of the power line.

This electrical scheme represents a set of circuits having as passive elements only condensers, and solving of such a problem supposes using Kirchhoff's theorems either for determining the electric charge of the condensers or determining the voltages at which these condensers are

*i*

*r*

which allows of the establishment of the following correspondences: value <sup>1</sup> *<sup>C</sup>* is analogous to value R, and value Q is analogous to value I and, as to voltage U, in order to have the same sense in both cases, the sense of current I has to correspond to the sense of electrostatic field E between the condenser armatures, as shown in Fig.11.

**Figure 11.** The correspondence between the analogous values of the theory of electrostatics and that of electro-kinet‐ ics.

Based on the analogies between the electrostatic and electro-kinetics values, there has been designed the equivalent electro-kinetics scheme, as shown in Fig. 12.

**Figure 12.** The analogous electro-kinetics scheme of the electric (capacitive) coupling for a phase of the disconnected circuit.

In order to determine the voltages induced through the electric (capacitive) coupling there can be applied the method of Kirchhoff's theorems in the case of the analogous electro-kinetics scheme in Fig. 12. The following system of equations result, where the currents of the edge circuits are unknown.

$$\begin{aligned} E\_1 - E\_2 &= \left( R\_{10} + R\_{12} \right) \cdot I\_1 - \left( R\_{20} + R\_{22} \right) \cdot I\_2 \\ E\_2 - E\_3 &= \left( R\_{20} + R\_{22} \right) \cdot I\_2 - \left( R\_{30} + R\_{32} \right) \cdot I\_3 \\ E\_3 &= \left( R\_{30} + R\_{32} \right) \cdot I\_3 + R\_e \cdot I \\ I\_1 + I\_2 + I\_3 &= I \end{aligned} \tag{7}$$

been transformed into constant values on small time intervals, *Δt<sup>k</sup>*

circuit, if there are assigned 2 ÷ 3 time interval values, *Δt<sup>k</sup>*

**Circuit B [km]**

**Active circuit voltage**

> 228 225 232

> 230 235 225

> 230 235 225

**Table 4.** Comparison between the measured voltages induced electrically and the calculated ones.

The mathematical model presented above also allows of the determination of the maximum values of the voltages induced electrically in each of the phases of the disconnected circuit by assigning to "time" discrete values around the maximum of the sinusoidal function of the

Knowing, by measuring the voltages induced electrically in the conductors of the disconnected

realized in MATHCAD, from relations (9), there result immediately the calculation values of the voltages induced electrically (capacitive). These values are given in table 4, comparatively with the measured values. The measured values and those obtained through calculation are very close, this fact demonstrating the validity of the mathematical model developed and used for determining the voltages induced electrically in the circuits of the disconnected power lines.

7 - 11 49.876 25.455 237.5 8.87 2.7 4.4 **8.823 2.755 4.415**

8 - 9 25.455 11.249 236.9 12.7 20.2 12.3 **12.726 20.324 12.278** 11 - 12 16.688 43.897 225 1.9 3.35 2.35 **1.845 3.419 2.427** 9 - 11 25.455 7.422 236.8 19.4 23.4 18.2 **19.433 23.259 17.906**

7 - 8 18.675 18.675 237 8.9 4.42 6.25 **9.076 4.573 6.278** 4 - 5 30.730 30.730 226.5 3.03 7.94 5.9 **3.154 7.998 5.876** 4 - 1 72.867 72.867 234 11.1 3.7 5.4 **11.221 3.768 4.798**

8 - 9 maximum load 25.455 11.249 236.9 12.7 20.2 18.3 **12.643 20.324 12.278**

**Measured voltage induced electrically in the disconnected circuit**

**U [kV] UR [kV] US [kV] UT [kV] Er [kV] Es [kV] Et [kV]**

Experimental Determinations and Numerical Simulations of the Effects of...

10.4 3.6 5.1 **10.422 3.619 4.948**

6.55 5.82 5.41 **6.582 5.865 5.452**

8.2 2.8 4.2 **8.206 2.845 4.210**

the analogies mentioned above become valid.

**Circuit A [km]**

4 - 6 116.550 116.550

1 - 2 53.719 24.620

2 - 3 53.719 55.173

inductor voltage.

**Overhead power line**

, where: *k = 1...100*, for which

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361

**Calculated voltage induced electrically in the disconnected circuit**

, and if we use a calculation program

After solving system of equations (7), considering the analogies *<sup>I</sup>* <sup>≡</sup>*Q* and*Rei* <sup>≡</sup> <sup>1</sup> *Ci*<sup>0</sup> + *Cip* , there results the value of the voltage induced electrically in each of the three conductors of the disconnected circuit of the high voltage overhead power line with double circuit, namely:

$$\mathcal{U}\mathcal{U}\_{fi} = \mathcal{R}\_{ei} \cdot I\_{i\prime} \text{ respectively } \mathcal{U}\_{fi} = \frac{\mathcal{Q}\_i}{\mathcal{C}\_{i0} + \mathcal{C}\_{ip}} \tag{8}$$

There should be mentioned that the analogies between electrostatic and electro-kinetics values are valid only in the case of d. c. circuits. But, in the present case, the analyzed circuits are in a. c. because voltage sources E1, E2 and E3 are sinusoidal alternative voltages, having expres‐ sions:

$$\begin{aligned} E\_1 &= \sqrt{2} \cdot \mathcal{U}\_{f\mathbb{R}} \sin \left( \boldsymbol{\phi} \cdot \boldsymbol{t} + \boldsymbol{\phi} \right) \\ E\_2 &= \sqrt{2} \cdot \mathcal{U}\_{f\mathbb{S}} \sin \left( \boldsymbol{\phi} \cdot \boldsymbol{t} + \boldsymbol{\phi} - \frac{2 \cdot \pi}{3} \right) \\ E\_3 &= \sqrt{2} \cdot \mathcal{U}\_{f\mathbb{T}} \sin \left( \boldsymbol{\phi} \cdot \boldsymbol{t} + \boldsymbol{\phi} - \frac{4 \cdot \pi}{3} \right) \end{aligned} \tag{9}$$

where *ω* =2⋅*π* ⋅ *f* - is the angular frequency of the sinusoidal wave of the phase voltage and *ϕ* =0 is the initial phase difference, considered null because the relative positions of the voltage phasers versus the fixed reference axis of the phasers are not known.

For the analogies presented above be valid, "time" must to be considered a constant. But time, *t = const*., represents the very moment of measuring the capacitive voltage induced for each phase of the disconnected circuit. Therefore it is necessary to know the measuring moment of the voltage induced for each of the three phases, separately. To determine the measuring moment there has been considered a period of the sinusoidal voltage wave, which at the frequency of *f = 50 Hz* has duration *T = 0.02* seconds. Duration *T*, of the sinusoidal voltage wave has been divided into 100 discrete and constant time intervals, each *Δt = 0.0002* seconds and thus, through the digitization of time, the variable values of the alternative current circuit have been transformed into constant values on small time intervals, *Δt<sup>k</sup>* , where: *k = 1...100*, for which the analogies mentioned above become valid.

scheme in Fig. 12. The following system of equations result, where the currents of the edge

( ) ( ) ( ) ( )

*e*

results the value of the voltage induced electrically in each of the three conductors of the disconnected circuit of the high voltage overhead power line with double circuit, namely:

, respectively *<sup>i</sup>*

There should be mentioned that the analogies between electrostatic and electro-kinetics values are valid only in the case of d. c. circuits. But, in the present case, the analyzed circuits are in a. c. because voltage sources E1, E2 and E3 are sinusoidal alternative voltages, having expres‐

0

3

p

> p

3

*i ip*

*C C* = × <sup>=</sup> <sup>+</sup> (8)

(7)

, there

(9)

*Ci*<sup>0</sup> + *Cip*

1 2 10 12 1 20 22 2 2 3 20 22 2 30 32 3


*EE R R I R R I EE R R I R R I*

After solving system of equations (7), considering the analogies *<sup>I</sup>* <sup>≡</sup>*Q* and*Rei* <sup>≡</sup> <sup>1</sup>

*fi ei i fi*

*<sup>Q</sup> U RI <sup>U</sup>*

( ) <sup>1</sup>

= × ×+

<sup>2</sup> 2 sin

æ ö <sup>×</sup> = × ×+ - ç ÷

w

w

 f

> f

è ø

 f

<sup>4</sup> 2 sin

æ ö <sup>×</sup> = × ×+ - <sup>ç</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup>

w

where *ω* =2⋅*π* ⋅ *f* - is the angular frequency of the sinusoidal wave of the phase voltage and *ϕ* =0 is the initial phase difference, considered null because the relative positions of the voltage

For the analogies presented above be valid, "time" must to be considered a constant. But time, *t = const*., represents the very moment of measuring the capacitive voltage induced for each phase of the disconnected circuit. Therefore it is necessary to know the measuring moment of the voltage induced for each of the three phases, separately. To determine the measuring moment there has been considered a period of the sinusoidal voltage wave, which at the frequency of *f = 50 Hz* has duration *T = 0.02* seconds. Duration *T*, of the sinusoidal voltage wave has been divided into 100 discrete and constant time intervals, each *Δt = 0.0002* seconds and thus, through the digitization of time, the variable values of the alternative current circuit have

2 sin

*EU t*

*EU t*

*fR*

*fS*

*fT*

*EU t*

phasers versus the fixed reference axis of the phasers are not known.

( )

*E R R I RI*

= + ×+ ×

3 30 32 3

123

2

3

*III I*

++=

circuits are unknown.

360 Computational and Numerical Simulations

sions:

Knowing, by measuring the voltages induced electrically in the conductors of the disconnected circuit, if there are assigned 2 ÷ 3 time interval values, *Δt<sup>k</sup>* , and if we use a calculation program realized in MATHCAD, from relations (9), there result immediately the calculation values of the voltages induced electrically (capacitive). These values are given in table 4, comparatively with the measured values. The measured values and those obtained through calculation are very close, this fact demonstrating the validity of the mathematical model developed and used for determining the voltages induced electrically in the circuits of the disconnected power lines.


**Table 4.** Comparison between the measured voltages induced electrically and the calculated ones.

The mathematical model presented above also allows of the determination of the maximum values of the voltages induced electrically in each of the phases of the disconnected circuit by assigning to "time" discrete values around the maximum of the sinusoidal function of the inductor voltage.

Another important observation is required, namely, that for each phase of the disconnected circuit there has been adopted a different value for the digitized time because the measure‐ ments have been made separately for each of the three phases of the respective circuit.

### **3.3. The mathematical modeling of the magnetic (inductive) coupling**

The magnetic (inductive) coupling is generated by the electric currents varying in time running through the three conductors of the active circuit of the high voltage overhead power line with double circuit and creating variable magnetic fields which induce electromotive voltages in the conductors of the disconnected circuit. The mathematical expressions of the electric currents running through the conductors of the active circuit are given by relations (10), namely:

$$\begin{aligned} i\_R &= \sqrt{2} \cdot I\_{f\mathbb{R}} \sin \left( \boldsymbol{\alpha} \cdot \boldsymbol{t} + \boldsymbol{\varphi} \right) \\\ i\_S &= \sqrt{2} \cdot I\_{f\mathbb{S}} \sin \left( \boldsymbol{\alpha} \cdot \boldsymbol{t} + \boldsymbol{\varphi} - \frac{2\pi}{3} \right) \\\ i\_T &= \sqrt{2} \cdot I\_{f\mathbb{T}} \sin \left( \boldsymbol{\alpha} \cdot \boldsymbol{t} + \boldsymbol{\varphi} - \frac{4\pi}{3} \right) \end{aligned} \tag{10}$$

But, as the electromotive voltage induced in each of the three conductors of the disconnected circuit represents the contribution of all the three magnetic inductive fields generated by the variable currents of the active circuit, the mathematical expression for each phase of the

> ( ) ( ) ( )

> > **Measured voltage induced magnetically in the disconnected circuit**

Experimental Determinations and Numerical Simulations of the Effects of...

**US [V] UT [V]**

(13)

363

**Ut [V]**

**Calculated voltage induced magnetically in the disconnected circuit**

http://dx.doi.org/10.5772/57044

**Us [V]**

**Ur [V]**

1400 400 1440 **1401 404.28 1444**

180 71 1260 **181.052 71.05 1261**

310 80 270 **311.44 81.009 270.174**

*r R Rr S Sr T Tr s R Rs S Ss T Ts t R Rt S St T Tt*

> **Active circuit current**

> > **I [A]**

440 480 460

212 237 218

212 237 218

**Table 5.** Comparison between the measured voltages induced by magnetic coupling and the calculated ones

This simple calculation algorithm has lain at the basis of designing the calculation program in MATHCAD, by means of which there were determined analytically, the voltages induced

**UR [V]**

25.455 11.249 236.9 156.9 68 10.2 39 **67.888 10.365 39.073**

7 - 11 49.876 25.455 237.5 265.75 320 21 120 **317.193 21.014 120.452**

8 - 9 25.455 11.249 236.9 74.826 34 10.2 18.5 **33.415 7.18 18.808** 11 - 12 16.688 43.897 225 181.68 120 26 122 **120.222 25.88 120.363** 9 - 11 25.455 7.422 236.8 92.542 11 4.3 13.2 **11.279 4.28 13.584**

7 - 8 18.675 18.675 237 71.77 63.6 13 42 **62.844 13.182 42.529** 4 - 5 30.730 30.730 226.5 14.54 20.8 3.6 17 **20.862 3.671 17.091** 4 - 1 72.867 72.867 234 497.03 1020 400 940 **1021 357.93 941.014**

*U j iM iM iM U j iM iM iM U j iM iM iM*

=- × × × + × + × =- × × × + × + × =- × × × + × + ×

w

w

w

**Active circuit voltage**

> **U [kV]**

228 225 232

230 235 225

230 235 225

**Circuit [km]B**

disconnected circuit will be:

**Circuit [km]A**

4 - 6 116.550 116.550

1 - 2 53.719 24.620

2 - 3 53.719 55.173

**Overhead power line**

8 - 9 maximum load

where φ represents the phase shift between the voltages and the currents of the active circuit and the phase shift is known because powers *P* and *Q* which charge the active circuit are known. (Table 2).

In the mathematical model called "network model", the magnetic (inductive) coupling can be represented by the mutual inductance between the conductors of the two circuits having the general expression:

$$M\_{12} = \frac{\mu\_0}{4\pi} \int\_0^l \frac{dl\_1 dl\_2}{\sqrt{\left(l\_1 - l\_2\right)^2 + d\_{12}^2}}\tag{11}$$

where, *l* 1and *l* <sup>2</sup> are the lengths of two conductors in parallel and *d*12 is the distance between them.

After the expansion in series of the radical and neglecting the superior rank terms, relation (11) becomes:

$$M\_{\vec{n}} = \frac{\mu\_0}{2\pi} \cdot l \cdot \ln\left(\frac{D\_{cp}}{d\_{\vec{n}}}\right), \text{ where } i \in \left(R, S, T\right), \text{respectively } \ k \in \left(r, s, t\right) \tag{12}$$

But, as the electromotive voltage induced in each of the three conductors of the disconnected circuit represents the contribution of all the three magnetic inductive fields generated by the variable currents of the active circuit, the mathematical expression for each phase of the disconnected circuit will be:

Another important observation is required, namely, that for each phase of the disconnected circuit there has been adopted a different value for the digitized time because the measure‐ ments have been made separately for each of the three phases of the respective circuit.

The magnetic (inductive) coupling is generated by the electric currents varying in time running through the three conductors of the active circuit of the high voltage overhead power line with double circuit and creating variable magnetic fields which induce electromotive voltages in the conductors of the disconnected circuit. The mathematical expressions of the electric currents running through the conductors of the active circuit are given by relations (10),

2 sin( )

<sup>2</sup> 2 sin

= × ×+ - ç ÷

w

w

 j

 j

è ø æ ö

æ ö

 j

è ø

<sup>4</sup> 2 sin

= × ×+ - ç ÷

w

where φ represents the phase shift between the voltages and the currents of the active circuit and the phase shift is known because powers *P* and *Q* which charge the active circuit are

In the mathematical model called "network model", the magnetic (inductive) coupling can be represented by the mutual inductance between the conductors of the two circuits having the

> ( ) 0 1 2 <sup>12</sup> <sup>0</sup> <sup>2</sup> <sup>2</sup>

After the expansion in series of the radical and neglecting the superior rank terms, relation (11)

( ) ( ) <sup>0</sup> ln , where ,, ,respectively , , <sup>2</sup>

*M l i RST k rst*

æ ö = ×× ç ÷ <sup>Î</sup> <sup>Î</sup> ç ÷

1 2 12

<sup>2</sup> are the lengths of two conductors in parallel and *d*12 is the distance between

ò (11)

*ll d*


3

p

3

p

(10)

(12)

**3.3. The mathematical modeling of the magnetic (inductive) coupling**

*R fR*

*iI t*

= × ×+

*iI t*

*iI t*

4 *<sup>l</sup> dl dl <sup>M</sup>*

*cp*

*D*

*d*

*ik*

è ø

p=

m

*S fS*

*T fT*

namely:

362 Computational and Numerical Simulations

known. (Table 2).

general expression:

1and *l*

*ik*

m

p

where, *l*

becomes:

them.

$$\begin{aligned} \mathbf{J}I\_r &= -\mathbf{j} \cdot \boldsymbol{\alpha} \cdot \left( \dot{\mathbf{i}}\_R \cdot \mathbf{M}\_{Rr} + \dot{\mathbf{i}}\_S \cdot \mathbf{M}\_{Sr} + \dot{\mathbf{i}}\_T \cdot \mathbf{M}\_{Tr} \right) \\ \mathbf{J}I\_s &= -\mathbf{j} \cdot \boldsymbol{\alpha} \cdot \left( \dot{\mathbf{i}}\_R \cdot \mathbf{M}\_{Rs} + \dot{\mathbf{i}}\_S \cdot \mathbf{M}\_{Ss} + \dot{\mathbf{i}}\_T \cdot \mathbf{M}\_{Ts} \right) \\ \mathbf{J}I\_t &= -\mathbf{j} \cdot \boldsymbol{\alpha} \cdot \left( \dot{\mathbf{i}}\_R \cdot \mathbf{M}\_{Rt} + \dot{\mathbf{i}}\_S \cdot \mathbf{M}\_{St} + \dot{\mathbf{i}}\_T \cdot \mathbf{M}\_{Tt} \right) \end{aligned} \tag{13}$$


**Table 5.** Comparison between the measured voltages induced by magnetic coupling and the calculated ones

This simple calculation algorithm has lain at the basis of designing the calculation program in MATHCAD, by means of which there were determined analytically, the voltages induced through magnetic (inductive) coupling in the disconnected circuits of 220 kV power lines with double circuit from the Banat area – Romania. The results are presented in Table 5, in com‐ parison with the values of the voltages measured on the ground..

**d.** If the grounding loops are closed at both ends through short-circuit devices, the voltages induced through magnetic (inductive) coupling can force high electric currents, which are

Experimental Determinations and Numerical Simulations of the Effects of...

http://dx.doi.org/10.5772/57044

365

**e.** Besides a number of other factors (including weather) which influence the voltages induced electrically or magnetically, an important factor is the length of the portions of parallelism between the active line and the disconnected one. For lengths of parallelism greater than 20 km, the value of the voltages induced increases, practically, linearly with the length of the portion of parallelism, a phenomenon observed in figures 6 and 7. **f.** The original mathematical models, designed to simulate the phenomena of electric and magnetic coupling between the conductors of the high voltage overhead power lines with double circuit lead to results very close to those obtained directly through measurements on the ground, in real operating circumstances. Therefore the mathematical models are useful instruments for studying the phenomena of electromagnetic interferences at low

frequency in the case of power lines operating on parallel neighboring paths.

[1] Surianu F.D. Electromagnetic Compatibility. Applies in Electric Power Systems. (in Romanian) Orizonturi Universitare, ISBN 973-638-244-3, ISBN 978-973-638-244-4, Ti‐

[2] Surianu F.D. An apparatus for signalizing the induced currents in the disconnected circuits of double circuit h.v. overhead lines, Scientific Bull. of Politehnica University of Timisoara, Romania, Serie Energetics, Tom 50(64), Fasc.1-2, Nov., 2007, ISSN

[3] Munteanu C., Topa V., Muresan T., Costin A.M., Electromagnetic Interferences be‐ tween HV Power Lines and RBS Antennas Mounted on HV Tower, Proceedings of the 6th International Symposium on Electromagnetic Compatibility, EMC Europe,

[4] TTU-T and CIGRE, Protection Measure for Radio Base Stations Sited on Power Line

[5] Kenedy Aliila Greyson, Anant Oonsivilai, Identifying Critical Measurements in Pow‐ er System Network, Proceedings of The 8th WSEAS International Conference on

very dangerous for the operating staff.

Politechnica University of Timisoara, Romania

misoara, Romania;

1582-7194, pp. 615-620;

2009, Eindhoven, Holland, pp. 878-881;

Towers, Recommendation K 57, 2003;

**Author details**

**References**

Flavius Dan Surianu

This case brings about an important observation. Considering phase shift φ, between voltages and currents, through which the charging with power of the inductor circuit is indirectly expressed, the mathematical model has required the representation of the currents through momentary values instead of effective ones. But this leads to an additional unknown, which is time, *t*, namely the measuring moment of the voltages induced for each phase. It means that this case also requires the digitization of period *T = 0.02* seconds in 100 time intervals, *Δt<sup>k</sup> = 0.0002* seconds and the search of the moment in which the measurement of the voltage induced through magnetic coupling for each of the phases of the disconnected circuit was performed.

Comparing the measured values of the voltages induced through magnetic (inductive) coupling with the calculated ones there has been observed a very good concordance, which demonstrates an appropriate mathematical approach of the physical phenomena which lead to the magnetic (inductive) coupling between the conductors of the high voltage overhead power lines.
