Electr Eng (2013) 95:221–227 DOI 10.1007/s00202-012-0255-5

ORIGINAL PAPER

Simplified procedure to estimate the resistance parameters of transmission lines Sérgio Kurokawa · Gislaine Aparecida Asti · Eduardo Coelho Marques Costa · José Pissolato

Received: 1 July 2011 / Accepted: 23 September 2012 / Published online: 10 October 2012 © Springer-Verlag Berlin Heidelberg 2012

Abstract An alternative and simplified procedure is described to estimate the longitudinal resistances of transmission lines based on the real-time load profile. This method proposes to estimate the resistance parameters from the synchronized measurements of complex currents and complex voltages at the sending and receiving ends of transmission systems. The synchronized measurements can be in practice obtained using phasor measurement units (PMUs). Keywords Transmission line theory · Parameters estimation · Power systems · Phasor measurement units.

1 Introduction It is well known that the self and mutual impedances of overhead lines can be calculated from the solutions of the Maxwell’s equations for the boundary conditions at the contact surfaces of the conductor, air and ground. The expressions obtained to calculate these electrical parameters are S. Kurokawa Unesp-Universidade Estadual Paulista, Ilha Solteira, SP 15385-000, Brazil e-mail: [email protected] G. A. Asti Firb-Faculdades Integradas Rui Barbosa, R. Rodrigues Alves, 756, Andradina, SP 16900-000, Brazil e-mail: [email protected] E. C. M. Costa (B) · J. Pissolato Unicamp-University of Campinas, Av. Albert Einstein, 400, Campinas, SP 13081-970, Brazil e-mail: [email protected] J. Pissolato e-mail: [email protected]

functions of the frequency, soil and cable general characteristics, magnetic permeability and dielectric permittivity [1]. In the most usual procedures to calculate the line parameters, there are some explicit and implicit features which infer in physical approximations. They are usually based on the transmission line geometry and simplifications concerning the electromagnetic fields coupled to the system. The first simplifying consists to assume a plane soil surface, the line cables are horizontal and parallel among themselves, the distance between any pair of conductors is much greater than the sum of their radii and the electromagnetic effects of structures and insulators are neglected. Another assumption usually applied to the electrical parameters calculation is the quasi-stationary approach for the electromagnetic fields coupled to the system [2]. For practical reasons, the most of the methods applied to calculate the multiconductor line parameters assumes a frequency-independent soil conductivity and neglects the soil dielectric permittivity. These approximations restrict the validation of the parameters calculation to a limited range of frequencies, which varies with the geometric and physical characteristics of the line [2]. Furthermore, the soil resistivity is variable not just with the geological characteristics of the soil, but also along the seasons and with the weather conditions [3]. Therefore, based on the related features and approximations associated with the most of the classical ethodologies, the calculated electrical parameters can present significant inaccuracies which can result errors in the protection and monitoring of electrical power systems. A possible and direct solution for this subject-matter is to estimate the transmission line parameters from the voltage and current synchronized measurements. Transmission line protective relaying algorithms usually require transmission line parameters as inputs and thus the

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accuracy of these values represents an fundamental role in ensuring the reliable performance of the system protection [4]. Furthermore, the precise knowledge of the line parameters are essential to detecting and locating faults along the transmission line, as previously proposed by several authors [5]. The statements presented in this paragraph and above are some of the many motivations to use online methods based on synchronized measurements to estimate the exact line parameters. The synchronized currents and voltages at both terminals of transmission lines can be obtained in real time based on techniques using the global positioning system (GPS) and phasor measurement units (PMUs) [6,7]. In addition, there are currently several new technologies to perform real-time simulations, e.g., the real-time digital simulator (RTDS) and the object virtual network integrator (OVNI), which also simulate the measurements obtained using PMUs [7–9]. In Ref. [6], an approach was presented to live line measuring of the inductance parameters of multiconductor transmission lines with mutual inductance. The hardware structure of the measuring system was given and a real-time digital simulator RTDS was used to validate the accuracy of the estimated inductance parameters. Similarly, the method proposed in this paper is applied to estimate the resistance parameters of transmission lines based on simulated synchronized currents and voltages at the terminals of the system. The development is described in detailed along this paper, the reference values of current and voltage at both terminals of the line are properly simulated using an electromagnetic transient program (EMTP), and based on a procedure using modal analysis techniques, the resistance matrix is estimated. The proposed method is applied to estimate the longitudinal resistance of an untransposed 440-kV three-phase line, taking into account a variable load profile, and a relative error, calculated based on the exact reference values of the resistance matrix, is proposed to evaluate the accuracy of the estimated values.

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Firstly, considering a generic overhead transmission line with n phases, it is possible to define the p.u.l. resistance and inductance matrices as follows: ⎤ ⎡ R11 (ω) R12 (ω) · · · R1n (ω) ⎢ R21 (ω) R22 (ω) · · · R2n (ω) ⎥ ⎥ ⎢ (1) [R] = ⎢ ⎥ /km .. .. .. .. ⎦ ⎣ . . . . Rn1 (ω) Rn2 (ω) · · · Rnn (ω) ⎤ ⎡ L 11 (ω) L 12 (ω) · · · L 1n (ω) ⎢ L 21 (ω) L 22 (ω) · · · L 2n (ω) ⎥ ⎥ ⎢ (2) [L] = ⎢ ⎥ H/km .. .. .. .. ⎦ ⎣ . . . . L n1 (ω) L n2 (ω) · · · L nn (ω) In Refs. (1) and (2), matrices [R] and [L] are frequency dependent, as denotes the angular frequency ω. The generic terms Rii (ω) and Ri j (ω) in (1) are the real terms of the self impedance of the phase i and the real-term of the mutual impedance between phases i and j, respectively. In Ref. (2), terms L ii (ω) and L i j (ω) are the self inductance of the phase i and mutual inductance between phases i and j, respectively. Usually, in the classical transmission line theory, the transversal parameters of the line are represented just by a constant capacitance matrix. The shunt and mutual conductances are conventionally neglected [10]. Thus, for a multiconductor transmission line, the p.u.l. transversal capacitance matrix is expressed as follows: ⎡

C11 C12 ⎢ C21 C22 ⎢ [C] = ⎢ . .. ⎣ .. . Cn1 Cn2

⎤ · · · C1n · · · C2n ⎥ ⎥ . ⎥ F/km .. . .. ⎦ · · · Cnn

(3)

In (3), a generic term Cii is the apparent capacitance of the phase i and Ci j is the apparent capacitance between phases i and j.

3 Description of the proposed estimation method 2 Transmission line electrical parameters Transmission lines performance and wave propagation are characterized by their distributed longitudinal and transversal parameters. The longitudinal parameters of multiconductor transmission lines are represented by frequency-dependent resistances and inductances per unit length (p.u.l.) while the transversal parameters are represented by p.u.l. conductances and capacitances, usually considered to be constant in transmission line theory and modeling [10]. The frequency dependence of the longitudinal electrical parameters is because of the current return through the soil and the skin effect on the line wires [10,11].

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A simplified description of a three-phase system is given in the Fig. 1. A three-phase AC generator is connected at the sending end of the line while a generic three-phase load is connected at the receiving end. Initially, a generic singlecircuit three-phase transmission line is linking the generation to the load Z Load as in Fig. 1. The complex voltages at the sending end of the line are represented by VA1 , VA2 and VA3 and the complex currents at the sending end are given by IA1 , IA2 and IA3 . Similarly, the voltages at the receiving end are described by VB1 , VB2 and VB3 and the complex currents through the balanced threephase load Z Load are: IB1 , IB2 and IB3 .

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Sending end

223

Receiving end

IA1

VA1

Phase 1

VB1

IA2

VA2

Phase 2

VB2

VA3

Phase 3

VB3

IA3

IB1

Step 6: Once the line parameters matrices are known, the longitudinal resistances of the line can be properly estimated.

IB2

Z Load IB3

Three-phase line

ground Fig. 1 Currents and voltages on a three-phase transmission system

The complex voltages and currents can be measured using synchronized PMUs [7]. Thus the proposed methodology takes into account that the voltages and currents are simultaneously obtained at the sending and receiving ends of the transmission system. However, in the current approach, these values are calculated from computational simulations and by this means, the line electrical parameters are previously known. Therefore, with the previous knowledge of the exact line parameters, since the currents and voltages are obtained by the line modeling using these parameters, they are considered as reference values to evaluate the accuracy of the resistance parameters estimated using the proposed method. Another feature of the proposed estimation method is the explicit use of modal techniques to decouple the phases into their exact propagation modes. The electrical parameters for a single-phase line can be easily calculated from the two-port equations in the frequency domain [13]. On the other hand, the same procedure is not a trivial task concerning multiconductor transmission lines because of the electromagnetic coupling among the phases and estimation of the mutual parameters. The proposed methodology can be properly described into the following steps: Step 1: Measurement of the phase currents and the phase voltages at the sending and receiving ends of the line. Step 2: Conversion of the voltages and currents from the phase domain to the modal domain. Step 3: The propagation functions and the characteristic impedances of each propagation mode are calculated as a function of the modal currents and the modal voltages. Step 4: Based on the propagation functions and the characteristic impedances obtained in the third step, the impedance and the admittance matrices are calculated in the modal domain. Step 5: The modal impedance and modal admittance matrices are transformed to the phase domain.

3.1 Measuring (simulating) the phase currents and the phase voltages In Ref. [7], it is mentioned that complex voltages and currents at both terminals of the line can be measured simultaneously using synchronized PMUs. Thus, the proposed development supposes that the voltages and the currents are known or previously measured. In the current development, these values are calculated based on simulations using the EMTP [12]. This means that the transmission line is modeled based on the previous knowing of its electrical parameters (to simulate the synchronized currents and voltages at the line terminals) and then the estimated resistance parameters can be compared with the real reference values first used in the line modeling. Thus, the performance of the proposed estimation method can be precisely evaluated. The synchronized current and voltage vectors at the sending end (terminal A) and at the receiving end (terminal B) of the line are given as follows: ⎡ ⎡ ⎤ ⎤ VA1 VB1 (4) [VA ] = ⎣ VA2 ⎦ ; [VB ] = ⎣ VB2 ⎦ VA3 VB3 ⎡ ⎡ ⎤ ⎤ IA1 IB1 [IA ] = ⎣ IA2 ⎦ ; [IB ] = ⎣ IB2 ⎦ (5) IA3 IB3 The vectors expressed in (4) are the complex voltages at the sending and receiving ends of the line, [VA ] and [VB ], respectively. In the same way, the complex currents at both terminals of the line are expressed in (5), [IA ] and [IB ], respectively. 3.2 Calculating phase voltages and phase currents to the modal domain The relationship of voltages and currents, between the phase domain and the modal domain, are given as follows [13]: [E] = [TCk ]t [V ] [Im ] = [TCk ]

−1

[I ]

(6) (7)

In (6) and (7), [E] and [Im ] are the vectors with the complex voltages and complex currents, respectively, in the modal domain. The terms [TCk ]t and [TCk ]−1 are transposed and inverse matrices of the Clarke’s matrix [TCk ], respectively. Emphasizing that the constant Clarke’s matrix is a direct solution for the proposed estimation procedure. Considering that the line parameters are unknown, this means that the frequency-dependent transformation matrix is unknown

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as well. Thus, the use of the Clarke’s matrix is the unique solution for the proposed development. The Clarke’s matrix is given as follows: ⎤ ⎡ 2 √ √1 0 6 3 ⎢ 1 1 1 ⎥ (8) [TCk ] = ⎣ − √6 √2 √3 ⎦ 1 1 1 √ √ √ − −

propagation mode can be obtained just knowing the modal currents and modal voltages. Since the modal impedances and admittances of the modes α, β and zero are known, the line electrical parameters can be easily obtained converting these modal parameters to the phase domain using the Clarke’s matrix as a modal transformation matrix, such as in (9)–(12).

Based on (6) and (7), the complex voltages and currents at both terminals of the line are transformed from the phase domain to the modal domain by the following expressions:

3.4 Calculating the modal impedance and modal admittance matrices

[E A ] = [TCk ]t [VA ]

(9)

[E B ] = [TCk ]t [VB ]

(10)

Based on the transmission line theory, the propagation function and the characteristic impedance of the kth propagation mode are defined [14]:

(17) γk = Z k Yk  Zk Zck = (18) Yk

6

2

3

[IAm ] = [TCk ]

−1

[IA ]

(11)

[IBm ] = [TCk ]

−1

[IB ]

(12)

The vectors [E A ] and [E B ] are the complex voltages in the modal domain at the sending and receiving ends, respectively. The vectors [IAm ] and [IBm ] are the complex currents in the modal domain at the sending and receiving ends of the line, respectively.

Z k = γk Zck γk Yk = Zck

3.3 Calculating the propagation functions and the characteristic impedances For the kth mode of a generic multiconductor line, the twoport equations for the frequency-domain currents and voltages are given as follows [13]: E Ak = E Bk cosh(γk d) − IBk Zck sinh(γk d) E Bk sinh(γk d) IAk = −IBk cosh(γk d) + Zck

(13) (14)

In (13) and (14), γk and Zck are the propagation function and the characteristic impedance of the kth propagation mode, respectively [10]. The term d is the line length expressed in kilometers. The terms E Ak and E Bk are complex voltages at sending and receiving ends of the kth mode, respectively. The elements IAk and IBk are complex currents at the sending and receiving ends of the kth mode, respectively. From (13) and (14), the propagation function and the characteristic impedance for the kth propagation mode can be expressed as:  E Ak IAk − E Bk IBk 1 (15) γk = Arc cosh d E Bk IAk − E Ak IBk E Bk sinh(γk ) (16) Zck = IAk + IBk cosh(γk ) From (15) and (16), the propagation functions and the characteristic impedances for each mode of the multiconductor line can be calculated. Based on these propagation characteristics, the impedances and admittances of each

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where Z k and Yk are the longitudinal impedance and transversal admittance of the kth mode, respectively. From (17) and (18), Z k and Yk can be simplified as follows: (19) (20)

Extrapolating the simplification presented above to a three-phase representation, the modal parameters can be expressed by the following diagonal matrices, [Z m ] and [Ym ], which are the longitudinal impedances and transversal capacitances in the modal domain, respectively: ⎡ ⎤ Zα 0 0 (21) [Z m ] = ⎣ 0 Z β 0 ⎦ 0 0 Z0 ⎡ ⎤ Yα 0 0 [Ym ] = ⎣ 0 Yβ 0 ⎦ (22) 0 0 Y0 Knowing the impedances and admittances in the modal domain, some electrical characteristics of the transmission system can be calculated in the phase domain with good accuracy by inverse modal transforms using the Clarke’s matrix. 3.5 Converting the modal parameters to the phase domain Assuming that the impedance and admittance values in the modal domain are previously known, the same phase values can be calculated using modal transformations [15]: [Z ] = [TCk ][Z m ][TCk ]−1 [Y ] = [TCk ]

−1

[Ym ][TCk ]

(23) (24)

Matrices [Z] and [Y] are the longitudinal impedance matrix and transversal admittance matrix of the line in the

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(25)

[Y ] = j ω[C]

(26)

The sixth step, described in the introduction of the Sect. 3, consists in to extract the real components of the matrix [Z ], which is the resistance matrix [R].

1 3.6 m

[Z ] = [R] + j ω[L]

5 (7.51; 36)

4

phase domain. The impedance and admittance matrices can be expressed in terms of (1) to (3):

2

3

(9.27; 24.4)

4 Validation of the proposed estimation method The estimation of the self and mutual resistances of a multiconductor transmission line is not a trivial procedure. The self resistance is composed of two components: the skin-effect resistance and the earth return resistance. The first component is intrinsically associated with the magnetic field inside each conductor, which means a major current density through the vicinity of the external surface of the cables with the frequency increment. The resistance associated with the earth return frequency-dependent impedance is a function of the soil resistivity, permeability and permittivity. The exact evaluation of these referred parameters is a very complex process, taking into account that they vary along the seasons and with weather conditions [2]. The mutual resistances of a multiconductor overhead line are only composed of the earth return resistances (soil effect), as described in the last paragraph. However, the mutual resistance representation and physical approaches are not also a trivial modeling, several approximations are required to calculate and model these electrical parameters in transmission line modeling [15]. To verify the performance of the proposed estimation method, a conventional 440-kV transmission line is considered. This line is untransposed and characterized by a vertical symmetry plane. The tower structure of this line, with geometrical position of the phases and the shield wires, are given in Fig. 2. The bundled conductors are composed of four Grosbeak subconductors and the shield wires are EHSW-3/8” conductors. Based on the structural and the physical descriptions given above, the 440-kV line was modeled using a frequencydependent line model available in the EMTP, the J. Marti Model [12,14]. The synchronized currents and voltages were simulated using this model and the line was modeled considering the following resistance matrix at the fundamental frequency: ⎤ 0.0603 0.0580 0.0580 [Rref ] = ⎣ 0.0580 0.0604 0.0581 ⎦ 0.0580 0.0581 0.0604

Fig. 2 Conventional 440-kV transmission line

The same resistance matrix is considered as reference to measure the relative errors associated with the values estimated using the proposed methodology. The synchronized currents and voltages at both terminals of the line are calculated considering several load profiles, thus the relative errors can be obtained as a function of the load apparent power and power factor. The exact concept of the relative error adopted in this paper to evaluate the performance of the proposed estimation method is the simple relationship of the estimated resistance Rest over the reference resistance Rref (Rest /Rref ). In the following figure, the maximum relative errors associated with the p.u.l. self resistances are described as a function of the load profile: Figure 4 describes the maximum relative errors associated with the estimated p.u.l. mutual resistances as a function of the load power demand and with a variable power factor: Figures 3 and 4 show minor relative errors for apparent power values up to 3 MVA, however, major variations can

⎡

/km

(27)

Fig. 3 Relative error of the self resistances as a function of the load profile

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Fig. 4 Relative error of the mutual resistances as a function of the load profile

be observed above 10 MVA with power factor values close to unitary. The estimation method shows more accurate for lower inductive power factors, as described the relative errors measured considering constant power factor values of 0.76 and 0.86, emphasizing that for an inductive power factor of 0.76, the relative error is practically constant and the resistance values can be accurately estimated up to 100 MVA. This statement can be better verified based on the following analysis, considering the relative error as a function of the power factor and constant values for the power load demand at the receiving end of the line: Based on the curves presented in the Fig. 5, it is possible to conclude that, in general, the proposed estimation method presents good accuracy for load demands above 100 kVA. Concerning load profiles up to 100 kVA, the estimation method has a better performance for lower inductive power factor values, as shown the curve related to 10 kVA in the Fig. 5. The more accurate resistances estimated applying the proposed estimation methodology are given in the following matrix, taking into account a high load demand (above 100 MVA) and power factor up to 0.85. ⎡

⎤ 0.0660 0.0615 0.0615 [R] = ⎣ 0.0615 0.0661 0.0616 ⎦ 0.0615 0.0616 0.0661

/km

(28)

Fig. 5 Maximum relative error of the estimated resistances as a function of the power factor for constant power load demands

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The estimated resistances in (28) are approximated to the reference values given in (27), proving the good performance and precision of the proposed estimation method.

5 Conclusion A simplified methodology to estimate the resistance parameters of transmission lines was developed. The procedure is based on the line modal decoupling into its exact propagation modes using a known transformation matrix, emphasizing that the exact transformation matrix is unknown, since the frequency-dependent parameters of the line are also unknown in a practical case. However, in the estimation method validation, the current and voltage measurements at both line terminals were previously simulated using a computational line model, thus the same line electrical parameters of the modeled transmission line were used as reference values to calculate a relative error to evaluate the method performance. The estimation method showed a variable performance as a function of the load profile at the receiving end of the line, i.e., the total apparent power and power factor are intrinsically related to the accuracy of the estimated values of the resistance parameters. However, the methodology presented a good performance for medium and high power demands with an inductive power factor up to 0.85. These load profiles are commonly observed in conventional power transmission systems, thus the proposed methodology can be easily applied to high-voltage transmission lines with good accuracy, emphasizing that the knowledge of these parameters is an important feature to estimate the p.u.l. active losses of transmission systems and their propagation characteristics. Furthermore, the resistance parameters are also an important information to set protective relaying and detecting/locating faults in transmission systems. As a future proposal, the current estimation methodology could be extended to estimate other line parameters, e.g., the inductances and the transversal capacitances. However, a few modifications and analysis have to be performed to evaluate the accuracy and robustness of the proposed estimation

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method applied for these additional measurements. This further proposal could represent an efficacious method to estimate all electric parameters of transmission lines for variable load conditions. Acknowledgments “Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/Processo 4570-11-1)” and the National Council of Technological and Scientific Development (CNPq).

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