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TRANSACTIONS ON ELECTRICAL ENGINEERING CONTENTS Pástor, M., Dudrik, J.: Stability of Grid-Connected Inverter with LCL Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 – 96 Bendík, J., Cenký, M., Eleschová, Ž.: 3D Numerical Calculation of Electric Field Intensity under Overhead Power Line Using Catenary Shape of Conductors . . . . . . . . . . . . . . . . . . 97 – 103 Cintula, B., Eleschová, Ž., Beláň, A., Janiga, P.: Comparison of Reconfigurations Using Deterministic Approach for Global Assessment of Operational State in Power System . . . . . . . . 104 – 111 Eleschová, Ž., Ivanič, M.: Impact of Asymmetry of Over-Head Power Line Parameters on Short-Circuit Currents . . . . . . . . 112 – 115 Murín, J., Hrabovský, J., Gogola, R., Goga, V., Janíček, F.: Numerical Analysis and Experimental Verification of Eigenfrequencies of Overhead ACSR Conductor . . . . . . . . . 116 – 121 Vol. 5 (2016) No. 4 ERGO NOMEN pp. 91 – 121 TRANSACTIONS ON ELECTRICAL ENGINEERING Publisher: ERGO NOMEN, o.p.s., K13114 FEE CTU in Prague, Technicka 1902/2, 166 27 Praha 6, Czech Republic E-mail: [email protected] Editorial Office: PIVONKA Pavel BAUER Jan HAVLICEK Radek KOBRLE Pavel MERICKA Jiri NOVA Ivana VONDRICH Jiri ZDENEK Jiri Periodicity: Language: Scope: On-line version: Quarterly English International scientific journal of electrical engineering www.transoneleng.org ISSN 1805-3386 Each paper in the journal is evaluated by two reviewers under the supervision of the International Editorial Board. International Editorial Board Editor in Chief: Prof. LETTL Jiri, Czech Technical University in Prague, Czech Republic Members: Prof. BAUER Palo, Delft University of Technology, Netherlands Prof. BRANDSTETTER Pavel, VSB-Technical University of Ostrava, Czech Republic Prof. DOLEZEL Ivo, The Academy of Sciences of the Czech Republic, Czech Republic Prof. DUDRIK Jaroslav, Technical University of Kosice, Slovakia Prof. NAGY Istvan, Budapest University of Technology, Hungary Prof. NOVAK Jaroslav, University of Pardubice, Czech Republic Prof. ORLOWSKA-KOWALSKA Teresa, Wroclaw University of Technology, Poland Prof. PEROUTKA Zdenek, University of West Bohemia, Czech Republic Prof. PONICK Bernd, Leibniz University of Hannover, Germany Prof. RICHTER Ales, Technical University of Liberec, Czech Republic Prof. RYVKIN Sergey, Russian Academy of Sciences, Russia Prof. SKALICKY Jiri, Brno University of Technology, Czech Republic Prof. VITTEK Jan, University of Zilina, Slovakia Prof. WEISS Helmut, University of Leoben, Austria Responsibility for the contents of all the published papers and technical notes is upon the authors. Template in MS WORD and basic typographic rules to be followed see www.transoneleng.org. Copyright: ©2016 ERGO NOMEN, o.p.s. All right reserved. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 91 Stability of Grid-Connected Inverter with LCL Filter Marek Pástor 1) and Jaroslav Dudrik 2) 1) 2) Dept. of Electrical Engineering and Mechatronics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Košice, Slovakia, e-mail: 1) [email protected], 2) [email protected] Abstract — The paper analyses oscillations in a singlephase grid connected inverter with the LCL output filter. Passive and active damping techniques are designed and compared by simulations. The inverter is controlled by a proportional resonant controller. Keywords — control, dc/ac converter, filtering I. INTRODUCTION Grid connected pulse width modulated (PWM) voltage source converters require an inductive filter for its operations. These converters are used in many applications such as the active power filters, PWM rectifiers or PWM inverters. To decrease the weight and volume of the filter it is desirable to use a higher-order filter. Usually an LCL filter is used [2–14]. The LCL filter has a drawback of possible resonance if excited at a resonant frequency. This resonance introduces the system instability and increases THD of current. To suppress the oscillations a damping technique is used. There are two ways of the oscillation damping in the LCL filter: the passive and active damping. Both of them have its advantages and disadvantages and are still analysed and designed. The passive damping [1] uses damping resistors which introduces additional system losses. An active damping is used to remove these extra losses but keep system stability. The active damping of the LCL filter is studied in the active power filters [2–4], PWM rectifiers [5–7], PWM inverters [8–12] and in general grid interacting converters [13–15]. This paper analyses various damping techniques for the single-phase grid connected inverter with the LCL filter. Several damping techniques are analysed and designed for a single phase inverter with the proportional-resonant (PR) controller. II. LCL FILTER The grid-connected voltage source inverter (VSI) needs an inductive load. This load is provided mainly by the output filter. The filter topology depends on the harmonics attenuation requirements. The harmonics are produced by the pulse-width modulation (PWM). Usually higher order filters, such as the LCL filter (Fig. 1), are used. The LCL filter is a third order filter with attenuation of 60 dB/dec. The advantage of the high attenuation of a third order system has a drawback of resonance. The LCL filter has three resonant frequencies defined by the reactive components. The grid influences the resonant frequency of the LCL filter as well. Connecting the LCL filter to the output of a PWM modulated inverter causes driving the LCL filter input with a spectrum of various highfrequency voltages. TELEN2016011 DOI 10.14311/TEE.2016.4.091 Fig. 1. LCL filter topology and its dynamical model (RS and RG are parasitic resistances). The inverter output voltage VS with a frequency equal to a resonant frequency of the LCL filter defined by (1) causes a resonance in the output current IG of the LCL filter. The resonance causes system instability and increases the THD of the grid current and thus it is undesirable. 1 2 LS LG LS LG C The example of the harmonic spectrum of the LCL filter output current is shown in Fig. 2. The resonance peak is clearly visible and its contribution to the THD of the grid current IG is more significant than the contribution from the inverter switching frequency. Suppression of the switching frequency is guaranteed by a proper design of the LCL filter reactive components. Suppression of the LCL filter resonance is guaranteed by a proper damping technique design. However, as it is shown, some damping techniques influence the harmonics suppression. f0 I GVS Fig. 2. Example of LCL filter output current (with rms value 1.409 A) spectrum without proper oscillation damping. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 The attenuation of the switching frequency in the voltage VS with respect to the grid current IG is described by the transfer function (2). IG ( s) 1 VS ( s) s3 LS LGC s 2 LS CRG RS CLG s RS CRG LS LG RS RG A. Passive Damping The passive damping is ensured by adding a dissipative component to the LCL filter topology. The properly placed resistor will decrease the resonance peak. The damping resistor is usually placed in series with the filter capacitor (Fig. 3). b) dynamical model Fig. 4. LCL with various damping resistor values. B. Active Damping The frequency characteristics of the system can be changed also by modifying the controller. The proper transfer function of the controller can damp the LCL filter oscillations. The main idea is to remove the undesired frequencies from the inverter output voltage by modifying the PWM modulation signal. This approach is called the active damping. C. Active Damping – Notch Filter There are several ways how to achieve the active damping. Probably the most straightforward one is to remove the resonant frequency from the inverter output voltage by a filter (Fig. 5). Fig. 5. Simplified control structure of PR current control with active damping by notch filter. Fig. 3. LCL filter with passive damping resistor. The damping resistor will change the transfer function of the LCL filter (3). IG ( s) 1 sCRC VS ( s) s3 LS LG C s 2 LS CRG RS CLG RC CLG LS CRC frequency of the LCL filter is not too far from the switching frequency, this decrease is minimal. III. RESONANCE DAMPING There are two ways how to damp the LCL filter. The so-called passive damping employs an extra resistor added in series with a filter capacitor. This resistor influences the transfer function (2) and suppresses the resonance by modifying the transfer function for frequencies around and above the resonant frequency defined by (1). The frequency characteristics of the system can be changed also by modifying the controller. The proper transfer function of the controller can damp the LCL filter oscillations. The main idea is to remove undesired frequencies from the inverter output voltage by modifying the modulation signal of the PWM. This approach is called an active damping. a) LCL filter with damping resistor 92 The suitable type of the filter is a notch (negative peak) filter. The simple notch filter consists of the LC resonant tank (Fig. 6). s RS CRG LS LG RS RC C RC CRG RS RG The damping resistor adds losses to the LCL filter. The value of the resistor is usually chosen as one third of the capacitor reactance at the resonant frequency: RC 1 1 3 2 f0 I C GVS The advantage of the passive damping is its robustness. However there are also disadvantages. The damping resistor has losses which decrease the overall system efficiency. The losses on the damping resistor RC connected in series with the capacitor can be calculated [16] where h represents harmonic order: PRC RC iS h iG h b) notch filter frequency characteristics Fig. 6. Notch filter. 2 h The second main disadvantage is decrease of the higher frequency damping (Fig. 4). If the resonant TELEN2016011 DOI 10.14311/TEE.2016.4.091 a) notch filter topology The transfer function of a notch filter is defined by (6). Fnotch ( s) LnCn s 2 1 LnCn s 2 RnCn s 1 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 93 where: IV. CONTROL SYSTEM DESIGN Ln 1mH 1 Cn 2 4 Ln f 02I V G S Rn 2 f n 2 LnCn 1 2 f nCn 2 The frequency fn defines the notch filter bandwidth around the LCL filter resonant frequency f0IGVS. The advantage of the active damping by the notch filter is its sensorless concept. When compared to the passive damping, the system transfer function for higher frequencies is not influenced by inserting the notch filter. However, the design of the notch filter depends on the LCL filter parameters. D. Active Damping – Virtual Resistance Another active damping method uses the concept of a virtual resistor. The virtual resistor method is based on the LCL filter capacitor current sensing and multiplying of this current by the virtual damping resistor resistance. The resulting virtual voltage is then subtracted from the inverter PWM modulating voltage (Fig. 7). The virtual resistor value can be calculated using (2). As can be seen from Fig. 8, the virtual resistor does not change the system frequency characteristic for higher frequencies than the f0IGVS resonant frequency. The transfer function of the LCL filter damped by virtual damping resistor RC is changed to: TABLE I. LCL FILTER PARAMETERS LCL filter parameter Apparent power S Switching frequency fSW Current ripple of IS Resonant frequency f0IGVS Inductance LS Inductance LG Capacitance C Value Unit 1.2 5 20 2 3.6 1.8 5.3 kW kHz % kHz mH mH µF B. Proportional Resonant Controller The inverter is a single-phase system. It is thus beneficial to use the proportional resonant (PR) controller (Fig. 9). This approach will remove the need to transform the single-phase system into the rotating reference frame in dq coordinates for the PI controller. Fig. 9. PR controller. IG ( s) 1 VS ( s) s3 LS LG C s 2 LS CRG RS CLG RC CLG s RC RG C RS RG C LS LG RS RG A. LCL Filter Parameters The performance of three damping methods is compared by simulation. The system consists of a singlephase PWM voltage source inverter with switching frequency of 5 kHz and dc link voltage of 420 V. The inverter is connected to the grid through the LCL filter with parameters shown in Table I. Fig. 7. Simplified control structure of PR current control with active damping by virtual resistor. The PR controller has a transfer function (11). FPR ( s) K P 2 K I PR s s 2 2PR s g2 The frequency ωPR defines the PR controller bandwidth around the grid frequency ωg. To design the PR controller it is necessary to know the transfer function from the inverter voltage VS (manipulated variable) to the grid current IG (controlled variable). The grid voltage VG is a measured disturbance and is compensated in the PR controller. The grid current is a measured controlled variable. The high frequency transfer function from VS to IG of the LCL filter is: IG (s) VS ( s ) 1 s3 LS LGC s 2 LS CRG RS CLG s RS CRG LS LG RS RG The PR controller controls the grid current IG with the grid frequency and thus generates the manipulated variable VS with the grid frequency. The transfer function (3) is simplified by omitting the high-frequency terms: Fig. 8. Frequency characteristics of LCL filter with active damping by virtual resistor. TELEN2016011 DOI 10.14311/TEE.2016.4.091 IG ( s) VS ( s) 1 s LS LG RS RG LL The LCL filter is therefore simplified to a first order system with the time constant of: Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 And its gain: LS LG RS RG TLCL 1 RS RG The PR controller is designed to compensate the time constant TLCL. The proportional gain of the PR controller is set to (τ is the time constant of the required control dynamics of the whole controlled system): K LCL KP TLCL K LCL KP TLCL The integral gain of the PR controller is set to: KI 94 A. Passive Damping Method The damping resistor RC for the LCL filter specified in Table I calculated using (2) is 5 Ω. Figure 11 shows frequency characteristics of the undamped LCL filter and LCL filter damped by passive damping resistor. TABLE II. PR CONTROLLER PARAMETERS PR Controller Parameters Time constant TLCL Gain KLCL Proportional gain KP Integral gain KI Frequency ωPR Grid frequency ωG Value Unit 13.5 2.5 5.4 400 1 314 Ms rad/s rad/s Fig. 11. LCL filter frequency characteristics with passive damping. The designed passive damping causes a positive gain margin of 13.4 dB and a positive phase margin of 90.2 deg. (Fig. 12). The stability of the system is thus ensured. V. COMPARISON OF DAMPING METHODS The stability of the system created by PR controller and LCL filter is checked by gain and phase margins obtained from Bode characteristics. The system is considered stable if the gain margin is at least 10 dB and the phase margin is at least 60 deg. Besides the control loop, the damping of the LCL filter frequency characteristic is compared as well. The frequency characteristics of the open control loop with designed PR controller and undamped LCL filter is shown in Fig. 10. It has a gain margin of -46.7 dB and phase margin of -89.7 deg. and is clearly unstable. The negative peak in phase at 315 rad/s is caused by resonant frequency of PR controller. Fig. 12. Frequency characteristics of open control loop with PR controller and LCL filter with passive damping. Figure 13 shows the dynamic response of the grid current. The stability is clearly visible in the time domain as well without influencing the system dynamic behaviour. Fig. 10. Frequency characteristics of open control loop with PR controller and undamped LCL filter. Fig. 13. LCL filter output current and its reference signal with passive damping. TELEN2016011 DOI 10.14311/TEE.2016.4.091 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 B. Notch Filter The simulated notch filter transfer function is: 6.36 109 s 2 1 6.36 109 s 2 4.664 105 s 1 To suppers the resonant frequency the notch filter has a narrow bandwidth of 1 kHz. Figure 14 shows the frequency characteristics of the undamped filter and the filter damped with the notch filter. The resonant peak is completely removed. Fnotch ( s) 95 C. Virtual Resistance To create a damping in the controlled system a virtual damping resistor with resistance of 10 Ω (two times the passive damping resistor) was chosen. Figure 17 shows the frequency characteristics of the undamped LCL filter and the LCL filter damped with the virtual resistor. Fig. 17. LCL filter frequency characteristics with virtual resistance damping. Fig. 14. LCL filter frequency characteristics with notch filter damping. The notch filter damping (Fig. 5) causes a positive gain margin of 17.2 dB and a positive phase margin of 87.5 deg (Fig. 15). After checking the stability of the virtual damping method only a 8.89 dB gain margin was observed (Fig. 18). The system is stable but it is advisable to use a three times the value of calculated passive damping resistor (4) to ensure a gain margin at least 10 dB. Fig. 18. Frequency characteristics of open control loop with PR controller and LCL filter damped with virtual resistor. Fig. 15. Frequency characteristics of open control loop with PR controller and LCL filter damped by notch filter. Fig. 16. LCL filter output current and its reference signal with notch filter damping. TELEN2016011 DOI 10.14311/TEE.2016.4.091 The smaller gain margin caused by the proposed virtual damping resistor is also visible in Fig. 19 as damped oscillations during a transient period. Fig. 19. LCL filter output current and its reference signal with passive damping virtual resistance damping. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 VI. CONCLUSIONS The paper describes the oscillation phenomena in the grid-connected inverter with the LCL filter. The LCL filter resonance problem is analysed in the single-phase inverter with the PWM control. The PR controller is used to control the grid current. The controller is designed and verified by simulation. It is shown that the PR controller by its own cannot ensure the stability of the system. Thus three different techniques are analysed and designed to suppress the LCL filter oscillations. The passive damping technique has the advantage of robustness and simplicity. Unfortunately the passive damping increases the system losses and the LCL filter capabilities of harmonics damping are reduced. The two active damping techniques remove these disadvantages. The notch filter at the output of the controller can be realised in a digital form. Thus no additional hardware costs are included. The notch filter needs to be tuned for a particular filter and its resonant frequency and is thus less robust. The additional current sensor is required for the virtual resistor damping technique. The capacitor current sensing will increase the robustness. The simple passive damping technique can be used to calculate the value of a virtual damping resistor. Choosing of a suitable damping technique depends on an application and usually an active damping together with passive damping is used. ACKNOWLEDGMENT The authors wish to thank the project VEGA 1/0464/15 for its support. The work was supported by project FEI-2015-3. REFERENCES [1] [2] [3] [4] Cuili Chen; Zhiqiang Wang; Yulong Zhang; Guofeng Li; Yan Wu, "A novel passive damping LCL-filter for active power filter," in Transportation Electrification Asia-Pacific (ITEC AsiaPacific), 2014 IEEE Conference and Expo , pp.1-5, Aug. 31 2014Sept. 3 2014, doi: 10.1109/ITEC-AP.2014.6940684 Guohong Zeng; Rasmussen, T.W.; Lin Ma; Teodorescu, R., "Design and control of LCL-filter with active damping for Active Power Filter," in Industrial Electronics (ISIE), 2010 IEEE International Symposium on , pp.2557-2562, 4-7 July 2010, doi: 10.1109/ISIE.2010.5637575 M. Routimo, H. Tuusa, "LCL Type Supply Filter for Active Power Filter - Comparison of an Active and a Passive Method for Resonance Damping," in Power Electronics Specialists Conference, 2007. PESC 2007. IEEE , pp.2939-2945, 17-21 June 2007, doi: 10.1109/PESC.2007.4342488 Wenqiang Zhao; Guozhu Chen, "Comparison of active and passive damping methods for application in high power active power filter with LCL-filter," in Sustainable Power Generation and Supply, 2009. SUPERGEN '09. International Conference on , pp.1-6, 6-7 April 2009, doi: 10.1109/SUPERGEN.2009.5347992 TELEN2016011 DOI 10.14311/TEE.2016.4.091 96 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] C. Liu, Z. Zhao, T. Lu; L. Yuan, "Design and implement of an active damping LCL-filter for three-level voltage source PWM rectifier," in Electrical Machines and Systems (ICEMS), 2011 International Conference on , pp.1-5, 20-23 Aug. 2011, doi: 10.1109/ICEMS.2011.6073657 M. Liserre, A.D Aquila, F. Blaabjerg, "Genetic algorithm-based design of the active damping for an LCL-filter three-phase active rectifier," in Power Electronics, IEEE Transactions on , vol.19, no.1, pp.76-86, Jan. 2004, doi: 10.1109/TPEL.2003.820540 C. Wessels, J. Dannehl, F.W. Fuchs, "Active damping of LCLfilter resonance based on virtual resistor for PWM rectifiers — stability analysis with different filter parameters," in Power Electronics Specialists Conference, 2008. PESC 2008. IEEE , pp.3532-3538, 15-19 June 2008, doi: 10.1109/PESC.2008.4592502 Ch. Bao, X. Ruan, X, Wang, W. Li, D. Pan, K, Weng, "Step-byStep Controller Design for LCL-Type Grid-Connected Inverter with Capacitor–Current-Feedback Active-Damping," in Power Electronics, IEEE Transactions on , vol.29, no.3, pp.1239-1253, March 2014, doi: 10.1109/TPEL.2013.2262378 X. Lu, K. Sun, L. Huang, M. Liserre, F. Blaabjerg, "An active damping method based on biquad digital filter for parallel gridinterfacing inverters with LCL filters," in Applied Power Electronics Conference and Exposition (APEC), 2014 TwentyNinth Annual IEEE , pp.392-397, 16-20 March 2014, doi: 10.1109/APEC.2014.6803338 Chenlei Bao; Xinbo Ruan; Xuehua Wang; Weiwei Li; Donghua Pan; Kailei Weng, "Design of injected grid current regulator and capacitor-current-feedback active-damping for LCL-type gridconnected inverter," in Energy Conversion Congress and Exposition (ECCE), 2012 IEEE , pp.579-586, 15-20 Sept. 2012, doi: 10.1109/ECCE.2012.6342769 Yong Shi; Jianhui Su, "An active damping method based on PR control for LCL-filter-based grid-connected inverters," in Electrical Machines and Systems (ICEMS), 2014 17th International Conference on , pp.944-948, 22-25 Oct. 2014, doi: 10.1109/ICEMS.2014.7013604 Sowjanya, M.L.; Babu, B.C., "Comparative analysis of LCL filter with active and passive damping methods for grid-interactive inverter system," in Students' Technology Symposium (TechSym), 2014 IEEE , pp.350-355, Feb. 28 2014-March 2 2014, doi: 10.1109/TechSym.2014.6808074 Wenli Yao; Yongheng Yang; Xiaobin Zhang; Blaabjerg, F., "Digital notch filter based active damping for LCL filters," in Applied Power Electronics Conference and Exposition (APEC), 2015 IEEE , pp.2399-2406, 15-19 March 2015, doi: 10.1109/APEC.2015.7104684 Pena-Alzola, R.; Liserre, M.; Blaabjerg, F.; Yongheng Yang, "Robust design of LCL-filters for active damping in grid converters," in Industrial Electronics Society, IECON 2013 - 39th Annual Conference of the IEEE , pp.1248-1253, 10-13 Nov. 2013, doi: 10.1109/IECON.2013.6699311 Orellana, M.; Grino, R., "On the stability of discrete-time active damping methods for VSI converters with a LCL input filter," in IECON 2012 - 38th Annual Conference on IEEE Industrial Electronics Society , vol., no., pp.2378-2383, 25-28 Oct. 2012, doi: 10.1109/IECON.2012.6388871 R. Teodorescu, M. Liserre, P. Rodriguez, Grid Converters for Photovoltaics and Wind Power Systems, John Wiley, 2011, doi: 10.1002/9780470667057 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 97 3D Numerical Calculation of Electric Field Intensity under Overhead Power Line Using Catenary Shape of Conductors Jozef Bendík 1), Matej Cenký 2) and Žaneta Eleschová 3) 1) 2) 3) Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Institute of Power and Applied Electrical Engineering, Bratislava, Slovakia, e-mail: 1) [email protected], 2) [email protected], 3) [email protected] Abstract — This article presents a superposition method in combination with the Coulomb’s law and the Method of image charges for calculation of the electric field distribution generated by high voltage overhead power lines above a flat surface in every dimension. Such calculations are required to ensure the operational safety of people exposed to the action of external electric field as well as to reduce the cost of people protection. The method provides options for calculation of the field around the wire of a general shape. This substantial improvement of the method could be applied to eliminate the usual error in the calculation created using approximation of catenary shape conductors by infinite straight conductors. The method has been extensively tested on a set of shapes with known analytical solutions. It has been shown that the numerical solution converges uniformly to the analytical solution and the accuracy depends only on the number of finite elements. according to the international standards. The reference levels for the Slovak and Czech Republic are shown in Table 1. [4], [5]. Several national and international organizations have formulated guidelines establishing limits for the occupational and residential EMF exposure. The exposure limits for EMF fields developed by ICNIRP – formally recognized by World Health Organization (WHO), were developed following reviews of scientific literature, including thermal and non-thermal effects. The standards are based on evaluations of biological effects that have been established to have health consequences [3] [6]. TABLE I. LEGISLATIVE VALID REFERENCE LEVELS IN SLOVAK AND CZECH REPUBLIC FOR EXPOSURE TO TIME-VARYING ELECTRIC AND MAGNETIC FIELDS (UNPERTURBED RMS VALUES) Keywords — electric field, FEM, power transmission line, charge I. INTRODUCTION The overhead transmission lines are source of magnetic as well as electric field. This electromagnetic field (EMF) is of low frequency and it is time-varying [1]. It is believed that the influence of the EMF field is harmful only in certain way to the human health so the effects of the field have to be taken in consideration in the power transmission line project design [2]. An increasing demand on operational safety in the vicinity of high voltage lines calls for a more precise project preparations. The project preparation mainly involves numerical calculations, the result of which can facilitate and cheapen particular project activities. An exposure to EMF field causes flow of induced currents in living organisms, and can have other unpleasant effects on human body [3]. In many cases this considerations has not been proven, but studies show a potential risk. According to this health risks non-governmental organization International Commission on Nonionizing Radiation Protection (ICNIRP) established for population reference levels for the exposure to time-varying electric and magnetic fields shown in Table 1 [2]. These reference levels can vary TELEN2016018 DOI 10.14311/TEE.2016.4.097 The main conclusion from the WHO reviews is that the electromagnetic field exposures below the limits recommended in the ICNIRP international guidelines do not appear to have any known consequence on health. In the past years the European Commission gave a new recommendation, based on the ICNIRP study, to establish that all European Union (EU) states observe standard reference levels of the exposure to the field. Although reference levels vary in different countries they cannot be lower than the EU standards. The best way to deal with these standards is to have a truth worthy method for the calculation of the electric field. From Maxwell`s equations with a combination of the method of image charges dependence of electric field from the position of the conductor, the observer and its mirror image can be derived. This complex approach gives in combination with finite element method tool for calculation of the electric field intensity. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 98 TABLE II. REFERENCE LEVELS FOR EXPOSURE TO TIME-VARYING ELECTRIC AND MAGNETIC FIELDS (UNPERTURBED RMS VALUES) [2] A common simplification in the evaluation is to calculate the electric field intensity with the real conductor approximation by the infinite straight conductor placed in the lowest height of the sag. However this condition does not reflect most of real situations. This simplification can be applied only on symmetric spans. II. TERMINOLOGY A. Quasi-static Field The time varying EMF is called quasi-static if we can neglect time changes of the EMF spread by finite speed. For a harmonic EMF field, which spread in the air and which variables vary with angular frequency ω, the quasistatic criterion can be expressed by Eq. 1, where σ is enviromental conductivity, f is frequency [7]. For air σ gets values from 3,10e-15 to 8,10e-15 [S/m]. The criterion is valid for both sides of the interval. For the quasi-static field applies, that we can neglect time derivations in I. and II. Maxwell’s equation [7]. Three orthogonal components of a vector may be phasors with different magnitude and phase angles. These components are called phasor-vector (Eq. 7). In this article, a vector is indicated with an arrow and phasorvectors with a hat over the arrow [8]. In many cases the single RMS value is necessary to evaluate the field. This value is calculated from magnitudes values of the phase-vector components as follows: C. Catenary Shape of Conductors Conductor attached on two sides, in this case on transmission towers, will form curve in shape of a catenary.[1] The catenary can be considered as symmetric, if conductors are at their ends in the same height above a flat surface, or asymmetric if not. Figure1 shows the general asymmetric span, where A is the length of the span. V1 and V2 are the heights at each end of the catenary above the flat ground. A1 is the distance from the beginning of the catenary to the middle of the span. The equations simplify to the form: B. Phasors and Vectors The EMF field near transmission lines are described in this article using phasors and vectors. A vector is characterized by a magnitude and angle in space or by three spacial components, Eq. 5 [1]. Fig. 1. Asymmetric span. If A1 is exactly half of the A then the catenary is symmetric. A1 is in general calculated as follows: A phasor on the other hand is a quantity with a sinusoidal time variation described by a magnitude and a phase angle (Eq. 6). The angle φ describes a phase shift [1]. TELEN2016018 DOI 10.14311/TEE.2016.4.097 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 The constant c is the parameter defining the shape of the catenary. The parameter h is the height at the lowest point of the sag and it is calculated as follows: 99 real one. The mirror conductor has the opposite charge as the real one, Fig. 3. The height of the catenary y at the place x is calculated as follows: III. METHODOLOGY From the III. Maxwell’s equation we can derive the Coulomb’s law, in the integral form Eq. 12. Let the point of an observer P be the point where we want to calculate the electric field intensity. The vector 𝑟⃗ starts at the conductor element dl and points to the observer P. The vector 𝑟⃗ can be written also as: where ⃗⃗⃗⃗ 𝑟𝑝 vector points from the coordinate system origin to the observer P and ⃗⃗⃗⃗ 𝑟0 points from the coordinate system origin to the conductor element dl, Fig.2. The Coulomb’s law equation in the analytical form is after a substitution as follows: Fig. 3. Example of mirror conductors above the conductive plate with the potential φ = 0 [7]. In this new model the charge distribution in the conductors remained unchanged and the potential on the boundary plate is also zero as in the basic model. The solution of the new model will have the same solution as the initial boundary value problem. The electric field intensity 𝐸⃗⃗ in point of the observer P, Fig. 3 equals the superposition of electric intensities created by each conductor, real and mirror one, Eq. 15. B. Change from Derivations to Differences The catenary shape of the conductor makes the analytical calculation of Eqs. 14 and 15 too complicated. To overcome this issue a numerical method based on the superposition of finite elements has to be used. The core of this method is to change the derivations to the differences to determine the length of the conductor element ∆l. Fig. 2. Positions of vectors 𝑟⃗, ⃗⃗⃗⃗, 𝑟0 and ⃗⃗⃗⃗ 𝑟𝑝 in relation to the observer P and catenary. A. Method of Mirror Images It is not possible to find out 𝐸⃗⃗ in a dielectric environment, where conductors hang above a conductive plate, just by Eq. 14. The electric field in this model is created not only by charges in conductors but also by charges created by electrostatic induction in the conductive plate (terrain) with potential φ = 0. The charges distribution and density in this plate is uneven. Solving this problem is done by the method of image charges. The method creates a new mirror conductor axially symmetrical according the boundary plane to each TELEN2016018 DOI 10.14311/TEE.2016.4.097 It is important to realize that the linear charge density changes with every one element due to a different distance from the ground. The vectors ⃗⃗⃗⃗⃗⃗ 𝑟0𝑛 and 𝑟⃗⃗⃗⃗⃗⃗⃗ 0´𝑛 will determine the conductor element position and its image according to the coordinate system beginning. We can determine values of the vectors in X and Y directions by Eq. 11, value in the Z direction equals to the overhang of the conductor on a transmission tower. The electric field intensity 𝐸⃗⃗ can be now determined for one conductor over a flat surface as follows: Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 100 coefficient of the conductors k and j, ℎ𝑘 is height of the conductor k element above ground, 𝑅𝑘 is radius of the conductor k, 𝐷𝑘𝑗 is distance between the elements of the conductors k and j in the same distance X from of the coordinate system beginning, 𝐷´𝑘𝑗 is the distance between the element of the conductor k and mirror element of the conductor j in the same distance X from the coordinate system beginning. A conductor of any given shape can be by this method "chopped" to finite elements, Fig. 4. Fig. 4. Visualization of calculation of electric field intensity by the Finite element method and method of mirror images. C. Calculation of Linear Charge Density It is necessary to evaluate the linear charge density τn for each conductor element ∆l. This is possible from known voltages on each conductor and from geometry of each catenary element ∆l. In general for an infinite straight conductor linear charge density can by calculated by Eq. 18, where [𝜏𝑛 ] is one 1D matrix of the linear charges densities at k conductors, [𝑃𝑘𝑘 ] is 2D matrix of Maxwell’s potential coefficients, by the unit [F/m], finally [𝑈𝑘 ] is 1D matrix of voltages at k conductors. When calculating the field created by the conductor catenary shape, the matrix [𝑃] will be different for every conductor element ∆l. For the nth element from the system of k conductors we can write: The result is 2D matrix [𝜏𝑘𝑛 ] consisting linear charge densities on conductor k for every nth element of the catenary. The components of matrix [𝑃𝑘𝑗 ] for every element can be determined by following Eqs. 20 and 21. The matrix [𝑃𝑘𝑗 ] is symmetric and for components not in diagonal equals 𝑃𝑘𝑗 = 𝑃𝑗𝑘 . Distances between conductors are shown in simple example in Fig. 5 where 𝑃𝑘𝑘 is the self-potential coefficient of the conductor k, 𝑃𝑘𝑗 is the mutual potential TELEN2016018 DOI 10.14311/TEE.2016.4.097 Fig. 5. Conductor positions and their mutual distances for calculation of coefficients 𝑃𝑘𝑗 . D. Generalization So far we have analyzed the calculation of 𝐸⃗⃗ only as a stationary field formed by a constant voltage U. In the calculation of harmonically oscillating field where the intensity 𝐸⃗⃗̂ is the phasor-vector unit we shall use as input ̂. phasor of effective phase voltage of each conductor ⃗⃗⃗⃗⃗ 𝑈 𝑘 The linear charge density will also harmoniously change so Eq. 22 can be rewritten as follows: The final equation for the electric field intensity under a power transmission line consisting of k conductors in shape of catenary at point of the observer P with use of FEM method is as follows: Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 101 IV. ANALYTICAL VERIFICATION The numerical method has been tested on a set of conductor shapes which analytical solutions are known [9]. It was shown that the numerical solution converges uniformly to the analytical solution and the accuracy depends only on the number of finite elements, Fig. 4. It is important to note that the conductor segment vector ∆l is in each case computed differently according to the conductor shape. In both cases the τ = 1.10e-4 C/m and the analytical results are RMS values [10]. Fig. 7. Circle loop conductor. Fig. 6. Dependence of the numerical error from the length of the element for given conductor shapes. Fig. 8. Two infinite straight conductors. V. FIELD UNDER POWER LINE In this farther example of results it is shown the calculation of the electric field intensity under the power line of the type 2×400 kV DONAU with two ground wires, Fig. 9. The distance between towers is 350 m. The parameters of towers for calculation are in Tab. 3. The lowest conductor is set to be 11.5 m above the ground. This minimum distance is no longer determined by the standard, which is 8 m above the ground, but according to the value of the field. It can vary approximately from 10 m to 12 m according to the mutual phase location, terrain curvature and many more factors. The current in each phase bundle is 2400 A and the phase voltage is set on 420 kV. Circle loop conductor: Conductor in shape of a circle loop with the radius R = 100 m, Eq. 24. Point of the observer is in distance 10 m from the circle at the central axis, Fig. 7. Two infinite straight conductors: The distance between conductors is a = 10 m, Eq. 25, and the point of the observer is placed at 𝑟𝑝𝑥 = 5 m, 𝑟𝑝𝑧 = 0 m from the coordinate system beginning, Fig. 8. Fig. 9. Transmission tower type 2×400$ kV DONAU with two ground wires (green). Figure shows positions of phases L1 (white), L2 (black) and L3 (red). TELEN2016018 DOI 10.14311/TEE.2016.4.097 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Fig. 10. 𝐸𝑅𝑀𝑆 shown in 3D graph in horizontal plane in a constant height above ground, 1.8 m . The axis Z and colours display the field value. TABLE III. HANGING POINTS OF THE CONDUCTORS ON TOWER 102 Fig. 11. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line. Calculation for the lowest distance of the conductors to the ground 𝐸𝑅𝑀𝑆 is indicated by colour from dark to red. VI. SHIELDING EFFECT OF THE GROUND WIRES In past years it have been discussions in the power line community about the posibility of shielding the electromagnetic field by ground wires placed beneath the phase conductors. As it is shown in Figs. 12. and 13., the transmission tower Portál 400 kV with minimum distance of the phase conductors above the ground 12 m , such effect can be achieved, Fig. 13. Figures 11 and 10 show results that are calculated using the EMFTsim ultimate software (all graphical results are made by Dislin graphical library) which was built for this specific type of calculations by the authors of this article. Fig. 12. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line of the type Portál 400 kV. Calculation for the lowest distance of the conductors to the ground. 𝐸𝑅𝑀𝑆 is indicated by colour from dark to red. TELEN2016018 DOI 10.14311/TEE.2016.4.097 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 103 including human itself. This means more study is needed to be done for ensuring operational safety of the transmission line but also for reducing the project costs. For a complex computation of the electromagnetic filed a simulation software EMFTsim Ultimate was developed. All graphical result in this paper are made by this software. ACKNOWLEDGEMENT This research was supported by Nadácia Tatra banky. REFERENCES Fig. 13. 𝐸𝑅𝑀𝑆 in a vertical plane crossing the transmission line of the type Portál 400 kV . Calculation for the lowest distance of the conductors to the ground. 𝐸𝑅𝑀𝑆 is indicated by colour from dark to red. As it is shown, the electric field can be dramaticly reduced, however shielding in reality will be highly unpractical, due to additional costs for development of new transmission towers. Also to maintain addition seperation distances between shielding wires and phase conductors and ground will in reality result into a transmission tower raising. VII. CONCLUSION Enumeration of the electromagnetic field generated by a power transmission line is one of crucial conditions for completing total project of the line. This article explained a numerical method for calculation of the electric field intensity in a complex way. The described method also tries to eliminate most of common approximations. As the paper shows, this method can be fully applied for any conductor shape. The results shown in this paper were verified not only analytically, but also using calculations of similar computation softwares. The value of electric field intensity is effected also by ground and all subjects TELEN2016018 DOI 10.14311/TEE.2016.4.097 [1] EPRI, “AC transmission line reference book – 200 kV and above.” [2] WHO, “Environmental Health Criteria 238 – Extremely Low Frequency Fields.” [3] D. M. Repacholi and E. Vandeventer, “WHO Framework for Developing EMF Standards,” pp. 1–13, October 2003. [4] “Vyhláška Ministerstva zdravotníctva Slovenskej republiky o podrobnostiach o požiadavkách na zdroje elektromagnetického žiarenia a na limity expozície obyvateľov elektromagnetickému žiareniu v životnom prostredí,” Zbierka zákonov č. 534/2007, pp. 3812–3816, 2007. [5] Sbírka zákonů České republiky, ročník 2008. “Nařízení vlády o ochraně zdraví před neionizujícím zářením,” Částka 1, pp. 2–29, 2008. [6] ICNIRP, “Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz).” Health physics, vol. 99, no.6, pp. 818–36, Dec. 2010. [7] D. Mayer and J. Polák, Metody řešení elektrických a magnetických polí, 1983. [8] EPRI, “AC Transmission Line Reference Book – 345 kV and above,” Chapter 7, pp. 329–417, 1982. [9] D. Mayer, Aplikovaný elektromagnetizmus: úvod do makroskopické teorie elektromagnetického pole pro elektrotechnické inženýry, 2012. [10] J. Bendík, M. Cenký, and Ž. Eleshová, “Complex calculation of intensity of electric field under power transmission line using catenary shape of conductors and flat surface”, in Power engineering 2016: Energy-Ecology-Economy 2016: 13th International scientific conference. Tatranske Matliare, Slovakia. May 31 – June 2, 2016. 1. vyd. Bratislava: Slovak University of Technology, 2016, s. 139–144. ISBN 978-80-89402-85-4 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 104 Comparison of Reconfigurations Using Deterministic Approach for Global Assessment of Operational State in Power System Boris Cintula 1), Žaneta Eleschová 2), Anton Beláň 3) and Peter Janiga 4) 1) 2) 3) 4) Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Institute of Power and Applied Electrical Engineering, Bratislava, Slovak Republic, e-mail: 1) [email protected], 2) [email protected], 3) [email protected], 4) [email protected] Abstract — This article deals with the analysis of impact reconfiguration on the power system operational state. The stated analysis assess results of the simulated calculations of N-1 events with the aim to obtain a more complex view of the security criterion N-1 use in comparison with the current methods and procedures being in practice. Methodologies based on the deterministic approach arising from calculations of steady states with the global assessment of the power system operational states are presented. The article objective is to comprehensively compare the selected operational states especially different reconfiguration variants in a power system. various steady states with regard to changes of load, transit, topology and power system development. In general, the deterministic approach is possible to define as a principle where effect of each kind is possible to determine entirely and definitely. Subject of the deterministic approach for a classification of the operational state is the assessment of consequences after N-1 events (contingencies) on the basis of limitations and criteria determined in advance. The proposed methodology using the deterministic approach is based on the following procedure: Keywords — security criterion N-1, deterministic approach, reconfiguration, global assessment, operational state, power system Step 1 Simulation calculations of the N-1 security criterion (internal contingency), i.e. repeated calculation of the reference steady state for outage of each power line within the transmission power system. The aim of the calculations is to obtain the loading values in N state and N-1 states of all power lines (after all contingencies) in the responsibility area. I. INTRODUCTION Currently a great attention is paid to the increase of security and operation of power systems. Several large system failures occurring worldwide in the previous decades affirm significance and need to develop this philosophy. There are examples of it such as the power system failures of blackout type in USA, Italy or a splitting into islanding operations in a synchronically interconnected power system in Europe. The reason for occurrence of such failures is a conjunction of several events; nevertheless, all the cases show the only common violated indicator which is a failure to meet the performance of the security criterion N-1. Nowadays, within synchronously interconnected system the European Awareness System is used for exchange of online information evaluating the security criterion N-1 analysis among other operating quantities. Based on the monitoring of the operational states, particular categories of the operational states are distinguished by means of the traffic light. In order to comply with the security criterion N-1as one of the input parameters to determine the overall operational state of the power system: this criterion is assessed “binary”, i.e. “condition of criterion N-1 is met” or “condition of criterion N-1 is not met”. II. METHODOLOGY OF GLOBAL ASSESSMENT The methodology of global assessment of the power system operational state is based on an analysis of simulation results of the security criterion N-1 considering TELEN2016019 DOI 10.14311/TEE.2016.4.104 Step 2 Based on the contingency simulation results in the responsibility area LODF values (Line Outage Distribution Factor) are subsequently calculated. The LODF value determines the percentage of power flow on the present power line that will be shown up on other transmission power lines after the outage of this line. Simply, LODFs are a sensitivity measure of how a change in a line status affects the flows on other lines in the power system. For the purpose of methodology proposal the LODF is calculated according to the following formula: PVy PVy LODFVxVy n 1 Vx n .100 P n (1) where PVyn-1 is a loading of the assessed power line “Vy” after outage of the power line “Vx“, PVyn – a loading of the assessed power line “Vy“ in time without power lines being turned off (state N), PVxn – a loading of the power line “Vx“ in time without power lines being turned off (state N). Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Within the contingency list (N-1 events) all power lines in the responsibility area are included while the ODF calculation is performed only for loading changes of the system power lines, i.e. loading changes of the radial power lines (radial feeder to generation and consumption) are not considered due to an high impact of the result distortion. Among the partial distortion causes of the methodology results are included: • Small loading changes of the radial power lines after the outage of the system power lines – consequence of the voltage fluctuation in the end node of the power line. • Significant loading changes after the outage of the radial power lines (accepted in cases for parallel connection of the radial power lines). Step 3 Further, in the methodology of global assessment only positive loading changes are considered. It is necessary to distinguish significant loading changes from the less significant applying the data filter. The data filter of increased loading changes is defined by common meeting of the below stated conditions where (2) defines a minimum loading change of the power line after contingency (N-1 event) against state N and (3) a minimum loading value of the given power line after contingency (N-1 calculation) with respect to a nominal load. PVN 1 0,05 (2) IVN 1 0,3.I nV (3) PVN Step 4 An overall assessment of the power system operational states is based on perspectives defining four weight factors (WF): Perspective on power lines loading N VF1 (4) Vi 1;51 i 1,2,3,4,5,6,7,8,9,15,16 Vj 1;51 Where i is a p.u. value of the power line loading, Vi – number of the system power lines in the responsibility area, Vj – total number of the power lines in the responsibility area, N – state N. ii. WF2 is determined by a p.u. value of the most loaded power line after all contingencies calculation in the responsibility area related to the nominal loading. TELEN2016019 DOI 10.14311/TEE.2016.4.104 Vi max iVj N 1 VF 2 (5) Vi 1;51 i 1,2,3,4,5,6,7,8,9,15,16 Vj 1;51 Where i is a p.u. value of the power line loading, Vi – number of the system power lines in the responsibility area, Vj – total number of the power lines in the responsibility area, N-1 – state after all contingencies. System perspective on the number of the most affected and effecting power lines iii. WF3 is determined by a p.u. value of the number of the affected power lines after all contingencies calculation related to the number of the system power lines. nV _ POVP nV _ CPSV VF 3 (6) Where nV_POVP is a number of the affected power lines, where the affected power line is considered a power line with at least one significant positive loading change after any contingency, nV_CPSV – number of the system power lines in the responsibility area. iv. WF4 is determined by a p.u. value of the number of the effecting power lines after all contingencies related to the total number of the power lines. nV _ PVPL nV _ CPV i. WF1 is determined by a p.u. value of the most loaded power line in the steady state (N state) related to the nominal loading. Vi max iVj 105 VF 4 (7) Where nV_PVPL is a number of the effecting power lines, where the effecting power line is considered a power line which outage will cause at least one significant positive loading change of any assessed power line, nV_CPV– total number of the power lines in the responsibility area. For the above described p.u. values weight factors values determining their sizes and severity levels (TABLE I, TABLE II, TABLE III) are appointed. The definition of the severity levels considers for the power line loading an uncertainty of mathematical models (model accuracy, scheduled loading, etc.) and the system perspective is based on the analysis of a large number results of the simulation calculations of the steady states. A proposal of weight factor values can be adjusted in accordance to the purpose of assessment, e.g. the operational planning and real time operation for on-line monitoring of the power system operation, defence plan or development of the power system. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 TABLE I. SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF1 EXPRESSING POWER LINES LOADING IN STATE N Intervals (p.u.) of Power Line Loading in State N WF1 <1;∞) 1 < 0.8 ; 1 ) 0.8 < 0.6 ; 0.8 ) 0.6 < 0 ; 0.6 ) 0.5 TABLE II. SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF2 EXPRESSING POWER LINES LOADING AFTER CONTINGENCY Intervals (p.u.) of Power Line Loading after Contingency WF2 <1;∞) 1 < 0.9 ; 1 ) 0.8 < 0.8 ; 0.9 ) 0.6 < 0 ; 0.8 ) 0.5 TABLE III. SIZES AND SEVERITY LEVELS OF WEIGHT FACTORS WF3, WF4 EXPRESSING SYSTEM PERSPECTIVE ON NUMBER OF AFFECTED AND EFFECTING POWER LINES 106 TABLE V. SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF2 EXPRESSING POWER LINES LOADING AFTER CONTINGENCY CONSIDERING SENSITIVITY ANALYSIS Sensitivity Analysis Intervals (p.u.) of Power Line Loading after Contingency <1;∞) a WF2 1a 1b < 0.9 ; 1 ) 0.9 < 0.8 ; 0.9 ) 0.7 < 0.7 ; 0.8 ) 0.5 < 0 ; 0.7 ) 0.4 Criterion N-1 performance is not met for 1 power line b Criterion N-1 performance is not met for 2 power lines at least TABLE VI. SIZES AND SEVERITY LEVELS OF WEIGHT FACTORS WF3, WF4 EXPRESSING SYSTEM PERSPECTIVE ON NUMBER OF AFFECTED AND EFFECTING POWER LINES CONSIDERING SENSITIVITY ANALYSIS Sensitivity Analysis Intervals (p.u.) of Number of Affected and Effecting Power Lines WF3, WF4 < 0.75 ; 1 > 1 Intervals (p.u.) of Number of Affected and Effecting Power Lines WF3, WF4 < 0.7 ; 0.75 ) 0.9 < 0.75 ; 1 > 1 < 0.65 ; 0.7 ) 0.8 < 0.65 ; 0.75 ) 0.8 0.7 < 0.5 ; 0.65 ) 0.6 < 0.6 ; 0.65 ) < 0.5 ; 0.6 ) < 0 ; 0.5 ) 0.5 < 0.4 ; 0.5 ) 0.5 < 0 ; 0.4 ) 0.4 Step 5 The limit of the weight factor values is reassessed on the basis of the sensitivity analysis for considering of severity extent of the particular weight factors. Based on the sensitivity analysis of several steady states results the weight factor intervals are reassessed. Furthermore, the inter-levels of weight factors are expressing an approximation of p.u. values from the margin of the nearest worse level of the weight factors (TABLE IV, TABLE V, TABLE VI). The result of the sensitivity analysis consideration is more precise partial as well as the overall assessment of the power system operation state. In this manner it is possible to prevent determination of a less serious state to be serious and clearly differentiate more serious state from other states. TABLE IV. SIZES AND SEVERITY LEVELS OF WEIGHT FACTOR WF1 EXPRESSING POWER LINES LOADING IN STATE N CONSIDERING SENSITIVITY ANALYSIS 0.6 Step 6 Eventually, the overall assessment of the operational state is determined based on the product of all weight factors. The proposed methodology includes two exemptions for the global assessment of the operational state. By the exception it is meant the direct determination of the overall assessment of the operational state, i.e. it determines the overall assessment as “emergency” in the case of validity of any following conditions: • If the loading value of the most loaded power line in state N is higher than 100 %, • If at least two WFs equal 1, then the overall assessment of the operation state determined by the product of all WFs is one state worse. TABLE VII. CLASSIFICATION OF OVERALL GLOBAL ASSESSMENT OF POWER SYSTEM OPERATIONAL STATE <1;∞) 1 < 0.9 ; 1 ) 0.9 Overall Assessment of Power System Operational State Normal < 0.0256 ; 0.05 > Green < 0.8 ; 0.9 ) 0.8 Alarm ( 0.05 ; 0.2058 > Yellow < 0.7 ; 0.8 ) 0.7 Alert ( 0.2058 ; 0.5832 > Orange < 0.6 ; 0.7 ) 0.6 Emergency ( 0.5832 ; 1 > Red < 0.5 ; 0.6 ) 0.5 < 0 ; 0.5 ) 0.4 Sensitivity Analysis Intervals (p.u.) of Power Line Loading in State N TELEN2016019 DOI 10.14311/TEE.2016.4.104 WF1 Intervals of Products Size (WF1 – WF4) Color Determination Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Fig. 1. Flowchart of methodology of global assessment. III. RECONFIGURATION ASSESSMENT Among the measures to solve situation of the N-1 criterion meeting possible implemented by transmission system operators (TSOs) belong as follows: • cancellation of the scheduled maintenance or break of all works in real-time operation utilizing standby time, • coordinated topology changes (network configuration), • use of phase shifter transformers, • contracted generation re-dispatch within the TSO own control areas, • reduction of interconnection capacities, • manual load shedding (consumption limiting plan in SR). The reconfigurations are effective remedial actions to restore the N-1 criterion meeting and therefore the objective of the article is to focus on the reconfiguration 107 and comprehensively compare selected different reconfiguration variants in the power system. Verification of the proposed methodology using the deterministic approach was gradually performed for a large number of different operational states. The aim of this chapter is to verify the methodology for system topology changes and result comparison with the steady state before the reconfiguration. Simulations are performed by means of the mathematical model of the power system in accordance with the following topologies: • Reference model of the power system (Fig. 2), • Reconfiguration in the substation Rz18_1 (Fig. 3), • Reconfiguration in the substation Rz14 (Fig. 3), • Reconfiguration in the substation Rz6_1 (Fig. 3), • Reconfiguration in the substation Rz6_1+Rz14 (Fig. 3), • Reconfiguration in the substation Rz18_1+Rz6_1+ Rz14 (Fig. 3), • Reconfiguration in the substation Rz6_2 (Fig. 4), • Reconfiguration in the substation Rz18_2+Rz34 (Fig. 5), • Reconfiguration in the substation Rz18_1+Rz6_1 (Fig. 3), • Reconfiguration in the substation Rz18_1+Rz6_2 (Fig. 4). Based on the result analysis of the steady state before the reconfiguration values for loading of the system power lines in the state N and after all contingencies (N-1 simulations), as well as values of number of the affected and effecting power lines under the present methodology are determined (TABLE VIII). Partial results: the most loaded power line in the state N (70,7 %), the most loaded power line after all contingencies (101,6 %), number of the significantly affected power lines (52,5 %) and number of the significantly effecting power lines (58,8 %). The steady state before the reconfiguration is close beyond the N-1 security criterion meeting. Above, there are stated only summary results, but the methodology also provides detailed identification of the affected and effecting power lines. Fig. 2. Reference model topology of the power system. TELEN2016019 DOI 10.14311/TEE.2016.4.104 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Fig. 3. Reconfiguration in particular substations (and its combinations). Fig. 4. Reconfiguration in particular substations (and its combinations). Fig. 5. Reconfiguration in particular substations (and its combinations). Fig. 6. Impact of reconfiguration on power flow change through profiles. TELEN2016019 DOI 10.14311/TEE.2016.4.104 108 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 109 Fig. 7. Impact of reconfiguration on transit and losses. TABLE VIII. GLOBAL ASSESSMENT RESULTS OF STEADY STATE BEFORE RECONFIGURATION AND DIFFERENT RECONFIGURATION VARIANTS IN THE POWER SYSTEM Variant Steady State before Reconfiguration: Pgen=4241MW, Transit=2044MW, Pcon=4577MW Reconfiguration in Rz18_1 Reconfiguration in Rz14 Reconfiguration in Rz6_1 Reconfiguration in Rz6_1+Rz14 Reconfiguration in Rz18_1+Rz6_1+Rz14 Reconfiguration in Rz6_2 Reconfiguration in Rz18_2+Rz34 Reconfiguration in Rz18_1+Rz6_1 Reconfiguration in Rz18_1+Rz6_2 Note: * - product result has only information character TELEN2016019 DOI 10.14311/TEE.2016.4.104 p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis p.u. Values Weight Factor Values Sensitivity Analysis WF1 WF2 WF3 WF4 0,707 0,6 0,7 0,677 0,6 0,6 0,695 0,6 0,6 0,740 0,6 0,7 0,744 0,6 0,7 0,726 0,6 0,7 0,637 0,6 0,6 0,668 0,6 0,6 0,701 0,6 0,7 0,572 0,5 0,5 1,016 1 1 0,886 0,6 0,7 1,032 1 1 1,030 1 1 1,034 1 1 1,059 1 1 0,931 0,8 0,9 0,948 0,8 0,9 0,950 0,8 0,9 0,852 0,6 0,7 0,525 0,6 0,6 0,625 0,6 0,7 0,550 0,6 0,6 0,575 0,6 0,6 0,625 0,6 0,7 0,625 0,6 0,7 0,625 0,6 0,7 0,575 0,6 0,6 0,625 0,6 0,7 0,650 0,8 0,8 0,588 0,6 0,6 0,569 0,6 0,6 0,667 0,8 0,8 0,667 0,8 0,8 0,647 0,6 0,7 0,647 0,6 0,7 0,667 0,8 0,8 0,686 0,8 0,8 0,569 0,6 0,6 0,647 0,6 0,7 Overall Assessment Alert Alarm Alert Alert Alert Alert Alert Alert Alert Alarm WF Product 0,222* 0,216* 0,252 0,213* 0,130* 0,176 0,263* 0,288* 0,288 0,292* 0,288* 0,336 0,311* 0,216* 0,343 0,311* 0,216* 0,343 0,247* 0,230* 0,302 0,250* 0,230* 0,259 0,237* 0,173* 0,265 0,205* 0,144* 0,196 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 110 TABLE IX. SUMMARY OF POWER GENERATION, CONSUMPTION AND TRANSMISSION – STEADY STATE BEFORE RECONFIGURATION AND DIFFERENT RECONFIGURATION VARIANTS Reconfiguration in Substation/Values Steady State before Reconfiguration Total Generation 4240,6 [MW] 4576,8 Total Load [MW] 67,36 Total Losses [MW] Total Load 4644,16 Considering Total Losses [MW] -336,2 Balance [MW] 2044,16 Transit [MW] Power Flow: Assessed Power System -1686,55 Neighbouring Control Area 1 [MW] Power Flow: Assessed Power System -841,46 Neighbouring Control Area 2 [MW] Power Flow: Assessed Power System 1432,13 Neighbouring Control Area 3 [MW] Power Flow: Assessed Power System 612,03 Neighbouring Control Area 4 [MW] Notes: Pgen – Generation in Responsibility Area Transit – Transit through Responsibility Area Pcon - Consumption in Responsibility Area Rz18_1 Rz14 Rz6_1 Rz6_1 +Rz14 Rz18_1 +Rz6_1 +Rz14 Rz6_2 Rz18_2 +Rz34 Rz18_1 +Rz6_1 Rz18_1 +Rz6_2 4240,6 4240,6 4240,6 4240,6 4240,6 4240,6 4240,6 4240,6 4240,6 4576,8 79 4576,8 69,26 4576,8 71,1 4576,8 70,97 4576,8 86,05 4576,8 80,27 4576,8 70,54 4576,8 85,34 4576,8 89,8 4655,8 4646,06 4647,9 4647,77 4662,85 4657,07 4647,34 4662,14 4666,6 -336,2 1948,69 -336,2 2042,98 -336,2 2008,68 -336,2 2006,64 -336,2 1861,64 -336,2 1959,78 -336,2 2009,81 -336,2 1888,38 -336,2 1881,34 -1686,55 -1594,92 -1547,64 -1545,75 -1614,41 -1365,03 -1639,05 -1620,72 -1455,9 -677,34 -853,52 -868,34 -868,06 -669,48 -1011,22 -777,5 -689,2 -851,44 1514,45 1441,83 1368,17 1362,81 1403,62 1408,5 1453,71 1441,55 1484,66 434,24 601,15 640,51 643,83 458,02 551,28 556,1 446,83 396,68 TABLE X. SUMMARY OF VOLTAGE PHASE CHANGES IN PARTICULAR SUBSTATIONS AFTER RECONFIGURATION Steady State before Reconfiguration: Pgen=4241MW, Transit=2044MW, Pcon=4577MW Reconfiguration in Rz18_1 Reconfiguration in Rz14 Reconfiguration in Rz6_1 Reconfiguration in Rz6_1+Rz14 Reconfiguration in Rz18_1+Rz6_1+Rz14 Reconfiguration in Rz6_2 Reconfiguration in Rz18_2+Rz34 Reconfiguration in Rz18_1+Rz6_1 Reconfiguration in Rz18_1+Rz6_2 Transit [MW] Transit Decrease [MW] Voltage Phase in 1st Substation on BusBar x_a [°] Voltage Phase in 1st Substation on BusBar x_b [°] Voltage Phase in 2nd Substation on BusBar x_a [°] Voltage Phase in 2nd Substation on BusBar x_b [°] Voltage Phase in 3rd Substation on BusBar x_a [°] Voltage Phase in 3rd Substation on BusBar x_b [°] Δ Voltage Phase in 1st Substation [°] Δ Voltage Phase in 2nd Substation [°] Δ Voltage Phase in 3rd Substation [°] 2044,16 - - - - - - - - - - 1948,69 -95,47 -11,898 -22,741 - - - - 10,843 - - 2042,98 -1,18 -12,281 -15,864 - - - - 3,583 - - 2008,68 -35,48 -9,282 -14,91 - - - - 5,628 - - 2006,64 -37,52 -9,147 -15,123 -15,307 -14,695 - - 5,976 0,612 - 1861,64 -182,52 -10,427 -23,953 -6,851 -17,189 -17,444 -12,547 13,526 10,338 4,897 1959,78 -84,38 -19,978 -7,706 - - - - 12,272 - - 2009,81 -34,35 -14,445 -17,699 -19,23 -15,932 - - 3,254 3,298 - 1888,38 -155,78 -11,249 -23,653 -8,004 -15,436 - - 12,404 7,432 - 1881,34 -162,82 -15,112 -24,876 -19,057 -7,705 - - 9,764 11,352 - Results of the power line loading in the state N, after all contingencies, number of the affected and effecting power lines are determined by the values of weight factors: WF1WF4. Based on their product the overall assessment of the power system operation for a particular reconfiguration stated in TABLE VIII is determined. The global assessment of the simulated operational states is classified as “alarm” for the reconfiguration in the substations Rz18_1 and simultaneous reconfigurations in Rz18_1+Rz6_2.Other simulated operational states of the reconfigurations are classified as “alert“. Based on the summary of the survey results (TABLE IX) it is not TELEN2016019 DOI 10.14311/TEE.2016.4.104 obvious decrease of the power transit through the responsibility area for all reconfiguration variants. A detailed overview of the transit changes through the power system and power flows in particular profiles after a reconfiguration is shown in Figs. 6 and 7. The primary objective of the reconfiguration is to ensure regular N-1 security criterion meeting. According to the results in TABLE VIII the reconfiguration objective is not met for variants in the substation Rz6_1 (one overloaded power line), Rz14, Rz6_1+Rz14 and Rz18_1+Rz6_1+Rz14Rz6_1 (two overloaded power lines). Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 111 One of the serious disadvantages is the significant increase of losses in the transmission system in the control area. Stated fact was determined by all simulated operational states (Fig. 7.). Other potential disadvantages include the reliability reduction of the transmission system and distribution system in the control area, maintenance restriction and reconstruction work restriction. There is also a potential problem with synchronization conditions in case of return to the base state, but this case was not confirmed by simulated operational states (TABLE X). Under the current rules of the power system control the operation is considered secured after the restore of the N-1 security criterion meeting. However, this idea has not to be correct, as confirmed by the results of the global assessment according to the proposed methodology. Based on the analysis of the weight factor values for all reconfiguration variants it is possible to establish that in all stated cases the weight factor values VF3 and VF4 are considerably increased, i.e. there is a significant increase of the number of the affected and effecting power lines. That means, the power system is substantially “vulnerable“, e.g. in case of other events the situation could lead to the “emergency” of the power system. This fact is confirmed also by the weight factor product, which is for the above mentioned reconfigurations higher than in the steady state before the reconfiguration. Although, the security criterion N-1 is met after the reconfiguration, in fact the operational state of the power system is worse than before the reconfiguration. Eventually, results obtained by applying the methodology of the global assessment using the deterministic approach do not lead only to classification of the overall operational state, but also provide a comprehensive view on the power system operation expressed by the weight factors. power system, preparation of defence plans as well as with a proposal of power system development plans. A significant advantage of the proposed methodology is definitely an optimization possibility on the basis of weight factor size in accordance with various levels of a dispatching regulation. It is possible to consider a more conservative perspective for a scheduled operation, defence plan and development of the power system, with a strict modification of single weight factor intervals especially for the reason of reserve due to the mathematic models uncertainty. For the operational regulation it is possible to consider a sensitive margin modification of single weight factor intervals as far as a dispatcher should be informed about a warning state only in case of a higher risk during the power system operation. Results of the global assessment of the power system topology changes refer to identification of the suitability and successfulness of the reconfigurations regard to the restore of the N-1 criterion meeting. The added value of the methodology is to provide comprehensive information about the power system operational state after the reconfiguration (restoration of the N-1 meeting). IV. CONCLUSION The submitted paper deals with the methodology of the global assessment of the power system operational states using the deterministic approach and the N-1 security calculations. The present methodology is based on more complex results using of the N-1 security calculations. The proposed methodology does not give only the answer whether the N-1 criterion is met but also how the criterion is met or not met. Based on the results it is possible to differentiate and compare the power system states where N-1 is not met and to formulate which “bad“ state is a worse one or, vice versa, to compare the states when N-1 is met and formulate which “good“ state is a better one. Its strong side is universality approved by verification of a large number of operational states and its universal characteristic is a definition of the weight factors by means of which the operational states are classified. Furthermore, sizes and severity levels of the weight factor need to be adjusted for a given regulation area where the methodology will be used. The weight factors need to be identified on the basis of sensitivity analysis of results of a large number of steady states due to a uniqueness of each power system characterized by e.g. extensity, geographical location, structure of power resources and many other attributes. In accordance with the universality of the proposed methodology it is a wide utilization in practice, especially in the dispatching regulation, operational regulation of the [1] P. Kundur, at al, “Definition and classification of power system stability,” IEEE Transactions on Power System, vol. 19, no. 3, August 2004, pp. 1387 - 1401. [Online]. Available: IEEE Xplore Digital Library, doi: 10.1109/TPWRS.2004.825981 [Accessed: 25 July. 2016]. [2] J. Machowski, J. W. Bialek, J. R. Bumby, Power System Dynamics: Stability and Control, 2nd ed., John Wiley & Sons, Ltd.: Chichester, 2008. [3] Continental Europe Operation Handbook: Operational Security. Brussels - Belgium: ENTSO-E, 2009. [4] Continental Europe Operation Handbook: Appendix 3: Operational Security. Brussels - Belgium: ENTSO-E, 2009. [5] Network Code on Operational Security. Brussels - Belgium: ENTSO-E, 2013. [6] Pravidlá prevádzkovania prenosovej sústavy : Technické podmienky prístupu a pripojenia, pravidlá prevádzkovania prenosovej sústavy. Bratislava: SEPS, a.s., 2014. [7] Supporting Document for the Network Code on Operational Security, 2nd Edition Final. Brussels - Belgium: ENTSO-E, 2013. [8] Continental Europe Operation Handbook: Appendix – Policy 5: Emergency Operations: Operational Security. Brussels - Belgium: ENTSO-E, 2010. [9] B. Cintula, Ž. Eleschová, A. Beláň, “Global assessment of power system operational state after reconfiguration,” Power engineering 2016 - Control of Power Systems, vol. 12, pp. 35-41, May 2016 [12th International scientific conference]. [10] M. Straka, “Rekonfigurácia prenosovej sústavy Slovenskej republiky,” Elektroenergetika, vol. 7, pp. 101-104, Sep 2013 [7th International scientific symposium]. [11] M. Jedinák, S. Prieložný, R. Šmidovič, “Topology Changes in the Transmission System of the Slovak Republic,” Power engineering 2012 - Control of Power Systems, vol. 10, pp. 1-9, May 2012 [10th International scientific conference]. TELEN2016019 DOI 10.14311/TEE.2016.4.104 ACKNOWLEDGMENT These publications are the result of implementation of the project: “Increase of Power Safety of the Slovak Republic” (ITMS: 26220220077) supported by the Research & Development Operational Programme funded by the ERDF. REFERENCES Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 112 Impact of Asymmetry of Over-Head Power Line Parameters on Short-Circuit Currents Žaneta Eleschová 1) and Marián Ivanič 2) 1) 2) Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Institute of Power and Applied Electrical Engineering, Bratislava, Slovakia, e-mail: 1) [email protected], 2) [email protected] Abstract — This paper analyses the impact of asymmetry of over-head power line parameters on short circuit currents when three-phase fault and phase-to-ground fault occur. The calculation results with consideration of an asymmetry of the power line parameters are confronted with the calculation in accordance with the Slovak standard STN EN 60909 which does not consider asymmetry of equipment parameters in the power system. The calculation of short-circuit conditions was carried out for two types of 400 kV power line towers on which is a considerably different arrangement of phase conductors. Keywords — over-head lines, power line parameters, asymmetry, short-circuit current. I. INTRODUCTION Short-circuit currents are addressed according to the Slovak standard STN EN 60909. The calculation according to this standard does not consider asymmetry of power system parameters and uses the method of symmetrical components instead. The objective of this paper is to compare the calculation considering asymmetry of the power system parameters with the calculation in accordance with the standard. Over-head power lines represent a dominant part of the power system and if the phases are not transposed on towers, one of the equipment in the power system contains a considerable asymmetry of electrical parameters. Therefore, the calculation of short-circuit conditions was carried out on a simplified model: an ideal source and power line. The power line was modelled for two types of 400 kV power line towers with a considerable arrangement of phase conductors. The following values of the short-circuit current have been evaluated from the simulation results: RMS value which represents the initial shortcircuit current I k , peak value with consideration of the DC component maximum value which represents peak short-circuit current ip. II. RESULTS OF SHORT-CIRCUIT SIMULATIONS WITH CONSIDERATION OF ASYMMETRY OF POWER LINE PARAMETERS The calculation of the power line parameters as well as short-circuit simulations were carried out in EMTP (Electro-Magnetic Transient Program). A simplified model for calculation of the short-circuit currents is depicted in Fig. 1. TELEN2016020 DOI 10.14311/TEE.2016.4.112 Fig.1. Simplified model in EMTP. A. Input Parameters The power line was modelled with the following two types of towers: Fig. 2. Types of towers. The first type of tower represents a deployment of conductors equally distant but in a different height above the ground. The conductors of the second type of tower have the same height but marginal conductors are placed in a double distance. The following table illustrates the conductor parameters: TABLE I. PARAMETERS OF CONDUCTORS ACSR conductor [Ω/km] [-] [mm] Ground conductor 120 AlFe6 0,225 0,809 16 Soil resistivity was 100 Ω/m. Phase conductor 450 AlFe6 0,065 0,818 29,76 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Parameters of the asymmetrical (untransposed) power line are provided in Table II and parameters of the transposed power line in Table III which indicates values in symmetrical component systems while the length of the power line is 60 km. TABLE II. PARAMETERS OF UNTRANSPOSED LINES Tower Phase R [Ω] Type 1 113 currents in individual phases while untransposed power line was considered, calculated according to the following relation: I I (1) UL TL 100 I TL UL – untransposed line TL – transposed line Type 2 L1 L2 L3 L1 L2 L3 L1 3,65 2,51 2,35 3,70 2,43 2,35 L2 2,51 4,08 2,51 2,43 3,79 2,43 L3 2,35 2,51 3,65 2,35 2,43 3,70 TABLE IV. RMS VALUES OF SHORT-CIRCUIT CURRENTS Tower type Type 1 Type 2 TL 14,2 13,4 L1 13,8 12,9 I [kA] X [Ω] k [nF] L1 26,29 7,84 8,62 26,21 8,60 6,47 L2 14,8 14,8 L2 7,84 22,85 7,84 8,60 25,35 8,60 L3 14,1 13,0 L3 8,62 7,84 26,29 6,47 8,60 26,21 L1 -2,6 -4,0 L2 4,3 10,4 L3 -0,4 -2,9 L1 683,74 -97,22 -61,25 672,61 -84,04 L2 -97,22 695,02 -97,22 L3 -61,25 -84,04 690,60 -97,22 683,74 -19,34 -84,04 -19,34 -84,04 672,62 TABLE III. PARAMETERS OF TRANSPOSED LINES IN SYMMETRICAL COMPONENTS Tower Type 1 Type 2 R0 [Ω] 8,76 8,60 R1 [Ω] 1,36 1,34 X0 [Ω] 41,40 41,77 X1 [Ω] 17,05 18,03 C0 [nF] 518,25 555,05 C1 [nF] 773,70 742,06 B. Results of Short-Circuit Simulations As mentioned above, the model in EMTP consisted of an ideal power source and power line. The source voltage was established in accordance with the Slovak standard STN EN 60909 as follows: cU n 3 , where c = 1,05 and Un = 400 kV. It was made a simulation of the three-phase fault and phase-to-ground fault at the end of the power line while the RMS value and peak value were evaluated. C. Results for Three-Phase Fault The three-phase fault is a symmetrical type of fault. The short-circuit current is the same in all three phases when asymmetry of power system parameters was not considered. Following the simulations results, it is evident that the short-circuit currents are different in case of this symmetrical fault when the asymmetry of the power line parameters was considered, the short-circuit currents are different also in case of this symmetrical fault. The shortcircuit current RMS values are included in Table IV. For the transposed (symmetrical) power line, the RMS value is 14,2 kA for the first type of tower and 13,4 kA for the second type of tower. Table IV includes also values of relative deviations from RMS values of the short-circuit TELEN2016020 DOI 10.14311/TEE.2016.4.112 δ [%] The biggest difference in RMS values of the shortcircuit current (10,4 %) is in the middle phase for tower of the second type, as can be seen from the results. The peak values of the short-circuit currents were also evaluated while the development of the maximum DC component value was considered. The DC component is dependent on the moment of the short-circuit occurrence. The time behaviour of the short-circuit currents with the maximum DC component are shown in the following figures. Fig. 3. The time behaviour of short-circuit current upon three-phase fault. The amplitudes of the short-circuit currents are shown in the table below. TABLE V. PEAK VALUES OF SHORT-CIRCUIT CURRENTS WITH CONSIDERATION OF MAXIMUM DC COMPONENT Tower type Type 1 Type 2 TL 35,9 34,2 L1 34,9 32,8 L2 36,4 37,3 L3 35,9 33,4 L1 -2,9 -4,1 L2 1,3 9,0 L3 -0,1 -2,4 Imax [kA] δ [%] Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 114 There are differences even in peak values of the shortcircuit currents, in percentage terms they are a bit different than in case of the RMS values. The difference is caused by different time constants of the DC components in individual phases, i.e. different decay. It means that the asymmetry of the power line parameters has the impact not only on the AC component value of the short-circuit current but also on time constant of the DC component of the short-circuit current. D. Results for Phase-to-Ground Fault The short-circuit current for the transposed line in case of the phase-to-ground fault is the same for every single phase. For the untransposed line, simulations of the phaseto-ground fault were carried out for each phase separately. TABLE VI. RMS VALUES OF SHORT-CIRCUIT CURRENTS Tower type Type 1 Type 2 TL 9,5 9,3 L1 9,2 9,2 L2 10,4 9,5 L3 9,2 9,2 L1 -3,9 -1,0 L2 8,8 2,1 L3 -3,9 -1,0 I [kA] δ [%] The highest difference is in the middle phase for the tower of the first type which has the biggest height above the ground but is the closest to the ground conductor. The results of the short-circuit currents for the phase-to-ground fault are almost identical for the tower of the second type. Similar results are valid also for the peak values of the short-circuit currents and again (as for three-phase fault), the percentage differences of the peak values are not the same as for the RMS values. TABLE VII. PEAK VALUES OF SHORT-CIRCUIT CURRENTS WITH CONSIDERATION OF MAXIMUM DC COMPONENT Tower type Type 1 Type 2 TL 22,1 21,7 L1 21,3 21,4 L2 24,4 21,9 L3 21,3 21,4 L1 -3,8 -1,2 L2 10,2 1,1 L3 -3,8 -1,2 Imax [kA] δ [%] Fig. 4. The time behaviour of short-circuit current upon phase-to-ground fault. III. CALCULATION ACCORDING TO THE SLOVAK STANDARD STN EN 60909 The transposed line parameters of EMTP, i.e. from Table III were considered for calculation of the shortcircuit impedance. The initial short-circuit currents and peak short-circuit currents for the three-phase fault and phase-to-ground fault were calculated. The calculation according to the Slovak standard STN EN 60909 is based on the method of symmetrical component systems. The initial short-circuit current is given by: Three-phase fault: I k3 cU n (2) 3 Z1 Phase-to-ground fault: I k1 c 3U n Z1 Z 2 Z 0 c 3U n 2Z1 Z 0 Z1 – positive sequence short-circuit impedance Z2 – negative sequence short-circuit impedance Z0 – zero sequence short-circuit impedance The value of the peak short-circuit current is determined according to the following relation: i p 2 Ik K K 1,02 0,98e 3R k Xk A. Calculation for Tower Type 1 TABLE VIII. SHORT-CIRCUIT IMPEDANCES positive sequence zero sequence 17,103 Ω 85,46° 42,317 Ω 78,05° TABLE IX. SHORT-CIRCUIT CURRENTS FOR THREE PHASE FAULT TELEN2016020 DOI 10.14311/TEE.2016.4.112 (3) I k3 14,18 kA K ip 1,792 35,93 kA (4) (5) Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 TABLE X. SHORT-CIRCUIT CURRENTS FOR PHASE TO GROUND FAULT I k1 9,526 kA K ip 1,641 22,11 kA B. Calculation for Tower Type 2 TABLE XI. SHORT-CIRCUIT IMPEDANCES positive sequence zero sequence 18,08 Ω 85,74° 42,65 Ω 78,37° TABLE XII. SHORT-CIRCUIT CURRENTS FOR THREE PHASE FAULT I k3 13,41 kA K ip 1,804 34,21 kA TABLE XIII. SHORT-CIRCUIT CURRENTS FOR PHASE TO GROUND FAULT I k1 9,25 kA K ip 1,654 21,64 kA The above mentioned results prove that the results of the short-circuit currents are the same as the results of simulations for the transposed power line which verifies the results of the simulation. IV. CONCLUSION This paper deals with the impact of an asymmetry of over-head power line parameters on the short circuit currents in its individual phases. The objective of the paper is to compare calculation of currents with consideration of asymmetry of the power line parameters, with calculation in accordance with the valid standard STN EN 60909 on which the asymmetry of parameters is neglected. The important result is that the short-circuit currents in some of the phases are higher than the shortcircuit current calculated in accordance with the standard considering untransposed power line. Taking into consideration the three-phase fault, for the second type of tower the difference was up to 10,4 % for the middle phase. The positive sequence impedance applies only with three-phase fault, therefore compared to the transposed line the biggest difference of the shortcircuit current is in the middle phase which has the shortest distance against other two phases. The distances of the phase conductors on the first type of tower are approximately the same and the currents in phases within the three-phase fault are approximately the same as well. The calculation of the phase-to-ground fault proved that TELEN2016020 DOI 10.14311/TEE.2016.4.112 115 the biggest difference (8,8 %) was for the first type of tower, again in the middle phase. Also the zero sequence is included into calculation for the phase-to-ground fault which depends on the distance of the phase conductors from the ground conductor. The phase conductors on the second type of tower have approximately the same distance from the ground conductor and the short-circuit currents in individual phases for the phase-to-ground fault are approximately the same. The arrangement of conductors on the first type of tower is different, the middle phase is the closest to the ground conductor and the short-circuit current for phase-to-ground fault is the highest in this phase. The other important observation from the results is that the asymmetry of parameters influences time constant of the DC component and so influences other short-circuit quantities such as: peak short-circuit current, thermal short-circuit current, unsymmetrical breaking current. Thus, these quantities are influenced by various values of the initial short-circuit current in individual phases on one hand and by various time constants of the DC component for individual phases on the other hand. This implies that correction factors taking into account asymmetry of parameters should be defined when calculating short-circuit currents. ACKNOWLEDGMENT These publications are the result of implementation of the project: “Increase of Power Safety of the Slovak Republic” (ITMS: 26220220077) supported by the Research & Development Operational Programme funded by the ERDF. REFERENCES [1] D. Reváková, Ž. Eleschová, and A. Beláň, Prechodné javy v elektrizačných sústavách. Bratislava: Slovenská technická univerzita v Bratislave, 2008. [2] M. Ivanič and Ž. Eleschová, “The faults with consideration of line asymetry parameters,” in Power engineering 2016: Control of Power Systems 2016: 12th International scientific conference. Tatranské Matliare, Slovakia. May 31 - June 2, 2016. 1. vyd. Bratislava: Slovak University of Technology, 2016, s. 73-77. ISBN 978-80-89402-84-7. [3] STN EN 60909-1, “Výpočet skratových prúdov v trojfázových striedavých sústavách.” [4] Š. Fecko, D. Reváková., L. Varga, J. Lago, and S. Ilenin, Vonkajšie elektrické vedenia. Bratislava: STU FEI, 2010. [5] A. J. Schwab, Elektroenergiesysteme. Erzeugung, Transport, Übertragung und Verteilung elektrischer Energie. Berlin Heidelberg, Springer, 2012. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 116 Numerical Analysis and Experimental Verification of Eigenfrequencies of Overhead ACSR Conductor Justín Murín 1), Juraj Hrabovský 1), Roman Gogola 1), Vladimír Goga 1) and František Janíček 2) 1) Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Department of Applied Mechanics and Mechatronics, Bratislava, Slovakia, e-mail: [email protected] 2) Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Department of Electrical Power Engineering, Bratislava, Slovakia, e-mail: [email protected] Abstract — This contribution deals with the modal analysis of ACSR conductor using the finite element method (FEM) and experimental measurements of eigenfrequencies. In numerical experiments for the modelling of the conductor the material properties of the chosen conductor crosssection are homogenized by the Representative Volume Element (RVE) method. The spatial modal analysis of the power line is carried out by means of our new 3D FGM beam finite element and by standard beam finite element of the commercial software ANSYS. Experimental measurements are also carried out for verification of the numerical calculation accuracy. Keywords — ACSR conductor, finite element method, modal analysis, experimental measurements I. INTRODUCTION Vibration of overhead power lines is a very dangerous problem because it can cause collapse of overhead power lines or collapse of the whole transmission system. The overhead power lines are exposed to dynamic loads (air flow, ice-shedding, etc.) in addition to the static ones. From the mechanical point of view the conductor is a 3D system, so it can vibrate in longitudinal, horizontal and vertical directions. The torsional vibrations are possible as well. For calculation of eigenfrequencies and eigenmodes the numerical methods are the most effective, over all the finite element method. For the modal analysis the beam finite element is preferable. The material of the conductor is inhomogeneous, therefore simplified models obtained by homogenization of material properties are used [1, 2, 3]. The heterogeneous cross-sections of several ACSR conductors are shown in Fig.1. II. HOMOGENIZATION OF MATERIAL PROPERTIES One important goal of mechanics of heterogeneous materials is to derive their effective properties from the knowledge of the constitutive laws and complex microstructural behaviour of their components. The methods based on the homogenization theory (e.g. the mixture rules [5]) have been designed and successfully applied to determine the effective material properties of heterogeneous materials from the corresponding material behaviour of the constituents (and of the interfaces between them) and from the geometrical arrangement of the phases. In this context, the microstructure of the material under consideration is basically taken into account by the Representative Volume Element (RVE). The homogenization techniques derived at our department (Department of Applied Mechanics and Mechatronics) for modelling the Functionally Graded Material (FGM) [1, 2] can also be used for homogenization of the ACSR conductors. In case of the conductor, the material properties vary layer-wise in the radial direction (the longitudinal variation is not assumed). The effective homogenized material properties (electric conductance, thermal conductance, thermal expansion, stiffness) are calculated from condition, that the relevant material property of the cross-section with real construction (Fig. 2) is equal to the material property of the homogenized cross-section. Fig. 2. Conductor cross-section. Fig. 1. Construction of ACSR conductor. Results of the modal analysis are obtained using the commercial finite element software ANSYS and by a new 3D finite element [4]. An experimental measurement was done to verify and to compare the effectiveness and accuracy of each numerical calculation. TELEN2016021 DOI 10.14311/TEE.2016.4.116 The real cross-section parameters of the ACSR conductors are: Ri is pitch circle of the kth layer, di is wire diameter, φi is the angle of circumferential position of the wire, zi and yi are the distances of the wire from the centre of the conductor cross-section. These distances of each wire can be calculated as follows: Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 yi Ri sin i 117 n (1) zi Ri cos i E LNH th Then the quadratic moment of the i wire cross-sectional area Ai di2 / 4 according the axis y and according the axis z can be calculated by equations [1]: I yi d i 4 64 zi d i 2 2 I zi 4 d i 4 64 yi 2 d i 2 4 (2) (3) Since the Young’s modulus multiplied by the crosssectional area defines the axial stiffness and multiplied by the quadratic moment of the cross-section area defines the bending stiffness, we have to distinguish homogenized effective Young’s modulus for axial loading E LNH and homogenized effective Young’s modulus for bending M H E L y and E LM z H . We assume that the maximum and minimum elasticity moduli for lateral and transversal bending can be calculated by equations: n M yH E L max , I yi zH E LMmax i 1 Here, Ei is the elasticity modulus of the ith wire, and n is number of the wires. The effective elasticity modulus for lateral and transversal shears M H y E L min i 1 k zsm A n H , GLz Ei I zi i 1 n I zi 4 4 Fe d Fe E Fe n Al d Al E Al (4) n 64 4 Fe d Fe E Fe 4 n Al d Al E Al Where Gi Ei / 21 i is the shear modulus of the ith n wire, A Ai is the cross-sectional area of the whole i 1 real cross-section and i is its Poisson’s ratio. Again, k ysm,i and k ysm are the shear correction factors for the ith wire and the whole cross-section, respectively. These constants have to be calculated by a special method [4]. The effective elasticity modulus for torsion is: GLM x H x Gi I pi i 1 n I pi LNH (5) i Ai i 1 n Ai (11) i 1 n and the effective mass density for torsional vibration is: i 1 i I pi n M H y y E L max E L min 2 (6) E LM z H zH zH E LMmax E LMmin 2 (7) The effective elasticity modulus for axial loading is: TELEN2016021 DOI 10.14311/TEE.2016.4.116 (10) n I zi M H k zsm A The effective mass density for axial beam vibration is: M yH (9) k 1 i 1 I yi i 1 k zsm,i Gi Ai n where nFe is the number of steel wires and nAl is the number of aluminium wires. The maximum elasticity modulus represents the case, when all wires are fixed together (e.g. after several years of lifetime), and the minimum elasticity modulus represents the case, when wires can slide over each other. In practice the effective elasticity modulus for lateral and transversal bending is assumed as average value of the maximum and minimum elasticity moduli [4]: EL H GLz k zsm,i Gi Ai n zH E LMmin n i 1 n 64 (8) Ai n Ei I yi i 1 n i 1 n i 1 The polar moment of the wire cross-sectional area to origin of the coordinate system x, y is: I pi I yi I zi Ei Ai LM x H i 1 n I pi (12) i 1 where, i is the mass density of the ith wire. III. 3D FGM BEAM FINITE ELEMENT EQUATIONS Let us consider a 3D straight finite beam element (Timoshenko beam theory and Saint-Venant torsion theory) of a doubly symmetric cross-section. The nodal degrees of freedom at the node i are: the displacements ui, vi, wi in the local axis direction x, y, z, and the crosssectional area rotations – x,i , y ,i , z ,i . The degrees of freedom at the node j are denoted in a similar manner. The internal forces at the node i are: axial force Ni, transversal forces Ry,i and Rz,i, bending moments My,i and Mz,i, and Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 torsion moment Mx,i. Establishing of the local 3D FGM N i B1,1 R y ,i Rz ,i M x ,i M y ,i S M z ,i N j Ry , j Rz , j M x, j M y , j M z, j 0 B2 ,2 0 0 0 B3,3 118 beam finite element equations is presented in [4]: 0 0 B1,7 0 0 0 B2 ,6 0 0 B3,5 0 0 B4 ,4 0 B5 ,5 0 0 Y B6 ,6 M 0 0 B2 ,8 0 0 0 0 B3,9 0 B3,11 0 0 0 0 0 B5 ,9 B4 ,10 0 0 B5 ,11 0 B6 ,8 0 0 0 B7 ,7 0 0 0 0 0 0 0 B9 ,9 0 B9 ,11 M B8 ,8 E T 0 B10,10 R 0 B11,11 Y 0 ui B2 ,12 vi 0 wi 0 x ,i 0 y ,i B6 ,12 z ,i 0 uj B8 ,12 v j 0 wj 0 x , j 0 y , j B12,12 z , j (13) In (13), the terms Bi,j contain the linear and linearized geometric non-linear stiffness terms – containing the axial force effect on the flexural beam stiffness matrix K and consistent mass matrix M [4]: B K 2M (14) where is the natural frequency. The shear correction is accounted as well. The global stiffness matrix of the beam structures can be established by classical methods. Establishing of the local and global stiffness matrices as well as the whole solution procedure were coded by the software MATHEMATICA [12]. IV. NUMERICAL SIMULATIONS AND EXPERIMENTAL MEASUREMENTS For the numerical simulations and experimental measurements the single power line with the span length L = 19.9 m and the height difference between the points of attachment yh = 0.8 m has been considered (Fig. 3). In this case the maximum deflection of the power line [6, 7] is minimal and therefore was not calculated, because the span is small. The constant tensile force in the conductor for each numerical calculations and experimental measurements were: FH1 = 1.65 kN, FH2 = 4.75 kN and FH3 = 6.68 kN. Fig. 4. Cross-section of the used ACSR conductor. Material properties of the material from which the conductor is made are [9, 10]: • Steel: elasticity modulus EFe = 207000 MPa, Poisson’s ratio Fe = 0.28, mass density Fe = 7780 kg.m-3; • Aluminium: elasticity modulus EAl = 69000 MPa, Poisson’s ratio Al = 0.33, mass density Al = 2703 kg.m-3. For numerical simulations a simplified model was used. For simplifying the model of the ACSR conductor the homogenized material properties are calculated [1, 2, 3]. The effective cross-sections of the conductor parts are: AFe = 7.07 mm2, AAl = 42.41 mm2 and the effective crosssectional area of the whole conductor is A = 49.48 mm2. The effective quadratic moments of the conductor crosssectional area are: Iz = Iy = 218.68 mm4. The effective circular cross-section of the conductor is constant with diameter def = 7.94 mm. The effective material properties of the used conductor are: ELNH 88714.29 MPa Fig. 3. Model of overhead power line for modal analysis. A symmetric conductor marked as AlFe 42/7 which is constructed from 1 steel wire in the centre of the conductor and 6 aluminium wires (see Fig. 4) has been used. The diameter of the steel wire is dFe = 3 mm and the diameter of the aluminium wires is dAl = 3 mm. The rated tensile strength (RTS) of the chosen conductor is FRTS = 15.27 kN [8]. TELEN2016021 DOI 10.14311/TEE.2016.4.116 M yH EL ELM z H 40704.43 MPa GLHy GLHz 34120.91 MPa M xH GL x 27503.49 MPa LNH 3460.49 kgm-3 M H L x 2795.31 kg.m-3 LNH 0.323 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 where E LNH is M yH EL , ELM z H the elastic modulus for tension, are the elastic moduli for bending around axis y and z, respectively. GLHy , GLHz are the effective shear moduli, GLM x H x is the effective elasticity modulus for torsion, LNH is the effective mass density for axial beam vibration, LM x H is the effective mass density for torsional vibration and LNH is the effective Poisson’s ratio. These calculated effective material properties have been used in the modal analyses of the single power lines. The first three flexural eigenfrequencies f [Hz] in the plane xy (vertical) and the first three flexural eigenfrequenciesf [Hz] in the plane xz (horizontal) have been found with a mesh 200 of BEAM188 elements of the FEM program ANSYS [11]. The same problem has been solved using the new 3D beam finite element (3D NFE) for the modal analysis of composite beam structures [4] with a mesh 80 of 3D FGM elements (the calculation is performed using the software MATHEMATICA). 119 Two bolted strain clamps were used for fixing the conductor on two ends of the span; two IEPE piezoelectric accelerometers with the range of 50 g (Fig. 5) were used for experimental modal analyses to determine the flexural eigenfrequencies. For scanning the signals from the accelerometers 2 way oscilloscope with USB connection to the PC was used. The range of the oscilloscope is 20 MHz. The tension in the conductor is measured with one load cell with sensing range Fmax = 10 kN (Fig 6), which is close to the conductor attachment point. Fig. 6. Attaching of the load cell to sensing the axial force in conductor. Fig. 5. Piezoelectric accelerometer attached on the conductor. The data from the accelerometers placed on the conductor is shown in Fig. 7. To obtain the frequency spectrum (Fig. 8) the Fast Fourier Transformation (FFT) of the measured data was realized by software LabView [13]. The flexural mode shapes were evaluated using software ANSYS. The results of numerical analyses and experimental measurements are presented in Tab. 1-3. First three flexural eigenfrequencies in horizontal and three flexural eigenfrequencies in vertical plane were investigated. Fig. 7. Measured data of acceleration of the overhead ACSR conductor. TELEN2016021 DOI 10.14311/TEE.2016.4.116 Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 120 1stflexural 2ndflexural 3rdflexural Fig. 8. Measured flexural eigenfrequencies of the used ACSR conductor for the tension FH = 6.68kN at times t = 3s and t = 7 s. TABLE I. FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE AT THE TENSION FH = 1,65KN fmeas [Hz] fans [Hz] f3D [Hz] ANS [%] 3D [%] horizontal 2,23 2,49 2,47 11,66 10,74 vertical 2,38 2,54 2,58 6,72 8,34 horizontal 4,85 4,98 4,94 2,68 1,85 vertical 4,91 4,98 4,94 1,43 0,61 horizontal 7,23 7,47 7,41 3,32 2,51 vertical 7,24 7,47 7,42 3,18 2,44 f [Hz] 1st 2nd 3rd TABLE II. FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE AT THE TENSION FH = 4,75 KN fmeas [Hz] fans [Hz] f3D [Hz] ANS [%] 3D [%] horizontal 3,88 4,31 4,20 11,08 8,34 vertical 3,97 4,32 4,21 8,82 6,08 horizontal 8,58 8,63 8,41 0,58 -2,01 vertical 8,73 8,63 8,41 -1,15 -3,69 horizontal 12,48 12,95 12,61 3,77 1,05 vertical 12,64 12,95 12,61 2,45 -0,21 f [Hz] 1st 2nd 3rd TABLE III. FIRST THREE HORIZONTAL AND VERTICAL MEASURED AND NUMERICALLY CALCULATED EIGENFREQUENCIES OF THE USED ACSR POWER LINE AT THE TENSION FH = 6,68 KN fmeas [Hz] fans [Hz] f3D [Hz] ANS [%] 3D [%] horizontal 4,51 5,13 4,98 13,75 10,46 vertical 4,72 5,13 4,99 8,69 5,62 horizontal 9,98 10,25 9,96 2,71 -0,16 vertical 10,23 10,25 9,96 0,20 -2,60 horizontal 14,67 15,35 14,95 4,64 1,89 vertical 14,97 15,38 14,95 2,74 -0,15 f [Hz] 1st 2nd 3rd TELEN2016021 DOI 10.14311/TEE.2016.4.116 USUM (AVG) RSYS=0 DMX =.767418 1 SMX =.767418 NODAL SOLUTION Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 STEP=1 SUB =5 FREQ=15.3547 USUM (AVG) RSYS=0 Y DMX =.775861 1 SMX =.775861 MN X Z NODAL SOLUTION STEP=1 SUB =6 FREQ=15.3831 USUM (AVG) RSYS=0 DMX =.767401 MN X Y SMX =.767401 MAR 22 2016 15:15:42 121 MX MAR 22 2016 15:15:58 nd a) 2 eigenmode in vertical plane MX Z b) 3rd eigenmode in horizontal plane MX 0 Y MN X .085269 Z .170537 .255806 .341075 .426343 .511612 .59688 .682149 .767418 c) 3rd eigenmode in vertical plane 0 Fig. 8. Eigenmodes of the used ACSR conductor for .517241 the tension FH = 6.68 kN. .172414 .344827 .689654 .086207 .25862 .431034 .603447 .775861 V. CONCLUSION REFERENCES [1] J. Murin, V. Kutis, Improved mixture rules for the composite In the presented contribution the numerical simulation (FGM's) sandwich beam finite element, Barcelona, Spain, 2007, pp. and experimental measurements of the.170534 selected ACSR 0 .341067 647-650..511601 .682134 conductor is presented. The numerical .085267 simulations .2558were .426334 .596868 .767401 [2] V. Kutiš, J. Murín, R. Belák and J. Paulech, „Beam element with done by the commercial software ANSYS and by our new spatial variation of material properties for multiphysics analysis of 3D FGM beam finite element which was implemented in functionally graded materials,“ Computers and Structures, 89, pp. the software MATHEMATICA. 1192 - 1205. Doi: 10.1016/j.compstruc.2010.10.012 [3] J. Hrabovský, Multiscale modelling and simulation of free vibration From the results shown in Tab. 1–3 it is obvious that of FGM beams, Dizertačná práca, Bratislava, 2013. the differences between the numerical simulations and [4] J. Murín, M. Aminbaghai, J. Hrabovský, V. Kutiš, J. Paulech, S. experimental measurements are very small. These results Kugler, „A new 3D FGM beam finite element for modal analysis,“ confirm the correctness of our procedure for rev. Proceedings of the 11th WCCM, Barcelona, Spain, 2014. homogenising the material properties of the ACSR [5] H. Altenbach, Mechanics of composite structural elements, Berlin: conductor as well as the efficiency and accuracy of a new Springer-Verlag, 2003. beam finite element for analysis of the composite [6] Š. Fecko, et. al., Elektrické siete: Vonkajšie silové vedenia, structures. Bratislava: STU v Bratislave, 1990. ACKNOWLEDGMENT This work was supported by the Slovak Research and Development Agency under the contract No. APVV-150326. This work was also supported by the Slovak Research and Development Agency under the contract No. APVV-0246-12 and APVV-14-0613, by Grant Agency VEGA, grant No. 1/0228/14 and 1/0453/15. Authors are also grateful to the companies SAG Elektrovod a.s. Bratislava, Elba a.s. Kremnica and Laná a.s. Žiar nad Hronom for sponsorship materials needed for measurements. TELEN2016021 DOI 10.14311/TEE.2016.4.116 [7] Š. Fecko, D. Reváková, L. Varga, J. Lago, S. Ilenin, Vonkajšie elektrické vedenia, Bratislava: Renesans, s.r.o., 2010. [8] STN EN 50182, Vodiče na vonkajšie vedenia. Vodiče koncentricky zlanovaných kruhových drôtov, 2001. [9] STN EN 50189, Vodiče na vonkajšie vedenia. Pozinkované oceľové drôty, 2001. [10] STN EN 60889, Tvrdo ťahané hliníkové drôty pre vodiče nadzemných elektrických vedení, 2001. [11] ANSYS Swanson Analysis System, Inc., 201 Johnson Road, Houston, PA 15342/1300, USA. [12] National Instruments Corporation, LabView, 11500 Mopac Expwy, Austin, 78759-3504 Texas. [13] S. Wolfram Mathematica 5, Wolfram research, Inc., 2003. Transactions on Electrical Engineering, Vol. 5 (2016), No. 4 Pástor, M., Dudrik, J.: Stability of Grid-Connected Inverter with LCL Filter The paper analyses oscillations in a single-phase grid connected inverter with the LCL output filter. Passive and active damping techniques are designed and compared by simulations. The inverter is controlled by a proportional resonant controller. Bendík, J., Cenký, M., Eleschová, Ž.: 3D Numerical Calculation of Electric Field Intensity under Overhead Power Line Using Catenary Shape of Conductors This article presents a superposition method in combination with the Coulomb’s law and the Method of image charges for calculation of the electric field distribution generated by high voltage overhead power lines above a flat surface in every dimension. Such calculations are required to ensure the operational safety of people exposed to the action of external electric field as well as to reduce the cost of people protection. The method provides options for calculation of the field around the wire of a general shape. This substantial improvement of the method could be applied to eliminate the usual error in the calculation created using approximation of catenary shape conductors by infinite straight conductors. The method has been extensively tested on a set of shapes with known analytical solutions. It has been shown that the numerical solution converges uniformly to the analytical solution and the accuracy depends only on the number of finite elements. Cintula, B., Eleschová, Ž., Beláň, A., Janiga, P.: Comparison of Reconfigurations Using Deterministic Approach for Global Assessment of Operational State in Power System This article deals with the analysis of impact reconfiguration on the power system operational state. The stated analysis assess results of the simulated calculations of N-1 events with the aim to obtain a more complex view of the security criterion N-1 use in comparison with the current methods and procedures being in practice. Methodologies based on the deterministic approach arising from calculations of steady states with the global assessment of the power system operational states are presented. The article objective is to comprehensively compare the selected operational states especially different reconfiguration variants in a power system. Eleschová, Ž., Ivanič, M.: Impact of Asymmetry of Over-Head Power Line Parameters on Short-Circuit Currents This paper analyses the impact of asymmetry of over-head power line parameters on short circuit currents when three-phase fault and phase-to-ground fault occur. The calculation results with consideration of an asymmetry of the power line parameters are confronted with the calculation in accordance with the Slovak standard STN EN 60909 which does not consider asymmetry of equipment parameters in the power system. The calculation of short-circuit conditions was carried out for two types of 400 kV power line towers on which is a considerably different arrangement of phase conductors. Murín, J., Hrabovský, J., Gogola, R., Goga, V., Janíček, F.: Numerical Analysis and Experimental Verification of Eigenfrequencies of Overhead ACSR Conductor This contribution deals with the modal analysis of ACSR conductor using the finite element method (FEM) and experimental measurements of eigenfrequencies. In numerical experiments for the modelling of the conductor the material properties of the chosen conductor cross-section are homogenized by the Representative Volume Element (RVE) method. The spatial modal analysis of the power line is carried out by means of our new 3D FGM beam finite element and by standard beam finite element of the commercial software ANSYS. Experimental measurements are also carried out for verification of the numerical calculation accuracy. _____________________________________________________________________________________________ TRANSACTIONS ON ELECTRICAL ENGINEERING VOL. 5, NO. 4 WAS PUBLISHED ON 31TH OF DECEMBER 2016