Modeling of power autotransformer

Transcription

1 Modeling of ower autotransformer VLADMÍR VOLČKO, ŽAETA ELEHOVÁ, ATO BELÁŇ, PETER JAGA, DOMK VGLAŠ, MROLAVA MTKOVÁ Deartment of Electrical Power Engineering lovak niversity of Technology in Bratislava lkovičova, 8 9, Bratislava LOVAK REPBL vladimir.volcko@stuba.sk, zaneta.eleschova@stuba.sk, anton.belan@stuba.sk, eter.janiga@stuba.sk dominik.viglas@stuba.sk, miroslava.smitkova@stuba.sk Abstract: - This article deals with the design of mathematical model of three-winding autotransformer for steady state analyses. The article is focused on model simlicity for the uroses of the use in comlex transmission systems and authenticity of the model taking into account different tyes of ste-voltage regulator. Key-Words: - mathematical model of three-winding autotransformer, ste-voltage regulators ntroduction owadays, there are many rograms used for steady state analysis of electric ower transmission systems. However, these rograms often do not distinguish between transformers and autotransformers. Autotransformers are almost exclusively used in electric ower transmission system and ignoring this fact may cause inaccuracies in simulations of steady states, esecially if the tertiary winding is loaded with a comensating element (reactor or caacitor). n this aer will derive the mathematical model of three-winding autotransformer for two different equivalent elements with consideration of the symmetric system and the symmetric transformer, constant frequency and ski saturation of the transformer core. Autotransformer Autotransformer does not have the rimary and secondary winding in contrast with transformer, but it has the serial and common winding. n no-load state, outut voltage corresonds to the common winding inducted voltage and inut voltage corresonds to sum of the serial winding inducted voltage and the common winding inducted voltage. The advantage of autotransformer in comarison with transformer is material saving and smaller dimensions at the same oerating arameters (voltage, ower). The serial winding of autotransformer has fewer turns than the rimary winding of transformer. Furthermore, outut current is equal to the sum of inut current and current inducted in the common winding, so the common winding of autotransformer could be dimensioned to smaller current than nominal outut current. Also, magnetic circuit of transformer is less massive because With autotransformers we distinguish the assing ower which the autotransformer is caable to transfer and the inner ower, for which the autotransformer is dimensioned. [].The issue of transformers and autotransformers is solved in detail by ref. [, ]. Fig. Three-winding autotransformer Three-winding autotransformers have windings connected mostly the way as shown on fig.. The common winding is connected in a grounded star and the tertiary isdelta. Desite the fact it concerns autotransformer, for comliance with convention, further we will not use term the inut/outut current (voltage, ower) but the rimary/secondary current (voltage, ower). te - voltage regulators t is essential to be able to control voltage in electric ower transmission system on each voltage level. B:

2 The use of regulating transformers is the main method of voltage regulation in distribution systems. There are many voltage regulator tyes and voltage regulator connections of regulating autotransformers which are described in detail in ref. [] but in rincile we distinguish three basic voltage regulator connections: Regulator located at the inut of autotransformer, Regulator located at the outut of autotransformer, Regulator located in the node of autotransformer. We can therefore say that by how many ercent the ratio between the rimary voltage and the secondary voltage and the ratio between the rimary voltage and the tertiary voltage changes, that many ercent the count of the common and serial winding turns changes.. Regulator located at the outut The voltage regulator located at the outut of autotransformer regulates voltage either by adding turns to the outut (fig. a) or by changing roortion between the serial winding turns and the common winding turns (fig. b).. Regulator located at the inut The voltage regulator located at the inut of autotransformer regulates voltage by change of number of the serial winding turns. The advantage of this tye of connection is that the regulator is dimensioned to the smallest ossible current. Great disadvantage is the level of voltage to which insulation of regulator must be dimensioned. Therefore, this connection of voltage regulator is not used in ractice with autotransformers. a) b) Fig. Regulator located at the outut For both cases is valid that the count of number of the common and serial winding turns is constant and does not change by regulation and so the ratio between the rimary voltage and the tertiary voltage is (see equation ) is constant, too. Fig. Regulator located at the inut The ratio between the rimary voltage and the secondary voltage may be calculated as follows:. Regulator located it the node Location of the regulator in the node of autotransformer does not have its equivalent among common tyes of transformers. imulating of such autotransformer by the model of common tye of transformer may cause mistakes. Regulator changes the ratio by change of number of the common winding turns. () And the ratio between the rimary voltage and the tertiary voltage as follows: () T Fig. 4 Regulator located in the node B:

3 For the change of number of the common winding turns by k %, the following relation is valid for the ratio between the rimary voltage and the secondary voltage: k /00 k /00 () And the ratio between the rimary voltage and the tertiary voltage shall be: T k /00 (4) We can therefore say that if the regulator is located in the node of autotransformer, not only the ratio between the rimary voltage and the secondary voltage is changed by regulation but also the ratio between the rimary voltage and the tertiary voltage does change while the change of articular ratio differs. Percentage change of winding turns is mostly unknown in case of voltage regulators. hange of the secondary voltage k u is usually identified on the namelate. t is ossible to rove (and from equation 4 to derive) that in order to reach the ste of regulation of the secondary voltage k u (with constant rimary voltage), the ste of change of the common winding turns shall be: k k (5) related to the rimary voltage. t is ossible to derive the relationshi between real imedances and substitute imedances but in this case there is no reason to do so. ubstitute imedances of transformer may be calculated from the system of three equations. The sum of two otional imedances Z, Z J equals to: Z u kj Z J (6) 00 Also the sum of two otional resistances of windings of the equivalent circuit equals to: R RJ PKJ (7) Magnetizing imedance may be calculated from values measured in no-load test: Z G i 0 M (8) M 00 M P (9) 0 The ratios are calculated on the basis of the tye of regulation according to the equations mentioned in art. t is essential to adhere to equation 5 for calculation between the voltage ste (stated on the namelate) and the ste of turns in case of regulation. 4 Equivalent circuit The equivalent circuit of three-winding autotransformer is based on T- element and is identical to the equivalent circuit of three-winding transformer. Fig. 4 Equivalent circuit of three-winding autotransformer (T- element) Real imedances of the serial, common and tertiary windings are relaced by substitute imedances of the rimary, secondary and tertiary windings, 5 Mathematical model n general, mathematical modeling and simulations as well as solving of non-linear models are dealt in ref. [, 4]. For requirements of modeling of steady state analysis under conditions as described in the introduction, for autotransformer it is enough to derive the square admittance matrix of comlex numbers [], with size x which meets the following condition:. (0) Where: [] vector of currents flowing into transformer [] vector of terminal voltages B:

4 [] admittance matrix Admittance matrix may be derived by means of one of the following substitute elements. 5. π - element The element is based on the equivalent circuit (fig. 4). Magnetizing admittance is not concentrated in the middle node (like in case of T- element) but divided into articular nodes of transformer. The next ste of transmission from T- element to π- element is the transformation of a star (connection of substitute imedances into a star) into triangle (see fig. 5). Fig. 5. Diagram of π- element The following equations are valid for articular nodes: 0 () / P 0 () 0 () / P For the urose of generalization, if we think for an ideal transformer with ratio P on inut, then the following is valid for each node A for ground current A0 : 0 A0 P A A (4) And the following is valid for each air of nodes A, B: AB AB P A P ) (5) ( A B B t is ossible to derive the basic equation of π- element of transformer from equations 5:. Where the admittance matrix is: (5) P P P PP (6) P PP And diagonal elements are: 0 P (8) 0 P (9) 0 (7) The advantage of π- element is that it works directly with admittances. t is suitable for the use in case when magnetizing admittance equals zero. The model working with imedance might be insoluble as transverse imedance would equal infinity in that case. 5. T - element T-element is based directly on the equivalent circuit. nder normal conditions the models transform T- element into three indeendent Γ-elements with exressed middle node. This method tasks calculation itself as each transformer adds a node into the diagram where it is necessary to calculate voltage. We chose a different aroach. For T-element it is easy to derive voltage equations leading to derivation of imedance matrix. f voltage in the middle node is 0, then the following is valid for this voltage 0 Z M (0) P P And the following is valid for articular loos of T- element: Z Z M () P P B:

5 P Z Z M P P P () P Z Z M P P P () From equations - it is ossible to derive voltage equation of transformer: Z. (4) Where the imedance matrix is: Z M Z M Z Z m P P Z M Z Z m Z M Z (5) P P PP Z M Z M Z Z m P PP P n this alication a simle system with autotransformer was simulated and the results were comared with solutions of two other commercially available software. The first software does not consider autotransformer and uses T-element slit into three indeendent Γ-elements with exressed middle node. The second software uses π- element and considers autotransformer. A rint-screen of alication with simulated system is on fig. 5. The autotransformer was fed by hard source 400 kv (ODE, or ZOL ) during simulation, constant load 50 MW (ODE, or ZOL ) was connected to the secondary terminals and two reactors kv, 40 MVA were connected to the tertiary terminals. Autotransformer arameters are listed in Tab.. Tab. Parameters of autotransformer Par. Value Par. Value 50 MVA ΔP K 606, kw ( ) With regard to the fact that simulating rograms work with the admittance matrix, it may be calculated by inversion of the imedance matrix: Z 50 MVA ΔP K 566,6 kw ( ) 00 MVA ΔP K 578, kw ( ) (6) 400 kv i 0 0,04 ()% The advantage of this model is higher reliability; however, this model cannot be used if we consider zero magnetizing admittance in transformer. 6 Model alication Model of autotransformer was rogrammed in rogramming language JAVA for demonstration and confirmation of mathematical models. The alication allows simulations of steady state analysis of simle electric ower transmission systems. kv ΔP 0 9,7 kw 4 kv te,5 % u K,9 % ( ) Max 8 u K 0,4 % ( ) Min -8 u K 5,74 % ( ) tye n node The simulations were executed for edge states of voltage regulator (ta: -8 and 8) and for middle state (ta: 0) while values on the secondary and tertiary terminals and values of demanded active and assive ower was monitoring. The results are listed in Tab.. The results of new model using T-element are labeled as ew T, the results of new model using π- element are labeled as ew π. The results of commercially available software using T-element are labeled as om T and commercially available software using π- element is labeled as om π. Fig. 5 Print-screen of simulated system B:

6 Tab. The results of simulations Ta: -8 ew T ew π om T om π [V] [V] P [MW] 50,87 50,87 50,5 50,87 Q [MVAr] 5,64 5,6 05,4,7 Ta: 0 [V] [V] P [MW] 50,8 50,8 50,5 50,8 Q [MVAr] 07,9 07,8 05,4 07,6 Ta: 8 [kv] [kv] P [MW] 50,74 50,74 50,50 50,47 Q [MVAr] 99,44 99,4 05,4 0,89 The first visible item in Tab. is that differences between the model using T-element and the model using π- element are minor. Then, the smallest differences are in the middle state when the regulator is set u for zero ta osition. Differences range on the level of tenths to hundredths of ercent and it is caused by different calculating rograms. ommercially available software using T- element in both edge states of the regulator held the constant tertiary voltage. t is caused by the fact that although the regulator was located in the node of the secondary winding, the software did not consider autotransformer. ommercially available software using π- element calculated similarly to new designed models. t roved that the tertiary voltage decreases as the secondary voltage increases. However, in comarison with new designed models, voltage changes against the middle osition of the regulator are smaller. As the software was different also in value of the secondary voltage (desite the fact that the ste of the regulation was defined on the level of, 5% and voltage has changed by less than % in the result), we assume that the software does not calculate the ste of voltage regulation to the ste of turns regulation (see equation 5). 7 onclusion t is ossible to model three winding autotransformer by means of T-element or π- element. Precision of both elements is comarable. n both cases it is imortant to take into account whether it concerns transformer or autotransformer and where ste-voltage regulator is located. At the middle osition of voltage regulator (ta: 0) were differences between individual simulation rograms very small the rinciles of simulations were similar, but at the side ositions of voltage regulator (ta: -8 and +8) were the differences between results considerable. ommercially available rograms were wrong. 8 Acknowledgements This work was done during imlementation of the roject Effective control of roduction and consumtion of energy from renewable resources, TM code , suorted by the Research and Develoment Oerational Program funded by the ERDF. This aer was suorted by the agency VEGA MŠVVaŠ R under Grant o. /00/. References: [] Hüttner Ľ., Klug Ľ.: Elektrické stroje; Vydavateľstvo T,. edition 005, B [] Jezierski E.: Transformátory: teoretické základy; Academia 97 [] Kutiš V.: Základy modelovania a simulácií; Katedra mechaniky FE T Bratislava; ; available in htt://aladin.elf.stuba.sk/katedry/kmeh/slovakve rsion/erson/kutis/zm.df [4] Beláň A.: Lectures of "Power ystem ontrol" B: