Modeling of nonlinear loads in high-voltage network by measured parameters

Transcription

1 nternational Conference on Renewable Energies and Power Quality (CREPQ 17) Malaga (Spain), 4 t to 6 t April, 017 Renewable Energy and Power Quality Journal (RE&PQJ) SSN X, No.15 April 017 Modeling of nlinear loads in ig-voltage network by measured parameters L.. Kovernikova 1, Luong Van Cung 1 Te Siberia Branc of te Russian Academy of Sciences Energy Systems nstitute, 130, Lermontov Str., , rkutsk (Russia) Pone: ext. 3, fax: , kovernikova@isem.irk.ru National Researc rkutsk State Tecnical University Hung Yen (Vietnam) cunglv@mail.ru Abstract. Te paper presents an algoritm for modeling nlinear loads connected to te des of ig-voltage network. Te algoritm is developed on te basis of measured parameters of armonic conditions. Te ig-voltage networks are extended. Many nlinear loads distributed across te territory of te region are connected to tem. An analysis of measured parameters of armonic conditions sows tat tey vary randomly. Te algoritm for modeling te nlinear loads is developed considering teir probabilistic properties. Te algoritm is illustrated wit an example of modeling te nlinear loads of a railway traction substation and an aluminum smelter sop tat are powered by a 0 kv network. Key words Harmonics, ig-voltage network, measurement, statistical analysis, nlinear load model 1. ntroduction Publications propose many ideas on modeling nlinear loads. Some general principles of modeling are presented in [1], []. Models of various electric equipment wit nlinear voltage-current caracteristics are presented in [3], [4]. n [5] autors propose a tecnique for modeling an aggregate nlinear load on te basis of measurements for distribution networks supplying power to consumers of industrial, commercial and residential sectors. Modeling of nlinear loads connected to te des of ig-voltage (110-0 kv) network is an unsolved problem. Te parameters of armonic conditions are largely determined by specific features of ig-voltage networks. Te ig-voltage networks are extended. Tey are spread over a large territory. Many ig-power nlinear loads are connected to te des of ig-voltage networks. Eac of te loads represents a facility tat as its own electric network of a lower voltage. Te network of te facility supplies power to different items of equipment including tat wit nlinear voltage-current caracteristics. An analysis of ig-voltage network conditions cant take into account and represent eac items of equipment of te facility in te calculation sceme. Te main teclogical electrical equipment of te facility is te source of armonics. Currents of tese armonics are drawn from te ig-voltage network to te medium- and low-voltage networks. Hig-voltage networks supply power to medium- and low-voltage networks. Tey contain also a great amount of electrical equipment wit nlinear voltage-current caracteristics. Harmonic currents from tese networks are drawn to te ig-voltage networks. Tus, te medium- and low-voltage networks can be considered as a nlinear load connected to te igvoltage network de. Taking into account te above specific features we can state tat measurements are te only way to receive information on a range of armonic currents, teir magnitudes and pases for modeling te nlinear loads connected to te des of ig-voltage networks. Duration of measurements sould be determined individually for eac certain case. Since te parameters of armonic conditions in ig-voltage networks vary randomly [6]-[8] it is necessary to analyze tem by te metods of matematical statistics. n tis paper we analyze te measured parameters of armonic conditions on te basis of data measured at connection des of a railway traction substation and an aluminum smelter sop. Based on te obtained results, we developed an algoritm to model te nlinear loads.. Matematical statement of te problem Te parameters of armonic conditions are calculated by solving a system of equations for eac armonic [9] U Z, (1) were - armonic order, U - column-matrix of dal voltage values to be determined, Z - a square matrix of self- and mutual impedances of network des, - column-matrix of armonic currents at network des ttps://doi.org/ /repqj RE&PQJ, Vol.1, No.15, April 017

2 representing load. Eac element i of matrix complex number i a jr. i i is a Te objective is to determine current by te measurement results. f network as one load wit nlinear voltage-current caracteristic tere will be problems in determining current. Wereas if te network as several nlinear loads connected to different des ten teir armonic currents are spread over te network and penetrate into te loads of virtually all des. n tis case tey are random values. Teir random nature is determined by varying topology of te network, equipment, wave and frequency properties of te network, pases of armonic source currents, voltage at nlinear equipment, etc. An equivalent circuit of ig-voltage network wit a nlinear load connected to its de is traditional and it is presented in Fig.1. Supply network S U PM L Nonlinear load Fig. 1. An equivalent circuit of a supply network and a nlinear load for te -t armonic. Te tations used in te sceme are: S - te -t armonic current pasor of te supply network, L - te -t armonic current pasor of nlinear load, PM te point of measurement. Current S is a resultant current of all nlinear loads available in te network except for current L. n Fig. 1, te armonic current drawn troug te point of measurement is determined by te vector sum of currents S and L, i.e. S L. () Based on te measurements at te connection de we obtained te effective values and pase angles ( U ) of current and voltage U. Te measurements were performed during 4 ours wit a time interval of 1 minute. As a result, we obtained an array of 1440 elements. Angles and U make it possible to determine te pase angle between current and voltage of te -t armonic by. (3) n [10] angle determines te directions of active and reactive power flows wit respect to te point of measurement. Te angle can also be applied to determine te directions of active and reactive components of U armonic currents. Te values of active ( a ) and reactive ( r ) currents are calculated according to [11], using cos sin. (4) a, r An analysis of directions of active and reactive currents allows us to make a conclusion weter te de is a generator de or a load de for a certain armonic in order to represent appropriately te model of current in te system of equations (1). Te previous analysis of measured data in [], [6]-[8] sows tat modeling a nlinear load sould include two stages: 1) analysis of directions of active and reactive armonic currents, ) development of models of active and reactive armonic currents based on te results of te first stage. 3. Analysis of directions of active and reactive armonic currents According to [10] te direction from network to load is assumed to be a positive direction of active current and active power. For reactive current we assume te same direction as for reactive power in [11], provided te load is inductive. Active and reactive currents are directed from network to load if angle lies in te interval from 0 to π/, i.e. in te first quadrant of a complex plane. Active current is directed from load to network and reactive from network to load, if angle lies in te interval from π/ to π, i.e. in te second quadrant. Bot currents are directed from load to network if angle lies in te interval from π to 3π/, i.e. in te tird quadrant. Active current is directed from network to load and reactive in an opposite direction, if angle lies in te interval from 3π/ to π, i.e. in te fourt quadrant. Tus, in eac quadrant tere is a certain direction of active and reactive armonic currents. n [8] based on te analysis of measurements we obtained tat at one and te same time instant active power of different armonics can ave opposite directions. Tis conclusion applies to currents as well. Te analysis of directions of active and reactive currents was made for one pase of a railway traction substation and an aluminum smelter sop for armonics 3, 5, 7, 9, 11, 13, 3, 5. Te results of te analysis are presented in Table. Notations of te table are: RTS - railway traction substation. ASS - aluminum smelter sop. Table. - Distribution of angles (%) RTS ASS Quadrants Quadrants ttps://doi.org/ /repqj RE&PQJ, Vol.1, No.15, April 017

3 Te Table presents te number of measurements in percentage of te total number of 1440 measurements tat correspond to te distributions of directions of active and reactive currents over te quadrants of te complex plane. Te data presented in te Table sow tat te distribution of current directions across te quadrants is different. Some armonic currents ave predominant directions, i.e. above 50% of measurement time. Tis applies to te currents of armonics 5, 7 and 3 of te railway traction substation and currents of armonics 3, 7, 9, 11, and 3 of te aluminum smelter sop. Te armonic processes are of random caracter and even insignificant canges in te operating conditions of te network can affect tem. Te predominant directions will t cange since tey are determined by te load conditions and network configuration. At armonic 5 of te railway traction substation and armonics 13 and 5 of te aluminum smelter sop tere are te most prounced directions altoug tey make up less tan 50%. At tese armonics tere is one more direction for wic te quantity of measurements considerably exceeds te remaining two. n suc cases it is suggested to develop te models of loads for two variants. Harmonics 3, 9, 11 and 13 of te railway traction substation and armonic 5 of te aluminum smelter sop ave two directions tat prevail but teir quantities differ little from one ater. n suc cases it is necessary to take two variants to develop a model. Based on te analysis of Table, to model armonic current we assume te variants igligted in bold. 4. An algoritm to model a current of one armonic Measured parameters U represent time series of random discrete values. Solving te system of equations (1) requires te values of active and reactive currents wit a probability of Te values of parameters wit a probability of 0.95 are used in te standard [1] in te assessment of voltage quality. To calculate currents wit a probability of 0.95 it is necessary to kw te distribution functions ( f ( x ), were x a or x r ). Te distribution function of a random value is determined on te basis of a probability density function. Tus, te aim of te algoritm is to identify te probability density functions of active and reactive armonic currents, and ten calculate te values of currents wit a probability of 0.95, using respective distribution functions. Figure presents a block-diagram of an algoritm for processing a time series of random values of active and reactive currents of one armonic. Te algoritm is based on an analysis of special literature, including [13]-[15]. Below te most important points of te algoritm are described in detail. Construction of a scatter plot (Step 1 in Fig.1) of te measured time series of random values makes it possible to visually determine te presence of abrmal elements wose values considerably exceed te values of te remaining elements. n [15] te abrmal elements are called outliers. Tey are well seen in te scatter plot. Te outliers can be replaced wit neigboring elements, a mean value of neigboring elements or by oter ways proposed in special literature [13], [14]. 7 Replacement of outliers Start Construction of a scatter plot Construction of a istogram Putting forward ypoteses about f(x) End Calculation of parameters f(x) Ceck for te presence of outliers Confirmation of ypotesis about f(x) Calculation x Special metods Fig.. A block-diagram of an algoritm for modeling a armonic current. Te construction of a istogram and visual analysis of its sape (Step ) make it possible to put forward te ypoteses on te identification of probability density function of current. n te event tat it is impossible to identify te probability density function by te istogram sape, ten special approaces sould be used (Step 4). Some approaces are presented in [], [13], [14]. f te ypoteses on te probability density function are put forward, ten eac ypotesis sould be tested furter by te algoritm. First of all, it is necessary to calculate te parameters (Step 5) for an analytical description of te probability density function according to [13], [14], by using an array of measured random values of current. Ten, by using special criteria developed for specific functions [13]-[15], ceck if tere are large outliers and very small outliers (Step 6). n te event tat tere are outliers tey sould be replaced as was indicated above. Ten te processing procedure is repeated starting wit Step since te replacement of series elements can cange te istogram sape. n te event tat tere are outliers, te ypotesis about te probability density function is cecked (Step 8), for example, by te goodness-of-fit tests of Pearson, Kolmogorov Smirv [13], [14]. n te case ttps://doi.org/ /repqj RE&PQJ, Vol.1, No.15, April 017

4 tat te first ypotesis is t confirmed, it is necessary to test te second ypotesis starting wit Step 3. f ne of te put forward ypoteses is confirmed by te tests, it is necessary to apply special metods (Step 4). n te case tat one of te ypoteses about te probability density function is confirmed ten te distribution function is used to calculate te value of current wit a probability of Te calculation of te current value wit a set probability completes te modeling of armonic current of nlinear load. 5. A case study on te algoritm application Te operation of te algoritm is illustrated by modeling te active current of te 5-t armonic of te railway traction substation. Step 1. We construct a scatter plot of te current (Fig.3). Te visual analysis sows tat tere are large outliers. Fig. 3. Scatter plot of active current a5. Step. We construct a istogram (Fig.4), make a visual analysis to identify te probability density function. Step 3. We put forward ypoteses about te probability density functions. Based on te analysis of te istogram sape two ypoteses are put forward. Te first ypotesis is tat te istogram sape is close to te Rayleig probability density function, i.e. x x f ( x,a ) exp( ), were x 0. a a Te second ypotesis is tat te istogram sape is close to te Weibull probability density function, i.e. 1 x f ( x ) x exp[ ( ) ], were x 0 0, 0. Step 4 is skipped. Step 5. For an analytical description of te Rayleig function we sould calculate one parameter а. t is calculated by an expression from [13]. ts value equals Step 6. To ceck if tere are outliers in te array of random values wic are described by te Rayleig x distribution function F( x,a ) 1 exp( ) we * 0.94 apply te Darling test [13]. Te calculations sow tat tere are outliers. Step 7 is skipped. Fig. 4. Histogram of current a5. Step 8. To confirm te correspondence between te istogram sape and te Rayleig density function we use te Pearson goodness-of-fit test [13]. After te necessary calculations are carried out, we obtain an experimental value of criterion эк equal to Critical value кр at a significance level of 0.05 and te number of degrees of freedom 18 is equal to Since эк > кр, te ypotesis about te Rayleig probability density function is t confirmed. We go back to Step 3. Step 3. We test te second ypotesis about te Weibull density function. Step 4 is skipped. Step 5. For an analytical description of te Weibull density function it is necessary to calculate two parameters and. Tey are calculated by te expressions from [13]. Te values and appeared to be equal to 1.3, and Step 6. To ceck if tere are outliers in te array of random values wic are defined by te Weibull x 1.81 distribution function F( x ) 1 exp[ ( ) ] we 1.3 apply te Darling test. Te calculations sow tat tere are outliers. Step 7 is skipped. Step 8. To confirm te correspondence between te istogram sape and te Weibull density function we apply te Pearson goodness-of-fit test. After te calculations we obtain tat te experimental value of criterion эк equals Te critical value кр at a significance level of 0.05 and te number of degrees of freedom 18 equals Since эк < кр, te ypotesis about te Weibull probability density function is confirmed. Step 9. To determine te value of current wit a probability of 0.95 we use te Weibull distribution function tat corresponds to te calculated parameters x and, i.e exp[ ( ) ]. By solving te 1.3 last equality we ave x Te active component of current wit a probability of 0.95 will t exceed.4а. Te reactive component of current, obtained by te described algoritm wit a probability of 0.95, will t exceed 3.05A. Tus, we obtain te 5-t armonic current ttps://doi.org/ /repqj RE&PQJ, Vol.1, No.15, April 017

5 equal to 5.4 j3. 05 А for solving te system of equations (1). Table presents te probability density functions and te values of active and reactive armonic currents for te variants igligted in bold in Table. Table. - Probability density functions of armonic currents RTS ASS a r a r Special Beta Exp Gamma Weibull Weibull Weibull Gauss Special Weibull Beta Beta Beta Gauss Rayleig Weibull Beta Beta Special Beta Weibull Beta Beta Exp Special Special Beta Beta Special Special Weibull Weibull Special Special Beta Weibull Weibull Beta Beta 3.1 Extremal Beta Special Notations of te table are: Exp exponential, Weibull, Gauss, Rayleig, Beta, Gamma, Extremal distribution functions, Special special metods. 6. Conclusion 1) Based on te measured parameters of armonic conditions at te des connecting nlinear loads to te ig-voltage network we proposed a metodological approac to modeling te nlinear loads. Te approac consists of two stages: an analysis of directions of active and reactive armonic currents, and development of models of active and reactive armonic currents, based on te analysis results. ) We propose an algoritm to model te armonic currents of nlinear loads. Te algoritm takes into account te probabilistic properties of te parameters of armonic conditions. 3) Te analysis of active and reactive currents of different armonics of te railway traction substation and aluminum smelter sop sows tat tese currents are described by te distribution functions of Gauss, Rayleig, Weibull, exponential, beta, extremal and gamma. References [1] EEE Power Engineering Society. Tutorial on armonics modeling and simulation [] Probabilistic aspects task force of te armonics working group subcommittee of te transmission and distribution committee, Time-varying armonics: part caracterizing measured data, EEE Trans. on Power Delivery, vol.13, No. 3, July 1998, pp [3] A. Mansur, W.M. Grady, A.H. Cowdury, M.J. Samotyj, An investigation of armonics attenuation and diversity among distributed single-pase electronic loads, EEE Transactions on Power Delivery, vol. 10, No. 1, January 1995, pp [4] Luis Sainz, Juan Jose Mesas, Albert Ferrer, Caracterization of n-linear load beavior, Electric Power systems Researc 78 (008), pp [5] Mau Teng Au, Jovica V. Milavic, Development of stocastic aggregate armonic load model based on field measurements, EEE Transactions on Power Delivery, vol., No. 1, January 007, pp [6] L.. Kovernikova, Results of te researc into te armonics of loads connected to te des of ig- voltage network, Proceeding of nternational Conference on Renewable Energies and Power Quality (CREPQ 14), Cordoba (Spain), 8t to 10t April, 014. [7] L.. Kovernikova, Analysis of probabilistic properties of armonic currents of loads connected to te ig voltage networks, Proceeding of nternational conference on renewable energies and power quality (CREPQ 15), La Coruña (Spain), 5t to 7t Marc, 015. [8] L.. Kovernikova, Researc into armonic power in te igvoltage networks, Proceeding of nternational conference on renewable energies and power quality (CREPQ 16, Madrid (Spain), 4t to 6t May, 016. [9] Arrillaga, Jos. Power system armonics / J. Arrillaga, N.R. Watson. -nd edit. Cicester: Wiley, 003. [10] R.H. Stevens, Power flow direction definitions for metering bidirectional power, EEE Transactions on Power Apparatus and Systems, Vol.10, No. 9, Sept. 1983, pp [11] A.E. Emanuel, Power definitions and pysical mecanism of power flow, Jon Wiley &Sons, 010. [1] State standard Power quality limits in public power supply systems. Moscow. Standartinform. 014 (in Russian). [13] Kobzar A.. Applied matematical statistics. For engineers and researcers. -nd edition, revised M.: FZMATLT, 01 (in Russian). [14] B.Yu. Lemesko, S.B. Lemesko, S.N. Postovalov, E.V. Cimitova, Statistical data analysis, simulation and study of probability regularities. Computer approac. Novosibirsk : NSTU Publiser, 011(in Russian). [15] J.O. rwin, On a criterion for te rejection of outlying observations. Biometrika, 195, Volume 17, ssue 3-4. ttps://doi.org/ /repqj RE&PQJ, Vol.1, No.15, April 017