In this paper, the authors propose the compute of reactive power allocation considering their losses and their cost using the distribution factors method. The comparison is performed using Bialek method, based on a topological approach for determining the contributions of generators (consumers) to individual power flow through network elements. The case study refers to the 12 buses test power system, heaving 6 P-U buses and 9 P-Q buses

Cristian CRĂCIUN

Oana DULCA

Petru ANDEA

Ştefan KILYENI

Directory of Open Access Journals

Annals of the Academy of Romanian Scientists Series on Engineering Sciences ISSN 2066 - 8570 Volume 4, Number 2/2012 33 COST ALLOCATION OF TRANSMISSION  SYSTEMS FOR REACTIVE POWER Petru ANDEA1, Oana DULCA2, Ştefan KILYENI3, Cristian CRĂCIUN4 Rezumat. În această lucrare, autorii propun determinarea alocării puterii reactive luând în considerare pierderilor şi costul acestora folosind metoda factorilor de distribuţie. Comparaţia se realizează folosind metoda Bialek ,bazată pe un metodă topologică pentru determinarea contribuţiilor generatoarelor (consumatorilor) la circulaţiile individuale de puteri prin elementele de reţea. Studiul de caz se referă la un sistem cu 12 noduri, având 6 surse şi 9 consumatori. Abstract. In this paper, the authors propose the compute of reactive power allocation considering their losses and their cost using the distribution factors method. The comparison is performed using Bialek method, based on a topological approach for determining the contributions of generators (consumers) to individual power flow through network elements. The case study refers to the 12 buses test power system, heaving 6 P-U buses and 9 P-Q buses. Keywords: cost allocation, power systems, reactive power, tracing method, distribution factors 1. Introduction Reactive power has a dominant effect on real energy transfer and an appropriate management of reactive power is very essential for supporting power system security. On the other hand, while reactive power production cost is highly dependent on active power generation, it is mainly confined to local consumption. Most researches have been focused on active power as the main good transacted in electricity markets and reactive power is studied less and superficial. As a result, to avoid market power and to maintain the secure operation of the system, a fair cost allocation method seems to be very essential.  Several methods have been developed to solve the allocation problem for reactive power costs. The Zbus method presents a solution based on Zbus matrix and considers the current injection at each bus [1], [2]. Methods based on proportional sharing principle provide efficient procedures for reactive power and reactive losses. References [3], [4], [5], [6] are example of these method.  These paper presents the distribution factors methods for reactive power allocation, considers active and reactive power losses. There are 3 categories of                                                  1Prof. Univ. Ph.D. Eng., University ”Politehnica” of Timisoara, Full member of AOSR. 2Academy of Romanian Scientists, Timisoara Branch, oana.pop@gmail.com. 3Prof.dr.eng., University ”Politehnica” of Timişoara, Romania, stefan.kilyeni@et.upt.ro. 4Ph.D. student, Eng., University ”Politehnica” of Timisoara.   34 Petru Andea, Oana Dulca, Ştefan Kilyeni, Cristian Crăciun  factors [7], [8]. Generation shift factors (AQ factors) refer to change of power flow through network elements due to changes of power generated distribution. Generalized generation distribution factors (DQ factors) measure the total use of transmission network facilities by generation injection. Generalized load distribution factors (CQ factors) measure the total use of transmission network facilities by loads. The distribution factors method was extended to calculate the AC power flow, heaving the object the calculus of the system cost allocation for the regime with active power losses [9]. The results are compared with Bialek method. The case study refers to the 12 buses test power system, heaving 6 P-U buses and 9 P-Q buses. 2. Method presentation Generation Shift factors for reactive power (AQ factors) reflect the modification of reactive power through network element, corresponding to change of reactive power generated (without changing the power system overall balance). They depend on the choice of reference bus and not on the operating regime. AQ factors are determined based on simplified reactive power flow (which implies neglecting resistances longitudinal and transverse conductances network elements, the renunciation of active power flow and consider all angles equal to 0 - the angle of slack bus voltage).  Q/ U B U    (1) where,  Q/U – vector of reactive power injected at system bus divided to voltage bus value; U – vector of nodal voltage;  B – matrix of nodal susceptances (the imaginary parts of nodal admittances matrix Yn.  If the voltage that appears in the left side of relation (1) is considered equal to 1 pu, the system (2) becomes linear:  U  Q B  (2) By solving the linear system (2) results the value of nodal voltage:   1  U B Q  (3) which means (noting with 1jib , j N,i N,    matrix elements 1B )  1j ji ii NU b Q , j N    ( )  (4) In same condition obtain:      Cost Allocation of Transmission Systems for Reactive Power 35  Q / U B U    (5) where  P / U  - vector of reactive power through the network elements divided by the voltage node value;  U  - vector of voltage of network elements from initial bus;  B  - diagonal matrix of longitudinal susceptances of network elements. If the voltage that appears in the left side of relation (4) is considered equal to 1 pu, the system (5) becomes linear:  Q B U    (6) Writing in extended variant the relation (6) lead to:  jk jk j kQ B U U , jk R    ( )  (7) Using the relation (5), relation (8) becomes:  1 1 1 1jk jk ji i ki i jk ji ki ii N i N i NQ B b Q b Q B b b Q , jk R                      ( ) ( ) ( )  (8) Relation (8) is linear and the modification of power through network element, jkQ , can be expressed without problems due to changing of power injected in bus i, iQ :  1 1jk jk ji ki iQ B b b Q     ( )  (9) Comparing the relations (9) and (1), the expression of AQ factors for network elements jk, corresponding changing of generated power in bus i:  1 1jk,i jk ji kiAQ B b b , jk R , i N \ e     ( )  (10) Analogous to generalized generation distribution factors for active power, DQ factors determine the impact of each generator on active power flow on network elements (so, they can have negative values).  They are determined in conditions of DC power flow too, being defined by the relation:  , jk jk,i gii NQ DQ Q , jk R   )(  (11) where,  , jkQ – reactive power flow on network elements jk;  giQ – power generated in bus i;    36 Petru Andea, Oana Dulca, Ştefan Kilyeni, Cristian Crăciun  jk,iDQ – DQ factor of a network elements jk, corresponding to power generated in bus i, heaving the expression:  0jk jk,i gii N\ejk,i jk,e jk,i jk,igii NQ AQ QDQ DQ AQ AQQ    ( ) (12) where, 0jkQ – reactive power flow on network elements jk from the previous iteration; e – slack bus. DQ factors reflect the utilization rate of electricity transmission capacity depending on reactive generated power (unlike the A factors, which indicated the incremental rate of use). They depend on network elements and operating regime and not on the choice of reference bus. Generalized load distribution factors (CQ factors) are very similar to DQ factors and determine the contribution of each load to network elements (so, they can have negative values). They are defined by the relation:  , jk jk,i cii NQ CQ Q , jk R   )(  (13) where, , jkQ – reactive power flow on network elements jk; ciQ – reactive power consumed in bus i; jk,iCQ – CQ factor of a network elements jk, corresponding to power generated in bus i, heaving the expression:  0jk jk,i cii N\ejk,i jk,e jk,i jk,icii NQ AQ QCQ CQ AQ AQP    ( ) (14) where,  0jkQ – reactive power flow on network elements jk from the previous iteration;  e – slack bus. CQ factors reflect the utilization rate of electricity transmission capacity depending on consumed power. They depend on network elements and operating regime and not on the choice of reference bus.  3. Description of test power system analyzed The test system with 12 buses is shown in Fig. 1. Bus 1 is slack bus. The network elements parameters are presented in Table 1. Table 2 contains the initial data of buses and the results of power flow for considered operating regime. Table 3 presents the results of the power flow for line system.     Cost Allocation of Transmission Systems for Reactive Power 37  Fig. 1. Test system with 12 buses, normal operation regime. Table 1. Network elements parameters Bus i Bus j R [u.r] X [u.r] B [u.r] L [km]  Bus i Bus j R [u.r] X [u.r] B [u.r] L [km] 1 2 0.00415 0.025 0.04 30  6 5 0.00554 0.03335 0.05379 40 1 6 0.00969 0.05838 0.0949 70  6 9 0.002075 0.0125 0.02 30 1 7 0.0166 0.1 0.16132 120  6 11 0.00692 0.0417 0.06725 50 2 8 0.00415 0.025 0.04 30  10 7 0.00554 0.03335 0.05379 40 3 7 0.00526 0.03169 0.0511 38  9 10 0.00277 0.01667 0.0269 20 8 3 0.00623 0.03752 0.06 45  10 11 0.00692 0.0417 0.06725 50 5 4 0.0083 0.05 0.08 60  10 12 0.00484 0.02912 0.047 34 7 4 0.00387 0.02335 0.03765 28  11 12 0.00346 0.0208 0.0336 25 11 4 0.0083 0.05 0.08 60        Table 2. Dates and results of buses power flow Number U [u.r] U [kV] d [grd] d [grd] PC [MW] QC [MVar] Pg [MW] 1 1.05 231 0 0 0 527.77 97.32 2 1.03468 227.629 -2.27 300 35 300 25 3 1.03182 227.001 -2.08 0 0 350 35 4 1.02382 225.241 -3.37 0 0 395.74 35 5 1.02747 226.043 -2.97 350 25 600 50 6 1.0088 221.937 -7.21 230 60 100 75 7 1.01245 222.739 -5.9 350 38 0 0 8 1.0224 224.929 -4.61 300 25 0 0 9 0.99914 219.81 -8.92 208 30 0 0 10 0.99724 219.392 -9.27 170 20 0 0 11 0.99757 219.466 -9.18 210 23 0 0 12 0.99332 218.531 -10.12 130 15 0 0   38 Petru Andea, Oana Dulca, Ştefan Kilyeni, Cristian Crăciun  Table 3. Power flow results for network elements Bus i Bus j Pij  [MW] Qij  [Mvar] maxijS  [MVA] ΔP  [MW] ΔQ  [MVAR] 1 2 178.3 36 75.8 1.27 3.19 1 6 235.9 44.1 100 5.14 20.69 1 7 113.6 17.3 47.9 2.11 -4.84 2 8 177 22.8 74.4 1.25 3.23 3 7 224.8 30.3 47.3 2.57 10.03 8 3 -124.3 -5.4 52.2 0.94 -0.78 5 4 15.5 0.7 7.5 0.05 -8.29 7 4 -198 -14.1 83.5 1.5 5.06 11 4 -208.1 -11.3 88.8 3.65 13.62 6 5 -231.5 -12.2 98.2 2.94 12.02 6 9 246.9 39.6 52.1 1.29 5.68 6 11 85.4 11 35.9 0.53 -3.7 10 7 -179.8 -13.1 76.1 1.83 5.45 9 10 37.7 3.9 15.9 0.05 -2.44 10 11 -3.7 -3.5 2.1 0.03 -6.68 10 12 51.2 2.9 21.5 0.15 -3.88 11 12 79.2 6.2 33.1 0.23 -2.01 4. Numerical results For the calculation we used Mathematica® environment, application software DFPQ (Distribution Factor for Active and Reactive Power) being developed by the author. Using the relations (12) and (13) we obtain DQ factors (Table 4) and CQ factors (Table 5). Table 4. Generalized generation distribution factors (DQ factors) Line j-k Djk,1 Djk,2 Djk,3 Djk,4 Djk,5 Djk,6 1-2 -0.449787 -1.79374 -0.23392 0.684412 0.83575 0.936693 1-6 -0.671947 -0.368645 0.402364 0.782211 0.783145 0.783769 1-7 -0.395048 -0.266395 0.0606469 0.304861 0.39825 0.460541 2-8 -0.289615 0.0238408 -0.17934 0.289781 0.367091 0.418657 3-7 0.0964449 0.0462214 -0.0814494 0.369913 0.218238 0.117071 3-8 0.733339 0.112715 -0.464943 -0.104448 -0.375162 -0.555729 4-5 0.0316719 -0.00615358 -0.102308 -0.246827 0.319719 0.0306042 4-7 -0.101657 -0.16929 -0.341218 0.26661 0.0623564 -0.0738809 4-11 -0.44855 -0.321843 0.000252289 0.313903 0.179898 0.0905158 5-6 -0.161478 -0.123652 -0.0274981 0.117021 0.550476 -0.16041 6-9 -0.823823 -0.558162 0.117163 0.554581 0.820685 0.998176 6-11 -0.260547 -0.185081 0.00675793 0.093705 0.261991 0.374237 7-10 -1.01215 -0.637673 0.31426 0.635423 0.601869 0.579489 9-10 -0.946538 -0.680877 -0.00555242 0.431866 0.69797 0.875461     Cost Allocation of Transmission Systems for Reactive Power 39 Line j-k Djk,1 Djk,2 Djk,3 Djk,4 Djk,5 Djk,6 10-11 0.336729 0.226573 -0.0534492 -0.271719 -0.290396 -0.302854 10-12 0.329447 0.23743 0.00351698 -0.178811 -0.194414 -0.20482 11-12 -0.282208 -0.19019 0.0437224 0.226051 0.241653 0.25206 Table 5. Generalized Load Distribution Factors (CQ factors) Line j-k Cjk,2 Cjk,5 Cjk,6 Cjk,7 1-2 2.09779 -0.531696 -0.632639 -0.253538 1-6 0.880437 -0.271353 -0.271977 -0.281812 1-7 0.532779 -0.131866 -0.194157 0.0397837 2-8 0.178612 -0.164639 -0.216204 -0.0225438 3-7 -0.137224 -0.309241 -0.208073 0.0552317 3-8 -0.644218 -0.156341 0.0242253 -0.110612 4-5 -0.0465516 -0.372424 -0.0833094 0.0983949 4-7 0.0863156 -0.145331 -0.0090936 0.345486 4-11 0.471525 -0.030216 0.0591657 -0.0140145 5-6 0.106293 -0.567835 0.143051 -0.0386532 6-9 0.929712 -0.449135 -0.626626 -0.0882986 6-11 0.295506 -0.151566 -0.263812 0.0063203 7-10 1.16546 -0.0740868 -0.0517062 -0.269525 9-10 1.1841 -0.194744 -0.372235 0.166093 10-11 -0.423646 0.0933229 0.105781 -0.0015303 10-12 -0.433347 -0.0015042 0.0089024 -0.0807386 11-12 0.322683 -0.10916 -0.119567 -0.0299256  Line j-k Cjk,8 Cjk,9 Cjk,10 Cjk,11 Cjk,12 1-2 1.4751 -0.903282 -1.2677 -0.889115 -1.04686 1-6 0.572644 -0.678025 -1.22477 -0.743912 -0.944268 1-7 0.402221 -0.36117 -0.58605 -0.352427 -0.44977 2-8 0.860515 -0.35446 -0.54062 -0.347223 -0.427805 3-7 -0.0862569 -0.169854 -0.118391 -0.22207 -0.178871 3-8 -0.0144048 0.311535 0.698395 0.254083 0.439213 4-5 -0.0081661 -0.0155522 0.0756824 0.0596382 0.0663233 4-7 0.154951 0.0423748 0.111677 -0.0279422 0.0302324 4-11 0.342942 -0.044516 -0.184123 0.118104 -0.007824 5-6 0.0679078 0.0752939 -0.0159407 0.0001035 -0.006585 6-9 0.660117 -0.11091 -0.762994 -0.591674 -0.663057 6-11 0.218922 -0.253335 -0.239227 0.0863527 -0.049306 7-10 0.785436 -0.305506 -0.647245 -0.359844 -0.479594 9-10 0.914508 -0.856518 -0.508603 -0.337283 -0.408666 10-11 -0.311859 0.156564 0.224944 0.427736 0.343239 10-12 -0.339968 0.0513236 0.108443 0.277844 0.623927 11-12 0.229304 -0.161988 -0.219108 -0.388508 0.265409 In order to make comparisons, it will be presented only the situations considered more representative. In this sense, it analyzed the contribution of source G4   40 Petru Andea, Oana Dulca, Ştefan Kilyeni, Cristian Crăciun  (QG4 = 35 MVar) to reactive power flow and reactive power losses through network elements.  Figure 2 shows the contribution of generator G4 to reactive power flow. The participation of this source leads to negative values on lines 3-8 (-3.66 Mvar), 4-5 (-8.64 Mvar), 10-12 (-6.26 MVar) and 10-11 (-9.51 Mvar). The most significant contribution is detected on line 2-8, for a value of 27.38 MVar. In contrast to distribution factors method, Bialek method leads to different values, the source G4 representing the contributions only on tracings 4-7-10-12 and 4-11-10-12.  Fig. 2. Contribution of source G4 to reactive power flow. After that it goes on to present the allocation of electricity transmission costs to energy market participants - in our case, for producers and consumers. It is considered as a basis for calculating a unit cost of transport lines 2 $/MW·km. In order to calculate the transfer cost by the MW-km method, the following formula will be used [12]:  ij ij ij uij LTij ij ij kk U ij Lc QC Ctuc Q    ( )( ) (16) where:  tuC  – transmission hourly cost for transaction u [$/h];  Lij – length of line ij [km];  cij – unit transmission cost of line ij [$/MW  km];      Cost Allocation of Transmission Systems for Reactive Power 41 Qij u – reactive power flow on line ij, due to transaction u [MW];  U – set of transactions,  L – set of lines. Using DQ factors, transmission costs allocated to generators (Table 6) are determined. Using CQ factors, transmission costs allocated to generators (Table 7) are determined. The values obtained are compared with those resulted with Bialek methods. All values are calculated in presents of power losses. Table 6. Transmission costs allocated to generators  Cost allocated to generator 1 [$] Cost allocated to generator 2 [$] Cost allocated to generator 3 [$] Cost allocated to generator 4 [$] Cost allocated to generator 5 [$] Cost allocated to generator 6 [$] Distribution Factors 297.34 32.34 66.41 354.92 633.91 887.59 Bialek 358.93 23.04 176.99 198.22 709.95 819.00 Table 7. Transmission costs allocated to consumers  Cost allocated to consumer 2[$] Cost allocated to consumer 5 [$] Cost allocated to consumer 6 [$] Cost allocated to consumer 7 [$] Distribution Factors 108.76 166.34 320.79 84.82 Bialek 16.48 0.00 262.93 146.23   Cost allocated to consumer 8 [$] Cost allocated to consumer 9 [$] Cost allocated to consumer 10 [$] Cost allocated to consumer 11 [$] Cost allocated to consumer 12 [$] Distribution Factors 33.63 240.34 258.16 185.53 141.63 Bialek 44.88 159.95 439.65 97.72 457.19 Conclusions Reactive power plays crucial role in power system security and reliability. An accurate reactive power allocation method is significant to have correct cost for each participant. The authors proposed a method for reactive power allocation using distribution factors, in presence of active and reactive power losses. Conclusions are achieved from study over more complex and different systems. The two methods compared have more differences, given by using different principles. Distribution factors method is based on superposition principle and produced negative flow. Bialek method uses the proportional sharing principle and the value obtained are always positive.    42 Petru Andea, Oana Dulca, Ştefan Kilyeni, Cristian Crăciun  R E F E R E N C E S [1] A. G. Conejo, F.D. Galiana, and I. Kockar, Z-bus loss allocation, IEEE Trans. Power Syst., Vol. 16, No. 1, pp. 105-110, February 2001. [2] A.J. Conejo, J. Contreras, D.A. Lima, A. Padilha-Feltrin, Zbus Transmission Network Cost Allocation, IEEE Transactions on Power System, Vol. 22, No. 1, pp. 342-349, February 2007. [3] J. Bialek, Topological Generation and Load Distribution Factors for Supplement Charge Allocation in Transmission Open Access, IEEE Transactions on Power System, Vol. 12, No. 3, pp. 1185-1193, August 1997. [4] D. Kischen, R. Allan, G.. Strbac, Contributions of Individual Generators to Loads and Flow, IEEE Transactions on Power System, Vol. 12, No. 1, pp. 52-66, February 1997. [5] I.D. Galiana, A.J. Conejo, A. Gil, Transmission Network Cost Allocation based on Equivalent Bilateral Exchanges, IEEE Transactions on Power System, Vol. 18 No. 4, pp. 1425-1431, November 2003. [6] H. A. Gil, F. D. Galiana and A. J. Conejo, Multiarea transmission network cost allocation, IEEE Transactions on Power System, Vol. 20 No. 3, pp. 1293-1301, Aug. 2005. [7] W. Y. Ng, „Generalized Generation Distribution Factors for Power System Security Evaluations”, IEEE Trans. Power App. Syst., Vol. PAS-100, pp. 1001-1005, Mar. 1981. [8] O. Pop, M. Nemeş, Reactive Power Allocation in Electrical Systems, 2nd International Conference on Modern Power Systems Mps 2008, 12-14 November 2008, Cluj-Napoca, Romania, pp. 58-61. [9] O. Pop, Contributions on the Tariff Access Assessment to the Transport System, Doctoral Thesis, Faculty of Electrical and Power Engineering, University “Politehnica” of Timisoara, 2009. 

COST ALLOCATION OF TRANSMISSION SYSTEMS FOR REACTIVE POWER

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals