Amercan Journal of Energy and Power Engneerng 017; 4(5): 6-1 http://www.aasct.org/journal/ajepe ISSN: 375-3897 Performance Evaluaton of the Voltage Stablty Indces n the Real Condtons of Power System Rahmat Allah Morad, Roohalamn Zenal Davaran *, Mohsen Safarzae Department of Electrcal and Computer Engneerng, Graduate Unversty of Advanced Technology, Kerman, Iran Emal address r.zenal@kgut.ac.r (R. Z. Davaran) * Correspondng author Keywords Voltage Stablty, Voltage Stablty Indces, Voltage Collapse, Modal Analyss Receved: May 1, 017 Accepted: June 14, 017 Publshed: August 3, 017 Ctaton Rahmat Allah Morad, Roohalamn Zenal Davaran, Mohsen Safarzae. Performance Evaluaton of the Voltage Stablty Indces n the Real Condtons of Power System. Amercan Journal of Energy and Power Engneerng. Vol. 4, No., 017, pp. 6-1. Abstract To prevent voltage collapse n the power systems, accurate estmates of the power system performance are necessary. In the prevous studes only by addng reactve loads to weakest buses, the voltage stablty of the power system lnes are evaluated. Whereas n the real condtons of the power systems, the actve and reactve loads are ncreased smultaneously and to provde actual predcton of lnes voltage stablty the effects of ncreasng both actve and reactve loads should be consdered. In ths paper, the performance of some most mportant lne voltage stablty ndces such as L mn, FVSI, LQP and LCPI are evaluated n real condtons of the power system, and t s shown that the results of these ndces are not dentcal. Whle that, these ndces provde smlar results as only reactve loads ncreased. In addton, to evaluate the obtaned results, by usng modal analyss method the lnes that have most partcpaton n voltage nstablty are detected. Ths study s nvestgated on the IEEE 14-bus test system. 1. Introducton The modern electrcal power system s one of the largest and most complex systems that have been operated wth a hgh level of relablty for more than one century [1]. The voltage stablty of a power system s one of the most mportant phenomena that should be analyzed carefully []. Snce the generatons and transmsson lnes have been utlzed near to crtcal lmts, voltage nstablty may occur n the power systems under heavly loaded condtons. In these systems because of nsuffcent power delvered to loads, some or all buses voltages decrease rapdly, whch may lead to severe blackout n a consderable part of the power system [3]. To evaluate the potentally of voltage nstablty and determnng the system loadng lmts, several ndces are ntroduced n the lteratures. In [4], the ndex L s defned to determne the weakest system bus. In ths ndex by ncreasng some of the power system loads, the nstablty voltage of system buses s evaluated. In [5], the ndex LQP s ntroduced and the performance of ths ndex s assessed aganst changes n reactve loads. In [6], the modal analyss method s used to dentfy the weak ponts of the power system. In [7], the ndex L mn s defned to study the voltage stablty of power system. For ths purpose, by ncreasng the reactve loads, the voltage stablty lmts of a power system s determned. In [8], to study the voltage stablty the ndex FVSI s defned and
7 Rahmat Allah Morad et al.: Performance Evaluaton of the Voltage Stablty Indces n the Real Condtons of Power System ts performance s compared wth the ndces LQP and L mn. Ths study s done on IEEE 30 bus test system and by ncreasng the reactve loads, the voltage stablty s evaluated. In [9], [10] the effcency of voltage stablty ndces s studed on IEEE 14 bus test system. For ths, by ncreasng the reactve loads, the voltage stablty ndces are obtaned, and the results are analyzed. In [11], the ndex LCPI s defned and ts performance s compared to the prevous ndces. To do so, by ncreasng the reactve loads of the power system, the ndces are obtaned and ther values are compared. In [1], the senstvty and accuracy of the voltage stablty ndces than reactve loads varatons are evaluated. In [13], based on maxmum loadng capablty of the bus a new voltage stablty ndex s proposed to predct the voltage collapse n the power system. Snce the voltage stablty s an mportant factor n optmal allocatng of Dstrbuted generatons (DGs) n the electrc grd, the performance of varous voltage stablty ndces on optmally utlzng DGs unts are evaluated n [14]. In [15], the senstvtes of voltage stablty ndces to the network dsturbances are assessed, and the obtaned results are valdated usng PSAT software. In [16], to analyss the voltage stablty of radal dstrbuton systems a stablty ndex named lne stablty ndcator (LSI) s developed. In [17], by usng the voltage stablty L-ndex the effect of the transformer tap changng on the voltage stablty s studed. In [18], to assessment the system voltage stablty a new technque s proposed based on the network response structural characterstcs. As seen, n the lteratures the lne voltage stablty ndces such as L mn, FVSI, LQP, and LCPI are only nvestgated nto reactve power varatons. Whle that, n actual condtons, both actve and reactve loads are changed. Although, the voltage of power system buses s more affected by reactve power than actve power, but n the heavly loaded power system, the effect of actve power on buses voltage can be notceable. Therefore, to thoroughly nvestgate the performance of voltage stablty ndces the varatons of both actve and reactve loads should be consdered. So, n ths paper the performance of voltage stablty ndces are evaluated by smultaneously ncreasng of the actve and reactve loads. Ths study s performed on the IEEE 14 bus test system. Ths paper s organzed as follows: In Secton, the voltage nstablty s defned and the mathematcal equatons of the voltage stablty ndces are provded. In Secton 3, the case study and the obtaned results are presented. The necessary dscusson s also provded n ths secton. Fnally, the conclusons are gven n Secton 4.. Voltage Stablty Indces The voltage stablty ndces can be used to nvestgate the voltage stablty of a power system. By these ndces, the crtcal buses and lnes n a power system are detected, and the voltage stablty margns of a system are obtaned [19]. The ndces used to examne the system voltage stablty are brefly descrbed n ths secton..1. P-V and Q-V Curves The P-V and the Q-V curves are the famlar method that be used to predct the voltage stablty n a power system. These curves are used to determne the loadng margns. Snce the power system load s ncreased slghtly, n each step, t s necessary to obtan the power flows untl the knee pont of the P-V curve s reached. The margn between the knee pont and the current operatng pont n the P-V curve s used as the voltage stablty crteron [0]. In addton, by usng the Q-V curve t s possble to obtan the maxmum reactve power that can be added to the weakest bus before reachng voltage collapse... Modal Analyss Modal analyss method can be used to study the voltage stablty of a power system. In ths method by lnearzed the power flow equatons, the egenvalues that be related to buses voltage are obtaned. The lnearzed steady state system power voltage equatons are gven by [5]: P Jpθ Jpv θ = Q JQθ J Qv V Where θ and V are the ncremental change n bus voltage angle and magntude respectvely. P and Q are relatvely the ncremental change n bus real power and reactve power njecton. By keepng P constant at each operatng pont, the ncremental relatonshp between Q and V s obtaned as below: (1) 1 V = J Q () R Where JR s named reduced Jacoban matrx and obtaned as: 1 J = ( J J. J. J ) (3) R Qv Qθ Pθ Pv The correspondng th modal voltage varaton s obtaned as below: v = 1 q λ (4) If all the egenvalues are postve, the system s consdered to be voltage stable and by evaluatng the mnmum postve egenvalues the potental for the voltage collapse can be predcted. In addton, by usng the partcpaton matrx, the weakest buses and lnes that have the maxmum contrbuton on crtcal egenvalues can be determned..3. L mn Index By usng the concept of the transmsson power, the lne
Amercan Journal of Energy and Power Engneerng 017; 4(5): 6-1 8 stablty ndex L mn s defned. The voltage quadratc equatons are stable f the dscrmnant of them s set to be greater or equal than zero. If the dscrmnant s smaller than zero, the roots wll be magnary, whch means that cause voltage nstablty n the system. The sngle lne of an nterconnected network s shown n Fgure 1. X X LQP = 4 P Q + j V V Where P s the actve power flow at the sendng bus. To mantan the voltage stablty of the power system the ndex must be less than unty. (7).6. LCPI Index By consderng the lne chargng reactance and relatve drecton of actve power flow n the lne wth respect to reactve power flow; an mproved Lne Collapse Proxmty Index (LCPI) s developed [11]. The exact model of a transmsson lne s usually descrbed by two port equvalent crcut usng ABCD matrx as: Fgure 1. Sngle-lne dagram of a transmsson lne. Based on the transmsson lne model shown n Fgure 1, the L mn ndex s defned as [7]: 4X Q j L mn = (5) V θ δ [ sn( )] Where, X s the lne reactance, Q j s the reactve power flow at the recevng bus; V s the voltage of sendng bus; θ s the lne mpedance angle, and δ s the voltage angle dfference between sendng and recevng buses. When the value of L mn for a lne s closed to 1, that lne s closer to be nstable. To mantan a secure condton, the L mn ndex should be less than 1..4. FVSI Index The lne stablty ndex FVSI s derved based on the current equaton n a transmsson lne. Ths ndex s defned as below [8]: 4 Z Q j FVSI = (6) V X Where Z s the lne mpedance. When FVSI ndex of a lne goes beyond 1, one of the buses connected to ths lne wll experence a sgnfcant voltage drop leadng to system voltage collapse..5. LQP Index The ndex LQP s defned to evaluate the voltage stablty n the transmsson lnes and obtaned as follows [4]: V A B V j = I C D I j Where A, B, C, and D are known as the transmsson parameters of the two-port network, and they can be expressed as: (8) ZY A = (1 + ) (9) B = Z (10) ZY C = Y (1 + ) (11) 4 D = A (1) Wth respect to the above equatons the ndex LCPI s defned as: 4A cos α( PB cos β + Q sn β) j j LCPI = (13) ( V cos δ ) Where a, and b are the phase angles of parameters A and B respectvely. To provde a secure condton, the value of LCPI ndex should be mantaned less than 1. 3. Smulaton Results The voltage stablty analyss s performed on IEEE 14 bus test system. The sngle lne dagram of the test system s shown n the Fgure, and ts parameters can be found n [1].
9 Rahmat Allah Morad et al.: Performance Evaluaton of the Voltage Stablty Indces n the Real Condtons of Power System Fgure. Sngle lne dagram of the IEEE 14 bus test system. At frst by usng the P-V and Q-V curves the loadng margn of the test system s obtaned. For ths purpose, by consderng the constant power factor for each bus and ncreasng the loads, the P-V curve s obtaned as Fgure 3. As seen, as the power system load s gradually ncreased, the buses voltage are decreased such that when the load factor s ncreased more than 1.779, the system experenced voltage nstablty. Fgure 3. P-V curves for IEEE 14 bus test system. By the Q-V curves the maxmum reactve power can be added to load buses before the system suffers a voltage collapse are obtaned, and the results are shown n Fgure 4. As seen from Fgure 4, the buses 1 and 14 have the lowest margn of reactve
Amercan Journal of Energy and Power Engneerng 017; 4(5): 6-1 10 power, whch ndcates that they are the most crtcal buses n the test system. Fgure 4. Margn of reactve power n IEEE 14 bus test system. To evaluate these results by usng the modal analyss method, the voltage stablty of the test system s analyzed at two operatonal Cases 1 &. In Case 1 and Case the loadng factor s consdered equal to 1.5 and 1.779 respectvely (see Fgure 3). The egenvalues for these cases are obtaned and the results are shown n Table 1. Table 1. Egenvalues of the IEEE 14 bus test system n operatonal Cases 1 &. Case 1 Case λ 1=59.13 λ 1=54.7 λ =36.97 λ =35.88 λ 3=19.98 λ 3=19.07 λ 4=17.67 λ 4=17.46 λ 5=14.18 λ 5=1.37 λ 6=.489 λ 6=.3 λ 7=10.6 λ 7=8.94 λ 8=5.15 λ 8=5.03 λ 9=7.099 λ 9=6.97 As seen, n these cases, all the egenvalues are postve and system s voltage stable. The results show that as the system approaches to nstablty (ncreasng the load factor) the magntude of egenvalues are decreased. At the crtcal operatonal case (Case ) the smallest egenvalue s equal to.3. The bus partcpaton factor for ths mode s shown n Fgure 5. The results show that bus 14 has the most partcpaton n ths egenvalue and accordngly bus 14 s the weakest bus on the test system. As mentoned before, the ndces L mn, FVSI, LQP and LCPI are used to detect the weak lnes of a power system. In the prevous studes to evaluate the performance of these ndces, by addng a bg reactve load to the weakest buses, the weakest lnes are detected. However, n the real condtons of a power system, the both actve and reactve loads are vared. Therefore, only consderng the varaton of reactve loads may be led to unrealstc results of system voltage stablty. In ths secton, at frst by ncreasng only reactve loads n weakest buses (buses 10, 11 and 14) the voltage stablty of system lnes are nvestgated and the results are shown n Table. The ndces results are just shown for two lnes that have the maxmum values. For example, the results show that when a reactve load as 0.73pu s added to bus 14, the lnes 9-14 and 13-14 are the weakest lnes of the test system. Fgure 5. Bus partcpaton factors of IEEE 14 bus test system at the crtcal egenvalue for Case.
11 Rahmat Allah Morad et al.: Performance Evaluaton of the Voltage Stablty Indces n the Real Condtons of Power System Table. Lne voltage stablty ndces for ncreasng only reactve loads n weakest buses of IEEE 14 bus test system. Load (p.u.) Q 10=0.95 Q 11=0.86 Q 14=0.73 Lne Voltage Stablty Indces L mn FVSI LQP LCPI 9-10 0.606 0.598 0.557 0.606 10-11 0.5801 0.5698 0.4840 0.5309 6-11 0.7835 0.8473 0.6846 0.7187 10-11 0.6408 0.688 0.5819 0.5577 9-14 0.8375 0.8867 0.7073 0.8004 13-14 0.7895 0.8341 0.6560 0.7479 To verfy the obtaned results of these ndces, by modal analyss the egenvalues of the IEEE 14 bus test system are obtaned and the partcpaton of lnes at the lower egenvalues are shown n Table 3. As seen when a reactve load s added at bus 14, the lnes 9-14 and 13-14 have the maxmum partcpaton n lower egenvalue (λ 3 =0.58). In the followng, the lne voltage stablty are assessed for two real condtons that defned before as Case 1 (ncreasng both of actve and reactve loads n all the system buses wth loadng factor equal to 1.5) and Case (ncreasng both of actve and reactve loads n all the system buses wth loadng factor equal to 1.779). The voltage stablty ndces L mn, FVSI, LQP and LCPI are obtaned for above consdered condtons and the results are shown n Table 4. Table 3. Branch partcpatons of IEEE 14 bus test system at crtcal egenvalue when only reactve loads ncreased. Addng reactve load to bus 10 as Addng reactve load to bus 11 as Addng reactve load to bus 14 as Base Condton Q 10=0.95p.u Q 11=0.86p.u Q 14=0.73p.u Crtcal egenvalues s λ 7=.71 Crtcal egenvalues s λ 5=0.3385 Crtcal egenvalues s λ 6=0.4945 Crtcal egenvalues s λ 3=0.58 Lne P Lj Lne P Lj Lne P Lj Lne P Lj 7-8 1 9-10 1 6-11 1 9-14 1 7-9 0.84 7-9 0.9 10-11 0.17 13-14 0.94 1-5 0.59 10-11 0.76 9-10 0.13 7-9 0.4 Table 4. Lne voltage stablty ndces for smultaneous ncreasng of actve and reactve loads n the IEEE 14 bus test system for two condtons load factor 1.5 (Case 1) and load factor 1.779 (Case ). Lne Case 1 Case L mn FVSI LQP LCPI L mn FVSI LQP LCPI 1-0.01 0.011 0.083 0.16 0.53 0.17 0.058 0.434 1-5 0.054 0.045 0.170 0.66 0.41 0.308 0.075 0.617-3 0.09 0.05 0.198 0.171 0.07 0.053 0.364 0.370-4 0.006 0.005 0.085 0.177 0.164 0.133 0.076 0.406-5 0.04 0.039 0.007 0.166 0.04 0.179 0.060 0.366 3-4 0.063 0.067 0.041 0.046 0.073 0.079 0.015 0.14 4-5 0.017 0.017 0.009 0.037 0.034 0.035 0.054 0.18 4-7 0.060 0.060 0.094 0.060 0.108 0.105 0.001 0.108 4-9 0.033 0.033 0.044 0.033 0.68 0.51 0.03 0.68 5-6 0.10 0.099 0.00 0.10 0.96 0.7 0.064 0.96 6-11 0.063 0.06 0.048 0.091 0.14 0.10 0.091 0.175 6-1 0.048 0.047 0.035 0.093 0.10 0.097 0.066 0.185 6-13 0.073 0.071 0.05 0.14 0.153 0.144 0.097 0.44 7-9 0.065 0.065 0.057 0.065 0.180 0.179 0.150 0.180 9-10 0.01 0.01 0.018 0.08 0.046 0.046 0.039 0.061 9-14 0.06 0.060 0.043 0.10 0.146 0.136 0.086 0.56 10-11 0.036 0.036 0.031 0.050 0.07 0.073 0.063 0.101 1-13 0.04 0.04 0.011 0.033 0.05 0.051 0.03 0.070 13-14 0.060 0.059 0.043 0.106 0.17 0.11 0.08 0.16 As seen, by consderng the real condton of power system and ncreasng the both actve and reactve loads, the values of almost lne voltage stablty ndces are ncreased. Although the results of some ndces n detectng the weakest lnes are not dentcal. For nstance, n Case whereas the ndces Lmn, FVSI and LCPI show that the lnes 1-5 s the weakest lne n real condtons, but by the ndex LQP the lne -3 s detected as a weakest lne. In addton, the value of ndex LCPI for the weakest lnes of the test system s notceably greater than other ndces. To evaluate these results by the modal analyss method the branch partcpatons at the crtcal egenvalue of Case are obtaned and the results are shown n Table 5. The results show that at ths crtcal egenvalue, the lne 1-5 has the maxmum partcpaton and so that the ndces L mn, FVSI, and LCPI truly detect the weakest lne of the test system. In addton, snce Case s near to voltage nstablty, so the hgh values of ndex LCPI shows that ths ndex can more accurately predct the voltage stablty of the power system lnes. Whle that when only the ncreasng of reactve power loads s consdered all the voltage stablty ndces have the smlar predcton of weakest lnes on the power system (see Table ).
Amercan Journal of Energy and Power Engneerng 017; 4(5): 6-1 1 Table 5. Branch partcpatons of the IEEE 14 bus test system at crtcal egenvalue for smultaneous ncreasng of actve and reactve loads for load factor 1.779 (Case ). Crtcal egenvalue s λ 6=.3 Lne P Lj 1-5 1 7-9 0.31 5-6 0.5 4-9 0.4-4 0. 4. Concluson In ths paper, the performance of lne voltage stablty ndces are evaluated n dfferent condtons. The results show that by addng only reactve loads to weakest buses the ndces predcton of weakest lnes are smlar. However, when the evaluaton s done on real condtons of power system as the both actve and reactve loads are ncreased n the all of the system buses, the ndces predcton of weakest lnes are not smlar. The analytcal results from modal analyss method show that the results obtaned from ndex LCPI are more accurate than other ndces. References [1] M. V. Suganyadev, C. K. Babulal, Fast Assessment of Voltage Stablty Margn of a Power System, Journal of Electrcal Systems, 014, Vol. 10, No. 3, pp. 305-316. [] M. R. Aghamohammad, S. S. Hashem and M. S. Ghazzadeh, A Novel Index for Onlne Voltage Stablty Assessment Based on Correlaton Characterstc of Voltage Profles, Iranan Journal of Electrcal & Electronc Engneerng, June 011, Vol. 7, No., pp. 131-140. [3] P. Kessel, H. Glavtsch, Estmatng the Voltage Stablty of a Power System, IEEE Transactons on Power Delvery, 1986, Vol. PWRD-1, No. 3, pp. 346-354. [4] A. Mohamed, G. B. Jasmon, S. Yusoff, A Statc Voltage Collapse Indcator usng Lne Stablty Factors, Journal of Industral Technology, 1989, Vol. 7, No. 1, pp. 73-85. [5] B. Gao, G. K. Morson, P. Kundur, Voltage Stablty Evaluaton Usng Modal Analyss, IEEE Transactons on Power Systems, 199, Vol. 7, No. 4, pp. 159-154. [6] M. Moghavvem and F. M. Omar, Technque for Contngency Montorng and Voltage Collapse Predcton, IEE Proceedngs onlne, 1998, Vol. 145, No. 6, pp. 634-640. [7] I. Musrn, T. K. A. Rahman, Novel Fast Voltage Stablty Index (FVSI) for Voltage Stablty Analyss n Power Transmsson System, Student Conference on Research and Development Proceedngs, Shah Alam, Malasa, July 00, pp. 65-68. [8] C. Res and F. P. Macel Barbosa, A Comparson of Voltage Stablty Indces, IEEE MELECON, Benalmádena (Málaga), Span, 006, pp. 1007-1010. [9] C. Res, A. Andrade and F. P. Macel Barbosa, Lne Stablty Indces for Voltage Collapse Predcton, Internatonal Conference on Power Engneerng, Energy and Electrcal Drves (POWERENG '09), 009, Lsbon, Portugal, March 18-0, 009, pp. 39-43. [10] R. Twar, K. R. Naz, V. Gupta, Lne Collapse Proxmty Index for Predcton of Voltage Collapse n Power Systems, Electrcal Power and Energy Systems, 01, Vol. 41, pp. 105 111. [11] N. A. M. Ismal, A. A. M. Zn, A. Kharuddn, S. Khokhar, A Comparson of Voltage Stablty Indces, 014 IEEE 8th Internatonal Power Engneerng and Optmzaton Conference (PEOCO014), Langkaw, the Jewel of Kedah, Malaysa, March 014, pp. 30-34. [1] T. Datta, P. Nagendra, S. Haldernee, D. S. Paul, Voltage Stablty Assessment of a Power System Incorporatng FACTS n Equvalent Mode, Journal of Electrcal Systems, 013, Vol. 9, No. 4, pp. 440-45. [13] R. Maharjan, S. Kamalasadan, Voltage Stablty Index for Onlne Voltage Stablty Assessment, 015 North Amercan Power Symposum (NAPS), Oct 015, pp. 1-6. [14] G. Nannapanen, T. Medallel Masaud and R. Challoo, A Comprehensve Analyss of Voltage Stablty Indces n the Presence of Dstrbuted Generaton, IEEE Conference on Technologes for Sustanablty (SusTech), July-Aug 015, pp. 96-10. [15] A. Oukennou and A. Sandal, Assessment and analyss of Voltage Stablty Indces n electrcal network usng PSAT Software, Eghteenth Internatonal Mddle East Power Systems Conference (MEPCON), Dec 016, pp. 1-6. [16] K. Chakraborty, G. Deb and S. Deb, Voltage stablty assessment n radal dstrbuton system by lne stablty ndcator (LSI) and ts mprovement usng SVC, IEEE Internatonal Conference on Power Electroncs, Intellgent Control and Energy Systems (ICPEICES), July 016, pp. 1-5. [17] T. Arya Vshnu Ram and K. M. Haneesh, Voltage stablty analyss usng L-ndex under varous transformer tap changer settngs, 016 Internatonal Conference on Crcut, Power and Computng Technologes (ICCPCT), March 016, pp. 1-4. [18] I. Adebayo, A. A. Jmoh and A. Yusuff, Voltage stablty assessment and dentfcaton of mportant nodes n power transmsson network through network response structural characterstcs, IET Generaton, Transmsson & Dstrbuton, 017, Vol. 11, No. 6, pp. 1398 1408. [19] C. Canzares, Voltage Stablty Assessment: Concepts, Practces and Tools, IEEE/PES Power System Stablty Subcommttee Specal Publcaton, August 00. [0] M. Heydarpour and A. Akbar Foroud, A New Framework for Congeston Management wth Exact Modelng of Impactng Factors, Iranan Journal of Electrcal & Electronc Engneerng, Dec. 01 Vol. 8, No. 4, pp. 39-340. [1] Power System Test Archve-UWEE, Unversty of Washngton. <http: //www.ee.washngton.edu/research/pstca>.