Static Voltage Stability Assessment Using Probabilistic Power Flow to Determine the Critical PQ Buses

Transcription

1 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 Statc Voltage Stablty Assessment Usng Probablstc Power Flow to Determne e Crtcal PQ Buses Farkhonde Jabar, Behnam Mohammad Ivatloo - Department of Electrcal and Computer Engneerng, Unversty of Tabrz, Tabrz, Iran Emal: farkhonde.jabar@gmal.com - Department of Electrcal and Computer Engneerng, Unversty of Tabrz, Tabrz, Iran Emal: mohammad@eee.org eceved: Aprl 4 evsed: July 4 Accepted: ovember 4 ABSTACT owadays, due to ncreased consumpton and operaton of electrc power systems close to er stablty boundares, power systems may become unstable durng severe dsturbances. So, t s very mportant to determne e stablty margn under dfferent condtons. In s paper, e statc voltage stablty of an nterconnected power system consderng load and generaton uncertantes s evaluated usng probablstc power flow. The Monte Carlo Smulaton meod s used to generate e probablstc power flow scenaros. Then, e expected statc voltage stablty ndex and probablty of stablty for all of e PQ buses are obtaned. Also, e standard devatons of e stablty ndexes are calculated. Fnally, e crtcal PQ nodes are determned under gven dsturbance. The study has been carred out on 39-Bus ew England and 8-bus IEEE test systems and results are presented. KEYWODS: Load Generaton Uncertanty, Monte Carlo Smulaton (MCS), Probablstc Power Flow (PPF), Statc Voltage Stablty (SVS).. ITODUCTIO Accordng to e defnton, power system stablty s e ablty of system to mantan an equlbrum state n e system under normal operatng condton and after a dsturbance. An nterconnected power system may become unstable due to several reasons such as earquakes, human operaton errors, control system falures, protecton system falures, etc. owadays, due to ncreased consumpton and operaton of electrc power systems close to er stablty boundares, power systems may become unstable durng severe dsturbances. So, t s very mportant to determne e stablty margn under dfferent condtons. In recent years, many meods have been proposed to assess e stablty of a large-scale power system. The stablty analyss n a power system s a complex ssue and ere are many approaches to solve s problem. These procedures can be dvded nto ree general categores []: a. Statc voltage stablty analyss b. Small dsturbance analyss c. Dynamc voltage stablty analyss Statc voltage stablty analyss s based on e soluton of conventonal power flow equatons and small dsturbance meods are based on lnearzed system dfferental equatons and dynamc voltage stablty approaches try to fnd why and how e voltage collapse has occurred. Because of nonlnear nature of dfferental equatons, t s not possble to determne e exact dstance to voltage collapse n dynamc meods. So, statc meods are used to evaluate e voltage securty margn. Statc voltage stablty of a power system has been studed n [] usng contnuaton power flow consderng e effect of contngences on Mega Watt Margn (MWM) and loadng pont. Modal based analyss and ts applcaton n e evaluaton of voltage stablty of bulk power system s reported n [3]. Ths meod makes use of e power system Jacoban matrx to determne e egenvalues necessary for e evaluaton of e voltage stablty of e power system. Ths meod was used to determne e components of e system at contrbute to nstablty rough e use of e partcpatng factors. A crteron for fndng e weakest bus n e system by usng Artfcal Immune System (AIS) clonal selecton algorm whch s supported by evaluatng egenvalues and er partcpaton factors s ntroduced n [4]. In [5], an Enhanced adal Bass Functon eural etwork (EBF) and Wnner-Take-All eural etwork (WTA) have been proposed to examne wheer e power system s secure under steady-state operatng 7

2 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 condtons. The voltage stablty assessment usng mxed statc and dynamc technques s dscussed n [6]. Also, several oer meods are provded to assess e dynamc voltage stablty and small dsturbance stablty. In recent years, many meods have been proposed to mplement e probablstc power flow. In [7], e use of Gaussan mxture models to represent non-gaussan correlated nput varables, such as wnd power output or aggregated load demands n e probablstc load flow s proposed. The man advantage of e Gaussan components meod s at e probablty densty functons of any varable s drectly obtaned. In [8], a probablstc load flow meod s dscussed at s based on e ataf transformaton and e Latn Hypercube Samplng. The man advantage of e proposed meod s at hgh accurate soluton can be obtaned w less computaton. A probablstc load flow meod based on polynomal normal transformaton (PT) and Latn hypercube samplng (LHS) s proposed n [9]. The correlaton between nput random varables has been taken nto consderaton. The proposed meod uses e statstcal moments and correlaton matrx of nput random varables nstead of er margnal dstrbuton functons and jont dstrbuton functons, whch are very dffcult to be obtaned, to establsh er probablty dstrbuton models by PT and LHS. The statstcal moments and probablty dstrbuton functons of node voltage and lne flow are calculated by Monte Carlo smulaton meod. As mentoned, t s very mportant to evaluate e statc voltage stablty of an nterconnected power system under dfferent condtons. So, s paper ntroduces a new meod based on modal analyss to determne e stablty ndex and probablty of statc voltage stablty for all of PQ buses n a large-scale power system consderng load and generaton uncertantes and probablty of power system condton. senstvty of ncremental change n bus real power njecton and ncremental change n bus voltage magntude, J Q s senstvty of ncremental change n bus reactve power njecton and ncremental change n bus voltage angle, J s senstvty of ncremental QV change n bus reactve power njecton and ncremental change n bus voltage magntude. Also, Jacoban matrx s defned as follows: JP J JQ JPV JQV () Jacoban matrx elements are obtaned from e followng relatons. The dagonal elements of s matrx can be obtaned as follows: P JP Q BV P JPV P GV V Q JQ P GV Q JQV Q BV V (3) (4) (5) (6) The non-dagonal elements of Jacoban matrx can be calculated as follows: P J P sn( ) cos( ) V k V k Gk k Bk k k P J PV V cos( ) sn( ) k V k Gk k Bk k V k (7) (8). VOLTAGE STABILITY EVALUATIO.. Determnstc ewton-aphson Power Flow In ewton-aphson approach, e lnearzed steady state system power flow equatons can be wrtten as follows []: Q J Q cos( ) sn( ) V k V k Gk k Bk k k Q J Q sn( ) cos( ) V V k V k Gk k Bk k V k (9) () JP JPV P JQ J QV V Q () In s equaton, P s ncremental change n bus real power njecton and Q s ncremental change n bus reactve power njecton, s ncremental change n bus voltage angle, and V s ncremental change n bus voltage magntude, J P s senstvty of ncremental change n bus real power njecton and ncremental change n bus voltage angle, J s PV.. Load Generaton Uncertantes Power flow analyss s used on mportant problems n power system operaton and plannng, such as evaluaton of system relablty and rsk management n electrcty market. So, t s very mportant to consder e power system uncertantes under dfferent operatng condtons. In a real power system, loads and generatons are uncertan. So n s paper, normal probablty dstrbuton functons are used to model e load and generaton uncertantes. 8

3 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4.3. Statc Voltage Stablty Based on Modal Analyss Statc voltage stablty s affected by bo actve and reactve power n an nterconnected power system. In order to reduce e computatonal burden, at each operatng pont actve power can be assumed constant. So, voltage stablty wll be evaluated by consderng ncremental relatonshp between Q and V []. Based on e above consderatons, e relaton between ncremental change n bus reactve power njecton and ncremental change n bus voltage magntude can be rewrtten as follows: Q J V () J J J J J () QV Q P PV ecent matrx s called e reduced Jacoban matrx whch drectly relates e bus voltage magntude and bus reactve power njecton as follow: V J Q (3) In (3), dagonal element of reduced Jacoban matrx s Q-V senstvty at bus. Because of nonlnear nature of Q-V relatonshps, e magntudes of e senstvtes do not provde a drect measure of e relatve degree of stablty. Therefore, e egenvalues of e reduced Jacoban matrx are used to determne e voltage stablty ndex. So, we have: J (4) In (4), s rght egenvector matrx of matrx J s dagonal egenvalues of matrx egenvector matrx of J J. J,, s left (5) V Q (6) V Q v q (7) - (8) V Q (9) V () Q () v q () In () and (), v and q are e vector of modal voltage varaton and e vector of modal reactve power varaton, respectvely. In (), s e egenvalue. If, e modal voltage and e modal reactve power varaton are along e same drecton and e system s stable. If, e modal voltage and e modal reactve power varaton are along opposte drecton and e system s unstable. If, e modal voltage collapses. 3. POPOSED POBABILISTIC METHOD 3.. Probablty of Power System Condton In an nterconnected power system, falure of equpments such as power transformers, transmsson lnes and generaton unts leads to protecton system operaton and wll affect e power system stablty. So, t s consdered e probablty of equpment outage as anoer factor to determne e probablstc nature of e statc voltage stablty ndex. Therefore, e probablty of power system condton s obtaned as follows: s s PC p ( outage ) p( outage ) (3) Where, P C s e probablty of power system condton and p( outage ) s e probablty of outage and s shows e number of smultaneous outages. Accordng to e recent relaton, e probablty of power system equpment smultaneous outages s smaller an e probablty of each element outage. In oer words, we have: s (4) P p( outage ) p( outage ) j,,...,s C j 3.. Statc Voltage Stablty Assessment The statc voltage stablty of an nterconnected power system s evaluated consderng e load and generaton uncertantes and e probablty of power system condton accordng to e followng steps:. = Enter e number of Monte Carlo Smulatons.. Determne power system condton and calculate P C. 3. k= 4. Generate U ( k ) rand(, k ) for,,...,, L L refers to e number of load buses. 5. Generate U ( k ) rand( j, k ) for j,,...,, G Gj refers to e number of generaton buses. 6. Calculate: L G 9

4 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 P ( k ) InverseCDF(' ormal ', U ( k ),, ) L L P L P L Q ( k ) InverseCDF(' ormal ', U ( k ),, ) L L Q Q L L P ( k ) InverseCDF(' ormal ', U ( k ),, ) G G P G P G j j j j Q ( k ) InverseCDF(' ormal ', U ( k ),, ) G G QG QG j j j 7. Update system topology and run probablstc power flow for k scenaro. 8. Modal analyss to obtan ( k n ) for n,,..., m, m shows e number of e PQ buses. 9. If real ( ( k )), calculate e Stablty Index for n n PQ bus n k scenaro. PC SI ( n, k ) for n,,..., m ( k ) (5) n. If k, go step. Else, k k and go step 4.. Calculate e expected values of stablty ndexes n scenaros: ESI ( n) SI ( n, k ) for n,,..., m (6) k. Obtan e standard devatons of stablty ndexes as follows: SD( n) SI (n, k ) - ESI (n) for n,,..., m (7) k 3. Determne Bnary Varables for all of e PQ buses n each scenaro: 4. f real( n, k ) BV ( n, k ) for n,,..., m f real( n, k ) j (8) 5. Calculate probablty of stablty as follow: SP (n) BV ( n, k ) for n,,..., m (9) k 3.3 Determnaton of Crtcal PQ Buses After evaluaton of power system stablty under normal operatng condton and after occurrng contngences, e crtcal PQ buses can be determned accordng to e followng ndex: ESI ( ) Under Contngences CBI ( )= for,,..., m ESI ( ) Under ormal Operaton (3) CBI () refers to e Crtcal Bus Index. The Crtcal Bus Index s between and. Whatever CBI () be larger, PQ bus wll be non-crtcal and whatever CBI () be close to zero, CBI () s zero, PQ bus wll be crtcal. If PQ bus wll collapse. 4. SIMULATIO ESULTS 4. Smulaton on 39-bus ew England test system The study has been carred out on e ew England 39- bus and 46- lne power system []. At frst, t s assumed at e power system s under normal operatng condton and e number of Monte Carlo Smulatons s and loads and generatons are uncertan. The probablty of normal condton s consdered.9. The standard devatons of load and generaton under normal operaton are assumed.99, e amounts of e loads and e generatons are consdered as e expected values of loads and generatons. The test system s shown n Fg.. Fg.. ew England 39-bus test power system

5 SD SI SD Voltage magntude n p.u SD Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 Voltage profle for tmes Monte Carlo Smulatons s shown n Fg.. The stablty ndex and e standard devatons of e stablty ndexes under normal operaton are shown n Fgs. 3 and 4, respectvely. The smulaton results shows at n an nterconnected power system under normal operatng condton, all of e egenvalues are postve and t s satsfed e statc voltage stablty constrant consderng load and generaton uncertantes. So, e probablty of stablty for all of e PQ buses s equal to umber of buses Expected of voltage magntude n p.u Voltage magntude for tmes MCS n p.u Fg.. Voltage profle under normal condton Stablty ndex n tmes MCS Expected of stablty ndex n tmes MCS Standard devaton of stablty ndex Fg. 4. Standard devaton of stablty ndex under normal operaton In order to assess e stablty of system under contngency, branch 8 to 9 and generator 34 have been out of e system. The number of Monte Carlo Smulatons s. Also, e probabltes of branch and generator outages are assumed. and.5, respectvely. The standard devatons of load and generaton under gven contngences are assumed.99. The amounts of loads and generatons are consdered as e expected values of loads and generatons. The smulaton results are shown n Fgs. 5 to x Stablty ndex n tmes MCS Expected of stablty ndex n tmes MCS Fg. 3. Stablty ndexes for scenaros under normal operaton Fg. 5. Stablty ndexes for scenaros under gven contngences 8 x -3 Standard devaton of stablty ndex Fg. 6. Standard devaton of stablty ndexes under gven contngences

6 CBI Probablty Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December Probablty of stablty As shown n Fgs. 5 and 6, 5, 4, 7, 8 PQ buses have e maxmum standard devatons, respectvely. Accordng to e Fgs. 7 and 8, 5 and 6 PQ buses wll be e crtcal PQ nodes w probablty of stablty.54 and.5, respectvely Fg. 7. Probablty of stablty under gven contngences Crtcal Bus Index 4. Smulaton on 8 bus IEEE test system Also, e study has been carred out on 8 buses IEEE [3]. At frst, t s assumed at e power system s under normal operatng condton and e number of Monte Carlo Smulatons s and loads and generatons are uncertan. The probablty of normal condton s consdered.87. The standard devatons of load and generaton under normal operaton are consdered.99. The amounts of loads and generatons are assumed as e expected values of loads and generatons. The test system s shown n Fg Fg. 8. Crtcal Bus Indexes under gven contngences Fg bus IEEE test power system

7 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 The smulaton results under normal operaton are shown n Fgs. to. Expected of voltage magntude n p.u Voltage magntude for tmes MCS n p.u umber of buses Fg.. Voltage profle under normal condton.7.6 Stablty ndex for tmes MCS.5 Expected of stablty ndex for tmes MCS SI.4 Voltage magntude n p.u Voltage magntude n p.u.5 In order to evaluate e stablty of system under contngency, branch 35 to 37 and generator 8 have been out of e system. The number of Monte Carlo Smulatons s. The probabltes of branch and generator outages are assumed. and.5, respectvely. The standard devatons of load and generaton under gven contngences are assumed.99. The amounts of loads and generatons are consdered as e expected values of loads and generatons. The smulaton results are shown n Fgs. 3 to 6. Expected of voltage magntude n p.u. Voltage magntude for tmes MCS n p.u umber of buses. Fg. 3. Voltage profle under gven outages x 6 Fg.. Stablty ndexes under normal condton Stablty ndex for tmes MCS Standard devaton of stablty ndex 5 Expected of stablty ndex for tmes MCS.5 SI SD Fg. 4. Stablty ndexes under gven outages Fg.. Standard devaton of stablty ndex under normal condton 3

8 CBI SD Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4.5 x -3 Standard devaton of stablty ndex Fg. 5. Standard devaton of stablty ndexes under gven outages Crtcal Bus Index Fg. 6. Crtcal Bus Indexes under gven outages As shown n Fgs. 4 and 5, 38 to 48 and 85 PQ buses have e maxmum standard devatons, respectvely. Accordng to e Fg. 6, 39 and 45 and 48 PQ buses wll be e crtcal PQ nodes w probablty of stablty. 5. COCLUSIO In s paper, e statc voltage stablty of an nterconnected power system consderng load and generaton uncertantes s evaluated usng probablstc power flow. The Monte Carlo Smulaton meod s used to generate e probablstc power flow scenaros. Then, e expected statc voltage stablty ndex and probablty of stablty for all of e PQ buses are obtaned. Also, e standard devatons of e stablty ndexes are calculated. Fnally, e crtcal PQ nodes are determned under gven dsturbance. EFEECES [] P., Kundur, Power System Stablty and Control, McGraw-Hll, Inc., ew York, 994. [] M. Alnezhad, M. Ahmad Kamarposht, Statc Voltage Stablty Assessment Consderng e Power System Contngences usng Contnuaton Power Flow Meod, IJEEE, Volume 3, Issue, pp. 36-3, Feb.. [3] F.O., Enemuoh, J.C., Onuegbu and E.A., Anaza, Modal Based Analyss and Evaluaton of Voltage Stablty of Bulk Power System, IJED, Volume 6, Issue, pp. 7-79, May 3. [4] S. Shrsha, P. L. V. Prasanna,. S. Mallkarjuna ao Evaluaton of Modal Analyss for Voltage Stablty usng Artfcal Immune System, IJCA Publshed by Foundaton of Computer Scence, Vol 46, o. 9, pp. 6-, May. [5] D., Seyed Javan, H., ajab Mashhad and M., ouhan, A Fast Statc Securty Assessment Meod Based on adal Bass Functon eural etworks usng Enhanced Clusterng, Internatonal journal of Electrcal Power and Energy Systems, Vol.44, Issue., pp , Jan. 3. [6] M., Hasan, M., Parnan, Meod of Combned Statc and Dynamc Analyss of Voltage Collapse n Voltage Stablty Assessment, IEEE/PES Transmsson and Dstrbuton Conference and Exhbton, Asa and Pacfc, pp. -6, 5. [7] G., Valverde, A.T., Sarc and T., Terzja, Probablstc Load Flow w on-gaussan Correlated andom Varables usng Gaussan Mxture Models, IET, Generaton, Transmsson and Dstrbuton, Vol. 6, Issue.7, pp. 7-79, July. [8] Y., Chen, J., Wen and S., Cheng, Probablstc Load Flow Meod Based on ataf Transformaton and Latn Hypercube Samplng, IEEE Transactons on Sustanable Energy, Vol.4, Issue, pp. 94-3, Aprl 3. [9] D., Ca, D., Sh and J., Chen, Probablstc Load Flow Computaton w Polynomal ormal Transformaton and Latn Hypercube Samplng, IET Generaton, Transmsson and Dstrbuton, Vol.7, Issue 5, pp , May 3. [] D.P., Koar, J.S., Dhllon. Power System Optmzaton, Prentce-Hall of Inda, nd Edton, 4. [] B., Borkowska, Probablstc Load Flow, IEEE Transactons on Power Apparatus and Systems, Vol. PAS-93, o. 3, pp , May 973. [] J.H., Chow, K.W., Cheung, A Toolbox for Power System Dynamcs and Control Engneerng Educaton and esearch, IEEE Transactons on Power Systems 7, Issue 4, pp , 99. [3] Power Systems Test Case Archve: 8bus.htm. 4

9 Majles Journal of Electrcal Engneerng Vol. 8, o. 4, December 4 [4] S., Hashem, M.., Aghamohammad, Wavelet Based Feature Extracton of Voltage Profle for Onlne Voltage Stablty Assessment Usng BF eural etwork, Internatonal Journal of Electrcal Power and Energy Systems Vol. 49, pp , July 3. [5] O.P., ah, A.K., Yadav, H., Malk and A., Azeem, Bhupesh Kr, Power System Voltage Stablty Assessment rough Artfcal eural etwork, Internatonal Conference on Communcaton Technology and System Desgn, Vol. 3, pp. 53-6,. 5