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Ranking algorithm is a search engine to index a list of its evaluation and ranking rules. Ranking algorithm to determine which results are relevant to a particular query.
Ranking algorithm is a search engine to index a list of its evaluation and ranking rules. Ranking algorithm to determine which results are relevant to a particular query.
Journal of ELECTRICAL ENGINEERING, VOL. 57, NO. 2, 2006, 116–119 COMMUNICATIONS AN EFFICIENT ALGORITHM FOR CONTINGENCY RANKING BASED ON REACTIVE COMPENSATION INDEX ∗ ∗∗ Kaanan Nithiyananthan — Neelamegam Manoharan ∗∗∗ — Velimuthu Ramachandran This paper describes a methodology for line outage ranking based on reactive compensation index (RCI). RCI has been used extensively at BC Hydro, Burby, Canada for assessing the voltage stability. A methodology has been developed to evaluate quickly RCI using the π section model for active power ﬂow and compensation injection theorem, RCI is speciﬁcally calculated, using only the results of base case load ﬂow, under each line outage condition. RCI approximates the distance between pre-contingency nose and line outage case nose by the total reactive injection required at the load buses to establish similar voltage levels for the two cases. Hence RCI has been used as severity index for ranking the line outages. Ranking results have been obtained for a sample test system and compared with those obtained using Performance Index method. K e y w o r d s: voltage stability; proximity indicators; ranking 1 NOTATION is a trend to utilize the severity index, which approxi- mates the distance of voltage collapse under line outage k, m buses to which j th line (outage)is connected condition [3, 4]. The power industry is facing an acute T Pkm Transmitted part of real power ﬂow shortage of reactive power due to a large demand for re- L Pk Loss part of real power ﬂow active power. This gives rise to a sharp decline in the rj , Xj Resistance and Reactance of j th line voltage and leads to a voltage collapse. In such a situa- ∆Pk c Compensation injection applied at k th bus tion the voltage magnitude alone is not a true indicator of ∆Pkm T T Change in Pkm due to compensation injection at the health of the power network. This has necessitated the th th k and m buses use of a voltage stability margin for ranking the line out- T ∆δnj Change in nth bus phase angle for the outage of age . It has been further stressed that for maintaining a j th line due to transmitted part of real power good voltage proﬁle the reactive power transfer must be ∆PkLc Compensation injection for outage simulation minimized. It will be quite straightforward to rank the ∆PkL Change in loss part of real power at k th bus line outage based on the maximum loadability criteria. bj Susceptance of j th line But the time required to calculate the loadability margin under each contingent condition will be long and the ap- gj Conductance of j th line plicability of the ranking list may be in doubt due to a ∆δrs,j Change in phase angle diﬀerence across ith line change in the loading pattern. Hence, the use of indica- for the outage of j th line tors or test functions, which shows indirectly the severity ∆qrs,j Change in reactive power ﬂow in ith line for the of line outages, is made for voltage security assessment. outage of j th line Jasman and Lee  were probably the ﬁrst to develop a contingency ranking algorithm based on a voltage sta- 2 INTRODUCTION bility criterion. This method is mainly applicable to the distribution system. Moreover, to develop the stability Line outage ranking is an integral part of on-line se- criterion the voltages of PV-base have been assumed to curity analysis. Conventionally, line outages have been be one per unit, which is not a realistic assumption. Arya ranked based on (i) line ﬂows (MW) and (ii) load bus et al  developed a line outage ranking algorithm based voltages under line outage condition. Based on the above on the estimated lower bound on the minimum eigen- considerations, two diﬀerent ranking lists are prepared value of the load ﬂow Jacobian. The lower bound of the which provide separate guidelines for rescheduling of ac- minimum eigenvalue has been estimated using Schur’s in- tive and reactive power control variables separately [1, 2]. equality for each line outage using the reduced load ﬂow Ranking of line outages not only requires a severity index Jacobian. Sub-Jacobian relating Q–V is quickly obtained but also a methodology for evaluating it. Nowadays there using linearized relations for every line outage conditions ∗ Brila Insitute of Technology and Science, Pilani, Dubai Campus Knowledge Village, Dubai, E-mail: firstname.lastname@example.org; ∗∗ CPCL Industrial Training Institute, Chennai 81, India, E-mail: haran mano email@example.com; ∗∗∗ Department of Computer Science and Engineering, Anna University, Chennai 25, India, E-mail: firstname.lastname@example.org ISSN 1335-3632 c 2006 FEI STU Journal of ELECTRICAL ENGINEERING 57, NO. 2, 2006 117 using the same base case pre-outage sub-Jacobian. The required reactive power injection at load buses will come change in the loading margin to voltage collapse, when from the artiﬁcial sources. line outages occur, has been estimated and used for line The real power ﬂows in a transmission line connected outage ranking by Green et al . The reactive support between buses k and m are given as follows . index and iterative ﬁltering technique has been used for voltage stability contingency screening and ranking by 2 Pkm = gJ VK gJ VK Vm cos δkm + bJ VK Vm sin δkm , (2) Vaahedi et al . Fu and Bose  ranked the line outages 2 Pmk = gJ Vm gJ VK Vm cos δkm bJ VK Vm sin δkm . (3) based on severity indices as encountered in dynamic se- curity analysis. These indices are based on the concepts of coherency, the transient energy conversion between ki- The transmitted and loss parts of pi section model (for real power) are given as follows . netic and potential energy, and their dot products, after the fault is cleared. But at present ranking of line out- T 2 2 Pkm = 0.5gJ (VK Vm ) + bJ VK Vm sin δkm (4) ages based on dynamic consideration is not performed and may get popularity in future. The objective of this L 2 2 Pk = 0.5gJ (VK + Vm )gJ VK Vm cos δkm (5) paper is to develop a line outage ranking algorithm which accounts for reactive power requirement as well as net- Where gj = rj /(rj + x2 ) and bj = xj /(rj + x2 ). 2 j 2 j work limitations under contingent condition. The reactive In π section model a half of the total real power loss compensation index fulﬁls both of the above requirements is lumped at each end and the remaining portion consists that are the main causes of voltage instability. of the transmitted part. For outage simulation two separate compensation in- 3 REACTIVE COMPENSATION INDEX (RCI) jections are required, ie, one for the transmitted part and the other for the loss part of real power. The operating point in base conditions is ﬁxed very Moreover, omitting the reactive power ﬂow equation, near to the critical loading condition. The base case load the incremental power ﬂow equation at solution points is ﬂow solution is obtained. Artiﬁcial shunt compensators written as follows: are added at every load bus. Under line outage condi- tion, the total sum of reactive power outputs of artiﬁcial [J1 ][∆δ] = [∆p] (6) compensators to maintain the load bus voltages to its pre- (or) [∆δ] = [S][∆p] (7) outage values is termed as RCI. RCI is deﬁned as follows: NB where, J1 is the load ﬂow sub-Jacobian at the solution RCI = Qc nj (1) point relating the phase angle changes to the active power n=1 injection change and [S] = [J1 ]−1 . Using the compensa- where, RCIj is the reactive power compensation index tion injection theorem  the following equation is written for j th line outage; N B is the number of load buses; and for outage simulation (for transmitted part of real power) Qc is the reactive power supplied by the artiﬁcial shunt nj c T T compensator at nth load bus for the outage of j th line to ∆Pk = Pkm + ∆Pkm . (8) maintain the voltage proﬁle as in pre-outage condition. In fact, equation (8) expresses that when compensation c c injections ∆Pk and −∆Pk are applied to k th and mth 4 EVALUATION OF REACTIVE COMPENSATION INDEX USING buses for the outage simulation of j th line, the resultant LINE OUTAGE SIMULATION ﬂow in the line is equal to the compensation injections. T ∆Pkm is evaluated using the following linearized relation. The reactive power compensation index can be eas- T ily evaluated by running ac power ﬂow program under ∆Pkm = C1 ∆δkm . (9) each line outage condition and treating all load buses as PV-buses whose voltage magnitude is held constant C1 is obtained by diﬀerentiating equation (4) and is given in contingent condition to pre-outage value. But running as follows: the exact power ﬂow program in contingent condition (as C1 = bj Vk Vm cos δkm . there may be thousand contingent conditions in a prac- Now ∆δkm change in the phase angle across the j th line tical system) will be a time consuming process. Hence, a c is expressed in terms of compensation injections ∆Pk (at compensation injection based line outage is simulated to th the k bus) using equation (7) as follows: obtain the RCIj quickly. It is obvious from the deﬁnition of RCI that all bus c ∆δkm = (Skk Skm Smk + Smm )∆Pk . (10) voltage magnitudes are held constant and hence the re- active power ﬂow equations are not considered in post- Substituting equation (10) in equation (9), expression for T c outage compensation injection based power ﬂow study. ∆Pkm is written in terms of ∆Pk as follows: This requires that one calculates only the active power T c compensation to evaluate the phase angle changes. The ∆Pkm = C1 (Skk − Skm − Smk + Smm )∆Pk . (11) 118 K. Nithiyananthan — N. Manoharan — V. Ramachandran: AN EFFICIENT ALGORITHM FOR CONTINGENCY RANKING . . . c Now compensation injection ∆Pk is obtained using equa- where expression for line outage sensitivity is identiﬁed tions (8) and (11) as follows: as follows: Snk + Snm c T Pkm βnj = . (22) ∆Pk = . (12) 1 − C2 (Skk + Skm − Smk − Smm ) 1 − C1 (Skk − Skm − Smk + Smm ) Using the superposition theorem, the total change in It is again stressed here that if the base case voltages phase angle δnj due to transmitted and loss parts is are close to unity and phase angles are small in the base written as follows: case pre-outage condition, then equation (12) can also be T L ∆δnj = ∆δnj + ∆δnj or directly obtained using the power ﬂow relations. But this (23) is not a reality in attempting the voltage stability related T L ∆δnj = αnj Pkm + βnj Pk . problems. Equation (12) can be used to evaluate the phase an- Equation (23) enables the calculation of the bus phase angles for contingent conditions with the knowledge of gle changes of nth bus for the outage of j th line using T αnj and βnj along with pre-outage real power ﬂows Pkm equation (7). L and Pk . Since the voltage magnitude of each bus is held T c ∆δnj = (Snk − Snm )∆Pk . (13) constant and the phase angles are known using equation (23), the reactive VAR injection at each bus from arti- c T Putting ∆Pk from equation (12) in equation (13), ∆δnj ﬁcial condensers can be calculated. Calculation of VAR is written as follows: Injection Required from Artiﬁcial Sources Connected at Bus. T T ∆δnj = αnj Pkm (14) Table 1. Line outage ranking results based on reactive Compensa- Snk − Snm tion index and PI method where αnj = . 1 − C1 (Skk − Skm − Smk + Smm ) The change in the phase angle due to the loss part Rank Based on of real power of the outaged line using simulation is also RCI index PIMethod obtained in a similar way. The compensation injection 1 7 22.4036 7 1.5000 equation for the loss part is written as follows: 2 2 4.1161 2 1.5000 3 1 2.4549 5 1.0000 LC L L 4 5 1.4534 1 1.0000 ∆Pk = Pk + ∆Pk . (15) 5 4 1.1688 4 1.0000 The change in the phase angle across the simulated out- 6 6 1.1150 6 0.5000 aged line ∆δkm is calculated using the following expres- 7 3 1.0963 3 0.0000 sion: LC The reactive power ﬂow in any transmission line con- ∆δkm = (Skk + Skm − Smk − Smm )∆)Pk . (16) nected to rth and sth bus is given as follows: L Linearized expression for ∆Pk is obtained using equation 2 (5) as follows: qrs = b′ vr bi vr vs cos δrs gi vr vs sin δrs . i (24) L ∆Pk = C2 ∆δ km (17) The change in the reactive power ﬂow from rth to sth where C2 is a constant and is evaluated as follows: bus is written as follows: L ∆qrs , j = C3 ∆δrs , , j (25) ∂Pk C2 = = gj VK Vm sin δkm . (18) where C3 = bj vr vs sin δrs gj vr vs cos δrs . ∂δkm Putting the value ∆qrs ,j from equation (23) the follow- Putting equation (17) in equation (16), the one obtains ing expression is obtained LC ∆δkm = C2 (Skk + Skm − Smk − Smm )∆?Pk . (19) ∆Qrs , j = C3 (∆δr , j − ∆δs , j) or T L T L ∆Qrs , j = C3 (αrj Pkm + βrj Pk − αsj Pkm − βsj Pk ), (26) Putting equation (19) in compensation injection equa- T L LC tion (15) and solving it for ∆Pk gives, ∆Qrs , j = C3 (αrj − αsj )Pkm + C3 (βrj − βsj )Pk . (27) L Pk Now reactive compensation required at rth bus from ar- LC ∆Pk = . (20) tiﬁcial condensers is given as: 1 − C2 (Skk + Skm − Smk − Smm ) LC QC = r−j ∆ qrs , j . (28) The change in the phase angle of any bus due to ∆Pk s is written as follows: where in equation (27) lies in the set of lines connected L L to r− th bus. Using equation (27) for all load buses the ∆δnj = βnj Pk (21) severity index RCIj as given in equation (1) is calculated. Journal of ELECTRICAL ENGINEERING 57, NO. 2, 2006 119 4 METHODOLOGY FOR non- iterative in nature and is based on eﬃcient evalua- LINE OUTAGE RANKING tion of RCI using line outage simulation. The validity of the methodology has been established by comparing the Computational sequence is as follows: results with those obtained using the Performance Index Step 1. The pre-outage base case load ﬂow results are ob- method established in an earlier paper . tained. Step 2. Sensitivity matrix [S] is obtained by solving equation (6). References Step 3. The contingent line, j = 1 is selected.  CASTRO, C. A.—BOSE, A. : Comparison of Diﬀerent Screen- Step 4. Coeﬃcient C1 and C2 as deﬁned in equations (9) and ing Techniques for the Contingency Selection Function, Interna- (17) are calculated. tional Journal of Electric Power and Energy System 18 No. 7 Step 5. Sensitivity coeﬃcients αnj and βnj as given in equa- (1996), 425. tions (14)and (22) for all buses, n = 1, . . . , N B are evalu-  ALBUYEH, F.—BOSE, A.—HEATH, B. : Reactive Power Con- ated. sideration in Automatic Contingency Selection, IEEE Transac- Step 6. Phase angle changes ∆δnj for all buses are obtained tion on PAS, PAS101 (January 1982), 107. using equation (23).  GREENE, S.—DOBSON, I.—ALVARADO, F. L. : Contin- gency Ranking for Voltage Collage via Sensitivities from a Single Step 7. ∆qrs j for all lines are obtained using equation (26). Nose Curve, IEEE Transaction Power System 14 No. 1 (Febru- Step 8. Reactive compensation VAR supplied by artiﬁcial ary 1999), 32. sources at each buses are calculated with the help of the  VAAHEDI, E.—FUCHS, C.—XU, W.—MANSOUR, Y.—HA- equation (27). MADANIZADEH, H.—MORISON, G. K. : Voltage Stability, Step 9. RCIj as deﬁned in equation (1) is calculated. Contingency Screening and Ranking, IEEE Transactions on Step 10. Another contingent line, ie j = j + 1 is selected. Power System 14 No. 1 (February 1999), 256. Step 11. If j > N L , then go to Step 13, otherwise repeat  JASMON, G. B.—LEE, L. H. C. C. : New Contingency Ranking Technique Incorporating a Voltage Stability Criterion, Proceed- from Step 4. ing of IEE, part C 140 No. 2 (March 1993). Step 12. All RCIj are arranged in descending order and  ARYA, L. D.—CHOUBE, S. C.—KORTHARI, D. P. : Line ranking list for line outage step is prepared. Outage Ranking Based on Estimated Lower Bound on Minimum Eigenvalue of Load Flow Jacobian, Journal of The Institution of Engineers (India) 79 pt EL (December 1998), 126. 5 RESULTS AND DISCUSSIONS  GREENE, S.—DOBSON, I.—ALVARADO, F. L. : Sensitivity of the Loading Margin to Voltage Collapse with Respect to Arbitrary Parameters, IEEE Transactions on Power System 12 The algorithm developed in this paper has been im- No. 1 (February 1997), 262. plemented on a 6-bus 7-line IEEE test system . The  FU, C.—BOSE, A. : Contingency Ranking Based on Severity system has PV-buses number one and two. Both PV-bus Indices in Dynamic Security Analysis, IEEE Transactions on voltages were ﬁxed at 1.0 pu. To obtain the ranking of Power System 14 No. 3 (August 1999), 980. line outages using RCI methodology developed in this  ARYA, L. D. : Contingency Evaluation using Newly Developed Line Outage Distribution Factors, Journal of The Institution of paper, 300 % loading of the base case has been consid- Engineers (India) 79 (December 1998), 135. ered. Table 1 shows the results of ranking for the 6-bus  LEE, C. Y.—CHEN, N. : Distribution Factors of Reactive Power test system. The same table also depicts the RCI value for Flow in Transmission Line and Transformer Outage Studies, each line outage. The top four ranked lines are 7, 2, 1, 5. IEEE Transactions on Power System, 7 No. 1 (1February 1992). Line outage ranking results for the same system were ob-  ARYA, L. D.—BIJWE, P. R.—KOTHARI, D. P. : Allevia- tained  using the Performance Index (PI) method. tion of Line Overloads and Voltage Violations By Corrective Rescheduling, Proceedings of IEE, part C 140 No. 4 (July 1993), Two results have been compared in the same table. The 249. results obtained by these two methods are in close agree- ment as it is clear from this Table. The advantage of the Received 21 May 2005 methodology developed in this paper is that it is non- iterative. Moreover, the reduced load ﬂow sub-Jacobian Kannan Nithiyananthan is currently working as a lec- is computed based on line outage simulation. It is again turer in the Department of Electrical and Electronics Engi- pointed out that results will be perfect in the case of the neering, College of Engineering, Guindy, Anna University, In- developed methodology if the base case operating point dia, where from he had received also his PhD in Electrical Engineering. is ﬁxed under stressed condition, whereas the results ob- Neelamegam Manoharan is currently working as prin- tained with PI method do not vary much with the base cipal, CPCL Industrial Training Institute, Chennai, India. He case operating point. has received his Bc and Ms degrees in power systems engi- neering, from College of Engineering Guindy, Anna University Chennai, India. 6 CONCLUSION Velimuthu Ramachandran is currently a professor of Computer Science and Engineering in College of Engineering, A methodology for line out ranking based on reactive Guindy, Anna University, India. He has received his Masters of compensation index (RCI) has been developed and im- Engineering and PhD in Electrical Engineering from College plemented on two sample test systems. The method is of Engineering, Guindy, Anna University, Chennai, India.
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