modern-power-systems-analysis-d-p-kothari-i-j-nagrath-

- 0 5 5lil. Postfault With the faulted line switched off, which it is obvious that the systernis unstable' J;=:::,J;2 = 1: :.: 1t 'x- '1s.i n dI = 2 . 0 s i n d Let us choose Al = 0.05 s The recursive relationshi'ps for step-by-step swing curve calculation are t reproducedbelow. ;100 Pa(n_r)=P^ - P** sin 4_r (iv) o L6n= L6n-t* (Lt)z 'oa(' n-l) (v) E M @ 6n= 6n-t + A,6n (vi) o80 Sincethereis a discontinuitiyn P, andhencein Po, the averagevalueof po o) mustbe usedfor thefirst interval. c(5 o ioo P P\"(0-)= 0 pu andPo (0*) = 0.9- 0.88sin 2I.64\"= 0.576pu Po(ouu.,u=s\"9l t#ZQ = 0.288pu fault cleared at 2.5 cycles L I tllt ll li 0i. 5 _ rl Sustained Fault o 0.1 0.2 0.3 0.4 0.6 f (s) -- Calculations are carried out in Table 12.2 in accordance with the recursive Fig. 12.40 Swingcuryesfor Example12.10for a sustainedfault andfor relationship(iv), (v) and (vi) above.The secondcolumn of the table showsP-* clearingin 2.5and 6.25cYcles the maximum power that can be transferredat time r given in the first column. Pn * in the caseof a sustainedfault undergoesa sudden change at t = 0* and !A A n^i-. l-., ^aia* t / v l l l P u t qr rrar vt irnr on c nv r{ ec rr r' rrri nr Vn nv ur r' n / a f n r q t r c t a i n a r l fattlt remains constantthereafter.The procedureof calculations is illustrated below aaDle aZ.Z fUllll'-Uy-Pulllt by calculating the row correspondingto t = 0.15 s. / f = 0. 05s ( 0 ' l s e c )= 3 1 . 5 9 \" t P^u sin 6 P\"=Prrr,*sin6 P,,= 0'9- P, a66 P.\"* = 0.88 sec pu Pu Pu deg deg sin d (0.1 s)= 0.524 2.44 0.368 0324 0.0 2.57 2r.64 P, (0.1 s) = P,,,us* in 6 (0.1 s) = 0.88 x 0.524 - 0.46I 0+ 0.88 0.368 0.576 4.8r P, (0.1 s)= 0.9 - 0.46I - 0.439 o^u, 0.368 0.361 0.288 3.92 2.57 2r.64 0.05 0.88 0.41 0.46r 0.539 2.68 7.38 74.21 ( At\\2 (0.1s)= 8.929x 0.439- 3.92 0.10 0.88 0.524 0.598 0.439 r.45 11.30 31.59 0 .t 5 0 . 8 8 0.680 0.736 0.301 0.55 13.98 42.89 Y PM, 0.20 0.88 0.837 0.838 0.163 0.18 15.43 56.87 0.25 0.88 0.953 0.879 0.06 0.426 15.98 72.30 6 (0.1ss)= Ad (0.1s) + qL P, (0.1s) 0.30 0.88 0.999 0.852 0.021 1.30 0.35 0.88 0.968 0.154 0.048 2.87 r6.r6 88.28 MU\\ 0.40 0.88 0.856 0.578 0.145 1 6 . 5 8 r04.44 0.45 0.88 0.657 0.32r r 7 . 8 8 r2r.02 = J.38\"+ 3.92\"= 11.33\" 0.50 0.88 20.75 138.90 d ( 0 . 1 5s )= d ( 0 . 1s ) + A d ( 0 . 1 5s ) 1s9.65 = 31.59\"+ 11.30=' 42.89\"

I f- Modernpo@is ltr -F . asut t.lat vFtlcaraeet e^.' tJ t:l t- o E n---t-- A,i, fryCIeS Time to clear f'ault= 2.5 = 0.05 s progressively greater clearing time till the torque angle d increases without 50 bound. In this example, however, we can first find the critical clearing angle using Eq. (12.67) and then read the critical clearing time from the swing curve P-u^ suddenly to 2.0 at t = 0.05-. Since the correspondingto the sustainedfault case.The values obtainedare: dupsbewrnetcdirnrcaeegesradgscsuouueremr.evsieasedtimthsoeaprsxleoai mmtmteuaedmiannisnccwolcFnionisgmgt. apon1lfe2t ft.3re4-7c0o.am5flr\"co0ubm.lu0at2wt ii5ohsnsicssahtt oarweb0sel e.h0afoi7snw5dncsi5tn.hf iaTnTthaateibhl ylereebg1seet2gno.ie3nfr.sTat ththooeer Critical clearing angle = 118.62 Critical clearingtime = 0.38 s Table 12.4 Computationos f swingcurvefor faultclearedat 6. 25cycles( 0. 125s)A, f = 0. 05s Table12'3 Computationosf swingcuryesfor fauttcleared at2.s cvcles P,no sin 6 P\"=P^ *sin6 Po= 0.9- P\" 466 (0 .0 5s ), At = 0.05s pu deg deg P,,,,,^ .sin.5 Pr,=P,rr.,*,tin5 pr,= 0.9- pu A6 6 0 2.44 0.368 0.9 0.0 . 2r.& pu pu pu deg deg 0.368 0.324 0+ 0.88 0.368 0.576 21.64 0.41 0.36r ouu, 0.524 o.46t 0.288 2.57 2.57 zt.U 0.680 1.36 0 2.44 0.368 0.9 0.0 21.64 0.05 0.88 0.767 1.53 0.539 4.81 7.38 24.2r 0.324 0.576 0.78 1.56 0. 0.88 0.368 21.64 0.10 0.88 0.734 1.46 0.439 3.92 11.30 3t.59 T 0.36 0.288 0.613 r.22 - 4.46 0.82 0.430 0.86 - 0.63 ouu, 0.368 0.54 2.57 2.57 21.64 0.15 2.O0 - 0.66 - 4. 10 7.20 42.89 0.986 u. z-1-1 u.4c)0 - 0.56 0.05 0.88 0.41 I.t2 0.08 0.20 2.00 - 0.327 - 5.66 r.54 50.09 T.I9 0.05+ 2.00 0.41 r.t9 0.31 24.21 0.25 2.00 0.04 - 5.89 - 4.35 51.63 I.t2 - 0.086 0.05uus 0.989 - 0.22 24.21 0.30 2.00 u.4-J4 - 5.08 - 9.43 47.28 0.82 - 0.29 0.615 - 0.29 2.767 5.33 24.21 0.35 2.00 - 2.92 - 12.35 37.85 - 0.22 - 0.767 4.56 29.54 0.10 2.00 0.493 - 0.089 0.40 2.00 0.35 - 12.00 25.50 0.i5 Z.UU 0.56 0.08 - 1.96 2.60 34.10 uA . 4A ?J Z . W' \\ N - r . 6/ - u.t - J l_J. ) u - 2.s8 36.70 o.20 2.00 0.s91 0.225 - 2.58 0.02 37.72 0.50 2.oo 5.37 - 1.96 - 2.56 34.16 0.25 2.00 0.597 - 0.79 _ 4.52 29.64 _ 5.31 0.30 2.00 0.561 T2.TO MULTIMACHINE STABITITY 0.35 2.00 0.494 From what has been discussedso far, the following stepseasily follow for determiningmultimachinestability. o.40 2.00 0.41 0.71 - 4.60 24.33 1. From the prefault load flow datadetermineE/ovoltage behind transient 0.45 2.C0 0.337 2.0 - 2.6 19.73 . reactancefor all generators.This establishesgeneratoremf magnitudes lEll which remain constant during the study and initial rotor angle 0.50 17.13 6f = lEt. Also record prime mover inputs to generators,P*o - PoGk Fault Cleared in 6.25 Cycles 2. Augment the load flow network by the generatortransientreactances. Shift network busesbehind the transient reactances. Time to clearfault= ua?t= 0.125s s0 3. End Inus for various network conditions*during fault, post fault (faulted line cleared),after line reclosure. 4. For faulted mode, find generatoroutputs from power angle equations (generalizedforms of Eq. (12.27))and solve swing equationsstep,by step(point-by-point method).

4ffi | Modernpower SystemAnalysis l,,m 5. Keep repeating the above step for post fault mode and after line Bus to bus 220kV, 100M VAbase HaIf line charging reclosuremode. Series Z 6. Examine d(r) plots of all generatorsand establish the answer to the stability question. The above stgpsare illustrated in the following example. Line 4-5 0.018 0.1r 0.113 Line 5-1 0.004 0.0235 0.098 Line 4-I 0.007 0.04 0.041 Trans;2-4 o.022 A 50 Hz, 220 kV transmissionline has two generatorsand an infinite bus as Trans:3-5 0.04 shown in Fig. 12.4I. The transformerand line data are given in Table I2.5. A three-phasefault occursas shown. The prefault load flow solution is presented Tabfe12.6 Bus dataand prefaulltoad-flowvaluesin pu on 220kV, in Table 12.6. Find the swing equation fbr each generatorcluring the fault 100M VAbase period. S.No. Voltage Bus Voltage Generation Load and Polar Upe Bus Form Real Imaginary o Vz=1.0328.2350 o) v iF1.O217.16o No. ef | 1.010\" Slack 1.00 0.0 - 3.8083 -0.2199 0 0 0.6986 0 0 l-lt-[(5, 2 t.0318.35\" PV t.0194 0.1475 3.25 0.3110 0 0 1.0 1.0 0.44 )vs=1.ollz 3 t.0217.16 PV 1.0121 0.1271 2.10 0 0.5 0.16 22.490 - iI' 4 1.017414.32\"PQ 1.0146 0.167 0 i6,uSe 5 1 . 0 1 1 2 1 2 . 6 9 \"P Q 1.0102 0.0439 0 ).5+j( 6) Solution Before determining swing equations, we have to find transient internal voltages. The current into the network at bus 2 basecion the <iatain Tabie i2.6 is 'rz- _ Pr-iQ, _ 3.25-i0.6986 v: r.o3l- 8.23519\" o E{= (1.0194+ j0.1475)+ 3.2s-j0.6986 x 0.06719V r . 0 3l - 8 . 2 3 5 1 9 ' Fig. 12.41 = 1.034092+9 j0.3632368 Data are given below for the two generatorson a 100 MVA base. = 1.0960333119.354398=\" 1.0960lo.337l tad I Gen 1 500 MVA, 25 kV, XJ = 0.067 pu, H = 12 MJAyIVA El = I.0 l0' (slackbus) Gen 2 300 MVA, 20 kV, X,j = 0.10 pu, H = 9 MJA4VA E4- Q.0I2r + j0.1271)+ 2 . 1 -j 0 . 3 1r x 0.1l9O\" Plot the swing curvesfor the machinesat buses2 and3 for the above fault r.021-7.15811' which is cleared by simultaneousopening of the circuit breakersat the ends of the faulted line at (i) 0.275 s and (ii) 0.0g s. = 1.0166979+ j0.335177= 1.0705 lI8'2459\" - 1.07110.31845rad The loads at buses4 and 5 are representedby the admittancescalculated as follows: y6rr.= '('91.0.1!7.:4,)0\" !(0.966- 1jo.4zsr)

PowerSystemStability I_ a,.grr, During Fault Bus Matrix v 0 . 5_ j 0 . 1 6 (0' 488-e prefaulr *Trl rr.off i 0'1s6 47) Since the fault is near bus 4, it must be short circuited to ground. The Ynus \"r\" during the fault conditions would, therefore,be obtained by deleting 4th row ;;'#;;\"J###:: and 4th column from the above augmentedprefault Y\".r. rnatrix. Reducedfault Load admittances,al matrix (to the generatorinternal nodes)is obtained by eliminating the new 4th row and column (node 5) using the relationship ttihhl:ee:Hrien,ftoe:Trren:'a|n,lovwoltd:a:e:g,se:i:gs:na:^an'tdle:ta,h:se:b,t1uras,:ennss2tienatnrredea3ca, ttchaetnacfiencstloietfoisotfuhsethmineatmecrhnaianclenhsoi.Tndheeuswswsbeeewtwewweglleerln,t, Y*iqn\"*1= Y*j@tat - Yt n(oltt) ynj(old Yrn @ta) )/ The reducedfaulted matrix ()'eusduring fault) (3 x 3) is given in Table I2.8, -Y2-^2= - @ = - i r r . 2 3 6 which clearly depictsthat bus 2 decouplesfrom the other busesduring the fault and that bus 3 is directly connectedto bus 1, showing that the fault at bus 4 Yzq= j11.236= yqz reducesto zero the power pumped into the system from the generatorat bus 2 and rendersthe secondgenerator at bus 3 to -eiveits power radially to bus 1. Ytt = io'04+io] = - i7'143 Table 12.8 Elementsof Yrus (duringfault) and Ysus(post fault) for Ex. 12.11,admittancesin pu. Yzs= j7.I43 = yst Reduced during fault Yu^ Y u = Y t q + y q t * y q s + Bo, LBo,' Bus ) ) * Yzq I = 0.966087-7 j0.425078+5 4.245- j24.2571+ 1.4488_ 2 s.7986-j35.6301 0 - 0 . 0 6 8 1+ j 5 . 1 6 6 l i 8 . 8 5 3 8 +j 0 . 0 4 1+ r O . 1 1 3 _j r t . 2 3 5 g - jrr.236 0 a 0 J - 0 . 0 6 8 1 7+5 . 1 6 6 1 0 0 . 1 3 6 2 ' -j 6 . 2 7 3 7 _ 6.659897_7 j44.6179 I Reducedpost Jault Yurt - 0.0901+ j6.O975 2 0 Y s s =Y r s * Y s q * Y s r * 92* Ur, - r.3932- jr3.873r - 0.2214+ j7.6289 2 *' r' 3\" ,5 0 . 1 5 9-1 j 6 . 1 1 6 8 J - 0.2214+ j7.6289 0.s- j7.7898 - 0.4889- j0.1565+ r.4488- j8.8538+ 7.039r_ j41.335 - 0.0901+ j6.0975 0 + /O.113+ j0.098_ j7.t42} Post Fault Bus Matrix _ 8.97695s_ js7.297202 Once the fault is cleared by removing the line, simultaneouslyopening the The completeaugmentepdrefaultlzuu,matrixis shown.iri'Table 12.1. circuit breakersat the either ends of the line betweenbuses4 and 5, the prefault T a b l e1 2 . T Y\"u5 has to be modified again. This is done by substituting Yqs= Ysq- 0 and T heau gme n t epdr e f a ubr tu sa d m i t t a n cmea t ri xf o r Ex . 12.11, subtracting the seriesadmittance of line 4-5 and the capacitivesusceptanceof admittanceinspu half the line from elements Yooand Ytt. Bus Y++lporrfau=lt)Y++(prefa-ultY) qs- 84512 rr.284_j6s.473 0 4.245 + j24.257 -7.039 + = 6. 65989- j44. 6179- 1. 448+ 78. 853- j0. 113 = 5.2III - j35.8771 2 0 _i1r.23sg j4r.35s Similarly, Ysr(oo*ra, ult=) 7.528I - i48.5563 3 4 06 0 j11.23s9 0 The reducedpost fault Y\"u5 is shown in the lower half to Table 12.8.It may -4.245+ j24.257 jll .2359 - j7.1428 be noted that 0 element appears in 2nd and 3rd rorvs. This shows that, 0 j7.t428 0 6.6598_j44.617 -1.4488 -7.039+ j41.355 0 +j8.8538 0 + j7.1428 -1.4488+ j8.8538 8.9769 + j57.2972

{92 | Modernpower SystemAnatysis PowerSystemStanitity I 491,1 l- Lr.r. ryr^ : ^ ^ l r'-r.uarryrtr'l-l-e generarorsI an0 Z are not Interconnectedwhen line 4-5 is It rnay be noted that in the above swing equations,P,, ntay be written in removed. generalas follows: During Fault Power Angle Equation Pn= Pr, - Pr.- P,r,ns* in (6- 7) Prz= 0 Solution of Swing Equation P,3= Re [YrrEr,El** El* \\F(]; sinceyy= 0 The above swing equations(during fault followed by post fault) can be solved = E{2 Gn + lEil lEil lrr,l cos(6zr_ Lzt) by the poinrby-point method presentedearlier or by the Euler's method = ( 1 . 0 7 1 )(20 .1 3 6 2+) I x 1 . 0 7 1 x 5 . 1 6 6 5c o s( d 3_ 9 0 . 7 5 5 \" ) presentedin the later part of this section.The plots of E and 4 are given in Fig. 12.42 for a clearingtime of 0.215 s and in Fig. 12.43for a clearing time P \" 3= 0 . 1 5 6 1+ 5 . 5 3 1s i n( h - 0 . 7 5 5 ) of 0.08 s. For the case (i), the machine 2 is unstable,while the machine 3 is stable but it oscillateswherein the oscillations are expectedto decay if effect Postfault Power Angle Equations of damperwinding is considered.For the case(ii), both machinesare stablebut p\"z= lE/P G22+ lElt lEll ly2Lcl os (dr, _ 0zr) = 1.0962x 0.5005+ I x 1.096x 7.6321cos({ - 9I.662\") the machine2 has large angular swings. = 0.6012+ 8.365sin (d, _ I.662) p,3 = tE{ 2q3 + Ell tElt t\\l cos ( 6r,_ 0rr) Machine1 is reference(lnfinitebus) - 7.0712x 0.1591+ 1 x 1.07rx 6.09gcos(dr- 90.g466\") = 0.1823+ 6.5282sin (d, _ 0.9466\") Swing Equations-During Fault #=ff (P^z-P,z)= = t:9/ e.2s - o) elecrdeg/sz 12 d , q _ 1 8 0f ( P n - P \" t ) a'r= k = t*:/ e . r - { 0 . 1 s 6 1+ 5 . 5 3 1s i n ( 6 t - 0 . 7 s s ) } l 9 = l.g43g - 5.531sin(d, - 0.755\")el lectd,eg/sz Y Swing Equations-Postfault ( 0 . 2 7 5 sf a u l tc l e a r e da t ) #={f p . 2 s - 1 0 . 6 0zr + 8 . 3 6 5s i n ( d )- t . 6 6 2 \" ) }eJr e cdt e g / s 2 F\\g.12.42 Swingcurvesfor machines2 and 3 of Example12.1for clearingat 0.275s. -d*td, =*11820f.10-{0.1823 + 6.5282sin(d3_ 0.g466.)}lelecdteg/sz If the fault is a transientone and the line is reclosed,power angle and swing dt' 9 equationsareneededfor the period afterreclosure.Thesecan be computedfrom the reduced Ysusmatrix after line reclosure.

494 Modernpower SystemAnalysis PowerSystemStabilitv l,,4PFr | (t2.74) Machine 1 is reference(lnfinitebus) . i * = #r7(.2P \" o r - P c o )k, = 1 , 2 ,. . . ,m 500 Initial state vector (upon occurrenceof fault) is M a c h i n e2 xoLk=fr= lEot at) x \" z k =0 c) The stateform of swing equations(Eq. (12.74)) can be solved by the many o available integration algorithms (modified Euler's method is a convenient L choice). q, Computational Algorithm for Obtaining Swing Currzes Using Modified Euler's Method o) ! o (t c 100 illtl _L__ I _ | _ l. Carry out a load flow studY (prior to disturbance) using specified 0.8 0.9 1.0 Faultcleared after4 cycles voltagesand powers. 2 . Compute voltage behind transient reactancesof generators(Eo*) using Eq. (9.31).This fixes generatore.mf magnitudesand initial rotor angle Fig. 12.43 Swing curves for machines 2 and 3 of lreference slack bus voltag\" Y?). Exampl1e2.11forclearinagt 0.0gs a Compute, Ysu5(during fault, post fault, line reclosed). J. Gonsideration of Automatic voltage Regulator (AVR) and 4. Set time count r = 0. Speed Governor Loops 5 . Compute generatorpower outputsusing appropriatgts\"us with the help of the geniral form of Eq. (12.27).This givet Pg},for / 1 /'). Note: After the occurrenceof the fault, the periodis divided into uniform discretetime intervals(At) so that time is countedas /(0),t(l), ......A typicalvalueol' lt is 0.05 s. 6. Computet(i[;'},i\\7)' ,k - - 1 , 2 , . . . , m l f i o m E q s .( 1 2 . 7 4 ) . 7. Corttputtchc I'ilslstutccstirrtirlcl'isu' t = ,{t+l)'prr state variable Formulation of swing Equations , [ , 0 * r=, , t l o + i [ , 0 a) t I = |.2. .... rrt The swingequationfor the hh generatoirs ,ff') - *V)+ *$'oA)t +d t ' = + (p ' ou o - p \" ) ;k - r , 2 ,. . .f,f i (r2.73) 8. Cortrputtehefirstestimateosf E^('+t) BQ+D= E? lcosx,(i*r)+7 sin *\\lf')) H k' g. ComputeP8;'); (appropriateY\"u5andEq. (12.72))' For the multimachinecase,it is more convenientto organiseEq. (12.73) is state variableform. Define 10. compute[t;{in'),it:o*t')k, = r,2, ...,mf fromEqs.(12.74). 11. Computethe averagevaluesof statederivatives xrk= 6r= lE*' xz*= 6t i[i,), urr=][iu,cl +;l[*t)] Then k=1,2,...,ffi i t*= xzt iL'),*=,*I*\\:|+,tt'i\"l

Power System Stabitity | 491;,, L1 'Lt . LA ompute tne lrnal state estimates for | = t\\r+t)- factors which affect transient stability and therefrom draw th\" .on\"llsions, regarding methodsof improving the transientstability lirnit of a system and *;Iu\" = *([)+ ill) uusa,t making it as close to the steadystate limit as possible. k = 7'2'\"'' frt For the caseof one machineconnectedto infinite bus, it is easily seenfrom ,&*r)- ,([l + if).^,rat Compute the final estimate for Eo at t = r('*l) using the angle through which it swings in a given time interval offering thereby a method of improving stability but this cannotbe employed in practice because BQ+t)= l4llcos xf;+r) + 7 sin *f1r) of economic reasonsand for the reason of slowing down the responseof the speed governor loop (which can even become oscillatory) apart from an 14. Print (\",9*t),*;:o*D); k = I,2, ..., m excessiverotor weight. 15. Test for time limit (time for which swing curve is to be plotted),i.e., With referenceto Fig. 12.30,it is easily seenthat for a given clearing angle, ch e c k i f r> rn n u rI.f n o t, r- r+ r and repeatfromstep5 above. the accelerating area decreasesbut the decelerating area increases as the Otherwise print results and stop. maximum power limit of the various power angle curves is raised, thereby adding to the transient stability limit of the system.The maximum steadypower The swing curves of all the machines are plotted. If the rotor angle of a of a system can be increasedby raising the voltage profile of the system and machine (or a group of machines) with r\"rp\".t to other machines increases by reducing the transferreactanceT. hese conclusionsalong with the various transientstability casesstudied,suggestthe following method of improving the without bound, such a machine (or grouf of machines) is unstable and transient stability limit of a power system. eventuallyfalls out of step. 1. Increaseof systemvoltages,use of AVR. The computationalalgorithm given abovecan be easilymodified to include excitation response,saturation of flux 2. Use of high speedexcitationsystems. simulation of voltage regulator, field paths and governor action 3. Reductionin systemtransf-ereactance. Stability Study of Large Systems 4. Use of high speedreclosingbreakers(seeFig. 12.32).Mo&rn tendency is to employ single-poleoperation of reclosingcircuit breakers. vmttwcTshoaouoheetnmbaetelhoslirpxmeosyuuttasdeisutttssapredna-ttomychiapowoeaiplsnshenrcuaoerdroeblxuermaensiftmnfyidnpilscaieceuttiytrxtieeeeeetnlbmerdsncabminr,onynsyued,aettaimenhlhdsllberoelayinenrwrsmyerggt<oteoaehilrsmfndmolrddencd.uT.raatslnhhbtcrirteeeh-eimeiitscnndiamthgeuocnsedimuhsiqytirodnuesfsroeequatrsoubspyfitstrohsieydcpetmsyeautenmleerlxaaynmitrsmretieassdricanncimlvndaueidolcqdwsdeufeuioeddidubvrlessliaentliyhlyndteseognitunevsacsmaaiedsnrk.detiTgeoutbh.Iauodnyeisfyl When a fault takes placeon a system, the voltagesat all busesarereduced. I2.T7 SOME FACTORS AFFECTING TRANSIENT STABILITY At generator terminals, theseare sensedby the automatic voltage regulators We have seenin this chapterthat the two-machinesystemcan be equivalently which help restoregeneratorterminal voltagesby acting within the excitation reduced to a single machine connected to infinite bus bar. The qualitative system.Modern exciter systemshaving solid statecontrolsquickly respondto emdcoqeunutletciivrmlmuaasliencionhinntingsoetrnheeseg-ysamstrtaadebcimnihl.giitInynseoyitfnshtfaeeinmmliatuessltttabiamburiastliicctsyhleyrsdnwtreeaesmwyhsncateavfmrneo.smbteudaeieatdwsiiolhy-emeaaxltcgehonirnditeehdmotrofoaarn bus voltage reduction and can achievefrom one-half to one and one-h'alfcycles atslrotaac3nIba-tspitlfhihieotaayrns'soseGbtfuefiavadeeuyfnnaltdsuetewlhtech,einsidcttoehyhotaphintesattrogthaafeennpsfsaeoeieurwtawnlteltloysar stnmfaaydbocsirtitltoeeistrmyssle.oaisIvcnneaagrtolrieyeousfan,rtotm\"llmy*euuta-sppfutfolesaeictnsnttteohodwewfbvevcyeiohertnyawhsveoiedoutsyeftseprpeleooettowcahftneeeadrrd (l/2-l]) gain in critical clearingtimes fbr three-phasteaults on the HT bus of the generator transformer. Reducing transfer reactanceis another important practical method of increasingstability limit. Incidentally this also raisessystem voltage profile. The reactance of a transmission line can be decreased(i) by reducing the conductorspacing,and (ii) by increasingconductordiameter(seeEq. (2.37)). Usually, however, the conductorspacingis controlledby other featuressuch as lightning protection and minimum clearanceto prevent the arc from one phase moving to another phase.The conductor diametercan be increasedby using material of low conductivity or by hollow cores. However, norrnally, the conductor configuration is fixed by economic considerationsquite apart from stability. The use of bundledconductorsis, of course,an effective means of reducing seriesreactance. Compensation for line reactanceby seriescapacitorsis an effective and economical method of increasing stability limit specially for transmission

'4lI I Modern power System Analysis I . I :.-^Ir i:: I.1.':/tlltt, eoadfvcifseectecravtnniovtculeetaasatgoenfesdtsmhedecrororprenirnootghbmlalieinncmea3-ls5itfoo0p-fkgapmrrrot.ouotTfnehdciteftaiivscueiletrsge.rlSeaeyeoirnfiegss,ecnrooierrnsmpcaoelmnvspoaelttnaiosgnabetepiocrnoo,fmhiloeewssm,evaoenrred, stzeof rotor reducesinertia constant,Iowering therebythe stability margin. The loss in stability margin is made up by such featuresas lower reactancelines, of compensationupon the occurrencec fastercircuit breakersand faster cxcitation systenrsas tliscussetlalreacly,and a faster system valving to be discussedlater in this article. Switehed serieseapaeitorssimultaneot A stage has now been reached in technology whereby the methods of and raise the transientstability limit tc irnprovinE=stability; discussetlabove, have been pushed to their limits, e.g., clearing times of circuit breakers have been brought down to virnrally limit. Switching shuntcapaciiorson ol irreducible valuesof the order of two cycles.With the trend to reducemachine inertias there is a constant need to determine availability, feasibility and ssrsehataqumbuneiitlrieeitnyldecrmliisemeatinhststerseian(esrteeoserteasEqbixxuilaiitrtmiyemdpleilfmesoritt1h.o2el.hT2reha)rutbipsnuugts{pteohorfeisesesMwscVi,tsacAipyha,ecrdciatosotinernstrgrioaeolrsefocfapsvprheoauflcnetaitrtgorceerasdppfuaoroncrfilitetlohesre.ss applicability of new methodsfor maintaining and/orimproving systemstability. Increasing the number of parallel lines A brief account of someof the recent methodsof maintaining stability is given often used to reduce transfer reactance.It between transmissionpoints is quite below: adds at the sametime to reliability HVDC Links of the transmissionsystem.Aclditional line circuits are not likely to prove Increaseduse of HVDC links ernploying thyristors would alleviate stability economical unit I aftet all feasibleimprovementshave beencarried out problems. A dc link is asynchronous,i.e., the two ac system at either end do first two circuits. in the not have to be controlled in phaseor even be at exactly the same frequency as they do for an ac link, and the power transmittedcan be readily controlled. As the majority of faults aretransientrn nature,rapid switching and isolation There is no risk of a fault in one system causingloss of stability in the other of unhealthy lines followed by reclosing has been shown earliei to be a great system. help in improving the stability marginr.ih. modern circuit breakertechnology has now made it possiblefbr line clearing to be done as fast as in two cycles. Breakingr Resistors arFneautcrtralutohnrsaesilnri et,ghnaawt tLghmGri ceehaftahtfmuouldrattsjihosheraairtvsyaesi doubfmet heterenaddtsnotesavoibeecinlcloitutpyfraeoluidmnlftositthrsae.s' eegWleeliintncheet irv-,at\"eotfos\"-rgri\"nbr.gou, \"lus\"e,npidtot oiilsneFiomni ;pa.met ulnezrid.nerig.7aI,att eniildsfy ol,lnt t he fault therewill now b For improving stability where clearing is delayedor a large tn)a i, suddenly an:d:b.1ud r,+u:+l^r_se^ ^.Lr,eni rromL - - - - zero power transfer for e a definite amountof power rransfer , lost, a resistive load called a breaking resistor is connected at or near the when the circuit breakerpole correspon a three-p hasefault. Also generatorbus. This load compensatesfor at leastsonreof the reduction of load the case of faulty line is opened,the on the generatorsand so reducesthe accelerationD. uring a f-ault,the resistors dingto the areapplied to the terminals of the generatorsthrough circuit breakers by means other two lines (healthyones)remainintact so that considlrablepower transfer of an elaboratecontrol scheme.The control schemedeterminesthe amount of continuesto take placevia theselines in comparisonto the caseof three-pole resistanceto be appliedand its duration.The breakingresistorsremain on for SsEsisswtiwxa,niaibgtttchmcilleheihptipryinlenoegplglber1iwors2sebh'w,l1aeeei2dtnamc.osthEphayitvnenteoegdpdnaiosstnwoewwdeehpirwrdeerteehncr alvylyotenhtsansheidtnfeesolgarstpioarsitgnn,beegoidflalfi.esTtucywphol tmoieunclsragelseersesfgaa,wairecninixntsisdpcgaheawcrinroneeislngllisiuslbuaue. esnfiqfntdirurceateri.deet.ne*ucntsdvclto,boessoyrdiittfnnamorgggeezlaeleaeaiddnrypiossipon.oislgInef.t a matter of cycles b<lthduring fault clearing and after system voltage is restored. Short Circuit Current Limiters and introducesthe as.socjatepcr!oblernsof overvoltagescausedby single pole Theseare generally used to limit the short circuit duty of distribution lines. opening owing to line capacitances.Methods Thesemay also be used in long transmissionlines to modify favourably the capacitivecoupling effects. are available to nullify these transferimpedanceduring fault conditions so that the voltage profile of the systemis somewhatimproved,therebyraisingthe systemload level durins the Recent Trends fault. crRaoetnicoseen(qstuctreRenntds=asvirni/nxgd)s,eisniwgmnhaioccfhhlianisregaemcamhlitfee, vrsneiazdtebo,yrwsetreeigndhdutctoainwngdarmcdoasscltoh. Rwineeedrausicrhtoigorant pcinirwctiuhthiet Turbine Fast Valuing or Bypass Valuing The two methods just discussedabove are an attempt at replacing the sysrem load so as to increase the electrical output of the generator during fault conditions. Another recent method of improving the stability of a unit is to decreasethe mechanicalinput power to the turbine. This can be accornplished

by rneansof fast valving. where the differenee between mechanicalinput and r--- - --l reduced electrical output of a generatorunder a fault, as sensedby a control scheme, initiates the closing of a turbine valve to reduce the power input. J- , t0 . 1 5 0'3 0.1 I Briefly, during a fast valving operation,the interceptor valves arerapidly shut 7 i-dT L_r6TT\\_ (in 0.1 to 0.2 sec) and immediately reopened.This procedure increasesthe | critical switching time lons enoush LI --z -r0fd' 1]-ui-rp l stablefor faults with stuck-breakerclearing times. The schemehasbeenput to usein somestationsin the USA. | I FUII Load Rejection Technique t T - t I - t -(b) Negativesequence network Fast valving combined with high-speedclearing time will suffice to maintain ; '' . l ] o ti ) go 'l' _ = C xo=0.0e15 stability in most of the cases.However, thereare still situationswhere stabilitv ao | l is difficult to maintain.In suchcases,the normal procedureis to automatically p 1' ot;,'r| \";*u1n\"\"| trip the unit off the line. This, however, causesseveral hours of delay before L| ----.-' |I i-r T60 I the unit can be put back into operation.The loss of a major unit for this length of tirne can seriouslyjeopardize the remainingsystem. networl To remedy thesesituations, a full load rejection schemecould be utilized Fig. 12.45 after the unit is separatedfrom the system. To do this, the unit has to be equipped with a large steam bypasssystem.After the systemhas recovered For an LG fault at P the sequencenetworkswill be connectedin series-as from the shock caused by the fault, the unit could be resynchronizedand shown in Fig. 12.46.A star-deltatransformationreducesFig. 12.38to that of reloaded. The main disadvantageof this method is the extra cost of a large Fig. 12.47from which we have the transfer reactance bypasssystem. Xr2(LGfaulQ= 0.4+ 0.4+ nOII''O = 1.45 0.246 The systetnshown in Fig. 12.44is loadedto I pu. Calculatethe swing curve I o'4 and ascertainsysternstability for: I (i) LG fault threepole switchingfolloweclhy reclosurel.inc lounclhealthy. .-t. (ii) LG fault singlepole switchingfbllowed by reclosure,line foundhealthy. Switching occurs at 3.75 cycles (0.075 sec) and reclosureoccllrs at 16.25 l E l = 1 ' 2( ) cycles (0.325 sec).All values shown in the figure are in pu. - I r-rd 60 0'4 P l. r.l =. ,1?.6,_7 X r =0 . 1 Fig.12.46 Connectioonf sequencenetworksfor an LG fault H =4 -'. -h'Ir-!.&-f! Xr=o3Oj f- ' ?*f ( / (,r 'lr Xt=0'15 L xo=o.t AY-l II rl /' X n _ _ _tJ - ftr f\\r Fig. 12.44 Fig.12.47 Transferimpedancefor an LG fault Solution The sequencenetworks of the systemare drawn andsuitablyreduced When the circuit breakerpoles correspondingto the faulted line are opened in Figs. I2.45a,b and c. (it correspondsto a single-lineopen fault) the connectionof sequencenetworks is shciwnin Fig. 12.48.From the reduced network of Fig. 12.49the rransfer reactancewith faulted line switched off is

-Y^ l.l ^ (fettlterl line nnen\\ -- 0v . - A t | A A1 -rr (l A -_ 1 .. I .o.ro \\-*--- v.aL V..+ L../-L Under healthy conditions transfer reactanceis easily obtained from the positive sequencenetwork of Fig. 12.45 a as Xrr(line healthy)= 0.8 Pettt = 0 PrN= Pd = 1.5 sin d Now o'1 P 46,- A6n-+, @LP'^a,(-n - l,), M H = 4.167MJA{VA Negativesequence Zerosequence 1,1= JU- - 4.63 x 10a sec2lelectricaldegree 1 8 0x 5 0 Flg.12-48 Connectionof sequencenetworkswith faultedlineswitchedoff Taking At = 0.05 sec lEl=1'z l v l= 1 ' o (at)' P P/ o:4 5.4 4.63xI0-4 Time when single/threepole switching occurs Fig. 12.49 Reducednetworkof Fig. 12.49givingtransferreactance = 0.075 sec (during middle of At) ., Time when reclosing occurs = 0.325 (during middle of at) Power angle eguations Table12.9 swing curyecalculation-threepoleswitching PreJault L P,u,o 6P\" P,, .5.4P,, ifi 6 (pu) (pu) elec deg p,= rE vr sin d= -7 . 2 x 1 S l I l d = l.) stn b sec ) elec deg elec deg X,, 0.8 1.5 0.667 r.0 0.0 41.8 0 0.827 0.667 0.552 0.448 1.2 1.2 41.8 Initialload= 1.0pu 0.224 N A 3.6 41.8 o* 0.827 0.682 0.564 0.436 43.0 Initialtorqueangleis givenby ouu, 0.05 0.075---+ 1 = 1.5 sin 5o 0.10 0.0 0.726 0.0 1.0 5.4 9.0 46.6 0.15 0.0 0.0 1.0 5.4 r4.4 55.6 or 6o= 47.8\" 0.20 0.0 0.0 1.0 5.4 19.8 70.0 o.25 0.0 0.0 1.0 5.4 2s.2 89.8 Duringfault 0.30 0.0 0.0 1.0 5.4 30.6 I 15.0 0.325--+ F'\",,= l . Z x l sind= 0.827sin lff 0.3s 1.5 0.565 0.85 0.15 0.8 31.4 r45.6 0.40 1.5 0.052 0.078 0.922 5.0 36.4 177.O I)uring single pole switching 0.45 1.5 - 0.55 - 0.827 r.827 9.9 46.3 2r3.4 0.50 1. 5 - 0. 984- r . 48 2. 48 13.4 59.7 259.7 Per=rr sin d= 0.985sin 0.s5 1 . 5 - 0 . 6 5 1- 0 . 9 8 1 . 9 8 10.7 10.4 3r9.4 +# 0.60 r.5 0.497 0.146 0.254 t.4 71.8 389.8 0.65 461.6 The swing curve is plotted in Fig. 12.50from which it is obviousthat rhe

\\4gqe!_lg',gf tyq]g1_Analysis PowerSystemStability | 505; 0.40 l.s 0.998 1.5 - 0.5 - 2.7 2.8 I 0.45 1.5 1.0 1.5 - 0.5 - 2.7 0.50 1.5 1.0 1.5 - 0.s - 2.7 0.1 86.6 0.55 1.5 0 . 9 9 8 51 . 5 - 0.5 -2.7 - 2.6 89.4 89.5 86.9 I 0.65 1.5 0.96 t.44 - 0.44 - 2.4 - 10.3 73.7 0.70 1.5 0.894 1.34 -0.34 -1.8 - 12.r 63.4 300! 0.75 1.5 - 13.0 51.3 I 0.80 1.5 0.781 t.l1 - 0.17 - 0.9 - 12.6 38.3 0.85 1.5 - 10.7 25.7 I \\ 0.90 1.5 0.62 0.932 0.068 0.4' - 7.4 15.0 0.95 1.5 - 3.r 7.6 t 250I I 1.00 1.5 0.433 0.6s 0.35 r.9 4.5 i 1.7 6.2 Ii r pote r.05 r.5 0.2s9 0.39 0.61 3.3 r2.4 i 6.2 22.3 I switchoff i 1.10 1.5 34.5 L 1.15 1.5 0.133 0.2 0.8 4.3 9.9 47.6 1.20 1.5 60.1 6 2ooi- / / l r.25 1.5 0.079 0.119 0.881 4.8 12.2 7r.O 1.30 1.5 79.6 q) 1.35 1.5 l3.l 95.6 r.40 1.5 gg.g Ig) 1.45 1.5 0.107 0.161 0.839 4.5 12.5 1.50 1.5 o 10.9 IE 0.214 0.322 0.678 3.7 8.6 o 6.0 (L .) 1 5 0 M A C H I N EU N S T A B L E 0.38 0.57 0.43 2.3 '3 . 3 \\ q) 0 . 5 6 6 0 . 8 4 0 .1 6 0 . 9 J) 0.738 1.11 - 0.11 - 0.6 0.867 1.3 - 0.3 - 1.6 to 0.946 t.42 - 0.42 - 2.3 0.983 1.48 - 0.48 - 2.6 100 0.997 1.5 - 0.s - 2.7 r I ,r l--,r- r ]...r r The swing curveis plotted in Fig. 12.51from which it follows thatthe sysrem is stable. o'5 1.0 Single pole switchoff Time (sec) /'/' | ' R/eclosure(faultcleared) Fig. 12.50 swingcuruefor threepoleswitchingwithreclosure M A C H I N ES T A B L E Table 12.10 swingcurvecalculation-singlepoleswitching A I BO I I t,,, I I',uu^ ,s:itth l'r, 5'4P,, Ab b (pu) elec deg elec deg SEC (pu) (pu) elec deg o) 0.0 4t.80 o 0 1.5 0.661 1.0 0.448 r.2 41.8 o) 0.224 1.2 3.6 41.8 o , 0.827 0.667 0.s52 0.436 2.4 43.0 o) 60 f o oory C) 0.05 0.827 0.682 0.s64 0.075--- L o o o) ra 0.10 0.98s 0.726 0.7Ls 0.285 1.5 5.1 46.6 i- l--r 0.15 0.98s 0.784 0.77 0.230 r.2 6.3 5r.7 o.20 0.985 0.848 0.834 0.166 0.9 7.2 58.0 05 o.25 0.98s 0.908 0.893 0.107 0.6 1.8 65.2 0.30 0.985 0.956 0.940 0.060 0.3 8.1 73.0 4325---+ Time (sec) 0.35 1.5 0.988 1.485 0.485 - 2.6 5.5 81.1 Fig. 12.51 Swingcurvefor singlepoleswitchingwithreclosure (Contd....)

506 | Modernpo@is I ina PROBTEIlIS from its prefault position, determinethe maximum load that could be transferredwithout loss of stability. t 2 . l A two-pole, 50 Hz, 11 kv turboalternatorhas a rating of 100 Mw, powel factor0.85 lagging.The rotor hasa momentof inertia of a 10,000 I2.8 A synchronousgenerator is feeding 250 MW to a large 5O Hz network over a doublecircuit transmissionline. The maximum steadystateDower 12.2 Two turboalternatorswith ratings given below are interconnectedvia a that can be transmitted over the line with both circuits in operation is short transmissionline. 500 MW and is 350 MW with any one of the circuits. Machine 1: 4 poIe, 50 Hz, 60 MW, power factor 0.g0 lagging, A solid three-phasefault occurring at the network-end of one of the lines Machine moment of inertia 30,000 kg-rn, causesit to trip. Estimate the critical clearing angle in which the circuit breakersmust trip so that synchronismis not lost. 2 pole, 50 Hz, 80 MW, power factor 0.85 lagging, moment of inertia 10,000 kg--' What further information is neededto estimate the critical clearing time? 12.9 A synchronousgenerator representedby a voltage sourceof 1.05 pu in Calculate the inertia constantof the single equivalentmachine on a base of 200 MVA. serieswith a transient reactanceof 70.15 pu and in inertia constantF/ = 4.0 sec,is connectedto an infinite inertia systemthrough a transmission t 2 . 3 Power station t hasfour identical generatorsetseachrated g0 MVA and line. The line has a seriesreactanceof70.30 pu, while the infinite inertia each having an inertia constant7 MJA4VA; while power station 2 has systemis representedby a voltage source of 1.0 pu in series with a three sets eachrated 200 MVA, 3 MJA4VA. The stations are locatld transientreactanceof 70.20 pu. close togetherto be regardedas a singleequivalentmachinefor stability studies.Calculatethe inertia constantof the equivalentmachine on 100 The generatoris transmitting an acti'repower of 1.0 pu when a three- MVA base. phasefault occurs at its terminals.If the fault is clearedin 100 millisec, determine if the system will remain stable by calculating the swing 12.4 A 50 Hz transmissionline 500 km long with constantsgiven below ties curve. up two large power areas 12.10 For Problem 12.9find the critical clearingtime from the swin! currrefor R = 0.11 f)/km L - 1.45mH/km a sustainedfault. C = 0.009 lFlkm G=0 l 2.l l A synchr t ) nougsener at or epr esent cbdy a volt ageof l. l5 pu in ser ies F i n d t h e ,s. f- -c' :. ,t,rJl v s l z f e c f e h i lrirtrrJr lr ir mr r rirtr i, ,f ,l rl ,,t/t | -_ ll/ | _- tnll t,\\/ /..^-^+^-+r with a transientreactanceis c<lnnectetdo a large power system with volt t t gc 1. 0 pu t hr ot lgh l powcr r r clwor k.Thc cquivalcntt lar r sient | v Rt ,\\rv A V \\Lt,|t:ltdlllr. transf'ereactanceX betweenvoltagesourcesis 70.50pu. After the occurrenceof a three-phasteo grounclfault on one of the lines What will the steadystatestability limit be if line capacitanceis also of the power network, two of the line circuit breakersA and B operate sequentiallyas follows with correspondingtransient transfer reactance neglected?What will the steadystatestability limit be if line resistance given therein. (i) Short circuit occurs at 6 = 30\", A opensinstantaneouslyto make X i s a l s on e g l o c tc dC' / o rn n rc l tot n tl rc rcsul ts. = 3.0 pu. (ii) At 6 = 60o,A recloses,X = 6.0 pu. t2.5 A power deficient area receives 50 MW over a tie line from another (iii) At 5=75o, A reopens. area.The maximum steadystatecapacityof the tie line is 100 MW. Find (iv) At d = 90o, B also opensto clear the fault making X = 0.60 pu the allowable sudden load that can be switched on without loss of stability. Check if the systenrwill operatestably. 1 2 . 6 A synchronous motor is drawing 30vo of the maximum steady state 12.12 A 50 Hz synchronousgeneratorwith inertia constant H = 2.5 sec and power from an infinite bus bar. If the load on motor is suddenly a transientreactanceof 0,20 pu feeds 0.80 pu active power into an increasedby 100 per cent,would the synchronismbe lost? If not, what infinite bus(voltage I pu) at 0.8laggingpower lactor via a network with is the maximum excursionof torque angle about the new steadystate an equivalent reactanceof 0.25 pu. r<ltorposition. A three-phasefault is sustainedfor 150 millisec acrossgenerator terminals.Determinethrough swing curyecalculationthetorque angle6, t 2 . 1 The transfer reactancesbetween a generator and an infinite bus bar 250 millisec, after fault initiation. o p e ri l ti n gi rf2 0 0 k V trn d e rv a r i ouscondi ti onson fhei nterconnectoarro: Pretault S0 0 per phase During fault m O per phase Postfhult 2ffi {) per phase

PowerSystemStabitity 5(D 12.13 A 50 Hz, 500 MVA,400 kV generator (with transformer)is connected I to a 400 kV infinite bus bar through an interconnector.The generator 12. Kundur,P., Power SystemStability and Control, McGraw-Hill, New York, 1994. hasF1- 2.5 MJA4VA, voltagebehind transientreactanceof 450 kV and is loaded 460 MW. The transfer reactancesbetween generator and bus 13. Chakrabarti,A., D.P. Kothari and A.K. Mukhopadhyay,Performance Operation and Cqntrol of EHV Power TransmissionSystems,Wheeler Publishing, New Delhi,1995. 14. Padiyar,K.R., Povter System Stability and Control, Znd 1 . 0p u lcatlons, Hy During fault 15. Sauer,P.W. and M.A. Pai, Power SystemDynamicsand Stabiliry, Prentice-Hall, Postfault 0 . 7 5p u New Jersey,1998. Calculatethe swing curveusing intervals of 0.05 secand assumingthat Papers the fault,is clearedat 0.15 sec. I2.I4 Plot swing curvesand checksystem stability for the fault shown on the 16. Cushing,E.W. et al., \"Fast Valving as an Aid to Power System Transient systemof Example 12.10for fault clearing by simultaneousopening of breakersat the endsof the faulted line at three cycles and eight cycles Stability and Prompt Resynchronisationand Rapid Reload After Full L,oad after the fault occurs.Also plot the swing curye over a period of 0.6 sec t if the fault is sustained.For the generatorassumeH = 3.5 pu, G = 1 pu Rejection\",IEEE Trans, L972, PAS 9I 1624. and carry out the computations in per unit. 17. Kimbark, E.W., \"Improvement of Power SystemStability\", IEEE Trans., 1969, P A S - 8 8 :7 7 3 . 18. Dharma Rao, N. \"Routh-Hurwitz Condition and Lyapunov Methods for the 12.15Solve Example 12.10for a LLG fault. TransientStability Problem\", Proc. IEE, 1969, 116: 533. 19. Shelton,M.L. et al., \"BPA 1400MW BrakingResistor\",IEEE Trans., 1975,94: 602. REFERNECES 20. Nanda,J., D.P. Kothari, P.R. Bijwe and D.L. Shenoy,\"A New Approach for Books Dynamic EquivalentsUsing Distribution FactorsBasedon a Moment Concept\", 1 StevensonW, .D., Elementsof Power SystemAnalysis, 4th edn., McGraw-Hill, Proc. IEEE Int. Conf. on Computers, Systemsand Signal Processi4g, Bangalore, New York, 1982. Elgerd, O.I., Electic Energy Systems Theory: An iniroduciion, 2nd edn., Dec. 10-12, 1984. \\ McGraw-Hill, New York, 1982. 2 r . Dillon, T.S., 'Dynamic Modelling and Control of Large Scale System\", Int. 3. Anderson,P.M. and A.A. Fund, Power SystemControl and Stability, The Iowa StateUniversity Press,Ames, Iowa, 1977. Journal of Electric Power and Energy Systems,Jan. 1982, 4: 29. 4. stagg, G.w. and A.H. o-Abiad, computer Methods in Power system Analysis, 2 2 . Fatel,R., T.S. Bhatti and D.P. Kothari,\"Improvementof PowerSystemTransient Chaps9 and 10, McGraw-HillBook Co., New York, 1968. stability using Fast valving: A Review\", Int. J. of Electric Power components 5. Crary, S.8., Power SystemStability, Vol. I (Steady State Stability), Vol. II (TransientStability), Wiley, New York, 1945-1947. and Systems,Vol. 29, Oct 2001, 927-938. 6 . Kimbark, E.W., Power SystemStability, Vols 1, 2 and3, Wiley, New york, 1948, 23. Patel,R., T.s. Bhatti and D.P. Kothari, \"MATLAB/simulink Basedrransient 7 . Veuikorz,Y.A., TransientPhenomenain Electrical Power System(translatedfrom Stability Analysis of a Multimachine Power System,IJEEE, Vol. 39, no. 4, Oct. the Russian),Mir PublishersM, oscow, 1971. 8. Byerly, R.T. and E.w. Kimbark (Eds.), stability of l^arge Electric power 2002,pp 339-355. Systems,IEEE Press,New York, 1974. L. A+. Patel R., T.S. Bhatti and D.P. Kothari, \"A Novel scheme of Fast valving 9. NeuenswanderJ, .R., Modern Power Systems,InternationaTl ext Book Co., 1971. t0. Pai, M.A., Power SystemStability Annlysis by the Direct Method of Lyapunov., Control\", IEEE Power EngineeringReview, Oct.2002, pp. 4446. North-Holland, Systemand Control Services,Vol. 3, 1981. 2 5 . Patel,R., T.S. Bhani and D.P. Kothari, \"Improvementof Power system Transient I 1. Fouad,A.A and V. Vittal, Power SystemTransientStability Analysisusing the stability by coordinated operation of Fast valving and Braking Resistor\", To TransientEnergy Function Method, Prentice-Hall,New Jersy, 1992. appearin IEE proceeriings-Gen.,Trans and Distribution.

PowerSystemSecurity Sll I Most of the security related functions deal with static \"snapshots\" of the power system.They haveto be executedat intervalscompatiblewith the rate of changeof systemstate.This quasi-staticapproachis, to a largeextent,the only practical approachat present,since dynamic analysisand optimization are conslder4bly mole {!fficu!! 4nd cqmpurallo44lly 1aqtelime corrsulurg, System security can be said to comprise of three major functions that are carriedout in an energycontrol centre:(i) systemmonitoring,(ii) contingency analysis,and (iii) comectiveaction analysis. Systemmonitoring suppliesthe power systemoperatorsor dispatcherswith pertinentup-to-dateinformation on the conditionsof the power systemon real time basisas load and generationchange.Telemetrysystemsrneasurem, onitor and transmit the data,voltages,currents,currentflows and the statusof circuit breakersand switchesin every substationin a transrnissionetwork. Further, other critical and important information such as frequency,generator outputs and transformertap positionscan also be telemeteredD. igital computersin a control centre then processthe telemetereddata and placethem in a data base 13.1 INTRODUCTION form and inform the operatorsin case of an overloador out of limit voltage. Important data are also displayedon large size monitors. Alarms or warnings In Chapter7, we have beenprimarily concernedwith the economical operation may be given if required. of a power system' An equally important factor in the operation of a power Stateestimation(Chapter14) is normally usedin such systemsto combine systemis the desire to maintain system security. System security involves telemetereddata to give the best estimate(in statisticalsense)of the curreltt practicessuitably designedto keep the systemoperatingwhen componentsfail. systemcondition or \"state\".Such systemsotten work with supervi$orycontrol systemsto help operatorscontrol circuit breakersandoperateswitches and taps Besideseconomizingon fuel costand minimizrngemissionof gases(co, cor, remotely.Thesesystemstogetherare called SCADA (supervisorycontrol and (aNcllooyxl l,\"asspeoscreu)o),rert\"ehpqeoupwiopewmr eesrynssdtytasetmme arisngsoehnI.ofeutlhwdeibtpheroolop. w\"err,aputriofobcnaaabsl licyli,at.ysdeioncfgu,f asr eyi ls,u,At.reenrsncoboplanectriakntsiooeunst_. d a t a a c q u i s i t i o n )s y s t e l n s . the systernas a whole or its tnajor parts may completely collapse .This is The second ma-ior security function is contingency analysis. Modern normally referred to as systemblackout. All these aspectsrequire security operationcomputershavecontingencyanalysisprogramsstoredin them.These loreseepossiblcsystetntroubles(outages)beforetheyoccur.They study outage c ons t r a i n epdo w e rs y s te mo p ti mi z a ti on(S C O). events and alert the operatorsto any potential overloadsor serious voltage vi ol ati tl nsF. or exalnplet,he sir nplesftir r m of cont ingencaynalysiscan be put i Since security and economy normally have conflicting requirements, it is together with a standard LF program as studied in Chapter 6, along with nappfopriateto treatthem separatelyT. he fina.laim of economyis the secur ity proceduresto set up the load flow dafa for each outageto be studied by the LF plogram. This allows the system operatorsto locate def'ensiveoperating eolumpneecrrtaigoteenntohcfyectohsneydsuittteiiolmitnysa.ctTohmmeipneaimmneuyr.mgTehnceoceyscnto,enrwgdiyitthmioanthnweailglgedumeaepranentnstydeosentdetamhlel(eEsvMeiavStei)oriintsyootoff stateswhere no single contingencyevent will generateoverloadsand/or voltage t iolat io n so f o p e ra ti n gl i rn i violation:;.This analysisthus evolves operatingconstraintswhich may be severeviolat io n sr e s u l t f i o m ts(b r a n c hf' l ow sand bus vol tagel i mi ts).The m ost cntpi oycdin t hc liD ( ccot r olnicdispat cha) nd UC ( unitcor nnr it r nclrpt )r ogr ar r r . cont ingenciesA. n irnportantpart of securitys tu dy, Thus contingencyanalysiscarricsout ornergcncyidentil'icationancl\"what if'' simulations. therefbre,moves around the power system'sability to withstanrjthe effectsof contingencies.A particular systemstateis said to be secureonly with reference The third major securityfunction, corrective action analysis,permits the to one or more specific contingency cases, and a given set of quantities operatorto changethe operationof the power systemif a contingencyanalysis monitoredfor violation. Most power systemsare operatedin such a way that program predicts a seriousproblem in the event of the occurrenceof a certain any singlecontingencywill not leaveother.o-pon\"nts outage.Thus this providespreventiveand post-contingencycontrol. A simple that cascadingfailures are avoided. heavily overloaded,so example of corrective action is the shifting of generationfrom one station to another.This may result in changein power flows and causinga change in loading on overloadedlines.

j5!2,1 M o d e rnP o @i s I srr Thresethreeftrnctionstogetherconsistoi a very compiex set of toois that heip in the secureoperationol'a power system. T3.2 SYSTEM STATE CLASSIFICATION o -oo o-= o o c o o oo 3 L o o- p bo . 6q Dyliacco [13] andfurtherclarified by Fink andCarlsenl23l in orderto define co, E8 EE relevant EMS (Energy ManagementSystem)functions.Stott et. al [15] have o (!6 also presenteda more practical static security level diagram (seeFig. 13.1) by L incorporating correctively secure(Level 2) andcorrectableemergency(Level4) q) security levels. B E o. In the Fig. 13.1, arrowed linep representinvoluntary transitions between q Fo r: .9 Levels 1 to 5 due to contingencies..Theremoval of violations from Level 4 -o= normally requiresEMS directed\"corrective rescheduling\"or \"remedial action\" zo bringing the systemto Level 3, from where it can return to either Level I or gH€ 2 by further EMS, directed \"preventive rescheduling\" depending upon the Eo oo oo_ desired operationalsecurity objectives. E ; .o. '=O- =o- .^;EEt b- E o Levels I and 2 representnormal power systemoperation.Level t has the o_o x ideal securitybut is too conservativeand costly. The power systemsurvivesany o of the credible contingencieswithout relying on any post-contingency corrective E o--c.Y action. Level2 is more economical,but dependson post-contingencycorrective rescheduling to alleviate violations without loss of load, within a specified abgo oo ! i e og, o period of time. Post-contingencyoperating limits might be different from their c o pre-contingencyvalues. o o o) = 9 8 (coJ 13.3 SECURITY ANALYSIS L oo P c 9 c System securitycan be broken down into two major functions that are carried EO =O (f ) out in an operationscontrol centre: (i) security assessmenta, nd (ii) security control. The tormer gives the security level of the systemoperating state.The dH d8 o latter determinesthe appropriate security constrainedscheduling required to optimally attainthe targetsecuritylevel. x g> co U) 'oE9 .b c) C) The security functions in an EMS can be executedin 'real time' and 'study' =-fEi. F O E o) (E modes. Real time application functions have a particular need for computing .= g, o speedand reliability. g)'E E8E c E 'fhe static securit.vlevel of a power systemis characterisedby the presence oo or rrtherwiseof emergencyoperating conditions (limit violations) in its actual cc o (pre-contingency)or potential (post-contingency)operating states. System FO o security assessmenits the processby which any such violations are detected. (g(J o o System e.:s^€sstlt€rrt involves two func tions: bod o o_ (i) systemmonitoring and (ii) contingencyanalysis.Systemmonitoring provides :85b b8 the operator of the power systemwith pertinent up-to-dateinformation on the d( l )d- .' o5 at currentcondition:;clf the power system.In its simplestform, this just detects 9c violations in the actual systemoperatingstate.Contingencyanalysisis much .=9= .d, 6H los i lr 5co g#EEz E C g (t) .= ('5 o 3 HE'= 9c =eee o y F.O d8 6- o L 0) o o) ac) o E o o) -o ::0 o 8s 9' : = (l)- -E -((o)[ oo - ,*^A 9i : >o ga dE r'/^ Y q) *, ^'kLua to >o )oc , _9!) Ail e ai L A>o zcof rxC . J3 C) J3(D

,.sfe ;l Modern power SvstemAnalvsis PowerSystemSecurity I_ srai Only a small proportionof work on optimal power flow (OPF)hastakeninto S L A C KB U S t +s*yts f+o*ys accountthe security constraints.The most successfulapplicationshave beento 1.02421-5.0\" 3 the security constrainedMW dispatch OPF sub-problem. The contingency- 1 . O 6 t O \"1 --.>40.7+j1.2 _ 39.5_i3.0:_ + 1 8 . 9- y 5 . 2 _ 1 6 . 9+ 1 3 . 2 - <_ 4 1 . 0 2 3 6 t - 5 . 3 \" ,f88.e-y8.6 constrainedvoltageivar reschedulingproblem, asof the writing of this text, still ,l -Ja.a--/e.8 + remains to be solved to a satisf desree. i6.3-j2.3 The total number of contingency constraintsimposed on SCO is enormous. --zt.s-i5.9 t The SCO or contingency constrainedOPF problem is solved with or without I z+.2*js.a 'L?:'-q''----'---/ first optimizing with respectto the basecase(precontingency)constraints.The t- az.s* l + -' u- \" ' ' generalprocedure adoptedis as follows: 1.0474t-2 -.i5;3n.7;-il'7-.-2+-*s 5 (i) Contingency analysis is carried out and cases with violations or near *S4.9 +17.3 Yi 1 . o 1 7 g t - 6 . 2 \" violations are identified. * o o* i t o ) , zo+1t o (ii) The SCO problem is solved. !G-) (iii) The rescheduling in Step 1 might have created new violations, and t40 +i30 therefore step 1 should be repeatedtill no violations exist. Fig. 13.2 Base Case AC Lineflowfor sample5 bussystem Hence,SCO representsa potentially massiveadditional computing effort. S L A C KB U S {+s*yts f+o+7s An excellent comprehensiveoverview of various available methods is presentedby Stott et. al [15]. 1 . 0 1 0 7 1 -5 . 9 \" 3 4 1.00682- 6.6\" There is still great potential for further improvement in power system security control. Better problem formulations, theory, computer solution 1.0610\" 1 ->48.6 + j5.2 -46.7 - j5.3<- + 38.5-10.6 -38.4-i1.1 :- { - r . o- 7a. s methodsand implementationtechniquesare required. +81.8-i 5.5 -- 3a?-/'er + T3.4 CONTINGENCYANALYSIS In the past many widespreadblackouts have occurredin interconnectedpower t- eo.o* i r . s- ; t . r systems.Therefore, it is necessaryto ensure that power systems should be 1.0468t-2. :63.1 + j1O.2 -<--- 61.6-i8.9 5 1.O114t-6.4\" operatecm! osf economic:r!lysuch that povrer is cle!i.rerecrleliably. Reliable rlr { 2s+ilo loo*lt o operation implies that there is adequatepower generationand the samecan be \\ lJ./ transmittedreliably to the loads.Most power systemsare designedwith enough t 4 0+ i 3 0 redundancyso that they can withstandall rnajortailure events.Here we shall Fig. 13.3 PostoutageAC LoadFlow(Linebetween2 and4 is open) sttrdy thc possiblcconsccprcncc:rsnclrcrncdialactionsrcquircd by two nrain f ailur eev e n ts ;l i n e .o rrta g easn c lg e n e ra t i ngrrni tfai l ures. S L A C KB U S t a sr r t s I a o, 7 s To explainthe problelnbriefly,we considerthefive-bussystemof Ref'erence (c) 0 . s e 3 5-/ i 1 i . 3- s 2 . 1 LI0J.The basecaseloadllow resultslbr theexamplearegiven in t ig. 13.2and, 1 ' 0 6 t 0 \"1_|l1_' _5 4r . 4+ 6.s\" o1 s 1 . 4+ i 7.3 -s 1.1 + 8 0 . e + 1 3 .|2 - i 9 . -7 | - s lt r r wir f ' lo w< tf2 4 .7MW a n d3 .6 MV AR on the l i ne f} om bus 2 to hus3. L.ct 1 qq.z _j1z .s tIlt;lf_ t ls lls s t llll cth a ta t p rc s c n lw, e i l rc o n l y i n tcresteicnl the MW l oadi ngof thel i ne. Let us examinewhat will h appenif the line from bu s2 to bus4 wereto open*. | +s,s* 1tz.t ':t't' - 52.s-i 13.0+ ' l' lr cr c s ulti n gl i n c l ' l o w sa r tcvl o l ta g c su l c s how ni n l ri g. 13.3.t t nrayb enotcd ':: )ll that the flow on the line 2-3 has increasedto 31.5 MW and that most of the other line flows are also changed.It may also be noted fhat bus voltage -5 magnitudesalso get aff'ectedp, articularly at bus4, the change is almost2To less 0.8994t-16.? from 1.0236to 1.0068pu. Supposethe line from bus 2 to bus 5 were to open. 1 2 0+ i 1 0 'foo*yro Figure 13.4showsthe resultingflows and voltages.Now the inaximum change t:fgl.g.j \",jus 5 which is almost 107oless. xSimulationof line outageis more complexthan a generatoroutage,sinceline Fig. 13.4 PostoutageAC LcadFlow(Linebetwee2n and5 is open) outageresultsin a changein systemconfigurations.

-;t'5- 18 | rr,roderPno@is 5LAUK HUS e t 4 s+ l 1 5 l+o*1s | 1 . o o 6 1_ts . 7 \" 3 1 1 . 0 6 t 0 \"1'l_l -!.6 +1t.^s _ 4s.9_j7.6 <_Ii __22.4_j2.6 _22 .3 +j0.7 <_ 4 1 . O 0 4 3 t - 6 . 1 \" {nze+iz1.7l L- I --i-1-I { t.s- 1t.t --- V_21.5_j4.8 -25.2-j4.4 + i '''\" 1 - r z o .-g1 t s I t (l - ' - ---t f-zs-7s.s GiveAlarmsignal GiveAlarmsignal . _- 5 o 9 9 5 6 / _ - 7. 1 \" 2 ..l ] -l- r _+ _53.6+16.8 <- -52.5 -16.5 {oo*iro u+ 1.o24st-3.7. i Y l , z o+ 1 l o Fig. 13.5 PostoutageAC LoadFlow(Generato2r outage,lostgenerationis pickedup by generator1) Figure 13.5is an exampleof generatoor utageanclis selectedto explainthe tact that generatoroutagescan also result in changesin line flows and bus voltages.In the exampleshowr in Fig. 13.5all the generationlost from bus 2 is picked up on the generatorat bus 1. Had there been more thanZgenerarors in the sample systemsay at bus 3 also,it on bus 2 is made up by an increasein was possiblethe loss of leneration generation at buses 1 and 3. The differencesin Iine flows anclbus voltagesvrouldshow how the lost gener.ation is sharedby the remaining units is quite significant. It is important to know which line or unit outageswill renderline flows or 1 ; --'t * r i A -. . . i u i o r \" . \" \" - i ;'{ o''flf'i' ' '' l voltagesto crossthe lirnjts. To find the eff'ectsof outages,contingencyanalysis techniques are empioyeci.Contingency analysis models single failure events no f i i ( i' e' one -l i n co u ta g c soiro l l c u n i t o u tu g cso) r nrul ti pl cccpri pnrcnfitri l urccvc^ts (failure of multiple unit or lines or their combination) one afteranotheruntil all l'r'.,o \"credible outages\"areconsidered.For eachoutage,all lines anclvoltagesin the netrvorkare checkedagainsttheir respectivelimits. Figure 13.6depictsa flow __l chart illustrating a simple method for carryingout a contingencyanalysis. -T-,vrri\\o=l_ation-?>*v-et_s-qjcJiventarn One of the important problems is the selection of \"all credible outages,,. Executiontime to analyseseveralthousandoutagesis typically I min basedon ,*\" I computerandanalyticaltechnologyasof 2000.An erpproxirnatrenodelsuclras DC load flow may be used to achievespeedysolution if voltage is aiso , N' - - o - i Ail --\\ required, then full AC load flow analysishas to be carried out. I FIg. 13.6 A simpletechniquefor contingencaynalysis

620j;:f uodern PowerSystemAnatysis I Y .FAUI-L'I(s ^I Ja . C? Df r E Ir r I I Ya r E ? r t Dr E l tI I I V I I limits and those violating their limit can be informed to the operator for necessarycontrol action. A .securityanalysisprogramis run in a load dispatchcentre very quickly to help The generationshift sensitivity factors are linear estimatesof the change in the operators.This can be attempted by carrying out an approximate analysis line flow with a changein power at a bus. Thus, the effects of simultaneous and using a computer system having multiple processorsor vector processors principle of superposition. Let us assumethat the loss of the ith generatoris to be madeup by governor y anarysts.I ne s may uate an equrvalent action on all generatorsof the interconnectedsystemand pick up in proportion shouldbe usedfor neighboursconnectedthrough tie-linos. We can eliminateall to their maximum MW ratings.Thus, the proportion of generationpick up from unit k (k * i) would be non-violation casesand run complete exact program for \"critical\" casesonly. (13.7) This can be achievedby using techniquessuch as \"contingency selection\"or \"contingency screening\", or \"contingency ranking\". Thus it will be easy to warn the operation staff in advanceto enablethem to take corrective action if one or more outageswiil result in seriousoverloadsor any violations.One of the simplest ways to present a quick calculation of possible overloadsis to employ network (linear) sensitivity factors.Thesefactors give the approximate changein line flows for changesin generationin the system and can be where calculatedfrom the DC load flow. They are mainly of two types: Pn,,,,,u=,^maximum MW rating for rnth generator 1. Generationshift factors 2. Line outagedistribution factors g*i= proportionality factor for pick up on kth unit when ith unit fails. Briefly we shall now describe the use of thosefactors without deriving them. Now, for checking the /th line flow, we may write Reference[7] gives their deri iation. j, = ff * 0r; APo, - E,lau, \\ri LPotl (13.8) The generation shift factorsl cr.,;are defined as: (13.4) In Eq. (13.8) it is assumedthat no unit will violate its maximdm limit. For where, unit limit violation, algorithm can easily be modified. Similarly the line outagedistribution factorscan be usedfor checking if the 4t = Changein MW power flow on hne I when a changein genera- line overloadswhen solllc of the lines are lost. tion, AP\", takes place at the ith bus The line outage distribution factor is defined as: Here, it is assumedthat LPotis fully compensatedby an equal and opposite d,,,= * (13.e) change in generation at the slack (reference)bus, with all other generators remaining fixed at their original power generationsT. he factor al,then gives the Ji sensitivityof the /th line flow to a changein generationat ith bus.Let us now study the outage of a large generating unit and assume that all the lost where generation(Pod would be supplied by the slack bus generation. Then dt,i = line outagedistribution factor when monitoring /th line atter an outageof ith line. APo,- -P1i Aft = changein MW flow on /th iine' (13.s) fi - precontingency line flow on ith line lf precontingencyline flows on lines / and i, tlrepower flow on line / with line andthe new power flow on eachline could be calculatedusing a precalculated i out can be found out employing \"d\" factors. set of \" d' factors as given below. ft = f i * dti APc, for all lines V / (13.6) ?,=ff *d,,,f,o (13.10) where, ft - power flow on /th line after the failure of ith generator Here, fi ^d foi =precontingency or preoutageflows on lines / and i respectively f i = power flow on /th line before the failure or precontingency power flow fr = powerflow on /th line with ith line out.

iii l ModerPno J\"srr a by precaiculating 'd' factors ali the lines for Thus one can check quickiy overloading for the outage of a particular line. This can be repeatedfbr the j=l j=2 j=3 j=4 j=5 j=6 j = 7 outage of eachline one by one and overloadscan be found out for corrective (line 1-2)(linel-3)(Iine2-3)(line2-4)(line2-5)(line3-4)(line4-5) 0. 0 1. 0001- 0. 3331- 0. 2685 - 0. 2094 0. 3735 0. 2091 action. It may be notedthat a line flow can be positive or negative. Hence we must (linel-2) check / agarnst- Jt ^u* as well as h **. Llne tlows can be louncl out usmg telemetry systemsor with stateestimation techniques.If the network undergoes =- J3 (\\ rlr \\irnw e L2- J-)3) --v0.4542 0.4545 0 0.4476 0.3488 -0.6226 -0.3488 = 4 (hne 2-4) 0.4418 0.6642 -O.44r8 any significant structuralchange,the sensitivity factors must be updated. = 5 (line2-5) -0.3634 0.3636 0.4443 0.0 = 6 (line 3-4) 0.0 0'3321 1.0 = 7 (line4-5) -0.1819 0.1818 0.2222 0.2835 0.5580 0.0 -0.5580 1.0002 -0.3321 0'0 0.5451 -0.5451 -0.6662 0.7161 0.18i6 -0.1818 -0.2222-0.2835 Find the generationshift factors and the line outagedistribution factors for the It has been found that if we calculate the line flows by the sensitivity five-bus sample network discussedearlier. methods,they come out to be reasonablyclose to the valuescalculatedby the Solution Table 13.1 gives the [x] matrix for the five bus samplesystem, full AC load flows. However, the calculations carried out by sensitivity together with the generation shift distribution factors and the line outage methodsarefasterthan thosemade by full AC load flow methodsand therefore distribution factors are given in Tables I3.2 and 13.3 respectively. are used for real time monitoring and control of power systems.However, where reactive power flows are mainly required, a full AC load flow method Table 13.1 X Matrixfor Five-busSampleSystem(Bus 1 as a reference) (NR/FDLF) is preferred for contingency analysis. 0 The simplestAC security analysisproceduremerely needsto run an AC load 0 0.05057 0.03772 0.04029 0.4714 flow analysisfor eachpossibleunit, line and transformeroutage.One normally 0 0.03772 0.08914 0.07886 0.05143 does ranking or shortlisting of most likely bad caseswhich are likely to result 0 0.04029 0.07886 0.09514 0.05857 in an overload or voltage limit violation and other casesneed not be analysed. 0. 0.04714 0.05143 0.05857 0.13095 Any good P1(performanceindex can be selected)is usedfor rankirig.One such P/ is (13.11) Table 13.2 GenerationShiftDistributioFnactorfor Five-busSystem For large n, PI will be a small numberif all line flows are within limit, and will be large if one or more lines areoverloaded. Bus I Bus 2 For rr = I exactcalculationsciur be donefbr P1.P1 tablecan be orderedfrom l=1(line1-2) - 0.8428 largest value to least. Suitable number of candidatesthen can be chosen for /=2(line1-3) - o.t572 further analysis[7]. /=3(line2-3) I=4(line2-4) o.0714 If voltagesare to be included,then the following PI can be employed. /=5(line2-5) 0.0571 /=6(line3-4) 0.0286 (13.12) I=l (line4-5) - 0.0857 - 0.0285 Here, Alvil is the difference betweenthe voltage magnitudeas obtained at the end of the lPlQ FDLF algorithm Alvl-u* it the valuefixed by the utility. Largest vaiue oi Pi is piaceciat the top. The security arraiysistrray rrow -ue startedfor the desirednumbel of casesdown the ranking list. $ummary and Further Reading: Reference [25] has discussedthe concept for screeningcontingencies.Such contingency selection/screeningtechniquesfonn the foundationfor many real- time computer security analysisalgorithms.

l#GrffiE IE!F}ILfr Reference[15] gives a broad overview of securityassessmenat nd contain an stability, the dynamics mainly involves the loads and the meansfor voltage bibliography covering the literature on\"security assessmenrup to control. Ref [11] provides a comprehensivelist of books, reports, workshops and technical papers related to voltage stability and security. ;;;;t]r\", Definitions: [2] Reference [11] gives an excellent bibliography on voltage stability. This topic is discussedbriefly in the next section. A power system at a given operating stateis small-disturbance voltage stable if, following any small disturbance,voltagesnear loads are identical or close 13.6 POWER SYSTEM VOITAGE STABILITY to the pre-disturbancevalues. The conceptof small-disturbance voltage stability is related to steady-statestability (Chapter 12) and can be analysedusing small- fornspaPaewyronocqostgiiwunbtlleiteilgeterhmierema(tdsossttyr,.thvansoaaanecnrshsrsdiiomrqobouuueoinssesetoshecnbuieozolseirennl)ctskstshautepteuragdeasblmcigeniwiliaoiatltisixytfltihynismoftattraurhhsabbmetaniylscssiptohitytehornonaessrctdrsttomrsiibctuntiaailoctrelrbtcnoilpiuoaollisoinltaily.wyoo.cEiefnbLlrlgegee-eautcehcnrctnrairkenoipiIJrgluiayam;gb*ttpiihrhiotlnoeiin\"tsp*idiaeiLu.;nsGrrnbro.diyub\"Ttsrlheeaeminetnhasxbemerclitaesriwitcsarnoknsotooroiitokoouwwsnntr signal (linearised)model of the system. uobavonfecopccilentveovarogapnoglttstlteiatranuaobggbgsllelejteaeeavlbceb(olitcleinelolttisardtadiytgctdeaoe)cibpssaliosilancidwttateoyiesabnirntliculwlsierbtvyhybruosen,asltnetenheadmcosgweaiwsen.iAaettdhnvhpideestorihtssu,weyriaesbsarftabaensnicmylottiestowuytrnerrodameefesaimrsuadnraispntoaisognrimwdtionitaseolslrimcahusoaeiytnvpsidenprteoioetmi!p-on.w\"lntatre;osrnriuetnamr\"diannaandgisnsafattafeaatnneritsdnrde. A power systemat a given operatingstateand subject to a given disturbance ts voltage stable if voltages near loads approachpost-disturbanceequilibrium mleaaIjdnosar tdoseyvqsoutelatmategfreae.iailnucsrtetivasebin.i-lipttyohweotewursoourilptdup.goT\"rthcf\"oryollma-\"p,sgee,nwehraicthorhsaavned transmission lines valuei. The conceptof voltage stability is related to the ffansientstability of a resulted in several power system.The analysis of voltage stability normally requiressimulation of the systemmodelled by non-linear diffdrential-algebraicequations. (i) south Florida, usA, system disturbance of 17 May r9g5, (transient, 4 sec) A power systemat a given operatingstateand subject to a given disturbance undergoes voltage collapse if post-disturbanceequilibrium voltages are below (ii) French systemdisturbancesof December acceptable limits. Voltage collapse may be total (blackout) or partial. The (longer term). voltage instability and collapse may occur in a time frame of a second.In this case the term transient voltage stability is used. Sometimes it may take up to tens of minutes in which casethe term long-term voltage stability is used. The term voltage security meansthe ability of a system, not only to operate stably, but also to remain stable following any reasonably crediblebontingency or adversesystemchange such as load increases[2]. Voltage stability involves dynamics, but load flow based static analysis methods are generally used for quick and approximate analysis. Figure 13.7depictshow voltage stability can be classifiedinto transient and long-term time frame l2l. 19, r97g and January 12, rgg7, Transientvoltage stability Longer-termvoltagestability (iii) swedish systemdisturbanceof December 27, rgg3 (longer term, 55 sec) Inductionmotor dynamics Increase in load/powertransfer (iv) Japanese(Tokyo) system disturbanceof July 23, 1gg7irorg\", term, z0 min) Generator/excitationdvnamics LTCtransf& Distvolt.Reg. (v) NREB grid disturbancein India in 19g4 and r9g7. Primemovercontrol Load diversity /thermostat (vi) Belgium, Aug 4, 1992.(longer term, 4.5 min) Mech. switchedcapacitors/reactors Excitationlimiting Gasturbinestart-up (vii) Baltimore, washington DC, usA, 5th July 1990 (longer rerm, insecure Under voltageloadshedding for hours) Hence, ', ;,ilir;;;,:;; ffiri#il;\";mitigation a full understandingof voltage stability phenomenaand designing schemesto prevent voltase instabilitv is nf o,raqr'or,,o+^,.+:r:a:^^ ;agestability. P r o t e c t i v e r e l a y i n g i n c l u d i n go v e r l o a d p r o t e c t i o n 'lef nptheenngoinmeeenrsa..BVeocltaadgseeoinfstthaibsi,livtvoltaangde 100 rterchangeably.by many .\"r\"ur\"h..r. Time-seconds- voltage instability or co[apse is a faster dynamic-processA. s opposedto angle Fiq. 13.7 Voltage stabilitVphenomenaand time responses

SPfti*f uooern po@is wffi;Ewe I ;;;''',h.;A#\";rivneotaelVctratloiigvnletekapegsodetaw' bsRetirloaittcbyooilrinatytanrndopglr.rloovetobosllrtteaamagbnseilgitnlsyeot,ar(.bmoair3lai-tsly.iyywnisecoxchcaorcsonunvcrooeiulrntnsa\")ghosee*taiastbtvt,aiillbiyltoiyliastaytdrreeaissrmseaeaofdsrfeeascnyotdsertldeolemabsdyss. desirable. Unitypowerfactor aSr':vre:re:^niar:evyirlsva,\"ptvo:ov:rs,'sY*stv:aib,rb'l,ieP'l3itwry'1liu.tI^hrnsouta^ut,la?laor:gil:sae^rgisonef,t1essr1yy-nsc2tcoeihntmirnauoenoncvidt*ee,rdosoltoyornsg.ate\"numtyrna,gvngeoslnlm\"tseaisrgtasaeitocbonorislili.latinypiessse. Voltage N o s e p o i n(t k n e e ) basicauy t of a load \\ Loaw of V6 and Pr\"r; samefor constZ load mspirmeoTtbuhhlloeaedmtesssdlao''rIw'esnenwaraiddpfodesrilhtmyioosutnssot'eoifndpv:otPoisml-ttVa-edgcfieosutlirlunovrsewbtsaainabngnicldiltlaoyena-advorufeltocaouigvfrtseve,eontsrwa.dponuVaorlitynchsgueerrlvdoloeaaasdsdasbrftuleeoialwuddsuyebp-dassfatoaerretde for othertypesof load, V degradationis faster. especiallyfor radialsystems. / curves (Fig. 13.g),eV curves (Fig. PplP6zy 'ig. 13.8) and methodsto quantify nose puted. Power flow analysis cletermines 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Nlwva!uoehsglaetv\"arreeagoarenftshelvooexowisnusltetoavrssfgoesoernptseeoetapiainsmcbtah,tihlilvrteeyaasulsuim.neeear.coiesacfatettlhpnoneetaoadr\"dssbs\"e.tlaelTpTLvhoaiel:ilinutiytrue'pms#(pimnaengiu.gll#oetinipni',ofl\"eaidnlodafil|cdoHawftloeIwss'Ios),.ltu:aAItbiJotlenHli;m;eiI;xi;t#;iisntgs. Fig. 13.8 PV curveswithdifferentloadpowerfactors Effective counter Measures to prevent or contain voltage Only the operating points above the critical points represent satisfactory Instability operating conditions. At the 'knee' of the V-P curve, the voltage drops rapidly with an increasein load demand. Power-flow solution fails to convefgebeyond this limit indicating instability. Operation at or near thel stability limit is impractical and a satisfactory operating condition is ensured by permitting sufficient \"power margin\". (i) Generatorterminal voltage shourdbe raised. 1400 j+Allowable range (ii) Generatortransformertap value may be increased. 1200 P> P, > P,, iNMW (iii) Q-injection should be carried out at an appropriate location. 1000 (iv) Load-endoLTc (on-loadtap changer)should be suitablyused. (v) For under voltage ffuo. Boo to' conditions,strategic load sheddingshould be resorted Prbliener(saiecS)ccsahytiFirbesncroetcdiaeerrtulwedmallaioosensrpnueegesetnianfacgonlftneriodsndnorlesciefsg._ersvhQtamctua\"-ts-rriefl!oilto'foninhna(wtadsgm'stnr-pa'-cuvda^rotpvoiyo^lnco'nfLbdvbarrgliae.retd,ianrom'cescucne'sao^,snrlrrmerrtauirapeLnueid-e-ddus\"i-ns*n.^ods'sg^seeua-,c,trtt.oiriyereobevsanycosa)emeliint\"nlavafdsolgyn.trse,pg*bhihc_.leuiieoe.nn,ianlntgltvadscypntoooalsemuorlewlatepi.rdnodeta.rnEcGadsoHnaenvsvdtnoimioetlitlnrioaisanmngsteseiiao.sosyn:n Systemcharacteristics (ii) isgFnohoveroosrulavulddepiba,slceI2otsxrsautn.ilsItofamsbstishlsyienicorcneoreimsalisnpaeeesssnh,rseoifasruttaaelntbigdynyegoloflsoirnscoacoinflacelarqelaiins,noeejueitrcnartkrive,oostnlhtp.aoelagfnceiemc,doprroueoparr.sctteoaTtnfhhcieises 0.9 0.951.0 1.05 Capacitor Vinpu characteristics Fig. 13.9 Systemand shunt capacitorsteady-stateQ-V characteristics, capacitorMVAr shown at ratedvoltage

Voltage Collapse M Voltage collapse is the processby which the seguenceof eventsaccompanying ; voltage instability le_adsto (v) Post-disturbanceMwA4vAR margins should be ffanslated to pre- the power system. It may unacceptablevoltag6 profile in a significant part of disturbanceoperatinglimits that operatorscan monitor. be manifested in ieverat different\"ways. Voltage (vi) Training in voltage stability basis (a training simulator) for control centre collapse may be characterisedas follows: and power plant operators should be i (iii) The voltage collapsegenerallymanifests itself as a slow decay of voltage. SUIvIMARY It is the result of an accumulative process involving the actions and interactions of many devices,controlJ, and protective iyrt\"*r. The time Power system security (including voltage stability) is likey to challenge frame of collapse in suchcaseswould be of the order of severalminutes. planneqs,analysts;researchersand operatorsfor the foreseeablefuture. As load Voltage collapse is strongly influenced by system conditions and grows, and as new transmission lines and new generations would be characteristics. increasingly difficult to build or add, more and more utilities will face the security challenge. (iv) Reactivecompensationcanbe made most effectiveby thejudicious choice of a mixture of shunt capacitors, static var system and possibly Deregulation and socio-economic ffends compounded by technological synetuonouscondensers. developmentshave increasedthe likelihood of voltage instability. Methods of Improvlng Voltage Stabllity Luckily many creative persons are working tirelessly to find new methods and innovative solutions to meet this challenge. voltage stability can be improved by adopting the following rneans: (i) Enhancingthe localisedreactivepower support(SVC) is more efl,ective REFERNECES and C-banks are lnore economical. ra-tS devices or synchronous condensermay also be used, Books (ii) Compensating the line length reduces net reactance and power flow L l.J. Narguthand D.P. Kothari, PttnterSystemEngincering,'fataMc0raw.Hill, New increases. Dclhi, 1994. (iii) Additional transmissionlinc muy be crectccl.It also improvesreliability. 2, C.W, Taylor, Power SystemVoltag,eStabiliry,McGraw-Hill, New york, 1994. (iv) 3. P. Kundur, Power SystemStability and Contol, Sections2.12, ll.2 and Chapter Enhancing excitation of generator, system voltage improves and e is suppliedto the system. 14, McGraw-Hill, New York, 1994. 4, T,J.E,Miller, Editor, ReactivePower Control in ElectricSyslens,John Wiley and (v) HVDC tie may be usedbetweenregional grids. (vi) By resorting to strategicload shedding, voltage goes up as the reactive Sons,New York, 1982. . burden is reduced. 5. A. chakrabarti, D.P. Kothari and A.K. Mukhopadhyay, Perforurutnce,operation Future Trends and Challenges and Control of EHV Power Transmission Systems,Wheeler Publishing, New Delhi. 1995. (i\\ Ontirnql uorir fr rir-6- nvr' F IE A/ - ' n - i, ,{^*,:^^^ 6. T.V. Cutsemand C. Vournas, VoltageStability of Electric Power Syslerns,Kluwer Academic Publishers,London, 1998. \\-./ nv r \\rlvv.ltvgD. 7. A.J. Wood and W.F. Wollenberg, Power Generation,Operation,and Control, Znd Edn, John Wiley, New York, 1996. (ii) Betterandprobabilisticloadmodelling. E. John J. Gratngerand W.D. Stevenson,Power SystemAnalysis,McGraw-Hill, New York, 1994. (iii) Dsiez.veeslyosptteemchsnF. ioqrueexsaanmdmploedn, eelwsfomr estthuoddyostfonoobnt-aliinnenaedrtwynoarkmeiqcusoivf alalergnrtr 9. G.L. Kusic, Computer-AidedPower SystemsAnalysis,Prentice-Hall,New Jersey, suitablefor voltagestabilityanalysis. 1986. 10. G.W. Stagg and A.H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968.

Papers t4 11. v. Ajjarapu and B. Lee, \"Bibliography on voltage stability\", IEEE Trans. on T4.I INTRODUCTION Power Systems,Vol. 13, No. 1, February 1998, pp lI5-125, State estimation plays a very important role in the monitoring and control of 12. L.D. Arya, \"Security ConstrainedPower System Optimization\", PhD thesis, IIT modern power systems.As in case of load flow analysis, the aim of state Delhi, 1990. estimation is to obtain the best possible values of the bus voltage magnitudes and angles by processing the available network data. Two modifications are, 13. T.E. Dyliacco, \"The Adaptive Reliability Control System\", IEEE Trans. on pAS, however, introduced now in order to achievea higher degreeof accuracy of the Vol. PAS-86, May 1967,pp 517-531 solution at the cost of some additional computations. First, it is recognised that (This is a key paper on system security and energy control system) the numerical values of the data to be processed for the state estimation are generally noisy due to the errors present. Second,it is noted that there are a 14. A.A. Fouad, \"Dynamic Security AssessmentPractices in North America\", IEEE larger number of variablesin the system (e.9. P, Q line flows) which can be Trans. on Power Systems,Vol. 3, No. 3, 1988, pp 1310-1321. measuredbut arc not utilised in the load flow analysis.Thus, the process involves imperfect measurementsthat are redundant and the process of 15. B. Stott,O. Alsac and A.J. Monticelli, \"security Analysis and Optimization\", proc estimating the system statesis based on a statistical criterion that estimates the IEEE, VoL 75, No. 12, Dec. 1987,pp 1623-1644. true value of the statevariables to minimize or maximize the selectedcriterion. A well known and commonly usedcriterion is that of minimizing the sum of the 16. Specialissue of Proc. IEEE, February2000. squares of the differences between the estimated and \"u1le\" (i.e. measured) 17. P.R. Bijwe, D.P. Kothari and L.D. Arya, \"Alleviation of Line Overloads and values of a function. voltage violations by corrective Rescheduling\",IEE proc. c, vol. 140, No. 4, Most state estimation programs in practical use are formulated as July 1993, pp 249-255. overdeterminedsystemsof non-linear equationsand solvedas weighted least- squares(WLS) problems. 18. P.R. Bijwe, D.P. Kothari and L.D. Arya, \"Overload Ranking of Line Outageswith postouragegeneration rescheduling\", Int. J. of Electric Machines and Power Stateestimatorsmay be both static and dynamic. Both have been developed Systems,Yol. 22, No. 5, 1994, pp 557-568. for power systems.This chapterwill introduce the basicprinciples of a static- state estimator. 19. L.D. Arya, D.P. Kothari et al, \"Post Contingency Line Switching for Overload Alleviation or Rotation\", Int J. of EMPg Vol 23. No. 3, 1995, pp 345-352. ln a power system, the statevariables are the voltage magnrtudes and phase angles at the buses.The inputs to an estimator are imperfect (noisy) power 20. P.R. Bijwe, S.M. Kelapure,D.P. Kothari and K.K. Saxena,\"Oscillatory Stability systemmeasurementsT. he estimatoris designedto give the \"best estimate\" of Lirnit Enhancementby Adaptive Control Rescheduliig, Int. J. of Electric Power the system voltage and phaseangleskeeping in mind that there are errors in the and Energy Systems,Vol. 21, No. 7, 1999,pp 507-514. measuredquantities and that there may be redundantmeasurements.The output data are then used at the energy control centres for carrying out several 21. L.D. Arya, S.C. Chaube and D.P. Kothari, \"Line switching for Alleviating Overloadsunder Line OutageConditionTaking Bus Voltage Limits into Account\", Int. J. of EPES, Yol. 22, No. 3, 2000, pp 213-ZZl. 22. P.R. Bijwe, D.P. Kothari and S. Kelapure, \"An Effective Approach to Voltage Securityand Enhancement\",'Int. J. of EPES, Yol. 22, No 7, 2000, pp 4g3-4g6. 23. L. Fink and K. carlsen, \"operating under sttress and Strain\", IEEE spectrum, March 1978,pp. 48-50. 24. S.M. Kelapure, \"Voltage Security Analysis and Enhancement\", Ph.D. thesis, IIT Delhi,2000. 25. G.C. Ejebe, et. al, \"Fast Contingency Screening and Evaluation for Voltage security Analysis\", IEEE Trans. on Power systems,vol. 3, No. 4, Nov. l9gg, pp 1582-1590. 26. T. Van Cutsen, Voltage Instability: \"Phenomena,Countermeasures,and Analysis Methods\", Proc. IEEE, Vol. 88, No. 2, Feb. 2000, pp 208-227.

or,01 - li,r:s, ys An lntroductionto State Estimationof PowerSystems s: :t ri Lf .u-rjll-:yt nalysts (Lnapter t eT^ s t u d i cssu c ha s c c o n o r n i c l i s p l r c h( C h a p r c7r ) , J =7'V /(\\ r|1+/. 4( .\\ 5 ) a lJ). J,, r4-2 LEASTsouARES ESTIMATION:THE BAsrc From Eqs. (14.31) and (14.4),one gets the following expressionfor the index: I7l- IeI soluTroN J =yt!- y'H*.- i<'H'y+ *.'HtH*. (r4.6) For minimizing J = f$), we must satisfy the tollowing condition. gswrenAaoussintocthidhdrwnorcrleaimrearl ltfssaivouetbeinlsvtecs,eittsohlsayreeanlxedimstntfhpelreiaoetahsmctsoieatadatrclhticcoinesoaftrnsildSceeuiaeanmolcsgifenttl-yirftosoihcbqnreaemurlema4vanra.toe3iwlorud,eine-dtegsebhreoleerynofipnereaegrrsnmoataoiblpmvtprliaoreareioymltariberoboeldnellf.eammtpfeoaoodryfwrbbaeeoensrtdhtusiomtyximlsaiasttvenei oedmdncywtsoo.tifratIhnyat e gradoJ = 0 \\r+.t) It is easyto check(see,e.g.[1]) that Eq. (1a.7)leadsto the following result. HtH*,-H'y-O (14.8) This equation is called the 'notmal equation' and may be solved explicitly Assume that x is a vector of n random variabres x1tx2, ..., x' that y is for the LSE of the vector i as another vector of m (> n) random variables !1, J2, ..., J^ and both are related as *. = (H,Il)-t Ht y (14.e) !=Hx+r (r4.1) Er ;il;il.] i lvvbwvienaaehcerleueitaaorserbertlsyilHarmerro(eaeiefsltaaretvarthdoeaek,rwdinslvtahooaeibwmcltelthneeoEet.dmrh)qiu.enaun-tvakrreeintxincooostwnofi(onrdl 4naviym.sel )creyet.nsopTsurrihgeoxesgneamevnnesxtdtcststthionhnareataxtvdhnradceerirlmpitairioebsenlasaeiesssznuwectrsroehtohromrmesueevepnvaatnteernuidcarmbatboeynlerryctischktiaolesml ln orclerto illustrate the methodof LSE, let us considerthe simple problem of The problem is basicallyto obtain thc bestpossibl evalue of the vector x estimatingtwo random variables x, and .rz by using the data for a thret: from the given valuesof the vector y. Sincethe variable r is assumedto be zero m ean,o n e ma y ta k eth e e x p e c ta ti o no f E q. (14.1)anc lget the rel ati on dimensionavl ector y. l-' 0.1 Assume H=10 ll Lrrl f = Hx (r4.2) The matrix Ht F1is then given bY whcre I , , = cxpcctcclvalue ol' x iurd y, respec:tively. u'u =12 I u n ai t s i n v er sies 2I) at hcetTumhaielsvasanhl uvoaewlsusoethfsaobtfuthst hevelooblatuadsgflevooswrl taamtgheeetsrht.oHhdaosnwotefhuceh\"irra, pat ntve.err6wa gcoeousul c.ldl l ibkeeutsoeedstot iemsatitmetahtee ( H'm=-Lf'l' ' -:':1 ':' y liOesnienptohsesuibsleeowfathyeomf oebthtaoidnoinfgltehaesbtseqsutpaoresessibtilmeeasttiiomn(aLteSoEf )t.hTeovedcetvoerlxofprtohmis l- u3 2131 rneth<tda, ssllmethat i It is easy to form the vector Hty and combining this with the itrverseof the equation (H'H), the following estimateof x is obtained. rcpresetttsthe desirecel stirnateof ,r so that y given by f rrt lat -. /1 t1\\/ r -l *^ =| \\Ltr)/r-\\rrt)\\/2-Yz) | j' = Hf, (r4.3) L - ( t/ 3 ) y r * ( 2 / 3 ) y r 1 ( r / 3 )l v - , representsthe estimateof the vectory. The effor ! of the estimationof y is then Weighted LSE given bv mI .. grsllll|:.iltr !rl D9. \\!+,2) ls ulrerr lr-rsrrsu r\\, (ri) rrru \\rr\\.rrrr(uJ r1,61ir Eilv€il lnq (r4.4) squaresestimate and is obtainedby minimising the index function that puts The estimatei is definectl o irc thc I-sE if it is cornputccbl y equalweightageto the effors of estimationof all componentsof the vector y. estimation index J given by r n i n i r n i z i n gt h e It is often desirableto put.dift'erentweightageson the ditt'erentcomponentsof y since some of the measurementsmay be more reliable and accuratethan the othersand these shouldbe given more importance.T'o achievethis we define the estimation index as

rvtouern Power I =l'wf / 1r . i4. 1 rnv\\i , An lntroductitoonStateEstimatioonf PoweSr ystems-T-l-\",3iiSl: \\ where W is a real symmetric weighting matrix of dimension m x ru. This is w =[tol .r I Il often chosen as a diagonal matrix for simplicity. L I It is relatively straightforwardto extendthe method of LSE to the weighted o.tj form of \"I and to derive the foilowing form of the normal equation. The matrir HLWH ts H|WH?_ H'Wy _0 G4.Ila) H,wH-l''' o'tl This leads to the desired weighted least squaresestimate(WLSE) 10.11.lJ *. = (H,WH)-t H,W y (14.11b) and the matrix HtW is obtained as This pertains to minimization as the hessian 2H|WH is a non-negative H ' | w = f o 'ot o t - l definite. L0 1 0.1j SomeProperties: Theweightedleastsquareesstimateof thevectorx is thenobtainedas(from Eq.(la.11b)) Rewriting Eq. (14.ilb) as i=ky (I4.I2a). where k = (H'WH)-' HtW. (r4.rzb) *^ = [ (lr/21) y, - (r}/Zt) y, (t0/2t) yr Here the matrix k dependson the value of H and the choice of w. L- ellr) y, ( 2 o t 2 ry), ( r /2 \\ y , I using Eqs. (14.1)and (r4.lzb) it is easyto get the relation as follows. ) r =f::;,7, ,,,wHx) +kr If this result is compared with the result in Example I4.1, the effect of introducing the weighting on the estimateis apparent.Note that the choice of Or X =r + kr (14.r3) W in this case suggeststhe data for y2 is consideredmore valuable and this resultsin the components of x being more heavily dependenton )2. and E{i} =E{xl (14.r4) The matrix ft is in this casefound to be (Eq. (1a.12b)) ln IJq. (14.14) it is assumedthat the error r is statistically independentof columns of H and the vector r has a zeromean. An estimate that satisfied r c = l r r / 2 1- r o / 2 r r o / z r f Eq. (l 4)4) is calledan unbiasedestimateT. his impliesthattheestimationerror ls zero on an averase. L- rlzt 20/21 r/21J If the covariance of the measurementerror is assumedto be R = L the covarianceof the estimationerror is obtained as (Ref. Eq. (14.15c)) x =kr (14.15a) P,= (1tr47|)*:_ .r 3: :41) The covarianceof the error of estimationis therefore given by (14.1sb) L-67 P, = KRK The choice of W above yields unacceptablylarge estimationerror variances. whereR is the covarianeeo1 the error vector r. Note that the covariancep\" is Let us now choosethe weighting matrix W = I. The matrix Kis then obtained a lneasureof the accuracyof the estirnaiionand a smaiier trace of this matrix indicatesa betterestimate.Eq. (14.15b)suggeststhat the bestpossiblechoice AS of the weighting matrix is to set w - R-1.The optimum value of the error covariance'matrix is then given by *=l''3 -1/3t/3f L- U3 2/3 1/3J The error covariance matrix is then given by P, - (HtR-rH)-l (14 15c) rD- x = \\/ rL/tn>\\ ) [ 6 - 3 ] L_ E 6 J t---*-- ** 1E t < a m p l e1 4 ; 2 The error variances are now seento be much smaller as is to be expected. Assumethat in the Example I4.1, we want to obtain the WLSE of the variable Non-linear Measurements x by choosingthe following weighting matnx The case of special interest to the power system stateestimation problem correspondtso the non-linearmeasuremenmt odel.

I Modern Power SystemAnalysis (14.16) An lrylfgductioton_StateEstimatioonf PowerSystems I S3T !=h(x)+ r 536 | I I I useful result in the sensethat it shows a mechanismfor improving on the initial where h(x) representsan rz dimensionalvector of nonlinear functionsof the estimafeby making use of the availablenleasurementsH. aving obtainednew variablex. It is assumedthat the componentsof the vector h(x) arecontinuous estimatei , theprocessof linearisationis repeatedas many times as desiredand in their arguments and therefore may be differentiated r.vith respect to the this leads to the {qllq*tng 4erylle {qryq qf the qqb]ro4 of thq 4on-linear estimationproblem. componentsof x. The problem is to extendthe methodof least squaresin order to estimatethe vector x from the datafor the vector y with thesetwo variables i ( t + 1 ) = i ( t )+ K ( t ) { y - h t t ( / ) l } (r4.23) being relatedthroughEq. (14.16). wherethe matrix K(D is deflnedas To mimic our treatmentof the linear measurementcase, assumethat i K(t)- LH/ wHil-' al w representsthe desiredestimateso that the estimateol the measLlremenyt could (r4.24) be obtainedusing the relation The index / representsthe iteration number and H lrepresentsthe value of the Jacobianevaluatedat x = i (l). Usualiy the iterative process is terminated I = h(i) (1.4.17a) wheneverthe norm of the differenceof two successivevaluesof the estimate This yields the error of estimationof the vectol y i 1t + l) - .i (/) reachesa pre-selecredrhresholdlevel. A flowchart for implementing the iterative algorithm is shown in Fig. 14.7. j =y_ h(3) (r4.r7b) A major sourceof computationin the algorithmlies in the needto updatethe In ordcr to obtain the WLSE of ,,r,we nrustcho<lsethe index of estimation Jacobianat every stageof iteration.As discussedearlierin Chapter6 (seeEqs. J as follows. (6.86)and (6.87))it is often possibleto reducethe computationsby holdingrhe value of H a constant,possibly after / exceeds2 or 3. T,ris is in general, J = U - h ( f i ) l ' W l J- h ( i ) l (14.18) permissible in view of the fact that the changein estimatetends to be rather small after a couple of iterations. The necessarycondition for the index ,I to have a minimum at .x,is given by Eq. (14.1e). ty-h(x)lH(i)=o (r4.re) whereH (fr) is the Jacobianof h (x) evaluatedat i. In generalthis non-linear equationcan not be solved for the desiredestimate,i. n way out of this difficulty is to make use of the linearisa-tiontechnique.Let us assumethat an a priori estimate xu of the vector x is available (say from the load flow solution). Using Taylor seriesapproximation,we get ! = h ( x d + H u & - x 1 ) +r (14.20) where 1/o standsfor the Jacobian evaluatedat x = x9 and the noise term r is now assurnedto includethe effects of the higher order terms in the'Taylor series.Equation(14.20)can be rewrittenas: A ) ' = y - h ( x o ) =H s A x + r (14.2I) i ,00\",\" ) eq(.+..227/ where Ay is the p^-rturbedmeasurementand Ax is the perturbed valueof the I vector r. An \\\\LSE of .r is then ea^silvobtaineda-sdiscussedearlierand this leads to the desired expressionfor the lineaized solurion of the non-linear c\"l::'1-:,=n) \\,*! -_I\\,,ll e s t j m a t . i opnr o b l e m . -''!' .i. = .rrr- [H,,' I+'HQ]-t H()' ]l' {-r - h t.t,,,)} r11.22) ./ ls :N- o \\ - . - 2 a . , I r is no r l i k e l v th a tth ee s ti ma te.i o b tai nedfrom E q. (11.22r i s goi ngto be r Yes t - r fI r t u c i r u s e s i l t . ' c . i r t t e n e r . r i . t l t r ' . r i , t ' r t t t ' t c ' s t i r l u t e . l ' , . rI l ) J \\ - I l r r t h ' c l o s e t i r t h f \\ (Dt-irnayl due rf rhe \\ect()r x. Hrt,*'everF, ,q.(14.22) prot'idesus wirh a very Stcp rig. r+.1

',:83--IE-, todern powerSystemAnalvsis _r-An tntroductionto State Estimationof Power Systems | 539 I It is thus apparentthat the problem of estimation of the power systemstate 'is a non-linearproblennand may be solvedusingeither the batchpr:ocessingor Considerthe simplecaseo1'ascalarvariablex and assunlethat the relationship sequentialprocessingformula [see Section 3.3 of Reference1]. Also, if the _xls glven Dy svstemis assumedto have reacheda steady-statecondition. the voltageangles Y = x 3+ r problem is then a static problem and the methodsof Sec. 14.2 may be used. if so desired.To develop explicit solutions, it is necessaryto start by noting the The JacobianH, is easily obtained in this caseand the iterative algorithm takes exact forms of the model equationsfor the componentsof the vector y (k). the explicit form Let P, arfi Qi denote the active and reactive power injections of ith bus. i (t + 1)= i (t) + t3i (Dl-z{y - ti (Dl3} These are related to the componentsof the statevector through the following wherewe haveused W = 1. equations. Let the correct value of x/ be eclualto 2 and assumethat due to the effect N (r4.2s) of r the measuredvalue of y is found to be 8.5. Also, assumethat the initial estimatex (0) is taken to be equal to 1. The table below gives the resultsof the P , =D I Yj | | Vjl lYij I cos (- 6t + 6, * 4ti) first few iterations. j --r , (t) N (r4.26) 0 1.0 e , = - D | % l I v j l l Y i j l s i n( - 6 , + 5 t + 0 ; ) I 3.5 2 2.56 j:1 _a ) 2 . t 6 wadhmeritetaInIcUeoI frtehperelin5eenctostnhneecutrinaggnthiteuldtheaanndd7tl/hUb- ruesperse.sTehnetastchteiveaanngdlereoafcttihvee It is apparentthatthe algorithmwould yield the correctsolutionafterseveral componentsof the power flow from the ith to the7th bus, on the otherhand are iterations. given by the tbllowing relations. 14.3 STATIC STATE ESTIMATION OF POWER SYSTEMS P,j= | vil I Vjl I YijI cos (d, - 6i* 0,) - l v i P l Y , , lc o s9 , , lrol-lr2l (r4.27) As notedearlier, for a systemwith N buses,the statevector x may be defined as t hc 2N - | v c c l tl rtl l ' l l tc N -- | v o l l agcangl cs62,..., 6\" and thc N vol tage Q , j = l % l I V j l l Y i j l s i n ( d '- 6 i * 0 t ) - I viP I Y,,I sin Iij magnitudes/1, v2, ..., v\". The load flow data,dependingon type of bus, are (14.28) generallycomrpted by noiseand the problem is that of processingan adequate set of availabledatain order to estimatethe statevector. The readily available Let us assumethat fhe vector y has the generalform data may not provide enough redundancy (the large geographical area over J = l P t . . . P N Q t . . . Q r u P r z . . . P u - l, N Qn ... Q7,J- 1, 7s, which the systemis spreadoften prohibits the telemeteringof all the available )2 b i nl vl 1 l , . . . . ,I y N I l / tnedsurementsto the centralcomputing station).The redundancyfactor, defined as the ratio m./nshould have a value in the range 1.5 to 2.8 in order that the (r4.2e) computedvalue of the statemay have the desiredaccuracy.It may be necessary to irleh-rdethe datafor the power flows in both the directions of some of the tie The Jacobian H will theh have the form (14.30) lines in order to increasethe redundancy factor. In fact, some 'psuedo Ht Hz measurementsw' hich represent he computedvaluesof such quantitiesas the H3 H4 activeandreactiveinjectionsat someremotebusesmay alsobe includerlin the Hs H6 vector y (k). H _ H7 Hs /ru-r o o IN where /\" is the iclentity matrix of dimension N, H, is the N x (N - l; submatrix of the partial derivatives of the active power injections wrt 6's, H2 is the

54q f ModernPo An tntroductionto State Estimationof Power Systems II 541' T N x N sub-matrixof the partial-derivativesof the active power injections wrt Equation(14.33)may be usedto determinethe Jacobianat any specifiedvalue of the systemstatevector. The injections only stateestimation algorithm is then | 7l' and so on. Jacobian H will also be a sparsematrix since I is a sparse obtained directly from the results of sec 14.2.Since the problem is non-linear, it is convenientto employ the iterative algorithm given in Eq. (14.22). matrix. ine. the submatricesH., andH^ becomenull Two specialcasesof interest are those correspondingto the use of only the active and reactiveiniections and the use of onl flows in the vector y. In the first case,there area total of 2N componentsof with the result that the linearised model equation rnay be approximatedas: y compared to the 2N - 1 componentsof the state x. There is thus almost no ^ fH, o-l ' redundancy of measurements.However, this case is very close to the case of (r4.34) load flow analysis and therefore provides a good measure of the relative rrweparrition,J\";,,1,o:'*^:::: strengthsof the methodsof load flow and stateestimation.In the secondcase, it is possible to ensurea good enough redundancyif there are enough tie lines A v=lf A^y' r, 1' r'.=j ' lArrl \"'l=, lr ol in the system. One can obtain two measurementsusing two separatemeters at Lor L ;j the two ends of a single tie-line. Since these two data should have equal magnitudes but opposite signs, this arrangementalso provides with a ready then, Eq. (14.34)may be rewritten in the decoupledform as the following two separateequationsfor the two partitionedcomponentsof the statevector check of metermalfunctioning.There are otheradvantagesof this arrangement as will be discussedlater. Alp= H1 Ax5+ ro (14.35) Alq= H4 Axr, + rn (r4.36) The Injections Only Algorithm In this case, the model equationhas the form Basedon thesetwo equations,we obtain the following nearlydecoupledstate !=hlxl+ r (14.31) estimation algorithms. with the componentosf thenon-linearfunctiongivenby i u Q + r ) = x d 0 + t H i0 w p H tU ) l - ' H , ( D{ t o - h r l i U ) l } .l= 0, 1,2, ... \\ (14.37) h , [ x ) =D l V i l l V j l l Y i j l c o s( 6 , - 6 i + 0 i ) ,i = I , . . . , N i, (/ + 1)= i, Q)+ tHl u) w, Hq(j)l-' nl u) wo ur- hqti (j)ll j:1 j=0 | 2 (14.38) Qa32a) where the subscriptsp andq are usedto indicatethe partitionsof the weighting N matrix W andthe non-linear function h (.) which correspondto the vectors yp - D t v * _ i l l v j l l y N _ i , ; l s i (n6 , - 6 i + 7 i i ) *rd lerespectively. As mentioned earlier, if the covariancesR* Rn of the j:1 effors r, and r(t areassumedknown, one should select Wo= R 1o1' dand Wn = i=N+1,N+2...,2N (r4.32b) The elements of the sub-matrices Hp H2, H, and Ho are then determined R q'N\" ote i truly decoupledbecausethe that Eqs. (14.37) and (14.38) are not easily as follows. partitions of the non-linear function dependon the estimateof the entire state H t ( i , j ) = l V i l l V j l l Y i j I s i n ( r { .- 6 t+ 0 t ) vector. It may be possible to assumethat vi U) = I for all i and 7 while' i = 1 , 2 , . . . ,N , Eq. (14.37) is being used in order to estimatethe angle part of the statevector. j=I,2,\"',N-1 similarly one may assume 6i U) = 0o for all i and 7 while using Eq. (14.38) Hz Q,i) = lvil lYij I cos (d,- 6i * 0;) i = 7,2, ..., N, in order to estimatethe_yllltagepart of the statevector. Suchapproximations j = 1,2,\"', N' allow the two equationsto be completely decoupledbut may not yield very good rf /r !\\ | ryt i l I I tr | | r, | /C a - n\\ . nA solutions.A betterway to decouplethe two equationswould be to usethe load flow sglutionsfor x, and x6 as therr suppose<iiyconsnnt vaiuesin Eq. (14.37) Ia3 \\tt J)= - | I vjl I IUI COS \\Oi-.Oj+ Aij) I = I, Z) ..., lY, j=I,2,\"',N-1 and Eq. (14.38) respectively. There are several forms of fast decoupled H q ( i ,j ) = l V i l l Y i j l s i n( { - 6 i + 0 , )i = I , 2 , . . . , N , estimationalgorithmsbasedon suchconsiderations(seee.g. [13], tl4l).A flow j = I,2, \"', N' chart for one schemeof fast decoupledstateestimation in shownin Fig. I4.2. (r4.33)

542 | Modern Po*er System Analysis An Introductionto State Estimationof Power Systems 1_| - 543 Fig.14.3 *11 I , o , r l l i 1 i *1 t ) - * ( j ) l l = , 'j Application of the LSE then yields the following expressionsfor the estimatesof the perturbationsin the three statevariablesaround their chosen on,^t ls > initial values: t\\u - \\, 2 an2 Ai, - AP, - AP, \\\\r---' A T ,= 0.78 AQz - 0.26AQr 1\"\", AV' = g39 AQ' + 0'14 AQr (, 3I'\\9) Theseequationsshouldbe usedin orderto translatethe measuredvaluesof Fig. 14.2 thc pcrturhat ionisn t he act iveanclr eact ivepowerinject ionsint o t heist inr at es of the perturbationsof the state variables. It is interestingto notethat for lhe sirnpleexample,partitionsHranrl Htare null matricesso thatthe decoupiedstateestimatorsarethe sameasthosegiven above. The Line Only Algorithm I This algorithm hasbeendevelopedin orderto avoid the needfor solving a non- j Example14.4 linearestimationproblem.which as seenearlier,requiressomeapproximation In order to illustrate an applicationof the injections only algorithm, considerthe or other.ln the line tlow only algorithm,the datator the activeandreactivetie \"imple 2-bus systemshownin Fig. 14.3. line flows areprocessedin order to generatethe vector of the voltagedifference A s s u m i n gl o s s l e slsi n e , 0 , j = 9 0 \" . A l s o l e t Y r r = j z z = 2 a n d y n = y z t = . The power relationsin this casewould be acrossfte tie-lines.Let z denotethis vector.A model equationfor this vector Pt= - | Yr| | V2llYrrl sin E may be expressedas Pz=lVtl lVzl I Y,rlsin 6, Q t = i r i l i i V r i ' - | y n l l y l | | V r l c o s$ z= Bx + r- (r4.39) Q z = l Y z z l V r l ' - l y t 2 l l y r | | v r l c o s6 I f we c h o o s eth ei n i ti a lv a l u e sl Vf l = l V ; l = 1, 6o2= 0\" ,thecorrespondi ng where B is the node-elernenitncidencernatrixand r- is the vectorof the errors power values arePf = Pf - 0, Qf = Qi = 1. The value of the Jacobianmatrix evaluatedat the above nominal values of the variablesturns out to be in the voltage el-ataS. ince this is a linear equation.one may use the WLSE techniqueto generatethe estimateas *,= lB, wBl- | Bt wz (r4.40) where the weighting rnatrix may be setequai to the inverse of the covariance of rrif this is known. The main problemwith Eq. (14.40)is thatthe vector z is not directly measurablebut needsto be generatedfrorn the tie line flow data.

r++ | Mocternpower System Analysis An lntroductiotno state Estimatioonf powersystems T-| 545 Network Obsenrability [17] I Considerthe staticWLSE formula lEq.(14.llb)l which servesas the srarting point fbr all the algorithrns.lnverse of information matrix Mn,, = Ht wH V,tdenotesthe voltageacrossthe line connectingthe ith ancltheTth buses,the shouldexist otherwisethereis no stateestimate.This will happenif rank of f/ followine relationholds. ls equal to n (no. ot statevartables).Since one can always choose a non- singular l4l,so if lt1has a rank n, the power network is said to be observable. Vii= Zii [@ii--j Qi;tV7- Vi Y,i] (14.41) Problem of lll-conditioning Here Z,.,standsfor the impedanceof the line. This shows that the vectorz is relatedto the vectorsx and ashion and one ffidy use the notation z = g (x, y) 04.42) In viewof thisnon-linearrelationE, cl.(14.40)maybeexpresseicnltheform i = LB,Wnyr B,W I (i, y) (14.43) Even if the given power system is an observable system in terms of the measurementsselectedfor the state estimation purposes,there is no guarantee This,beinga non-linearrelationshipc,an not be solvedexceptthrougha that the required inversion of the information matrix will exist. During numericaal pproach(iterativesolution)T. he iterativeform of Eq. (14.43)is multiplications of the matrices,there is somesmall but definite error introduced due to the finite word length and quantisation.Whether or not these errors i (j + r) = lBt wBll H w s [i (i), y], (r4.44) create ill-conditioning of the information matrix may be determined from a - /= 0 , 1 , 2 ,. . . knowledge of the condition number of the matrix. This numberis defined asthe ratioof the largestand the smallesteigenvaluesof the inforrnationrnatrix.The Note that the original problem of estimation of x from the datafor z is a linear maftix M becomesmore and more ill-conditioned as its condition number pr ob l e ms o th a t th e s o l u ti o ng i v e n by E q. (14.40)i s the opri nralsol rrti on. increasesin magnitude.Somedetailedresultson power systemstateestimation using Cholesky factorizationtechniquesmay be fbund in tl8l. Factorization However,the data for z needto be generatedusing the non-lineartransforma_ helps to reduceill-conditioning but may not reducethe computationalburden. tion in Eq' (14.42), which in turn has necessitatectlhe use of iterative Eq. A techrriqueto reducecomputationalburden is describeciln Ref. []91. (14.44)' Compared to the injections only iterative algorithm, the pr-ese1t algorithm has the advantageof a constantgain matrix [8, W B]-t gr I4zT. his resultin a considerablceomputationaslimplification.The conceptof decoupled estimationis easily extencledto the caseof the lirreflows [15]. T4.4 TRACKING STATE ESTIMATION OF POWER 14.6 EXTERNAL SYSTEM EOUTVAIENCTNG [20] SYSTEMS t16l One of the widely practicedmethodsusedfor computationalsimplification is to divide the given systeminto three subs Trackingthestateestimationof a given power systemis importantfor realtirne theseis referredto as the 'internal' sub ystemsas shown in Fig. 14.4.One of monitoringof the system.As is well known, the voltagesof all real systemvary systemand consistsof thosebusesin randomly with time and should therefore be considered to be stochastic which we are really interested.The secondsubsystemconsistsof those buses processesI' t is thus necessaryto make use of the sequentialestimation which are not of direct interest to us and is referred to as the 'external, techniquesof Ref. [1] in order to obtain the state estimateat any given time subsystem'Finally, the buseswhich provide links betweentheseinternal and point. The power relationsin Eqs. (14.25) and,(74.26) arestill valid bur must external subsystemsconstitutethe third subsystemreferredto as the 'boundary' be rewritten after indicating that the voltage rnagnitudesand angles are new functions of the discretetime index ft. subsystem. For any given power network, the identification of the three subsystemsmay be done either in a natural or in an artiflcial way. -t, /,/ \\ , I4.5 SOME COMPUTATIONAL CONSIDERATIONS B r r f h t h e q t a f i n q n d f h e rt rruqvorl\\zrlrn1o6 ov rc)frirmr ror qf ilnr u- l r C^ 1t r-5^\\*Ji l+ tLL*l^l l l . l J ai^,r^61^-r i- rL^ ---^--r: -- PIEi)t'f lttrlt lll tll€ pICL:CUlllg sectionsare computationally intensive, particularly for large power networks which may have nlore than200 importantbuses.It is, therefore,very important ,/ t o pa y a tte n ti o nfo s u c h c o m p u ta ti onailssuesas i l l condi ti oni ng,computer \\\" storageand tinnerequirementsH. owever,we needto first considerthe question Internal Boundary system system of existenceof a solution of the stateestimationproblem. Flg. 14.4

fl6 -l ModernPo An lntroductionto state Estimationof Power systems t{7 Bad Data Detection [231 I I To illustratethe simplification of the stateestimationalgorithm, considerthe linearisedmeasuremenet quationfor the injectionsonly case.Sincethe system A convenient tool fcr detecting the presenceof one or more bad data in the vector y at any given point of time is basedon the'Chi SquareTest'. To is partitioned into three subsystems,this equation may be written as appreciatethis, flrst nc:tethat the trrethodt-ll least square ensures that the A y i H,, H,, o A*, rflLW Ta: 1r (t)l- /ltgr haritrminimum Ayr, Hu, Huu Ho, A * , ,ol (r4.4s) value when x - i. Sincethe variable r is random,the minimum value ,I.in is ay\" 0 Hra H\"\" A * , ,\"] also a random quantity. Quite often, r may be assumed to be a Gaussian variable and then .I*1nwould follow a chi squaredistribution with L = m - n It may be noted that the internal me ..rrerrrertvector Ayt is not completely degreesof freedom. It turns out that the mean of ./,o1nis equal to L and its variance is equal to 2L. This implies that if all the data processedfor state inclepenclenot f the external subsystemstate Axu since Ay, dependson the estimatiorrare reliable,then the computed value of ../r,r,snhould be close to the averagevalue (=L). On the other hand, if one or ntore of the data for )' tre boundary subsystetn state Axr, and Ax6 dependson Axr. unreliable, then the assumptiorrsof the least squaresestimation are violated and the computed value of J*1nwill deviate significantly from I. Ayr= Ayt,i+ Aynr,+ Ayu, + ru (r4.46) It is thus possibleto developa reiiable schetrrefor the detectionof bad data where Ayo\" representsthe injections intc the boundary busesfrom the external buses,Ay66is the injection from the boundary buses znd Ay6, is the injection in y by computingthe valueof [y - h(i)l'w ly - h(r)], i being the estimate fiom the internalbuses.It is assurnedthat the term Ayu, (= Ht,, Axr), may be obtainedon the basisof the concernedy. If the scalarso obtainedexceedssome threshold Ti - cL, c:being a suitable nuutber,we concludethat the vector y approximatedas n A*, where A It estimatedfrom the relation includes some bad data.(Note that the datafor the.cornponenty;, i - 1,2, ..., nr will be consiclerebcal clif it deviatesfrom the tneanof r',by more thant 3e,. H = Ayt,nlAxl, (14.47) whereo, is the standarddeviationof r;). Caremustbe exercisedwhile choosing The conrponent Ayu\" may be estimated if the terms Ayy, and Ayuo ne the value of thresholdparameterc.If it is closeto 1,the testmaf producemany 'talse alal1s' anclil it is tt-rtl-ar rge, the testworrldfail to detect nlany bad data. computedas H,,,Ax,and H1,1A,x1,and then subtractedfrom the measuredvalue of Ayo. This woulc!resultin part of Eq. (14.45) to be rewritten as To select an appropriatevalue of c, we may start by choosing the l^;,)=',l;':,;;.);[l,;*],1 (r4.48) significancelevel d of the test by the relation' I'lJ (x) > cLlJ (.r) lbllows chi squaledistribution)= 6/ whcrc Ht,, = Ht,t,t U pcrprcscrnfthsc cff-ccifvc Jacohiltnif the hottnclitry subsysternthat accountsfor the eff'ectsof the external subsysternon the We rnay select,fbr example,d = 0.05 which correspondsto a 5Vofalse alarm boundarysubsystemE. quation(14.48)has a lower dimensionthan the original sitrrltign.[t is thcn possibleto find the valr.reof c hy making use of the table t nc a s u rc l n c rcttc l L rl ti o nu n d w o rrl rltl tcrcl orci rtvol vclrcss ctl tttl l tr(.tl i ol lTsl.tc X@).Once the valuc of c is determined,it is simple to cary out the test conceptof externalsystemequivalencingmay be employed with the line or whether or not .l(x\\ exceedscL. r n i x e dC a t as i t u a t i o n sa l s o . Identification of Bad Data [231 14.7 TREATMENT Ol.- BAD DATA 127,221 The ability to detectand identify bad measurementsis extremelyvaluableto a Once the presenceof bad data is detected,it is iraperative that these be load dispatchcentre.One or more of the datamay be affected by rnalfunction- identified so that they could be removed from the vectorof measurementsbefore ing of eitherthe measuringinstrumentsor the datatransmissionsystemor both. it is processed.One way of doing this is to evaluatethe componentsof the Transtlucerrsnay have becnwilcd incorrcctlyor tlto l.rattsduccirtscll rttaybe measuremenrtesidual !, = y, - h, (.x\\,i = 1,2,..., m. If we assLlmethat the malfunctioning so that it simply no longer gives accuratereadings. residualshave the Gaussiandistribution with zero meanand the varLance4, If suchfaulty daia are includedin the vector Ay, the estimationalgorithm then the magnitucieof the resiciuaiy, shouiciiie in therange- 3o i ( )i ( 3orwith 95Voconfidenceievel. Ihus, if any one of tlte cotnputedresidual turns out to will yielclunreliableestimatesof the state.It is therefbreimportant to develop be significantly larger in magnitudethan threetimesits standarddevia,tion,then techniquesfor detecting the presenceof faulty data in the measurementvector correspondingdata is taken to be a bad data.If this happenSfor more than one at any given point of time, to identify the fauity dataand eliminate thesefrom con)p ot r ctor (l'_yt,hcn t hc con'lponcnltur vingtlhc lar gcsrt csidualis assunt edt o the vector y befble it is processcdlbr stateestinration.[t is alstl ilttpoltant to be thc bad dataand is removedfrom y. The estimationalgorithmis re-runwith rnodify the estirnationalgorithmsin a way that will permit tnore reliable state the remaining data and the bad data detectionand identification tests are es t i ma ti o ni n th e p !e s e n c eo f b a d d ata.

,548 ,! ModernpowerSystemAnalysis An tntroductiotno State Estimationof PowerSystems It 549. - I performed againto find out if there are additionalbad data in the measurement The main advantageof the choice of the form (14.49) for the estimation set. As we will seelater bad rneasuremenct lataare detected,eliminatedand replacedby pseudoor calculatedvalues. index is that it is still a quadraticin the function g(.i) and so the LSE theory may be mimicked in orderto derive the following iterativeformuia for the state of Bad Data estimate. Hr(D CTWCH Q)Tfl (D WCtfiDI The procedures described so far in this section are quite tedious and time (74.5ra) consuming and rnay not be utilized to removeall the bad datawhich may be wherethe matrix C is diagonaal ndits elementsarecomputedas presentin the vectory at a given point of time. It may often be desirableon the other hand to modify the estimationalgorithms in a way that will minimise the Ct= l, fot l;lo, S ai ( 1 4 . s1 b ) influence of the bad data on the estimatesof the statevector. This would be = 0, for lilo,> a' possible if the estimationindex J(x) is chosento be a non-quadraticfunction. Comparing this solutionwith that given in Eq. (14.22),it is seenthat the The reasonthat the LSE algorithm does not perform very well in the presence main effect of the particularchoice of the estimationindex in Eq. (14.49) is to ensurethat the dataproducingresidualsin excessof the thresholdlevel will not of bad data is the fact that becauseof the quadraticnature of J(x), the index changethe estimate.This is achievedby the production of a null value for the assumesa large value for a datathat is too far removedfrom its expectedvalue. matrix C for large valuesof the residual. To avoid this overemphasison the effoneousdataanclat the sametime to retain the analytical tractabilityof the quadraticp: erlbrmanceindex, let us choose /(i) =s'(t) w sG) (14.49a) I4.8 NETWORK OBSERVABILITY AND PSEUDO- MEASUREMENTS where 8(t) is a non-linearfunction of the residual !. There may be several possiblechoicesfor this f'unction.A convenientform is the so-called'quadratic A minimum amount of data is necessaryfor State Estimation (SE) to be effective.A more analyticalway of determiningwhethera given data is enough flat' form. In this case,the componentsof the function g (y) are defined by the for SE is called observabilityanalysis.It forms an integralpart olany real time following relation. state estimator.The ability to perform state estimation depends on whether gi (j) = li, for j,lo, 1 a, = ai, ftx' l,/o,> u, (r4.49b) .sufficientmeasurementsare well distributedthroughoutthe system. When where a, is a pre-selectedconstantthreshold level. Obviously, the perform- sufficient measurementsare available SE can obtain the state vector of the ance indcx ./(\"r)may bc cxpressedas whole system.In this casethe network is observable.As explained earlier in m (14.s0) Sec. 14.5 this is true when the rank of measurentenJtacobiantlratrix is equal to the number of unknown state variables. The rank of the measurement I(i) = Dy, c;) Jacobian matrix is dependent on the locations and types of available .; -- - I rneasurementsas well as on the network topology. I An auxiliary problem in stateestimation is where to add additional dataor and eachcomponenthas a quadratic nature for small values of the residualbut pseudomeasurementsto a power systemin orderto improvethe accuracyof the has a constantmagnitudefor residualmagnitudesin excessof the threshold. calculatedstate i.e. to improve observability.The additional measurements Figure 14.5 showsa typicai variation of J, (.r) for the quadratic and the non- represent a cost for the physicat transducers,remote terminal or telemetry quadraticchoices. sy.stem,and software data processingin the central computer. Selection of pseudomeasurements,filling of missing data,providing appropriateweightage are the functions of the observability analysisalgorithm. UbsefvaDlllty Can I I - -r-- I Ul - -U- :t-r r l B^ Cl i^l u^ tl^u-lil-z^ i+tlt^L- l t r l l . TI lf 4^ -I.r' J ^i\"^+ Lr .^r^v^ t- r^ r. r . r r r l r .Y' ^v- .r'J De CIICCKCU Prv\\rr \\ small or zero during factoization, the gain matrix may be singular, and the Fig. 14.5 systemmay not be observable. To finil the value ol an injection without nteasuringit, we tlrust know the power systembeyondthe measurementscurrently being made.For example,we pormally know the generatedMWs and MV ARs at generators through telemetry channels (i.e. thesemeasurementswould generally be known to the

An introductiotno state Estimatioonf powqrsystems I stateestimator).If thesechannelsareout, we canperhapscommunicatewith the U'E It FSI operatorsin the plant control room by telephoneand ask for thesevaluesand - enterthem manually.Similarly, if we requirea load-busMW and MVAR for b Al !o L i , a pseudomeasurementw' e could use past recordsthat show the relationship lm --9 t, betweenan individual load and the total qyrtemload. We canestimatethe tstal I FG systetnload quite accuratelyby f)nding the total power being generatedand tr .Yt estimating the line losses.Further, if we havejust had a telernetry failure, we o could use the most recentlyestimatedvaluesfiom the estimator(a.ssgmingthat ]U =th i','is run periodic:ally)as pseudorlreasureftlentsT.hus, if required,we can give t- . the stateestimatorwith a reasonablevalue to use as a pseudomeasuremenat t drx any bus in the system. Pseudomeasurementisncreasethe dataredundancyof SE. If this approach og is adapted, care must be taken in assigning weights to various types of a measurementsT.echniquesthat can he usectl o cleternrinethe rnctcror.pseu4g measuremenltocationsfbr obtaininga completeobservabilityof the sysremare F I available in Ref. [251.A review of the principal observability analysisand CnF m et erp l a c c rn c nat l g ' ri th rn si s a v u i ra hrci n I(cr' .[261. atrJ I T4.9 APPLICATION OF POWER SYSTEM STATE I ESTIMATION I o I o 1f o o oo (! o E (E L oo o a (L c L oo U) n ^ I E vv o I a U) In real-timeenvironment he stateestimatorconsistsof different modulessuch a .nF as network topologyprocessoro, bservabilityanalysis,stateestimationand bad L data processing.The network topology processoris required for ail power I qA I- ' =L . o I systemanalysis.A conventionanl etwork topologyprogram usescircuit breaker 0-(E ooo= statusinformationandnetworkconnectivitydatato determinethe connectivity I o I of the network. t,- o q I sf I E; I ll- I L o oI Ua ) E i oo ,tl l_ X HL f I a (! l 6 ncl zi;i'o8>- , i o L>.! UiI o) LLI E Figtrrc 1r4.6is it schcrnaticliagtarnshowingthe info.rnationflow between I o) o the varior-rsfunctions to be performed in an operationscontrol centrecompurer system. The system gets information from remote terminal unit (RTU) that F cncodc lllcasLlrclllcntlt'ullsducc0r r-rtputasnclopcnccl/closcsctlaLusinlbmration into digital signalswhich are sent to the operationcentre over communications ,l circuits. Control centre can also transmit commands such as raiseflower to 'I generators and open/closeto circuit breakers and switches. The analog U) measurementsof generatoroutput would be rJirectlyusedby the AGC program (Chapter 8). However,restof the datawill be processedby the stateestimator COe befbre being usedfbr otherfunctions such as OLF (Optimal LoaclFlow) etc. Before running the SE, we must know how the transmissionlines are b 8 gN ' E C E E.E EEE 6Eo FO o r-nnnenfprl fLn\\ / fLl rror u lrn\\^/'cl u (^ r^ Jr r L rl i c r r r s r . u . ( r rl-.-,-u-s- -c l-i- l . e . n e l w o r K t o p o l o g y , lhrs kceps 6n changingand hcncel.hccurrertrt .clenteterebdreaker'/switcshtertums ustbe used to restructurethe electricalsystemmodel.This is called the network tr,tpop.tgy program or systemstatusproce,\\.toror netwc;rkconfigurator.

ilr l ModerPno An lntroductionto State Estimationof Power I Fig.P.14.2 The output of the stateestimatori.e. lvl, 6, P,j, Q,jtogether.with latestmodel 14.3Given a single line as shown in Fig. p 14.3, two measurementsare form the basisfor the economic dispatch (ED) or minimum emission dispatch available. using DC load flow, calculatethe best estimateof the power (MED), contingency analysisprogram etc. flowing throughthe line. Further Readin 4= 0 rad. The weighted least-squaresapproach to problems of static state estimation in l -[l power systems was introduced by Schweppe 11969-741. It was earlier originated in the aerospaceindustry. Since 1970s,state estimators have been Mzt 2 installed on a regular basis in nern'energy (power system or load dispatch) control centresand have proved quite helpful. Reviews of the stateof the art in stateestimationalgorithmsbasedon this modelling approachwere published by BoseandClements[27] andWu [28]. Reviewsof externalsystemmodelling are available in 1291.A generalisedstateestimatorwith integrated state,stutus and parameterestimationcapabilitieshas recentlybeenproposedby Alsac et al [30]. The new role of stateestimationand other advancedanalyticalfunctions in competitive energy marketswas discussedin Ref. [31].A comprehensive bibliography on SE from 1968-89 is availablein Ref. [32]. (/1 | t t----- y0.1pu (1ooMVAbase) ,r-1-r- Mt2 PROBLEMS 1 Fig.P. 14.3 14.1 For Ex. 6.6 if the power injected at busesare given as Sr = 1.031- Meter Full scale Meter Standard Meter j0.791,Sz= 0.5 + 71.0and 'S3= - 1.5- j0.15 pu. Consider Wt= Wz= (MW) Wz = l. Bus I is a referencebus. Using flat start, find the estimatesof Mrz Deviation (o ) Reading lV,l and {. Tolerance= 0.0001. Mzr 100 in full scale lAns: Vl = l/0\", V', = t.04223 10.4297\",V\\ = 0.9982ql-2.1864\"; 100 (Mw) Final values: Vt - 1.04./.0\", V2 = 1.080215l-1.356, Va- 1.03831 1 4 32 l- 3.736\"1. -26 14.2 For samplesystemshownin Fig. P. 14.2,assumethat the threemeters REFERECNES have the followine characteristics. Books Meter Full scale (MW) Accuracy (MW) o (pu) l. MahalanabisA, .K., D.P. Kothari and S.L Ahson, ComputerAidetl pr;wer System Mrz +8 Analysis and Control, Tata McGraw-Hill, New Delhi, 19gg. Mtt 100 +4 0.02 Mzz r00 + 0.8 0.01 2. Nagrath, I.J. and D.P. Kothari, Power SystemEngineering, Tata McGraw-Hill. 100 0.002 Ncw Dclhi, Iq94. Calculate the best estimate for the phase angles 4 *d d2 given the 3. Mtrnticclli' A., State li.rtinrutionin Eler:tric:Power SysremsA Gcnerali.recAl pprct- at:h, Kluwcr Academic Publishers,Boston, 1999. f,'11,...,: nmnnf t, | 1-rl l\\., W | | l5 | I lU{lDLll t/l I lUl lL,) 4. Kusic, G'L., Computer-AidedPower SystemsAnalysis,Prentice-HallN, .J. 19g6. 5. Wood, A'J. and B.F. Wollenberg, Power Generation,Operationand Control. Znd Meter Meusured value (MW) Ed.,JohnWiley, NY, 1996. Mtz 70.0 Mrz 4.0 Mt, 30.5

'l t ModernPowerS- ystemAnalysls I ;554.;l An Introductionto State Estrmationof Power Systems _TI -5- 55 I 26. Clements,K.A., \"Observability Methods and optimal Meter Placement\", Int. J. Elec. Power,Vol. 12, no. 2, April 1990,pp 89-93. j 6. Grainger, J.J. and W.D. Stevenson, Power System Analysis,McGraw-Hill, NY', 27. BoSe,A. and Clements,K.A., \"Real-timeModelling of Power Networks\", IEEE 1994. Proc., SpecialIssue on Computersin Power SvstemOperations,Vol. 75, No. 12, Dee 1987;aP 76ff7=1ffi2: 7. Deautsch,R., Estimation Theory, Prentice-HallInc' NJ, 1965 g. Lawson,C.L. and R.J. Hanson, Solving Least SquaresProblens, Prentice-Hall. 28. Wu, F.F., \"Power System State Estimation:A Survey\", Int: J. Elec. Power and Energy Syst.,Vol. 12, Jan 1990,pp 80-87. inc.,NJ., i974. g. SorensonH, .W., ParameterEstimation,Mercel Dekker, NY, 1980' 29. Wu, F.F. and A. Monticelli, \"A Critical Review on External Network Medelling for On-line Security Analysis\", Int. J. Elec. Power and Energy Syst.,Vol. 5, Oct Papers 1983,pp 222-235. 10. SchweppeF, .C., J. Wildes, D. Rom, \"PowerSystemStaticStateEstimation,Parts 30. Alsac, O., et. aI., \"GenetalizedStateEstimation\", IEEE Trans.on Power Systems, Vol. 13, No. 3, Aug. 1998,pp 1069-1075. l, ll and lll\", IEEE Trans', Vol. PAS-89, 1970,pp 120-135' ll. Larson,R.E., et. al., \"State Estimation in Power Systems\", Parts I and II, ibid-, 31. ShirmohammadDi, . et. al., \"TransmissionDispatchand CongestionManagement in the EmergingEnergy Market Structures\",IEEE Trans. Power System.,Vol 13, pp 345-359. No. 4, Nov 1998,pp 1466-1474. lZ. Schweppe,F.C. and E.J. Handschin,\"static State Estimationin Electric Power 32. Coufto,M.B. et. aI, \"Bibliography on Power System StateEstimation (1968- System\", Proc. of the IEEE, 62, 1975, pp 972-982' 1989)\",IEEE Trans.Power Sysr.,Vol.7, No.3, Aug. 1990,pp 950-961. 13. HorisbcrgcrH, .P.,J.C. Richarcal nd C. Rossicr,\"A Fast DccouplcdStaticStatc Estimatorfor Electric Power Systems\", IEEE Trans. Vol. PAS-95, Jan/Feb1976, pp2O8-215. 1 4 . Monticelli, A. and A. Garcia,\"Fast DecoupledEstimators\",IEEE Trans' Power Sys/,5, May 1990,pP 556-564. 1 5 . Dopazo,J.F. et. al., \"State Calculation of Power Systems from Line Flow MeasurcmentsP,artsI and ll\", IEEE Trans.,89, pp. 1698-1708,91, 1972,pp 145-151. 16. Dcbs, A.S. anclR.E. Larson,\"A Dynamic Estimatorfor Tracking the Stateof a PowerSystem\",IEEE Trans' 89, 1970,pp 1670-1678' iZ. K1u-pholz, G.R. et. al.,\"PowerSystemObservabilityA: PracticalAlgorithm Using Nctwork Topology\", IEEE Trans.99' 1980'pp 1534-1542' lg. Sirnoes-CostaA, and V.H. Quintana,\"A RobustNumericalTechniquefor Power SystemStateEstimation\",IEEE Trans. 100, 1981, pp 691-698' 19. Simgcs-CostaA, and V.H. Quintana,\"An OrthogonaRl ow ProcessingAlgorithm for Power System SequentialState Estimation\", IEEE Trans., 100, 1981' pp 3'79r-3799. 20. Debs, A.S., \"Estimationof External Network Equivalentsfrom Internal System Data\", IEEE Trans.,94, 1974,pp 1260-1268' 21. Garcia.A., A. Monticelli and P. Abreu, \"FastDecoupledStateEstimationand Bad i Dara Processing\",IEEE Trans. PAS-98, Sept/Oct 1979, pp 1645-1652. Ii 22. Handschin,E. et. al., \"Bad Data Analysis for Power System State Estimation\", IEEE Trans.,PAS-94, 1975,pp 329-337- .t2 r.^-r:6 rJ r or nl \"Flqd T)cre Detecfion and ldentification\". Int. J. EleC. POWer, 4J. r\\V6lrrrt Lt.r. eL. @e., Vol. 12, No. 2, April 1990,PP 94-103' 24. Me-Jjl,H.M. and F.C. Schweppe,\"Bad Data Suppressionin Power SystemState Estimation\",IEEE Trans. PAS-90, 1971' pp 2718-2725' 25. Mafaakher,F., et. al, \"Optimum Metering Design Using Fast Decoupted Estimator\",IEEE Trans. PAS-98, 7979,pp 62-68.

Compensatioinn lglver Systems |I 55? can be connecteciin the systemin two ways, in seriesand irr shunt at ihe line ends (or even in the midPoint)' Apart from the well-known technologies of compensation,the latest technologyof Flexible AC TransmissionSystem(FACTS) will be introduced towardsthe end of the chaPter. 15.1 INTRODUCTION 15.2 LOADING CAPABILITY For reduction of cost and improved reliability, most of the world,s electric There are three kinds of limitations for loading capability of transmission power systemscontinueto be intercor of diversity of loads,availabilityof sor system: to loads at minimum cost and pollr (i) Thermal (ii) Dielectric (iii) Stability deregulatedelectric service environme Thermal capabitity of an overhead line is a function of the ambient to the competitiveenvironmentof reli temperature, wind conditions, conditions of the conductor, and ground qgannaueboonantsndNleeuiettornhyatawicseltoeii-ystnfayoetg-sofgdcdueeoaalponmmycpnesqaplgy,rugnaa-.irdtnrteeeosiaerwrnmtssaeiealnlrwndpddcelargocmingruneanhstianittnnsodtugemsohe,rcfeataoorvwnsmehrdaiabpsytveee.heetIebinrnteeicponcsrlneeaauaaecultmesdseededotodoionnfndgtelptehuhmrsetoeisalvtiisnnrtiadiecdcnecrssesuotoamrhnipsetiyeist'nrarnsagsninnoeadsunlcnrvmmceeeiedbtsswsues.Isocitroteroinoksdf,, clearance. There is a possibility of converting a single-circuit to a double-circuit line to increasethe loadingcaPabilitY. Dieletric Limitations From insulationpoint of view, manylines are designed very conservatively.For a given nominal voltage rating it is often possible to increasenormaloperatingvoltagesby l07o (i.e. 400 kV - 440 kV). One should, however,ensurethat dynamic andtransientovervoltagesarewithin limits. [See Chapter 13 of Ref. 71. Stability IssuesT. here are certain stability issuesthat limit the tlansmission capability. These include steady-statestability, transient stability, dynamic stability, frequency collapse, voltage collapse and subsynchronousresonance. Severagl ooclbooksl, l, 2, 6,7 ,8) areavailableon thesetopics.The FACTS technology can certainly be used to overcome any of the stability limits, in which casethe final limits would be thermal and dielectric' ttoheacscoehmpirepovebenlteshmaistsioo'sbneijnericeptsoiv/wsehe. ur nsytcsotmempesinss,athtieornheafosrbee,eesnsinenutsiaeltfoorapllaesvtimataensyomyeeaorfs 15.3 LOAD COMPENSATION ssrmdsgny'yeuoaassnprIpyttnge,eehpirmmnaaaaocnivftsoopirdcorlthosvln,taewohsaepluelenftn'lscrrad-oeerhgslebAa'oseygylaecdstuiroldetneflel-armdlrmcnateuitarpn,gtcegtsgsiuiInitve.afvlbgae.egwsatlntieylhoeai,nienttfnehh.ltrcfrcehoaeeoaa,it,nridt^noes.ranii\"+sHilgssl-inon-tle^wieiafmis:;dec-:sevea;t:esiqnh_i.rutla,,TT-eanotrthiree:oresceaor,teacmr:icltic,vse#taioev'orljsxetn|pattiloeo;gyrlnwe:aatafeg,ftnrefhLce,,tel;wthfhreeeelpsqeepyucrsoetcrtwneieccnemaytrl Load compensationis the managementof reactive pcwer to improve power quality i.e. V profile and pf. Here the reactive power flow is controlled by pt oo awTlhel eirsvsiycasht etaetphmteesaprnirsodbdvleaevrmioostuoesfct ylpt opoewt hsecerl fsscytousmct elpymoeofnusvtaal itrniineocdgursaebmvoiecvteeh.socT,dahsiel.csf ce.rcc, ,oo,mmr r ppj \"ee, rnnrssuaa, utt oirnrrgs, installing shunt compensatingdevices (capacitors/reactorsa)t the load end bringing aboutproperbalancebetweengeneratedand consurnedreactivepower. This is most effective in improving the power transfer capability of the system anclits voltagestability.It is desirableboth economicallyand technicallyto operatethe systemnear uriity power factor. This is why someutilities impose a penalty on low pf loads. Yet another way of irnproving the system performa-neeis to operateit under nearbalancedconditions so as to reduce the ho* of legative sequencecurrents thereby increasing the system's load capabilityand reducingpower loss' A transm issiolnine hast hr cccr it icalloadings( i) nat ur aloading( ii) st eady- starestabilitylimit and (iii) thermallimit loading.For a compensatedline the naturalloading is the lowest and beforethe thermal loadinglimit is reached, steady-statestability limit is arrived.

esJ compensated,it will behave as a purely resistive element and would. cause I series resonance even at fundamental frequency. The location of series T5.4 LINE COMPENSATION Ideal voltageprofile for a transmissionline is flat, which can only be achieved capacitorsis decidedby economicalfactorsand severityof fault currents.Series by loading the line with its surge impedance loading while this may not be capacitor reducesline reactancethereby level of fault currents. achievable,the characteristicsof the line can be modi so that on on vanous lssuesln in series and shunt compensatorsnow follows. (i) Ferranti effect is minimized. (ii) Underexcitedoperationof synchronousgeneratorsis not required. 15.5 SERIES COMPENSATION (iii) The power transfer capability of the line is enhanced.Modifying the A capacitor in series rvith a line gives control over the effective reactance characteristicsof a line(s) is known as line compensation. betweenline ends.This effective reactanceis eiven bv Various compensatingdevicesare: Xr=X-X, o Capacitors where . Capacitors and inductors Xl = line reactance . Active voltage source(synchronousgenerator) Xc = capacrtorreactance When a number of capacitors are connected in parallel to get the desired capacitancei,t is known as a bankof capacitorss, imilarly a Uant<of incluctors. A bank of capacitorsand/or inductorscan be adjustedin stepsby switching It is easy to seethat capacitorrcduccsthe cffectivc line rcactance*. (mechanical). . This resultsin improvementin performanceof the systemas below. Capacitors and inductors as such are passive line compensators,while (i) Voltage drop in the line reduces(getscompensatedi).e. minimization of synchronousgeneratoris an active compensator.When solid-statedevices are end-voltagevariations. used for switching off capacitorsand inductors, this is regardeclas active compensation. (ii) Preventsvoltagecollapse. Before proceedingto give a detailedaccountof line compensatorw, e shall (iii) Steady-sttttpeower transferincreasesi;t is inversclyproportionalto Xl. briefly discussboth shuntand seriescompensation. (iv) As a result of (ii) rransientstability limit increasbs. The benefitsof the seriescapacitorcompensatorareassociatedwith a problem. Shunt compensation is more or less like load compensationwith all the The capacitive reactanceXg fbrms a seriesresonantcircuit with the total series advantagesassociatedwith it and discussedin Section 15.3. It needs to be reactance pointed out here that shunt capacitors/inductors can not be distributed uniformally along the line. Thesearenormally connectedat the end of the line X = Xt * X*.n * Xoun, and/orat midpoint of the line. The natural frequency of oscillation of this circuit is given by. Shunt capacitors raise the load pf which greatly increases the power transmitted over the line as it is not requiredto carry the reactivepower. There rfL^ - | is a limit to which transmittedpower can be increasedby shuntcompensation 2rJrc as it would require very iarge size capacitorbank, which would be impractical. For increasing power transmittedover the line other and bettermeanscan be '2n adopted.For example,seriescompensationh,igher transmissionvoltage,HVDC etc. where l= system frequency When switched capacitorsare employed for compensation,theseshculd be xReactive voltage drops of a seriesreactanceadded in a line is I2x disconnectedirnmediatelyunderlight loaccl onclitionsto avoicel r.cessivveoltage It is positive if X is inductive and negative if X is capacitive. So a seriescapacitive rise and ferroresonancein presenceof transformers. reactancereducesthe reactancevoltage drop of the line, which is an alternative wav of saying that The purposeof seriescompensationis to cancel part of the seriesinductive reactanceof the line using series capacitors.This helps in (i) increase of X't= \\- X,- maximum power transfer (ii) reduction in power angle for a given amount of I power transfer (iii) increasedloading. From practical point of view, it is ii ti ti

I ,, ModernPowerSystemAnalYsis rL =degree of compensation ' technology, the capacitanceof the series capacitancebank can be controlled X much more effectively; both stepwiseand smoothcontrol. This is demonsffated by the schematicdiagramof Fig. 15.2wherein the capacitoris shuntedby two I = 25 to J5Vo(recommended) nstors ln antl fc<f current in positive half cycle and the other in negativehalf cycle. In each half cycle when the thyristor is fired (at an adjustable angle), it which is subharmonicoscillation. Even though series compensationhas often been found to be cost-effective conductscurrent for the rest of the half cycle till natural current zeto. During the off-time of the thyristor current is conductedby the capacitorand capacitor compared to shunt compensation,but sustainedoscillations below the funda- voltage is vr. During on-time of the thyristor capacitor is short circuited i.e. mental system frequency can cause the phenomenon, referred to aS v, = 0 and current is conductedby the thyristor. The sameprocess is repeated subsynchronousresonance(SSR)first observedin 1937,but got world-wide in the other half cycle. This meansthat v, can be controlled for any given i, attention only in the 1970s, after two turbine-generatorshaft failures occurred which is equivalent of reducing the capacitanceas C = vJi.By this scheme at the Majave Generating station in Southern Nevada. Theoretical studies capacitancecan be controlled smoothly by adjusting the firing angle. pointed out that interaction between a series capacitor-compensatedline, oscillating at subharmonic frequency,and torsional mechanical oscillation of ^ o ' l l \" \" il--+{- i n s turbine-generatorset can result in negative damping with consequentmutual | reinforcementof the two oscillations.Subsynchronousresonanceis often not a - _ - _ _ - a , . C ur r erneuaci mtoirt major problem, and low cost countermeasuresand protective measurescan be | r' applied. Some of the corrective measuresare: l-1<-------r l (i) Detecting the low levels of subharmoniccurrents on the line by use of sensitiverelays, which at a certain level of currents triggers the action to ---. Cri i bypassthe seriescapacitors. t 7,c-----] (ii) Modulation of generator field current to provide increased positive Fig. 15.2 damping at subharmonicfrequency' Seriesincluctorsareneeclecfol r line compensationunderlight loadconditionsto Thyristors are now availableto carry large current and to withstand (during counter the excessivevoltage rise (Ferranti effect). off-time) large voltage encounteredin power systems.The latest device called a Gate Turn Off (GTO) thyristor has the capability that by suitable firing As the line load and, in particular the reactive power flow over the line circuit, angle (time) at which it goes on and off can both be controlled. varies, thereis needto vary the compensationfor an acceptablevoltage profile. The mechanical switching arrangementfor adjusting the capacitanceof the This meanswider range and finer control over capacitanceS. imilarly confrol capacitor bank in series with the line is shown in Fig. 15.1. Capacitanceis is possibleover seriesreactorin the line. varieclby openingthe switchosof individualcapacitancews ith thecapacitance C1, being startedby a bypassswitch. This is a step-u'isearrangement.The All controlled.\".?nd uncontrolled (fixed) series compensators require a whole bank can also be bypassedby the starting switch under any emergent protective arrangement.Protectioncan be provided externallyeither by voltage conclitionson the line. As the switchesin series with capacitorare current arrester or other voltage limiting device or an approximate bypass switch carrying suitablecircuit breakingarrangementare necessary.However, breaker arrangement.In no case the VI rating of the thyristors should be exceeded. switched capacitorsin seriesaregenerallyavoided thesedays the,capacitoris Dependingon (i) kind of solid-statedeviceto be used(ii) capacitorand/or either fixed or thvristor switched. reactor compensation and (iii) switched (step-wise) or smooth (stepless) control, severalcompensatronschemeshave beendevisedand are in use.Some of the more common compensationschemesare as under. (i) Thyristor Controlled SeriesCappcitor (TCSC) (ii) Thryristor\\Switched SeriesCapacitor (TSSC) (iii) Fllyristor controlled Reactor with Fixed capacitor (TCR + FC) (iv) GTO thyristor Coniiolled Series Capacitor (GCSC)

*562i I Modern Power SystemAnalysis I i;'r (v) Thyristor Controlied reactor (TCR) Capacitor and/or reactor seriescompensatoract to modify line impedance. An altcrnativoapproachis to introducca controllablevoltagcsourccin series with the line. This schemeis known as static synchronousseriescompensator (SSSC). SSSC has the capabitity to induce both capacitive and inductive voltage ln senes wrtn [lne, wrdenrngme operatmgreglon o called static var switchesor systems.It meansthat terminologywise SVC = SVS It can be used for power flow control both increasing or decreasingreactive flow on the line. Further this schemegives better stability and is more effective in dampingout electromechanicaol scillations. and we will use theseinterchangeably. Though various types of compensatorscan provide highly effective power Basic SVC Configrurations (or Desigrns) flow control, their operating characteristicsand compensating features are different. These differences are related to their inherent attributes of their Thyristorsin antiparallelcan be used to switch on a capacitor/reactour nit in control circuits; also they exhibit different loss characteristics. stepwise control. When the circuitary is designeclro adjust the firing angle, capacitor/reactournit actsas continuouslyvariablein the power circuit. From the point of view of almost maintenancefree operation impedance modifying (capacitorsand/or reactors)schemesare superior. The specific kind Capacitor or capacitor and incluctor bank can be varied stepwise or of compensator to be employed is very much dependent on a particular continuouslyby thyristor control. Severalimportant SVS configurationshave application. been devisedand are applied in shunt line compensationS. ome of the static compensatorsschemesare discussedin what follows. 15.6 SHUNT COMPENSATORS (i) Sutu.rutcdreuctlr As alreadyexplainedin Sec. 15.4and in Ch. 5 (Sec.5.10) shuntcompensators This is a multi-core reactor with the phase windings so arrangedas to cancel are connected in shunt at various system nodes (major substations) and the principal harmonics.It is consicleredas a constantvoltageieactive source. sometimesat mid-point of lines. Theseserveihe purposesof voltage control and It is almost maintenancefree but not very flexible with reipect to operating load stabilization. As a result of installation of shunt compensatorsin the characteristics. system,the nearbygeneratorsoperateat nearunity pf and voltageemergencies mostly clo not arise.The two kinds of compensatorsin use are: (ii)'fhyristor-coilrroll,cd reuctor (T'CR) (1) Static var compensators(SVC): These are banks of capacitors (some- A thyristor-controlled-reacto(rFig. 15.3) compensatorconsistsof a combina- t im e si n c l u c to rasl s ofg r u s eu n d e rl i ght l oad condi ti ons) tion of six ptrlseor twelveplusethyristor-controllercel actprswith a fixe<tshunt capacitorbank. The reactive power is changectby adjustingthe thyristor firing (ii) STATCOM: static synchronouscompensator angle. TCRs are characterised by continuous control, no transients and (iii) Synchronous condenser: It is a synchronousmotor running at no-load gencrationof harmonics'kT. he controlsystenrconsistsof voltage(andcurrent) and having excitation adjustableover a wide range. It f'eedspositive Power I - - Line oi.rt I =n..\"'xr.'t,orn, c.-- VARs into the line under overexcited conditions and negative VARS transformer I f when underexcited.(For details seeSec.5.10.) I i]l.: It is to be pointed out here that SVC and STATCQM are stgtic var generatorswhich are thyristor controlled. In this section SVC will be detailed -? {-rf '-- l I uvvv while STATCOM forrns a part of FACTS whose operafionis explainedin S e c .1 5 . l O . neaitr i )tl I I Statia VAR Compensator (SVC) Fixed I Thesecomprisecapacitorbank fixed or switched(controlled)or fixed capacito caoacitor bank and switched reactor bank in parallel. These compensatorsdraw reactiv I (leadingor lagging) power* from the line therebyregulatingvoltage,improv I *A rcrrctlrrcccor)ncctcd in shunt to tinc at voltage V draws reactive power Vzl] I Neutral It is negative (leading) if reactance is capacitive and positive (lagging) if reactance is Fig. 15.3 Thyristocr ontrolledreactor(TCR)with fixedcapacitor inductive. *Though ) -connected TCR's are usedhere, it is betterto use A-connectedTCR,s since it is better configuration.

Modern Power SystemAnalysis |; measuring devices, a controller for error-signal conditioning, a Iinearizing fe\" circuit and one or more synchronisingcircuits. (iii) Thyristor switcherl capacitor (fSC) system (e.9. line faults, load rejection etc) TSC/TCR combinations are It consistsof only a thyristor-switched capacitor bank which is split into a characterisedby continuouscontrdl, no transients,low generationsof harmon- numDeor unrtso equal ratrngsto achrevea stepwisecon ics, low losses,redundancy,flexible control and operation. '0d6' in Table15.1. Table 15.1 Comparisonof Static Var Generators Type of TCR.FC TSC-(TSR) TCR-TSC Var (1) (2) (3) Generator VI and VQ Max comp. current Max. Comp. current is Same as in proportionalto system (1) or (2) / characteristics is proportional to voltage. Damping systemvoltage. reactor Max cap. var output Max. cap. var output - Fig. 15.4 Thyristosr witchedcapacitor(TSC) decreaseswith the decreaseswith the As suchthey areappliedas a discretlyvariablereactivepower source,where this typeof voltagesupportis deemedadequateA. ll switchingtakesplacewhen squareof the voltage square of the voltage the voltageacrossthe thyristor valve is zero, thus providing almost transient tree switching.Disconnectionis eff'ectedby suppressingthe firing plus to the decrease. decrease. thyristors, which will block when the current reaches zero. TSCs are, charetcLorisbccyl stcp wisc control, no transients,vcry low htlrnronics,low Loss Vs var High lossesat zero Low lossesat zero Low lossesat losses,redundancyand flexibility. output. output. Losses ouput. Losses increase zero output. (iv) CombinedTCR and TSC Compensator decreasesmoothly step-like with cap. Lossesincrease A combinedTSC and TCR (Fig. 15.5) is the optimum solution in majority of with cap. output, output step-like with cases.With this, continuousvariablereactivepower is obtainedtirroughoutthe i n c r e i r s ew i t h cup. output, c ot t t l; lc tccro n l l o l rl n g c . F ru ' tl tc l rn o rcl i rl l control o1' botl r i nducti ve and inductive output smoothly with capacitivepartsof the cornpensatoris obtained.This is a very advantageous Hannorric I n t c r n a l l yh i g h hrtcrnallyvcry low ind. output ---{ generation (large pu TCR) Resonancemay Internallylow Requiressignificant necessiratetuning (small pu Neutral Max. theoret. filtering reactors TCR) Filtering delay l/2 cycle I cycle required I cvcle Trrnsir:nl Poor (FC ('iluscs Cun bc lrcutral. Sanrcas in (2) behaviour under systent transientover- (Capacitorscan be voltagr: disturbances voltagesin response switchedout to minimise to step disturbances) trattsicntovcr-voltages) T5.7 COMPARISON BETWEEN STATCOM AND SVC It mqw he nnterl trhr roqtL ri -r r tLhror v nr rnvrl r, l-l,a' ll ll r; l-r^t^/ q- . r \\^,*P^g-r, 4- ft:r^l r-E rdlrBc ^r -r- - r t - lt \"'*J ul ule v ll a characteristicand functional compensationcapability of the STATCOM and the Capacitor Neutral sVC aie similar l2l. However, the basic operating principles of the STATCOM, which, with a converterbasedvar generator,functions as a shunt- connectedsynchronousvoltage source,are basically different from thoseof the SVC, since SVC functions as a shunt-connectedc, ontrolledreactiveadmittance. Fig. 15.5 A combinedTCR/TSCcompensator This basic operational difference renders the STATCOM to have overall

tto .t lvlooernPower SystenrAnalysts t superior functional characteristics,better performance,and greater application flexibility as comparedto SVC. The ability of the STATCOM to maintain full * FACTStechnologyhavebeenproposedandimplementedF.ACTSdevicesca1 capacitle output current at low system voltage also makes it rnore effective be effectively used for power flow conffol, load sharing among parallel corridors, voltage regulation, enhancementof transient stability anOmitlgation than the SVC in improving the transient (first swing) stability. enablea line to carry power closer to its thermal rating. Mechanical switching Comparison between series and shunt compensation: has to be supplementedby rapid responsepower electronics.It may be noted that FACTS is an enabling technology, and not a one-on-one substitute for (i) Seriescapacitorsare inherently self regulatingand a control systemis not mechanicalswitches. required. FACTS employ high speedthyristors for switching in or out transmission (ii) For the same performance,series capacitorsare often less costly than line cornponents such as capacitors, reactors or phase shifting transformer for SVCs and lossesare very low. some desirable performanceof the systems.The FACTS technology is not a singlehigh-power controller,but rather a collection of controllers,wtrictr can be (iii) For voltage stability, series capacitors lower the critical or coliapse applied individually or in coordination with others to conffol one or more of the voltage. systemparameters. (iv) Seriescapacitorspossessadequatetimc-ovellt-radcapability. Before proceeding to give an account of some of the important FACTS (v) Series capacitorsand switched seriescapacitorscan be used to controi controllers the principle of operation of a switching converter will be explained, which forms the heart of thesecontrollers. loading of paralledlines to minimise active and reactivelosses. 15.9 PRINCIPLE AND OPERATION OF CON\\ZERTERS Disadvantagesof seriescompensation: Controllable reactive power can be generatedby dc to ac switching converters (i) Series capacitorsare line connectedand compensation is removed for which are switched in synchronismwith the line voltage with whicn tfr\" reactive outagesand capacitorsin parallel lines may be overloaded. power is exchanged.A switching power converterconsistsof an array of solid- stateswitches which connectthe input terminals to the output terminals. It has (ii) During tru.,vyioading,the voltage on one side of the seriescapacitormay no internal storage and so the instantaneousinput and output power are equal. be out-ofrange. Further the input and output terminations are complementar|, that is, if the input is terminated by a voltage source (charged capacitor or battery), output ii a (iii) Shunt reactorsmay be neededfor light load compensation. cuffent source (which meansa voltage sourcehaving an inductive impedance) (iv) Subsynchronoursesonancemay call for expensivecountermeasures. and vice versa. Thus, the converter can be voltage sourced (shunted by a capacitoror battery) or current sourced (shuntedby an inductor). Advantagesof SVC Single line diagram of the basic voltage sourcedconverter scheme for (i) SVCs control voltagedirectly. reactivepower generationis drawn in Fig. 15.6.For reactivepower flow bus (ii) SVCs controltemporaryovervoltagesrapidly. voltage V and converter terminal voltage V, arein phase. Disadvantagesof SVC Then on per phase basis i (i) SVCs havelimited ovcrloadcapability. r=v-v4 (ii) SVCs areexpensive. x'i The bestdesignperhapsis a combinationof seriesand shuntcompensation. Because of higher initial and operating costs, synchronous condensersare 'Fha rl vacav^L+t;v\" l^y -^'.'^- ^-,^f^^J^^ :^ normally not competitivewith SVCs. Technically,synchronouscondensersare ruv PrJw9r E^Urr4rrBtr l5 better than SVCs in voltage-weak networks. Following a drop in network voltage, the increasein condenserreactive power output is ilrslantaneousM. ost ' O=vI= v(v-vo) synchrono.uscondenserapplications are now associatedwith HVDC installa- X llons. 15.8 FLEXTBLEAC TRANSMTSSTONSYSTEMS (FACTS) The rapid developmenot f power electronicstechnologyprovidesexciting opportunitietso developnewpowersystemequipmenftor betterutilizationof

563 l ModernPo - I --S-_ys_teim--bus V capacitoris zero. Also at dc (zero fiequency) the capacitor doesnot supply any reactivepower. Thereibre,the capacitorvoltage cloesnot changeand the 'i ' Couplingtransformer capacitorestablishesonly a voltage level for the converter. The switching causesthe converterto interconnect he 3-phaselines so that reactivecurrent I can flow between thent. .J, The converterdraws a small amountof real power to provide for the internal X :l J,Transformelreakagereactance loss (in switching).If it is requiredto feed reai power to the bus,the capacitor is replace,lby a storagebattery. For this the circuit switching has to be Vd. modified ro create a phasedifference dbetween Vsand Vwith Vsleading V' Fig.15.6 Staticreactivepowergenerator The aboveexplainedconverteris connectedin shuntwith the line. On sirnilar lines a convertercan be constructedwith its terminalsin serieswith the line. The switching circuit is capableof adjusting Vo,the output voltageof the It has to carry the line current and provide a suitablemagnitude (may also be converter. For Vo1 V,1 lags V and Q drawn from the bus is inductive, while phase)voltage in serieswith the line. In such a connectionit would act as an for Vo> V, I leads V and Q drawn from the bus is leading. Reactive power drawn can be easily and smoothly varied by adjusting Voby changing the on- impedancemodifier of the line. time of the solid-state switches. It is to be noted that transformer leakage reactanceis.quite small (0.1-0.15 pu), which meansthat a amall differenceof 15.10 FACTS CONTROLLERS voltage (V-Vo) causesthe required ,1and Q flow. Thus the converter acts like a static synchronouscondenser(or var generator). The developmentof FACTS controllershas followed two different approaches. The first approachemploys reactive impedancesor a tap changing ffansformer A typical convertercircuit is shown in Fig. 15.7.It is a 3-phasetwo-level, with thyristor switches as controlled elements,the second approachemploys six-pulse H.bridge with a diode in antiparallel to each of the six thyristors self-commutatedstatic convertersas controlled voltage sources. (Normally, GTO's are used). Timings of the triggering pulses ate in synchronismwith the bus voltage waves. ln general,FACTS controllerscan be divided into tour categories. (i) series (ii) shunt (iii) combined series-series(iv) combined series-shunt Vo\" Vot Vo, controllers. [1' The generalsymbol for a FACTS controller is given in Fig. 15.8(a).which K showsa thyristor arrow inside a box. The seriescontroller of Fig. 15.8bcould be a variable impedance,such as capacitor,reactor,etc. or a power electronics Va\" basedvariable source.All seriescontrollersinject voltage in serieswith the line. If the voltage is in phasequadraturewith the line, the seriescontroller only C suppliesor consumesvariablereactivepower.Any otherphaserelationshipwill Fig. 15.7 Three-phasetw, o-levesl ix-pulsebridge involve real power also. Tlte shunt controllers of Fig. 15.8c may be variable impedance,variable sourceor a combination of these.All shunt controllers inject current into the system at the point of connection. Combined series-seriescontrollers of Fig. 15.8dcould be a combination of separateseriescontrollerswhich areconffolled in a coordinatedmanneror it could be a unified controller. Combined series-shuntcontrollers are either controlled in a coordinated ma-nneras in Fig. 15.8e or a unified Power Flow Controller with series and shuntelementsas in Fig. 15.8f.For unified controller, there can be a real power exchangebetween the seriesand shunt controllers via the dc power link. Storagesourcesuch as a capacitor,battery,superconductingmagnet,or any other sourceof energy can be addedin parallel through an electronicinterface to replenish the converter's dc storageas shown dotted in Fig. 15.8 (b). A

ff.*J ModernpowerSystemAnalysis CompensatiniopnoweSr ystems tffi*ffil T- controller with 'storage is much more effective for conffolling the system dynamics than the corresponding controller without storage. Llne Line Line (a)General symbolfor FACTScontroller(FC) (a) (b) / lun\"l <-oc ac lines Fig. 15.9 (a) STATCOMbasedon voltage-sourceadnd (b) current-sourced [,--TT- I (d) Unifiedseries- converters. l-FTc-l seriescontroller . A combination of STATCOM and any energy source to supply or absorb ( c ) S h u n tc o n t r o l l e r powei is called static synchronousgenerator(SSG). Energy sourcemay be a battery, flywheel, superconducting magnet, large dc storage capacitor, another rectifi erlinverter etc. Statlc Synchronous Series Compensator (SSCC) FCH Coordinated !-. It is a seriesconnectedcontroller. Though it is like STATCOM, bu(its output control voltage is in serieswith the line. It thus controlsthe voltage acrossthe line and I it henceits impedance. dc power Interline Power FIow Controller (IPFC) link (e)Coordinatesderies (f) Unifiedseries- This is a recently introduced controller 12,3]. It is a cornbinationof two or a n d s h u n tc o n t r o l l e r shuntcontroller more staticsynchronousseriesconlpensatorswhich are coupledvia a common dc link to facilitate bi-directional flow of real power betweenthe ac terminals Flg. 15.8 DifferentFACTScontrollers of the SSSCs, and are controlled to provide independent reactive series compensation for the control of real power flow in each line and maintain the The group of FACTS controllers employing switching converter-based desireddistribution of reactivepower flow amongthe lines. Thus it managesa synchronous voltage sourcesinclude the STATic synchronous COMpensator comprehensiveoverall real and reactive power managementfor a multi-line (STATCOM), the static synchronousseriescompensator(SSCC), the unified transmissionsystem. power flow controller (UPFC) and the latest, the Interline Power Flow Controller (IPFC). Unified Power FIow Controller (UPFC) STATCOM This controller is connectedas shown in Fig. 15.10. It is a combination of STATCOM and SSSC which are coupled via a iommon dc lin-k to -allow STATCOM is a static synchronousgeneratoroperatedas a shunt-connected bi-directional flow of real power between the series output terminals of the static var compensatorwhose capacitive or inductive output current can be SSSCand the shunt output terminals of the STATCOM. Theseare controlled i^;- or- - ^ - rrr- ^ l l --l oiieor! not epenaoen, toi rhe ac system voltage. The STATCOM, like its fo provide concurrent real and reactive series line compensation without an conventional countetpart,the SVC, controls transmission voltage by reactive external energy source.The.UPFC, by meansof angularly unconsffainedseries shunt compensation.It can be based on a voltage-sourcedoi curtent-sourced voltage injection, is able to control, concurrently/simultaneouslyor selectively, converter. Figure 15.9 shows a one-line diagram of srATCoM basedon a the transmission line voltage, impedance, and angle or, alternatively, the real voltage-sourcedconverterand a current sourcedconverter.Normally a voltage- source converter is preferred for most converter-basedFACTS controllers. STATCOM can bc designedto be an active filter to absorb systemharmonics.

rr$il#4 Modern - r' i: l l\"r'. I stability and cnnffol line flows. Voltage source converter based (self- commutated) HVDC system may have the same features as those of andreactive line flows. The UPFC may alsoprovide independentlycontrollabte STATCOM or UPFC. This systemalso regulatesvoltage and provides system shuntreactive compensation. A comparative perfonnance of major FACTS controllers in ac system is given in Table 15.2 U4l. STATCOM Table 15.2 A comparativeperformanceof major FACTS controller Fig. 1S.10 UnifiedpowerFlowControileUr pFC Typeof FACTS Load V Transient Oscillation Th yri s t or- co n tro II ed p h as e -sh ifti n g Tra n sfo rm er (TCp s r) Controller flow control control stability Damping This controller is also called Thyristor-controlled Phase Angle Regulator (TCPAR). A phaseshifting transformercontrolled by thyristor switchesto give SVC/STATCOM X XXX xxx XX a rapidly variablephaseangle. Th yristor- Con troll ed VoI tdgre R egu trator flCVR) TCSC XX x xxx XX A thyristor controlled transformer which can provide variable in-phase voltage with continuouscontrol. SSSC XXX x XXX XX Interphase Power Controller (IpC) A series-connectecdontrollerof active and reactivepower consisting,in each TCPAR XXX XX X XX phase,of inductive and capacitive branches subjected to separately phase- shifted voltages.The active and reactive power can be set inpedendentlyby UPFC XXX XXX XXX XXX adjustingthe phaseshifts and/or the branch impedances,using mechanical or electronicswitches. * a;v-slrong influence; xx-average influence; x-smail infruence Thyristor Controlled Braking Resistor ICBR) It is a shunt-connectedthyristor-switched resistor, which is controlled to aid sutyffvtARy stabilization of a power system or to minimise power acceleration of a generatingunit during a disturbance. Since the 1970s,energy cost, environmentalrestrictions,right-of-*ay difficul- Thyristor-controlled Voltage Limiter FCVL) ties, along with other legislative sociai and cost problems huu. postponed the A thyristor-switchemd etal-oxidevaristor(Mov) usedto limit the voltage construction of both new generationand transmissionsystemsin India as well acrossits terminalsdurinstransientconditions as most of other countries. Recently, becauseof adoption of power reforms or 'TIVDC restructuring or deregulation, competitive electric energy markets are being developedby mandatingopen accesstransmissionservices. It may be noted that normally HVDC and FACTS are complementary technologiesT.he role of HVDC, for economicreasonsi,s to interconnecat c In the late 1980s,the vision of FACTS was formulated.In this variouspower system?,wheraereliableac interconnectiown ould be too expensiveH. VDC electronicsbasedcontrollers (compensatorsr)egulate power flow andtransmis- sion voltage and through fast control aciion, mitigate dynamic disturbances. Due to FACTS, transmissionline capacitywas enhanced.Two typesof FACTS controllers were developed. One employed conventional thyristor-switched capacitorsand react<lrsa, nd quadraturetap-changingtransformerssuchas SVC and TCSC. The secondcategory was of self-commutatedswitching converters as synchronousvoltage sources,e.g. STATCOM, SSSC,UPFC and IpFC. The two groups of FACTS controllers have quite different operatingand perform- ance characteristics.The secondgroup usesself-commutated dc to ac converter. The converter, supported by a de power supply or energy storage de.,,icecan also exchangereal power with the ac systembesidesconmollingreactivepower independently. The increasing use of FACTS controllers in future is guaranteed.What benefits are required for a given systemwould be a principal justification for the choice of a FACTS controller. Its final form and operation will, ofcourse, depend not only on the successfuldevelopmentof the necessarycontrol and

r:5?4i I Modernpower System Analysis I communieationteehnologiesand protocols, but also on the final structureof ihre evolving newly restructuredpower systems. REFERNECES 16.1 INTRODUCTION Books Load forecastingplays an important role in power systemplanning, operation I ' Chakrabarti,A', D.P. Kothari and A.K. Mukhopadhyay, Performance Operation and control. Forecastingmeans estimating active load at various load buses and Control of EHV Power TransmissionSystems.Wheeler,New Delhi, 1995. aheadof actualload oacccuerrrteainnce'l.ePaladnntiimnge'anadlsoopecraallteiodnafol arepcpalisctaintgionirsitoefrvloaalsd. 2. Hingorani, N.G. and Laszlo Gyugyi, (Jnderstanding FACTS. IEEE press, New forecasting requires York, 2000. Nature of forecasts,lead times and applicationsare summarisedin Table 16.1\" 3. Song,Y.H' and A.T. Johns,FlexibleAC TransmissionSystems,IEEL, bndon, 1999. 4. Miller, T'J.E., ReactivePower Control in Electric SystemsJ, ohn Wiley and Sons, Table 16.1 NY, lgg2. Nature of Lead time Application 5. Nagrath' I.J. and D.P. Kothari, Electric Machines, 2nd edn, Tata McGraw-Hill. forecast Very short term A few seconds to Generation,distribution schedules, New Delhi, 1997. several minutes contingency analysis for system 6. Taylor, c.w., power system vortage stability, McGraw-Hill, singapore, 1994. Short term secunfy 7. Nagrath,I.J and D.P. Kothari, power S),stemEngineering,TataMcGraw_Hill, New Half an hour to Allocation of spinning reserve; Medium term a ferv hours operarionilI planning ild unit Delhi, 1994. Long term commitment; maintenancescheduling 8. Indulkar, C.S. and D.P. Kothari, Power SystemTransientsA StatisticalApproach, A few days to a t'ew'rveeks Planning for seasonal peak- Prentice-Hallof India, New Delhi, 1996. A t'ew months to wrnter. summer 9. Mathur, R.M' and R.K. Verma, Thyristor-BasedFACTI; Controllersfor Electrical a few years Planning generation growth TransmissionSystems.John Wiley, New york, 2002. A good t'orecastreflecting current and future r.rends,temperedwi.Ji good Papers judgement, is the key to all planning, indeedt.o financial success.The accuracy I0- Edris- A-, \"FACTS Technologv Development: An Update\".\\EEE Pott'er Engirteer- of a forecast is crucial to any electric utility, sinceit determinesthe timing and ing Rerieu'. \\'ol. 20. lVlarch 2@1i. pp f9. characteristicsof major system additions. A forecastthat is too low can result 1l - Iliceto F- and E. Cinieri, \"Comparative Analysis of Series and Shunt Compensation in low revenuefrom salesto neighbouringutilities or even in load curtailment. Schemesfor Ac Transmission svstems\". IEEE Trans. pAs 96 6). rg77. pp lglg- Forecasts that are too high can result in severe financial problems due to l8-10. excessive investment in a plant that is not fully utilized or operatedat low 12. Kimbark. E.W.. \"Hou, to Impror,e S.r'srem Stabilir-i, u,ithout Risking Suhs-rtchnrnous R('s(')nitncc\"'. IEEE lr.-rrs. P.-tJ-.96 t-i,I. Sept/Oc.t 1977. pp 160.S- 19. 13. CIGRE/ 'wG 38-01. .srrzricl/ar Conrp<,r-rdrorCr.IGRE/. prrr-is.i 986 l{. Ptrvlr.f).. \"Llsc'ofHDVC and F.ACTS\".IEEE procc'edingvso. l. .\\8. 2, Feb. 2000, pp 235-245. 15. Kinrbark,E.W. \"A New Look At Shunt Compensation.,IEEE Trans. Vol PAS- 102.No. l. Ja-n1983.nn 212-2!8

LoadForecastinTgechnique Llif,iffi T6.2 FORECASTING METHODOLOGY I capacityfactors.No forecastobtainedfrom analytical proceclurescan be strictly Forecasting techniquesmay be divided into three broad classes.Techniques r\"ii.d upon the judgement of the forecaster, which plays a crucial role in may be basedon extrapolation or on correlation or on a combination of both. ariving at an acceptable forecast. rrnlnlstlc, Choosing a forecasting technique for use in establishing future load requirements is a nontrivial task in itself. ing on nature o variations,one particularmethcd may be superior to another. Extrapolation The two approachesto load forecasting namely total load approach and Extrapolation techniquesinvolve fitting trend curves to basic historical data component approachhave their own merits and demerits.Total load approach adjusted to reflect the growth trend itself. With a trend curve the forecast is has the merit that it is much smootherand indicative of overall growth trends obtained by evaluating the trend curve function at the desired future point. and easy to apply. On the other hand,the merit of the componentapproach is that abnormal conditions in growth trends of a certain component can be Although a very simple procedure, it produces reasonableresults in some detected,thus preventingmisleadingforecastconclusions.There is a continuing instances.Such a techniqueis called a deterministicextrapolationsincerandom need,however, to improve the methodology for forecastingpower demand more accurately. errors in the data or in analytical model are not accountedfor. The aim of the presentchapteris to give brief expositionsof some of the Standard analytical functions used in trend curve fitting are [3]. techniques that have been developed'in order to deal with the various load (i) Straight line ! = a+ bx (ii) Parabola' forecasting problems. All of theseare based on the assumptionthat the actual (iii) S-curve !=a+bx+c*2 load suppliedby a given systemmatchesthe demandsat all points of time (i.e., (iv) Exponential (v) Gempertz !=a+bx+ci+dx3 there has not been any outagesor any deliberate sheddingof load). It is then !=ce& possibleto make a statisticalanalysisof previous load datain order to set up ! = In-r 7a + ced''1 a suitable model of the demandpattern.Once this hasbeendone,it is generally ' The most corlmon curve-fitting technique for fitting coefficients and exponents(a4) of a function in a given forecastis the method of leastsquares. possible to utilize the identified load model for making a prediction of the If the uncertainty of extrapolated results is to be quantified using statistical entities such as mean and variance,the basic techniquebecomesprobabilistic estirnated demand for the selectedlead time. A major part of the forecasting extrapolation. With regression analysis the best estimate of the model task is thus concernedwith that of identifying the bestpossiblemodel for the describing the trend can be obtained and used to forecast the trend. pastload behaviour.This is bestachievedby decomposingthe load demandat any given point of tirne into a number of distinct components.The load is dependenton the industrial, commercial and agricultural activities as well as the weathercondition of the system/areaT. he weather sensitivecomponentdepends Correlation on temperarure,cloudiness,wind velocity, visibility and precipitation. Recall Correlation techniques of forecasting relate system loads to various demo- graphic and economic factors. This approach is advantageousin forcing the the brief discussionsin Ch. 1 regarding the nature of the daily load curve which forecaster to understand clearly the interrelationship between load growth patterns and other measurable factors. The disadvantageis the need to forecast hasbeenshown to have a constantpari correspondingto the baseload and other demographic andeconomic factors, which can be more difficult than forecasting system load. Typically, such factors as population, employment, building variable parts. For the sakeof load forecasting, a simple decomposition may permits, business,weather data and the like are usedin correlation techniques. serveas a cdnvenient starting point. Let y(k) representthe total load demand No one forecasting method is effective in all situations. Forecasting (eitherfor the whole or a part of the system) at the discretetime k = l, 2,3, techniques must be used as tools to aid the planner; good judgement a:rd experiencecan never be completely replaced. ....It is generallypossibleto decomposey(k) into two parts of the form y(k)= ya(k)+ y\"(k) (16.1) where the subscript d indicates the deterministic part and the subscript s 16.3 ESTIMATION OF AVERAGE AND TREND TERMS indicates the stochasticpart of the demand.If k is consideredto be the present time, then y(k + j), j > 0 would represbnta future load demandwith the index The simplest possibleform of the deterministicpart of y(k) is given by 7 being the lead time. For a chosenvalue of the indexT, the forecastingproblem is then the sameasthe problemof estimatingthe valueof y(k +/) by processing adequatedata fbr the pastload dernand. ya &) = !-a + bk + e(k) (r6.2)

ModernPowerSystemAnalysis LoadForecastinTgechnique kI t5f,*A where larepresents the averageor the mean value of yd(k), bk representsthe it 'trend' term that grows linearly with k and e(k) representsthe error of I modelling the complete load using the averageand the trend terms only. The ln order to illustrate the nature of results obtainablefrom Eqs' (16'6a) and io(o1op6iup.6ruebt,r)i,oincnomninislliimdoenilrslitohrn.erTfcr.ulTaphtea=esshcaaonswdhntvhianeluatehmseoogufnratthpoefhasegolerfcicFtirriigtcta.ulr1ea6nl 'ael nrgwdyhthciceohnjsngudimuvesptd+thoiaenl I questionis one of estimatingthe valuesof the two unknown model parameters (toaA demand) in MWs 1n Punjab over a period of sevenyears starting from l 1968. A total of g5 data have been generatedfrom the graphsby sampling the it la aldb !o ensurea good model. As seenin Ch, 14, when little orlo st1listical graphsat intervals of 30 days.These have beensubstitutedin Eqs' (16'6a) and information is availableregardingthe error term, the methodof LSE is helpful. if O.OUiIn order to compute the avetageand the trend coefficients of the four T variables.The results are given in Table 16'2' I l If this methodis to be used forestimating yo and b,the estimationindex \"/is defined using the relation J - E{ez(D} (16.3) where E(.) representsthe expectationoperation. Substituting for e(A) from Table 16.2 Eq. (16.2) and making use of the first order necessaryconditions for the index J to have its minimum value with respectto ya md b, it is found that the following conditionsmust be satisfiedl2). E { y a - y a & )+ b k I = 0 (16.4a) Variable Average Trend Cofficient E {bkz- ya(k)k+ tdkl - 0 (16.4b) Population 13 million o.2 Industrial output Rs 397million Since the expectation operationdoesnot affect the constantquantities,it is Agricultural output Rs 420.9million 0.54 easy to solve these two equationsin order to get the desired relations. Load demand 8 5 5 . 8M W 0.78 r.34 ta= E{yd&)l- b{E(k)} (16.5a) b - lE{ya&)kl- yo E{kllt4{k2l (16.5b) If y(k) is assumedto be stationary(statisticsare not time dependent)one may involve the ergodic hypothesis andreplacethe expectatiori operationby the time averaging fonnula. Thus, if a total of N data are assumedto be avai.labLefor determining the time averages,the two relationsmay be equivalentlyexpressed as follows. (16.6a) (16.6b) - peprJlatioin millions lndustryin millionsof ruPees These two relations may be fruitfully employed in order to estimatethe average and the trend coefficient for any given load data. --- Agriculturme illionsof rupees - Loaddemandin MW Note that Eqs. (16.6a) and (16.6b) are not very accuratein casethe load data behavesas a non-stationaryprocesssince the ergodic hypothesisdoes not / - - -/ holcl for suchcases.It may still bepossibleto assumethat the dataover a finite window is stationaryand the entire set of data may then be consideredas the t,, juxtaposition of a number of stationaryblocks, each having slightly different statistics.Equations(16.6a)and(16.6b)may then be repeatecol ver the different Ii 1 6 t blocks in order to compute the average and the trend coefficient for each window of data. the data

ffi4 ModernpowerSystemAnalysis { I I -t--tr in m\" -aeYr tb-e assumedto be 600 I the load model ,II t I I y(k)-fu3,+e(k) (16.8a) I 400 ;: 3 i:l wherethe coefficient b,needto be estimatedfrom the pastload d /<and would model above is obviously a non-linear function of the time index 72 120 need L coefficients to be estimated. A much simpler approach to non-linear k (hours)-_--- modelling of the load is to introduce an exponential form (16.8b) 600 y(k)= c exp [bk]+ e(k) t uwtthrnaheknicngshoifvowienrnmnvsoed,ldatvhteia\"ns.teoIonxnupelyoliitnnthwereunor,tciuaafnolsmkremno,to.dhwAeenllmlhcaetohstehatfohtfideicsiaoerdfnedtLqsiSut.iiBEoreneidassliiesdaaedtsosvrialetyandektuaxecgtetiehnnoegdftenhbdaeettiunnorugaemlsreltboiamegdraiooltyeff I the model parametersfrom the given historical data. 400 i. 3 E 2oo o J 0 |6,4ESTIMATIoNoFPERIoDIGcoMPoNENTS 1 192 216 240 264 288 312 336 T;cbiluhyrrie'av.oedp,e'so,t;hweoerwmr niutnntiiin\"slittFiyuciugopr.rvaaterZgtr.eZoafapwntehtdrreiictohhldoegaopidfvoetm*riyoantyhowemcevoieaanklrtsiaar.ietnIitromsnisosomo.fcbetoshpneeesravridiecoetddirvtihcefaopctrootlwmregexpardomasniueplypnleptlsoltiaheinded k (hours)--------- variations are repetitive from day to day except for somerandom fluctuations' Iwentniegitseihrkeat dlwosaofeyesstkheilneeynvpntieheeraxwiott tdtShhseuatntacSdrutaurinvyngedbffaieooyhrmsaas,vureesnsahdyaoa,slytihdsaeadymicsflfi.iedsIr,tn,ins]irgicTghttnttyiofpiofceuaortnniotethdlyaSictfurtonhwdmeaavcyteuhtfrioolvlsremethiooerfwmtiihhtidhee Fig. 16.2 Hourlyloadbehaviourof Delhiovertwo consecutiveweeks Caution The 85 data, used in Example 16.r, aregenerally not adequatefor making superposedrandom variations' interval, statistical caiculations so that the vaiues given ubour may not be entirely If it is assumed that the load data are being sampled at an hourly adequate.In addition, the statistical characteristicsof the set of variables btih.efe\"gne'x.tqhp\"ereresusatertdeoinaZto7tetrarl\\ml6o8sfroa1fd6s8a.lAoFarosduurdiiteaartbasleienmrioeonsdeweplietfhorirotthdheseofluotnhadadatytmh(kee)nloitsaaldthfrpeeanqtugteeivrnnecmnybauyyl concernedmay have changed (i.e., the data may in fact be non-stationary)and this alsomav inffoduce someerror in the results.Finally, the graphsin Fig. 16.1 are actually based on half yearly data obtained from the planning commission documentand an interpolation processhas beenemployed in order to generate the monthly data. This may add some unspecified errors to the data which will also affect the accuracyof the estimates. Prediction of ya &+j) y(k) = y + t [a, sin iuk + b, cos iuk] + e(k) (15'9) Once the model for the deterministic component of the load has been i=l determined,it is simple to make the prediction of its future value. For the simple model in Eq. (76-2), the desired prediction is computed using the relation where L representsthe total number of harmonics present and a; and b; are the amplitudes of respectively the sinusoidal and the cosinusoidalcomponents' only v'(k + i) = T, + h(k+ i\\ in the model' /4 \\ r/ JA \"U'- (i6.7) r l n r r( l t J l l l l l l g r r ! r i n q n tt',o r m o n i c sn e e - dt o be included Jl lrq^r^\\ Once the harmonic load model is identified, it is simple to make a prediction More General Forms of Models of the future load ya(k + 7) using the relation Before leaving this section,it may be pointed out 9a&+ /) = h' (k+ l)i(k) (16.10) generalizedby including secondand higher order that the load model may be terms on the right hand side

'eifrz#d ModernPo@is LoadForecastinTgechnique -TH--ffi !6.5 ESTIMATION OF YS(^ft):TiME SERIESAPPROACH hv2 Auto-regressive Models 15(ft=) -t a,y, (k - r) +D b, w(k- ) + w(k) (16.14) i:l j:l Estimation of two structuralparametersn and rn as well asmodel parameters ap bi and the variance d of the noise term w(k) is required. Moie complex can be represented.The identification problem is solved off-line. The acceptable load model is then utilized on-line for obtaining on-line load forecasts.ARMA model can easily be modified to incorporate the temperature, rainfall, wind velocity and humidity data [2]. In some cases,it is desirable to show the dependenceof the load demand on the weather variables in an explicit manner. The time series models are easily generalized in order to reflect the dependenceof the load demand on one or more of the weather variables. The sequencye,(ft)is saidto s a t i s f ya n A R modelof ordern i.e.it is [AR(n)], 16.6 ESTIMATION OF STOCHASTIC COMPONENT: if it can be expressedas KALMAN FILTERING APPROACH n (16.11) The time series approachhas been widely employed in dealing with the load y,(k)= Do,r,(k - i) + w(k) forecasting problem in view of the relative simplicity of the model forms. i_l However, this method tendsto ignore the statisticalinformation about the load data which may often be available and may lead to improved load forecastsif utilized properly. In ARMA model, the model identificationproblem is not that simple. These difficulties may be avoided in some situationsif t'he Kalman filtering techniques are utilized. , lie inside the unit circle in the e-plane. Application to Short-term Forecasting The problem in estimating the value of n is refer-redrn qc tha ntnhtaw )^s An application of the Kalman filtering algorithm to the load forecasting problem has been first suggestedby Toyada et. al. [11] for the very short-term and short-term situations.For the latter case,for example,it is possibleto make useof intuitive reasoningsto suggestthat an acceptablemodel for load demand would have the form v,(k)= y'(k)+ v(li) (16.15) i,&) = -D a,y, (k- i) (16.r2) where y,(ft) is the observedvalue of the stochasticload at time ft, y,(k) is the (r 6 .l 3 ) true value of this load and u(fr) is the error in the observedload. In addition. the dynamics of the true load may be expressedas i:l y,(k +I) = y{k) + z(k)+ u1&) (16.16) The variance d of w(k) is then estimatedusing the relation where a(k) representsthe increment of the load demand at time k and u1@) n 2 -- ( 1r 'lln' n N representsa'disturbanceterm which accountsfor the stochasticperturbationsin \\ ) y,(k).The incrementai ioaciitseif is assumerito remain constanton an average F -Zrrt /,-rc \\tr ) at every time point and is modelled by the equation k:l Auto-Regressive Moving-Average Models z(k +l) = z(k)+ ur(k) (16.r7) In some cases,the AR model may not be adequateto represent-the observed wherethe term uz&) representsa stochasticdisturbanceterm. Ioad behaviour unlessthe order n of the model is madevery hrgfi. In such a case ARMA (n, m) model is used.

a-.,.\".lm In order to make use of the Kalman filtering techniques, the noise terms u(k), uz&) and u(k) are assumedto be zero meanindependentwhite Gaussian to use the solution of Eq. (16.18a)for the vector x (k+ d)to get the result sequences.Also, the model equations are rewritten in the form f;g+ d)= pdi(tcttc) (16.199) x (ft+l) = Fx(k)+ Gu(k) (16.18a) In order to be able to make use of this algorithm for generating the forecast k v(k of thg load v-(k + d\\. it is necessarv fhaf the nnisc cfeficrinc qnr{ o^-o nr}ra- wherethe vectorsx(ft)and u(ft) are definedas information be available. The value of R(ft) may often be estimated from a knowledge of the accuracyof the metersemployed. However, it is very unlikely x(k)= ly,(k)= (Df andu(k) = fur(k) uz&))r that the value of the covariance Q(k) will be known to start with and will therefore have to be obtained by some means. An adaptive version of the The matrices4 G andh' arethenobtainedfrom Eqs.(16.15)-(16.17e)asily Kalman filtering algorithm may be utilized in order to estimate the noise andhavethe followingvalues. statisticsalongwith the statevector x(k) tZ). Now let it be assumedthat both R(k) and Q(k) are known quantities. Let it also be assumed that the initial \" =[L1o 1 -cl ,=[ 1 1ol1'r =, ltl estimate i (0/0) and the covarianceP\"(0/0) are known. Basedon thesea priori u' Lo Lol information, it is possible to utilize Eq. (16.19a)-(16.19e)recursively to Based on model (16.18),it is possible to make use of the Kalman filtering processthe data for yr(l), yr(z), ..., yr(k) to generatethe filtered estimate ,(kl algorithm to obtaintheminimum varianceestimateof the vector x (k) basedon k). Once this is available, Eq. (16.19g) may be utilized to ger,eratethe desired the data y,(k): {y,(1), ),,(2)... y,(k)}. This algorithm consisrsof the following load forecast. equations. i (k/k)= i (ktt<- 1) + K\" (k) ty\"(ft)- h'ft(k/k-I)l (16.19a) i(ktk -1)= F i((k-L)/(k-r)) (16.19b) K,(k) = P,(k/k-l) hlh' P,(klk-I)h + R(k)l-r (16.19c) P-(k/k)= V - K,(k) h') P,(k/k-l) (16.19d) To illustrate the nature of the results obtainable through the allorithm just discussed,the data for the short term load behaviour for Delhi have been P,(k/k - l)= FP,(k-l/k-l)F'+ GQG-DG', (16.19e) processed.A total of 1030datacollectedat the interval of 15 minuteshave bee.n where, processed.It has been assumedthat, in view of the short time interval over which the total data set lies, the deterministic part of the load may be assumed Q(k) = covarianceof u(k) to be a constantmean term. Using the sample averageFormula (16.7) (with b = 0), we get y = 220 Mw. The data for yr(k) have then been generated by R(ft) = covarianceof y(ft) subtracting the mean value from the measured load data. ft(klk) = filtered estimateof x(ft) To processthese stochasticdata, the following a priori information have beenused: ft (k/k-L) = singlesteppretliction of x(ft) R(k)= 3.74, K.r(k) = filter gain vector of samedimension as a(ft) i (oro=) Pr(k/k) = filtering error covariance [;] Pr(k/k-l) = predictionerror covariance The results of applicationof the prodietionAlgonthnn (16.19) are shown in Fig. 16.3.It is noted that the elror of 15 minutesaheadload predictionis around F r onr E q. (1 6 .1 8 b )o b ta i nth e p re d i c ti o ni ((k+ 1)l k) 8 MW which is about 3Voof the averageload and less than 2Voof the daily From this the one step aheadload fbrecast is obtained as maximum load. Y \" ( f t +I ) = h ' i ( ( k + l ) l k ) (16.1e0 It may be noted that filtering implies removal of disturbanceor stochastic term with zero mean. It is also possibleto obtaina multi-step aheadprediction of the load from the multi-step ahead prediction of the vector x (k). For example, if the prediction