Power Systems Engineering Research Center

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1 Power ystems Engneerng Research Center oltage ecurty Margn Assessment Proect Fnal Proect Report Proect Team Garng M Huang Al Abur Texas A&M Unversty PERC Publcaton 0-35 November 00

2 nformaton about ths Proect For nformaton about ths proect contact: Garng M Huang Professor Electrcal Engneerng Department Texas A&M Unversty College taton, TX Phone: Fax: Emal: huang@ee.tamu.edu Power ystems Engneerng Research Center Ths s a proect report from the Power ystems Engneerng Research Center (PERC). PERC s a mult-unversty Center conductng research on challenges facng a restructurng electrc power ndustry and educatng the next generaton of power engneers. More nformaton about PERC can be found at the Center s webste: For addtonal nformaton, contact: Power ystems Engneerng Research Center Cornell Unversty 48 Phllps Hall thaca, New York Phone: Fax: Notce Concernng Copyrght Materal Permsson to copy wthout fee all or part of ths publcaton s granted f approprate attrbuton s gven to ths document as the source materal. Ths report s avalable for downloadng from the PERC webste. 00 Washngton tate Unversty. All rghts reserved.

3 ACKNOWLEDGEMENT The work descrbed n ths report was sponsored by the Power ystems Engneerng Research Center (PERC). We express our apprecaton for the support provded by PERC s ndustral members and by the Natonal cence Foundaton under grant NF EEC receved under the ndustry / Unversty Cooperatve Research Center program. The ndustry advsors for the proect were Man ubramanan, ABB Network Management; Don evck, Center Pont Energy; Bruce Detzman, Oncor. Ther suggestons and contrbutons to the work are apprecated.

4 EXECUTE UMMARY ncreasngly, wthn the deregulated power systems, voltage stablty ssues are becomng sgnfcant n the way we plan, operate and mantan the system. The nvolvement of newer players to the electrcty power busness has led to the prolferaton of ntra-area and nter-area transactons of electrcty n the transmsson network. Typcally, these transactons are of consderably shorter duraton and larger varety than that n vertcally ntegrated utlty (U) structures, where utltes controlled power generaton, transmsson and dstrbuton. Not only does ths lead to frequent and sgnfcant changes n system operatng ponts and load flow patterns, but also result n ncreasng volatlty n the system. Ths leads to potental securty and relablty degradaton n the exstng electrc power system operatons. One of the most affected securty degradaton, that s beng wtnessed, s the voltage stablty ssue. There s a need to evolve procedures to ncorporate voltage stablty effectvely nto the operatonal aspects of deregulated power systems. To acheve ths am one needs to quckly assess, from measurable quanttes, the operatonal state of the system from vewpont of voltage stablty. At the same tme n case of stablty problems, the responsblty evaluaton procedures need to be dstnctly dentfed n the sprt of deregulaton. The obectve of ths proect was to evolve a framework, wthn the context of the restructured power market operaton, to ncorporate voltage stablty assessment nto the power system securty, accountablty and utlzaton factors for control devces. n course of completng the obectve of ths proect, we have come up wth varous new and practcal algorthms and procedures that can effectvely address the ncorporaton of voltage stablty nto the deregulated power system operaton. We summarze the sgnfcant outcomes of our work as gven below. Dynamc Modelng of generators, governors, ULTC, swtched capactor and loads usng EUROTAG have been carred to study dynamc voltage stablty and the mportance of dynamc reserves to mantan stablty. [8] Detectng dynamc voltage collapse usng state nformaton has been nvestgated for a varety of dynamc dsturbances. [4] tatc modelng of FACT devces n nvestgatng voltage stablty studes also has been carred out. t s observed that usage of devces lke TCC and C could mprove stablty margn sgnfcantly. [3] A new way of usng bfurcaton analyss, usng the unreduced acoban matrx, [7] that avods sngularty nduced nfnty problem and beng computatonally attractve has been formulated. Wthn the context of deregulated market the responsblty evaluaton, n a potental voltage collapse, assumes sgnfcance. Usng the bfurcaton analyss a procedure to allocate contrbuton of generators, transmsson and control elements n voltage

5 stablty has been evolved. [6] Ths could be used as the bass for evaluatng the utlzaton factors and the prcng of control elements n power system. An algorthm to compute Optmal Power Flow ncorporatng voltage stablty has been proposed. [1] The voltage stablty constrant s computed from the power flow state varables and the network topology. Ths algorthm has been appled further to evaluate relablty ndces n plannng stages. [] The ncorporaton of voltage stablty enhancement devces lke FACT, nto the algorthm, has also been formulated. [3] The framework for transacton based power flow analyss for transmsson utlzaton allocaton has been proposed. [10] The methods to model transacton for both pool type and pont-to-pont long term blateral type has been desgned. Ths analyss has been used to address the approach to equtable loss allocaton n a deregulated market. [11] The approach has been appled to congeston management and responsblty evaluaton n the compettve, deregulated energy markets. [9] A new way to evaluate voltage stablty responsblty n a composte market model framework havng both the pool type spot market and the blateral long-term transactons has been devsed. [5] Ths decomposton approach has the potental to address voltage stablty usage, voltage securty prcng and responsblty settlement n a transacton based deregulated operaton of electrcal markets

6 TABLE OF CONTENT 1.0 OLTAGE TABLTY TUDE AND MODELNG UE 1.1 Typcal two bus system for voltage stablty studes 1..1 Test Case for a -bus system 1.. Effects tuded 1..3 oftware used for mulaton 1.3 Power factor ssues on statc voltage collapse lmts 1.4 Modelng of TCC n statc voltage stablty analyss 1.5 Modelng of C & ts effect on voltage stablty analyss 1.6 Modelng of load and ts effect on voltage stablty margns 1.7 ummary of observatons for the -bus case study.0 TABLTY NDEX FOR TATC OLTAGE ECURTY ANALY.1.1 oltage collapse pont at load bus usng a two bus model.1. Formulate a stablty ndcator..1 Numercal erfcaton.. Another Example: (ndex L for the TCC scenaro dscussed n 1.4).3 Extenson of the two bus voltage stablty ndex L theory to a Mult-bus system.4.1 Test Case.4. Detals of Case cenaros Presented.5 Results.6 ummary for the mult-bus case scenaros 3.0 DYNAMC OLTAGE TABLTY UE 3.1 nvestgaton towards applyng ndex L for dynamc voltage stablty studes mulaton set-up detals 3..1 Obectve 1: Contrbuton of other load buses towards ndex at local bus 3.. Observatons Obectve : The ndex L nformaton at frst dp on voltage after a load change 3.3. Observatons Obectve 3: ndex varatons at all buses followng a load change 3.4. Observatons Obectve 4: The ndex L nformaton mmedately after lne outage 3.5. Observatons Obectve 5: Approxmate L evaluaton after lne outage based on pre-outage data 3.6. Observatons 3.7 Publcatons 4.0 OLTAGE TABLTY CONTRANED OPF ALGORTHM Algorthm 4.1. Proposed ncorporaton of voltage stablty ndex constrants

7 4. A sample run 4.3 Observatons 4.4 Publcatons 5.0 TRANACTON BAED TABLTY MARGN AND UTLZATON FACTOR EALUATON 5.1 Theory behnd Transacton based power flow 5. Transcaton based power flow algorthm 5..1 Assumptons 5.. tep-wse procedure for decomposton 5.3 Transacton based voltage securty margn allocaton algorthm Test case for demonstratng voltage securty margn allocaton algorthm 5.4. Results for varous scenaros 5.5 Publcatons 6.0 CONCLUON 7.0 REFERENCE

8 1 OLTAGE TABLTY TUDE AND MODELNG UE 1.1 Typcal two bus system for voltage stablty studes We have taken a sample bus system whch s generally used to evaluate statc voltage stablty lmt and smulated the effects whch we were nterested n evaluatng. The followng subsectons gves detals of our case, the methods used for analyss and the results obtaned n our smulatons Consder the typcal representaton for studyng voltage stablty where the power transmsson model wth one end wthout voltage support s as shown n Fg 1. X 1 D As shown n the fgure D = * = e φ = (Cos φ + n φ) = P D ( 1 + β) (where β = Tan φ) Now we know that, P D = -P 1 = P 1 = ( 1 n θ 1 ) X Q D = -Q 1 = Q 1 = Cos θ 1 X X Elmnatng θ 1 and solvng the second order equaton we fnally get,

9 = 1 - β P D X + [ P D X ( P D X + β 1 ) ] ½ 4 As can be seen from the equaton, the voltage at the load pont s nfluenced by the power delvered to the load, the reactance of the lne and the power factor of the load. The voltage has two solutons, out of whch the hgher one s the stable soluton. The load at whch the term approaches zero, whch s reflected n the egen value analyss as approachng zero from a negatve value, ndcates the steady stage voltage collapse pont Test Case for a -bus system The test case used n our smulaton s gven n the followng dagram. The voltage at the generator bus (E) s taken as 1.0 p.u. The reactance of the lne s taken as 0.15 p.u. E X Gen Load 1.. Effects tuded Our frst effort was to study the effect of varous control apparatuses lke TCC & C and load condtons lke power factor, nature of the loads on the teady tate oltage tablty. The cases we have smulated are enlsted as follows: 1) Effect of Power Factor ) Effect of TCC 3) Effect of C 4) Effect of C poston 5) Effect of type of loadng 1..3 oftware used for mulaton We have used MATLAB based program for repeated power flows wth ncreased loadng at the load bus to get the voltage at the load bus and hence study voltage stablty. At the collapse pont the load flow program would fal to converge and gve a soluton. For C and types of loadng mpact on voltage stablty, we have used EUROTAG smulatons.

10 1.3 Power factor ssues on statc voltage collapse lmts t can be seen from the fgure that as the power factor degrades.e comes down, the voltage collapse occurs at lower power delvery.

11 1.4 Modelng of TCC n statc voltage stablty analyss TCC s a control devce whch s nstalled n the lne to control the lne mpedance thereby controllng the maxmum power loadng n the lnes. Generally one can control the lne mpedance upto + 50% usng the TCC. As far as steady state voltage stablty analyss s concerned, we can model TCC as a varable capactor n seres wth the lne mpedance t can be seen that lower the lne mpedance, further s the voltage collapse pont. Hence t can be nferred that by employng TCC to reduce the lne mpedance for long lnes, one can ncrease the voltage stablty margn at the load end of the lnes.

12 1.5 Modelng of C & ts effect on voltage stablty analyss C provdes voltage support to the lne. C s modeled as a P bus wth zero real power n the power flow analyss. t s observed that employng C.e supportng voltage n between the lne mproves the voltage stablty margn at the load end. Ths can be clearly seen n the result presented below. Moreover by placng C s at equal dstance between the lne, the collapse pont ncreases further. FGURE: (a) Wth & wthout C (b) wthout C, wth one C, wth C s

13 t s seen that placement does affect the voltage collapse pont. Ths can be seen n the followng output graph. The rght most curve corresponds to placng a sngle C closer towards the load end. The mddle curve corresponds to placng the C n the mddle whle the leftmost curve corresponds to placng the C towards the generatng end. Fgure: (a) Leftmost curve represent one C close to generator end (b) Centre curve represents one C n the mddle (c) Rghtmost curve represents one C close to load end

14 1.6 Modelng of load and ts effect on voltage stablty margns The load modelng equaton has been taken from reference (1).e the book by Carson Taylor, P = P 0 (U/U 0 ) Pv * (w/w 0 ) Pf Q = Q 0 (U/U 0 ) Qv *(w/w 0 ) Qf We have chosen three types of load, where the coeffcents are gven as follows Lghtng Pv=1.54, Pf=0.0, Qv=Qf=0.0, P.F=1.0 Central A/C Pv=0., Pf=0.9, Qv=., Qf = -.7, P.F.=0.81 Refrgerator Pv=0.8, Pf=0.5, Qv=.5, Qf=-1.4, P.F. = 0.84 To get the followng curves, the load was ncreased n all the 3 cases startng from 1MW. The voltage profle wth respect to tme s shown n the followng fgure Fgure: (a) Leftmost curve represents Central A/C (b) Mddle curve represents refrgeraton load (c) Rghtmost curve s the ncandescent Lghtng load

15 t can be seen from the above fgure that the AC load becomes unstable at a hgher voltage magntude compared wth the refrgerator and lghtng load model cases. Thus from vewpont of severty of the load on voltage stablty the AC load s the most severe, followed by the refrgerator and lastly the lghtng loads n case of the lghtng load t s seen to be stable at very low voltages. Ths can be explaned by the followng fgure. The hgher plot shows the power varatons wth tme whle the lower trace shows the voltage profle wth tme for the lghtng load case. (1) Lghtng t can be seen that ntally power ncreases to a peak wth decrease n voltage. However, after a peak the power decreases ndcatng stable operaton at the low voltage soluton usually represented n the lower half of the conventonal P curve representaton. 1.7 ummary of observatons for the -bus case study After runnng the smulatons and observng the voltage collapse profle the followng ponts became evdent to us. 1) As X ncreases (long lnes) collapse pont becomes nearer. Ths nfers that lkelhood of voltage collapse s more n case of loads suppled from generaton at a long dstance.

16 ) Lower power factor mpedance loads causes voltage collapse earler. Hence the lmt of voltage stablty margn for load buses operatng at lower power factors s less. 3) t s very clearly seen from our smulatons that use of C mproves the stablty margn.e the load at whch collapse pont occurs. However, t s also seen that employng C very near to the load mproves more the voltage stablty margn n comparson wth a case where t s employed farther from the load bus. 4) The effect of load type on voltage stablty was brought out dstnctly n our smulatons. t could be seen from the smulaton that voltage & frequency dependent load lke ar-condtoners and refrgerators affect voltage collapse sgnfcantly.

17 TABLTY NDEX FOR TATC OLTAGE ECURTY ANALY.1.1 oltage collapse pont at load bus usng a two bus model, G G G G1 Y Q Y L Y Q D Load Fgure 1: sngle generator and sngle load As showed n Fgure 1, one smple system s conceved where there s a load bus and a generator bus. We are nterested n ther voltage behavor. D D = D Y Q + ( D G ) Y L = (.1) D D = D Y Q + D Y L D G Y L = D Y D Y 11 (.) Y L Here Y 11 = Y Q + Y L and 0 = G (.3) Y L + Y Q olvng ths voltage problem (Because we would lke to fnd the effect or load on the voltage, we would lke to fnd soluton of n the equaton (.). Let D = a + Y 11 b. Then (.) s expressed as follows: D. We represent the magntude of D as D D = a + b = D + 0 D = D + 0D cos( δ 0 δ D ) + 0D sn( δ 0 δ D ) Y 11 (.4) a D cosδ = cos( δ 0 δ D ) = 0D (.5) b sn δ = sn( δ 0 δ D ) = (.6) 0 D (.5) + (.6), We wll get

18 4 0 D = ( a D ) + b = a ad + D + b (.7) olve ths equaton: D D = + a ± + a0 b = ( r ± r 1) (.8) 4 Y Because D = a + Y b, hence D D a = cos( φ + φ Y ), b = sn( φ ) D 11 + φ. Then D Y11 Y Y 0 0 D = + a ± ( + a) ( a + b 0 D 0 D = + cos( φ + φy ) ± cos( D 11 D Y Y + φ + φ 11 Y11 11 D = ( r ± r Y 11 1) 11 ) ) Y 11 D 11 (.9) 0 Y11 Here r s defned as r = + cos( φ + φy ) (.10) D 11 D.1. Formulate a stablty ndcator 4 0 We can see, when + a0 b = 0, the voltage at node bus wll collapse. Because 4 Y L 0 = G, so t can be specfed and supported by generator, hence Y L + Y 4 Q 0 When + a0 b 0, the voltage at node bus wll be sustaned; when a0 b < 0, the voltage cannot be sustaned, that s to say, the threshold of the 4 b 0 D voltage collapse s expressed a =, because = a + b, we can deduce the 4 0 Y 11 actve power P and reactve power Q of D ( D = P + ) as follows: D D D Q D PD = Re( Y 11( a + b)), QD = m( Y 11( a + b)), We can get the correspondng curve n D complex plane. Now, we can take ths curve as the borderlne of voltage collapse at load node. o we wll fnd an ndcator to reflect the proxmty to ths borderlne. 1 From equaton (.9), we can get when the voltage collapses, r = 1, = 1. D Y11 From the equaton of (.), we can get:

19 1 0 = 1 + (.11) D Y 11 D o, we defne an ndcator L for voltage collapse as: L = = D D Y 11 = 1 D Y11 When the load s zero ( 1 = 0 ), then L=0, f the voltage of bus 1 collapse, L=1. Here s an example of sngle generator and load system: (.1) f we consder ths problem from the vewpont of Jacoban matrx sngularty, we can get: f the voltage at load bus collapse, the Jacoban matrx wll be sngular, that s say, the determnant of the matrx wll equal to zero. From equaton (.5) and (.6), we can lst the power flow equatons for the above twobus system: f ( D, δ ) = 0D cosδ + D = a (.5 ) g( D, δ ) = 0 D snδ = b (.6 ) o, the correspondng Jacoban Matrx s as follows: D + 0 cosδ D0 snδ J = (.13) 0 snδ D0 cosδ When the determnant of Matrx J equals to zero, the voltage at load bus wll collapse: cos D δ D 1 det( J ) = D 0 cosδ + D0 = 0 = Re = 0 0 D Then we wll express = 1+ = 1 D + b 1 = b b = 1 b Actually, when we dvde equaton (.) by, where b s a real number, so we can get:, we can get: (.14) D Y = 1 + (.15) D Y 11 D From the above analyss, we confrm the ndcator of voltage stablty at load bus as equaton (.1).

20 ..1 Numercal erfcaton To drectly show how the ndcator L show the voltage stablty margn along wth the load changng at load wth bus, we adopt the two-bus system as smulaton obect n EUROTAG.

21 .. Another Example: (ndex L for the TCC scenaro dscussed earler) To show the above ndcator L functon, we wll apply TCC n the transmsson lne n a typcal two-bus system as the above studed case, we get a seres of ndcator L wth the change of transmsson reactance under the placement of TCC

22 .3 Extenson of the two bus voltage stablty ndex L theory to a Mult-bus system We use the and to express the crcut of n node system. = = G L GG GL LG LL G L G L Y K F Z H Here L means load, and G means generator. When we consder the voltage of load node, we know that + = G L F Z (.16) Carryng out the followng transformatons: = L G Z F, Multplyng at the both sdes of the equaton: = + L Z 0, Here = G F 0 + = L Z Z )) ( ( L Z Z Z Z + = ) ( L Z Z Z Z + = ) ( Let Z + = 1 Y, Then: L Z Z Y Y = ) ( 1 = = corr Y Y, n whch = L corr Z Z ) ( o, equaton (.16) can be converted to + + = + Y 0 (.17) Deduced as above, = G F 0, t can be regarded as an equvalent generator comprsng the contrbuton from all generators, lke the for the two bus case. 0 Z Y + = 1

23 corr + + = = L corr Z Z ) (, Ths part expresses the contrbutons of the other loads at the node. Ths case can be consdered as an equvalent system as a sngle generator and load system. Through the same analyss, we know that: 0 1 Y L + + = + = Thus, we get an ndcator to show the proxmty of a system away from voltage collapse.

24 .4.1 Test Case The WCC 9 bus system s taken as a sample system to llustrate the applcablty of the ndcator L to a mult- bus system. The test system s as shown n fgure.

25 .4. Case cenaros Presented Normal Loadng at Load buses are BU 5: MA BU 7: MA BU 9: MA BU1, BU, BU 3 are generaton buses Whle at BU 4, BU 6 & BU 8 there are no loads or generatons. Three Case scenaros has been smulated to study the steady state voltage collapse at load buses & ther respectve L ndex. Case 1: (a) ncrease loadng of Bus 5 from zero upto voltage collapse keepng the load at other buses fxed at the normal value, and observe the ndex L5. (b) Observe effect on ndex L7 at bus 7, when load at bus 5 s ncreasng & approachng collapse. (c ) Observe the effect on local ndex L6 at a bus 6 whch s connected to bus 5 but has no load or generaton on t. Case : ncrease loadng of Bus 7 from zero upto voltage collapse keepng the load at other buses fxed at the normal value Case 3: ncrease loadng of Bus 9 from zero upto voltage collapse keepng the load at other buses fxed at the normal value Note: Power factor s kept constant throughout the loadng of buses.

26 .5 Results The followng sub-sectons gves detals of the results obtaned from the smulaton..5.1 CAE 1(a) Result of ncreasng load at bus 5 and observng the ndex L. t s seen very clearly that ndex L approaches 1 at collapse pont. For ths smulaton the load at bus 7 s taken as MA and load at bus 9 s taken to be MA. The collapse occurs when the load at bus 5 s about MA

27 .5. CAE 1(B) Effect of ndex L at dstant load bus when loadng at local load bus Case: The ndex L at bus 7.e L(7) s nvestgated for ncreasng load at bus 5. The load at bus 7 s fxed at MA & bus 5 load s ncreased from 0 to collapse that occurs at MA. Result: t s found that L(7) ncreases margnally from to

28 .5.3 CAE 1(C) Effect of ndex L at adacent bus wthout load when loadng at local load bus Case: The ndex L at bus 6.e L(6) s nvestgated for ncreasng load at bus 5. There s no load at bus 6 & bus 5 load s ncreased from 0 to collapse that occurs at MA. Result: t s found that L(6) ncreases margnally from to 0.168

29 .5.4 CAE Result of ncreasng load at bus 7 and observng the ndex L. t s seen very clearly that ndex L approaches 1 at collapse pont. For ths smulaton the load at bus 5 s taken as MA and load at bus 9 s taken to be MA. The collapse occurs when the load at bus 7 s about MA

30 .5.5 CAE 3 Result of ncreasng load at bus 9 and observng the ndex L. t s seen very clearly that ndex L approaches 1 at collapse pont. For ths smulaton the load at bus 5 s taken as MA and load at bus 7 s taken to be MA. The collapse occurs when the load at bus 9 s about MA.

31 .6 ummary for the mult-bus scenaros 1) From the results of the smulatons t can be dstnctly observed that the ndex L for the bus n a mult-bus system approaches unty (1) at steady state voltage collapse pont. ) The ndex L for the system ncorporates the effect of the load at the bus t s calculated as well as the loadng n the other parts of the system. However, the effect of other loads depend on how the bus under consderaton s connected to the other buses. 3) There s no sgnfcant effect of the ndex for a load bus whch s not connected drectly to the bus whch s nearng voltage collapse. 4) For a bus lke BU 6 (whch has no load & no generaton) whch s connected to a load BU 5 on one end and Generator 3 on the other end, t s observed that the ndex L6 has only margnal change as BU 5 approaches collapse. Ths s because that BU 6 s supported by the Generator 3.

32 3 DYNAMC OLTAGE TABLTY UE 3.1 nvestgaton towards applyng ndex L for dynamc voltage stablty studes n ths sub-secton the applcablty of whether the ndex L can be used as an early ndcator to represent dynamc voltage stablty lmt s nvestgated. The stuatons nvestgated are (1) a large step load change, and () sudden loss of a transmsson lne mulaton set-up detals The WCC 9 bus system s taken as a sample system to nvestgate the effects of dynamc voltage stablty. The test system s as shown n fgure. Base Case Loadngs are as follows BU 5: BU 7: BU 9: MA MA MA The smulatons have been done n EUROTAG. Models for the excter and governor have been ncluded n all the generator models. The ndex L has been calculated on the same lne as has been dscussed for the statc voltage stablty case. t s however, to be noted that the voltage and the angles at all the load buses are not the same durng the dynamc tme frame of nterest. For smplcty, we have consdered all the loads to be voltage and frequency ndependent. The detals of the work carred out so far are gven n the followng sub-sectons. Work s stll under progress n ths drecton. nce the voltage at the generator buses s not held constant durng the dynamc stuaton the ndex L at each generator bus s evaluated, consderng other generator buses as constant P buses. The method of calculatng s smlar to the one adopted for load buses.

33 3..1 Obectve 1 To look at the contrbutons of other bus loads (7,9) on ndex L at a local bus (5) wth respect to tme. The table below s for a step change of load at bus 5 from to MA The ndex L s gven by L = * / (Y+ x 5 ) Where Y+ = for our case and * = con ( * + 9 *) Here 5 = Tme 7 * 9 * 5 L Observatons 1. As can be seen from the values of the contrbutons of dfferent buses, the varatons n the real power component s not much. However, the reactve power component swngs are perceptble. Ths s n lne wth the fact that voltage s related closely wth reactve power.. The ndex L changes accordng to the local voltage profle & settles down to a defnte value as the voltage settles down.

34 3.3.1 Obectve To observe whether the ndex calculated at the frst dp of the voltage at the bus where a step change n load has occurred, can gve any nformaton of dynamc stablty. [Note: n the smulaton only Governor & AR models have been ncluded. ULTC have NOT been ncorporated.] Motvaton: t was observed that the maxmum value of L ( when calculated over a tme perod) occurred at the frst trough n the voltage profle. The followng tables have been tabulated for a range of step changes and gves the value of ndex L at the frst bg dp & the fnal settlng value POWER (MA) FRT Neg PEAK (L) TEADY TATE (L) Observatons Lookng at the data of the power contrbutons from other load buses (Bus 7 & 9 n our case) to the ndex calculated at the referred bus (Bus 5 n our case) t s observed that the actve component remans substantal even at hgher loads compared wth the ntal value. However, consderable effect s reflected on the reactve power contrbutons. For example, at a load change from to at bus 5, the real part of 9 * changed from to However the reactve contrbuton dropped steeply from to Ths seems to brng out two facts That the voltage at the bus nearng voltage collapse s strongly nfluenced by the reactve power demand at ts bus. The effect of reactve power contrbutons of other load buses to the ndex s mnmal whch seems to support that voltage collapse starts of as a local phenomenon at a partcular overloaded voltage bus whch s nfluenced strongly by ts local reactve power requrement. t s observed that the largest value of the ndex, whch happens to occur at the frst trough of the voltage after the load change, approaches 1 as the dynamc voltage collapse pont. The fnal value of the ndex L matches wth the value whch was calculated for the steady state voltage stablty case.

35 3.4.1 Obectve 3: To nvestgate nto the profle of the ndex varatons at all load buses and all generator buses durng the dsturbance for a partcular step load change n one load bus. A load change from to was mposed on load bus 5 at tme=10sec. (a)the followng fgure shows the varatons of all the ndces wth respect to tme.

36 (b)the followng fgure ncludes the voltage & ndex varatons for all load buses w.r.t tme. (c) The followng fgure shows the voltage & ndex varatons at Gen bus 1 w.r.t tme.

37 (d) The followng fgure shows the voltage & ndex varatons at Gen bus w.r.t tme. (e)the followng fgure shows the voltage & ndex varatons at Gen bus 3 w.r.t tme.

38 3.4. Observatons The ndex at generator bus 1 changes to a peak of around from an ntal Lookng at the poston of bus 5 wth respect to the generator bus 1, ths seems reasonable. The load change affects the nearest generator the most. n ths case the lne 4-5 mpedance s less that lne 5-6 mpedance. o Load bus 5 s closer to generator 1 than generator 3. t s observed that near the dynamc collapse pont loadng of load bus 5, ( ) the ndex at load bus 9 umps from to thus decreasng ts voltage stablty margn. Ths mght be due to the fact that generator 1, whch has been affected by load bus 5 loadng, s the nearest connected generator to load bus 9. (The mpedance of lne 4-9 s less than mpedance 9-8) The ndex at load bus 7 changes from to around near the collapse pont of load bus 5.The dfferental change n the ndex value s less than that n case of load bus 9. Ths mght be because of the fact that Load bus 7 s supported by generator 3, whch s affected the least because of the dsturbance. The L ndex calculated at generator bus 3 durng the dsturbance has peaked only to from an ntal

39 3.5.1 Obectve 4 To observe whether the ndex calculated at the frst dp of the voltage at the bus where a loss of lne has occurred, can gve any nformaton of dynamc stablty The voltage profle at bus 5 ( havng loadng ) after loss of lne 4-5 The voltage profle at bus 7 after loss of lne 4-5 The voltage profle at bus 9 after loss of lne 4-5

40 The followng tables show the ndex calculated (a) at the tme of the largest dp (Negatve peak) n the voltage profle observed at BU 5, and (b) after the voltage oscllatons des down.e steady state. At Load Bus 5 POWER (MA) FRT Neg PEAK (L) TEADY TATE (L) At Load Bus 7 POWER (MA) FRT Neg PEAK (L) TEADY TATE (L) At Load Bus 9 POWER (MA) FRT Neg PEAK (L) TEADY TATE (L)

41 3.5. Graphcal Plots of the Result (a) The followng graph show the varatons of the ndex at the load bus 5 w.r.t to ts bus loadng. Two curves are shown (1) The peak ndex L whch occurs at the frst largest negatve dp n the voltage at bus 5 followng the loss of lne 4-5, and () The ndex L evaluated after the voltage stablzes down after the dsturbance.

42 (b) The followng fgure shows the varatons of ndex evaluated at bus 7 and bus 9 wth respect to the loadng at BU 5. Both the peak value evaluated at the frst largest negatve dp n the voltage at bus 5 followng the dsturbance and the steady state ndex are plotted.

43 3.5.3 Observatons t can be seen that the ndex calculated, based on the nformaton of the system at the frst largest negatve dp on bus 5, approaches 1 as the voltage collapse nears. However, t can be seen that the dfference between the peak ndex and the steady state ndex calculated does not vary much compared to the case of step load change, whch was dscussed n the earler secton. The bus 5 collapsed around a load value of dynamcally because of lne outage 4-5. The load at the steady state voltage stablty lmt evaluated for ths case was Thus the dynamc voltage stablty lmt calculated on the bass of lne outage s less than the steady state voltage stablty lmt. The ncreasng effect on the ndces of other load buses.e bus 7 and 9 are margnal only. The magntude of ncrease n the peak ndex value observed s defntely less than for the case dscussed (A large step load change) n the part 1 of ths report. Ths s because of the fact that the real load at bus 5 durng the present smulaton s only of the order of 0.9 p.u near collapse compared to the case of.7 p.u dscussed n the prevous scenaro.

44 3.6.1 Obectve 5 To nvestgate whether for the loss of lne case for dynamc stablty evaluaton, the ndex calculated on the bass of exact local Z term (consderng the loss of lne nformaton) but other mpedance terms remanng the pre-contngency value can stll gve suffcently accurate ndex calculaton Case cenaro: The followng tables tabulates the result of calculatng ndex based on (A) Consderng the effect of the lost lne 4-5 n calculatng the ZLL matrx, (tated as EXACT) and (B) Takng only the Z term takng the lost lne 4-5 nto consderaton whle the rest of the terms as the orgnal healthy state ZLL matrx (tated as APPROXMATE) At Load Bus 5 POWER (MA) Exact Peak ndex Appro.. Peak L Exact teady L Appro.. teady (L) At Load Bus 7 POWER (MA) Exact Peak ndex Appro.. Peak L Exact teady L Appro.. teady (L)

45 At Load Bus 9 POWER (MA) Exact Peak ndex Appro.. Peak L Exact teady L Appro.. teady (L) Observatons (1) t can be seen that the exact and the approxmate matches closely for all the load buses. Thus t can be concluded that the ndex s predomnantly dependent on the term Z of the Z LL matrx. () Thus f any local measurement at load buses can yeld ths value, then the ndex L calculated would be farly approxmate to the exact value, even f we cannot get the complete nformaton of all the healthy lnes n the network.

46 3.7 Publcatons The readers can refer to reference [4] for summary of the work that have been dscussed n the earler sectons of ths chapter. n ths proect the dynamc modelng of generators, governors, ULTC, swtched capactor and loads usng EUROTAG have also been carred to study dynamc voltage stablty. [8] The varous myths surroundng dynamc voltage stablty have been clarfed here. Moreover, the mportance of dynamc reserves of generator to mantan voltage stablty durng a dynamc dsturbance has been clearly brought out usng smulatons. nterested readers may refer to publcaton [8] for knowng the detals of the smulaton and the summary of the results.

47 4 OLTAGE TABLTY CONTRANED OPF ALGORTHM t has been seen from the prevous sectons that the voltage stablty ndex L represents n a way how far the load bus s from collapse pont. Ths feature can be exploted n developng a load curtalment polcy ncorporatng the securty feature of voltage stablty margn. The followng secton proposes a method to acheve ths Algorthm The followng steps explan the procedure of carryng out the OPF wth the ndex L at load buses as one of the constrants The obectve functon to be carred out s the mnmzng of load curtalment mnmze n =1 Load Curtalment For all buses from =1, n For each bus, the term Load-Curtalment s gven by the followng expresson Load Curtalment = P lreq P Where P lreq s the ntal load expectaton at bus before the OPF procedure, P l s the load demand, fnally possble to be met wthn the constrant specfed. For all the buses, the power flow equatons to be satsfed are, l P g P l n = 1 ( G cos δ + B sn δ ) = 0 Q g Q l n = 1 ( G sn δ B cos δ ) = 0 The mnmum and maxmum lmts on generators actve & reactve power output s gven by, P Q g mn g mn P g Q g P Q g max g max The transmsson lne constrants can be specfed by

48 P + Q max The load sheddng phlosophy can be smplfed f we assume that sheddng, s carred out n equal proporton of actve & reactve power. n other words, the power-factor of all the loads remans the same as the ntal value. Ths can be represented as P / P = Q / Q l lreq l lreq 0 Pl P lreq 0 Ql Q lreq 4.1. Proposed ncorporaton of voltage stablty ndex constrants (a) For all buses, consder the followng constrants, as usually used n normal OPF program. mn (b) For all the load buses (PQ) and buses where there are no loads and generators, use the followng addtonal constrant, based on local ndex calculaton L max 4. A sample run L L crt The WCC 9 bus system s taken as a sample system to llustrate the applcablty of the ndcator L to a mult- bus system. The test system was the same that was used n secton.4.1. Before runnng the OPF the followng loads were put BU 5: BU 7: BU 9: MA MA MA The followng table gves the result of the OPF run based on the proposed algorthm for the above case. t s observed that there s no load curtalment on BU 7 and BU 9. Only BU 5 has load curtalment. NOTE: All the P buses are held at =1.0 p.u L crt for all the load buses Load curtalment at BU

49 Wthout the constrant of voltage stablty ndex mposed on the load buses, the load curtalment value got at bus 5 after runnng OPF was found to be Observatons 1. For the above case f we choose any value of L crt ust above 0.1, the voltage stablty ndex constrant does not seem to effect the OPF for our load pattern and system chosen. Ths s because of the fact that the constrant mn s already volated and hence held constant at the volated bus. Hence, thereafter the algorthm stops from searchng a soluton based on the voltage stablty ndex constrant crteron.. Choce of a low value of L crt ncreases the load curtalment to be carred out. Hence, the above OPF algorthm encompasses the securty based feature of voltage stablty n the calculaton of load curtalment. 3. f the allowable mn for BU 5 was kept as 0.8 p.u, the load curtalment got by the ncorporaton of the stablty margn crteron, for a L crt of 0.3, was found out to be more than that calculated wthout usng t. 4.4 Publcatons For nterested readers, further smulaton and applcaton detals of the algorthm can be got from reference [1]. The authors have also formulated the algorthm to ncorporate FACT devces lke TCC, the detal of whch can be got from reference []. t s observed that loadablty of the system can be ncreased by usng these devces. The voltage stablty constraned OPF algorthm, developed n ths proect, blends tself effectvely to the steady state characterstc of the devces wthn ts formulaton. Applcaton of the algorthm to composte relablty analyss has been explored n reference [3]. Evaluaton of relablty ndces that ncorporates the steady state voltage stablty, could be acheved usng the algorthm.

50 5 TRANACTON BAED TABLTY MARGN AND UTLZATON FACTOR EALUATON 5.1 Theory behnd Transacton based power flow To derve a more complete formula of decomposton, we frst recall the coupled AC power flow equatons n the polar form as follows. the slack bus. where G Q G ( P ( Q G G P D Q ) D ) ( g ( g cosθ + b snθ b snθ ) = 0 cosθ ) = 0 for = 1,..., n ( s), s s [5.1] P, are actve and reactve power generatons at bus, P, Q are actve and reactve power loads at bus, θ s the voltage magntude and angle of bus, θ = θ θ, y = g + b s the branch admttance between nodes and. D D Let (,θ ) be the soluton of problem [5.1], several basc facts wth respect to a general transmsson system are observed below. (1) Lne resstance s consdered much smaller than lne reactance.e. r/x <<1, and voltage angle dfference across each branch s also assumed rather small. () Actve power of the system s strongly coupled wth voltage angle dfferences across each branch, or phase angles θ of each bus n reference to the desgnated slack bus. (3) Reactve power of the system s strongly coupled wth the voltage magntudes throughout the entre network.

51 (4) When absolute value of voltage angle θ s small enough throughout the system, nodal magnary current components are strongly coupled wth the voltage magntudes. Facts 1, and 3 are wdely recognzed, and used n DC flow analyss and other lnearzed flow models. n general Fact 4 holds on the condtons r/x 1/3 and θ π/9, whch s rather straghtforward from the nodal voltage and current equaton. 1 Partcularly, let Y be the (n n) nodal admttance matrx, and Z bus [ Y ], bus consder the followng nodal voltage equaton, = bus θ [ e ] = [ Z ] [ + ] where = Re( ), = m( ) bus R M, R bus M bus cosθ [ Z bus ][ M ] Approxmately, t s satsfed that for snθ [ Z bus ][ R ] r 1 x 3 Consder θ cosθ 1 for θ π/9, we derve e = cosθ + snθ cosθ. That s Z bus ][ ]. [ M n case of a constant Z bus heren, mmedately, t follows that the voltage magntudes strongly couple wth the magnary current components on the condtons r/x 1/3 M and θ π/9. These facts are used to explot relatons between a partcular nodal real power P and assocated nodal current. From Krchhoff Laws, a real power necton P can be expressed n terms of real and magnary current components. That s θ * P = Re[ e ( )] = cosθ Re( ) + snθ m( ) [5.] t seems mpossble to separate partcular contrbutons of P on real and magnary current components from [5.]. Fortunately, the facts as mentoned above enable us to make approprate approxmatons as follows. () uppose that the second term of [], snθ m( ) s small enough, P can be approxmately related to the real current component Re( ) by

52 P Re( ) [5.3] cosθ () The remanng terms ncludng all m( ) and resdual errors assocated wth Re( ) are approxmately attrbuted to the system reactve power, whch domnates the network voltage profles. Thus the current necton vector of the reactve power market s approxmated by Q = Re( k bus ) [5.4] Note that the accuracy of [5.3] depends on snθ m( ), whch s the approxmaton error. A relatve approxmaton error on a bass of P s ntroduced below. E P snθ sn( ϕ θ ) = snθ m( )/ P = [5.5] cosϕ Note that E P depends on both phase angle θ and power factor (PF) angle ϕ. To secure dynamc reactve power reserves, the normal PF on the demand sde s restrcted to a narrow margn (say cosϕ n laggng). two approxmaton error levels are marked: Level 1: E P 0.065, for 6.5 θ 15 deg. Level : E P 0.115, for 10 θ 0 deg. t s observed that gven a range of 10 θ 0 degrees, the largest approxmaton error s around 10 percentages. For examples, from a standard power flow soluton, a range of approprate reference angles to reduce approxmaton errors can be determned by where θ L, mn θ mn θ s θ L,max θ max [5.6] θ max,θ mn are the largest and smallest phase angles correspondng to a zero reference angle, θ θ are the lower and upper lmts of Level 1 or. L, mn, L, max

53 5. Transacton based power flow algorthm 5..1 Assumptons To begn wth the decomposton algorthm, we frst ntroduce some economc contexts and nvolved assumptons. (1) An energy market conssts of ndvdual energy schedulng coordnators C k, who are enttled to arrange MW exchange schedules and to choose loss supplers. () A C k may not mantan ts own reactve power balance. nstead, a separate market named Q, the central and O-dependent reactve power schedulng s responsble for the reactve power support. (3) A C k s also responsble for a porton of transmsson loss resultng from consumng reactve power support, whch ntroduces reactve power flows. 5.. tep-wse procedures for decomposton We shall derve the decomposton formulae on a general power system wth N-buses and L-branches. To smplfy our presentaton, we assume only two market players n the system: PX appears a central power exchange market; TX s a blateral transacton. The assumptons wll be relaxed later on. A system-wde reactve power market Q conducted by O s responsble for overall reactve support servces. tep1: elect an approprate angle for the slack bus from the gven power flow soluton, referrng to [5.6] tep : Decompose the nodal current vector based on TXs. From a known operatng pont (,θ ), the (n 1) nodal current vector s determned by θ 1e bus = [ Ybus ] Ebus, where Ebus =. [5.7] θ ne bus

54 where Y bus s the (n n) nodal admttance matrx, whch s nonsngular n consderaton of lne chargng and other shunt terms. Accordng to the proposed approxmaton formula [5.3], ndvdual market components. That s bus s decomposed nto where P PX C G. PX PX PG, PD, cosθ =.. PX PX PG, n PD, n n cosθn P TX 0 TX PG, k k cosθk = 0 TX PD, m m cosθm 0 Q = [5.8] k C k,, D, are the actve power generaton and load at bus, n assocaton wth PX or TX. TX s wth the source and snk buses at k and m respectvely. bus PX TX tep 3: Decomposed nodal voltage components mmedately follow from tep by Krchhoff Laws. E 1 = [ Y ] ( bus where the subscrpt symbol means PX, TX or Q ndvdually. ) * [5.9] Normally t s satsfed that E E 1 E 0 E 0 [5.10] Q bus PX TX tep 4: Compute branch current components *,, *, on any lnk between buses and by substtutng the decomposed bus voltage vectors E* of [5.9] nto the branch current equatons as follows. n terms of a transmsson lne or a transformer wth a rato 1.0, the decomposed branch current components drected from the buses to where are derved by = E, b0, + ( E, E, ) ( g + b ) [5.11] *, l = E, b0, + ( E, E, ) ( g + b ) [5.1] *, l

55 bl s the half lne shunt susceptance, the symbol means PX, TX or Q ndvdually. For other branches such as transformers wth non-standard ratos (.e., branch currents can be derved easly. t 1.0), ther tep 5: Decompose complex power flows over each branch. For examples, n terms of a branch between buses and, the complex power flow wth respect to from bus of the branch s Further, t can be rewrtten as * Ebus, = [5.13] = ( E + E + E = ( E Q, Q, PX, Q, + E PX, PX, ) * 3rd term Q, + E * 5th term Q, + E TX, * 1st term PX, + ( E + E PX, TX, ) ( + ( E Q, Q, * TX, + E + E * 6th term Q, + PX, TX, TX, ) + * nd term TX, * PX, TX, ) ) * 4th term [5.14] where E bus,, EPX,, ETX,, EQ, s the th element of the voltage vectors E, E bus PX, E, E TX Q ndvdually. We categorze terms of [5.14] as follows: The 1 st, nd and 3 rd terms are maor components attrbuted to PX, TX and Q markets respectvely. The 4 th term represents an nteractng component between energy markets PX and TX. The 5 th and 6 th terms represent nteractng component between PX/TX and Q markets separately. Evdently, the market players PX and TX account for self-nduced terms, and also take care of the nteractng cross terms. Moreover, there s a flexblty to allocate the nteractve component between PX and TX, whch can be desgned nto market rules. Therefore, we conclude complex flow decomposton formulae for one branch drected from to as follows. = PX + TX, + Q,, [5.15]

56 where f * * * = ( EQ, + E PX, ) PX, + E PX, Q, + f PXω TX ( E PX, TX, + E * PX, TX, PX, * * * = ( E + E ) + E + f ω ( E + E * TX, Q, TX, TX, TX, Q, TX TX TX, PX, TX, PX, * Q, = EQ, Q, PXω TX, ftxωpx are sharng factors mposed upon PX and TX for ther nteractve component. f 1. PXωTX + ftxωpx ) ) Along the same lne, the complex power flow wth respect to to bus of the branch can be decomposed nto the followng market components: + where = PX TX, + Q,, [5.16] * * * = ( EQ, + EPX, ) PX, + EPX, Q, + f PXωTX ( EPX, TX, + E * PX, TX, PX, * * * = ( E + E ) + E + f ω ( E + E * TX, Q, TX, TX, TX, Q, TX PX PX, TX, TX, PX, * Q, = EQ, Q, ) ) Further, the decomposed real flow, real loss, reactve flow and reactve loss components on the branch mmedately follow from the solved complex power flow components *,, *,. n partcular P P 1 = Re( *, *, ) [5.17] flow(*), = Re( *, *, ) [5.18] loss + (*), 1 Q flow = m( *, *, ) (*), [5.19] Q loss = m( *, *, ) [5.0] (*), where * means PX, TX or Q ndvdually. tep 6: Dstrbute the porton of transmsson loss arsng from reactve power delvery to the energy customers, n proporton to ther reactve power usage.

57 The ntent of reactve power schedulng s to balance the system reactve loads and MAr losses manly generated from nterzonal power transferrng. The transmsson loss ncurred from reactve power flow only takes up a few percentages of the system loss under the normal operatng condton. We therefore reallocate t between PX and TX n proporton to ther reactve power usage. D*, loss(*), N L P L Q) = Ploss( Q), L QDk, + Qloss( k ), k= PX, TX N L Q + Q (*, [5.1] where * denotes energy nterchange schedules PX or TX. Eventually, the transmsson loss charges to PX and TX are P L( PX ) loss( PX ), L( PX, Q) L = P + P [5.] P L( TX ) = Ploss( TX ), + PL ( TX, Q) [5.3] L Thus far, all transmsson losses are dstrbuted among energy transactons, ndependent of the reactve power market clearng system. tep 7: Adust loss shares among the market players by an teraton scheme. As only a relatvely small number of generators are desgned for load followng purposes n a power system, the loss generated from a PX or a blateral transacton s lkely suppled by a thrd party, not necessarly the same generator servng the load. Accordngly, an adustment process s needed to take care of the loss. For examples, suppose TX decdes to buy the loss from a thrd party generator (say s ), then a small amount of generaton from the suppler s s attrbuted to the TX, whch corresponds to the allocated loss reflected n [3]. Adust the current vector accordngly, and repeat teps through 6 agan. Ths adustment scheme can be extended for a PX market smlarly. Under normal operaton condtons, the loss adustment process converges n a few teratons. TX t s straghtforward to generalze to cases wth a large number of the TXs. For any C, the complex power flow contrbutons to one branch between the buses and are k