Some New Results on the Identification of Two-Area Power System Models with SVC Control

Transcription

1 009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 ThA03. Some New Results on the Identification of Two-Area Power System Models with SVC Control Aranya Chakrabortty, University of Washington, Seattle, WA, Abstract In this paper we study a measurement-based model identification problem for constructing dynamic equivalent models of two-area radial power systems with intermediate voltage control by Static VAr CompensatorsSVC. We consider two types of feedback mechanisms, namely, combined voltage and current feedback from the SVC bus, and supplementary current feedback from the transmission line. Given these system configurations, we first derive expressions for the aggregated equivalent reactance and machine inertias in each area, and then show how these parameters can be computed from the voltage and frequency measurements at three buses in the transfer path including the SVC bus. I. INTRODUCTION Large-scale interconnected systems such as electric power systems typically exhibit two time-scale behavior in their dynamic responses, the time-scale separation arises due to the differences in the strength of interconnections[3]. Strongly connected components in the system synchronize witheachotheroverafasttime-scaleandformanaggregate node, while the different aggregated nodes, which, by assumption are weakly connected, synchronize over a slower time-scale. In a recent reference [] we have developed a method, referred to as the Interarea Model EstimationIME method, for constructing a reduced-order dynamic model capturing this slow interarea motion for a two-area power system. Unlike typical model reduction approaches as in [8], [4] etc., the model parameters are assumed to be known, this method relies solely on dynamic electrical measurements available from specific parts of the system, and, hence, can equivalently be viewed as a model identification method. We showed that the construction of the twomachine interarea equivalent of the two-area system reduces down to the estimation of the equivalent reactance and machine inertia, internal to each area, upon we derived expressions by which these quantities can be computed from voltage and frequency measurements at any three buses on the path. In [] we extended the method to two-area systems with intermediate voltage support in the form of a Static VAr compensatorsvc [6]. However,[] studied the simple case the SVC was assumed to employ simple first-order proportional controllers with bus voltage feedback for voltage regulation. In actual power system operation, much more advanced forms of feedback may be used: for example, papers like [9] have used the current flowing through the SVC branch as an additional feedback signal for increasing the effectiveness of the regulation using a smaller gain. Similarly, the authors of[7] have considered several other supplementary feedback variables such as line current, bus frequency etc., to provide adequate damping to the transient interarea oscillations by the SVC in addition to voltage regulation. The objective of this paper is to extend the results of [] and develop the variant of the IME method when these complicated mechanisms of SVC feedback are employed in the system. In particular, we focus on two feedback mechanisms, namely the combined current and voltage feedback, and the line current supplementary feedback. Therestofthepaperisorganizedasfollows.InSection we describe the two-machine equivalent of a two-area radial power system with a SVC controller. In Sections 3 and 4 we derive the IME algorithm for SVC current feedback, and line current feedback, respectively. Section 5 concludes the paper. II. TWO-AREA POWER SYSTEM WITH SVC Consider the two-area power system with intermediate voltagesupportintheformofasvcinfigurea.inthis G G Gn 3 I I I c SVC Controller a Two-area power system with SVC G G Gn V V E jx T jx e c jx e V jx T E G I 3 I c SVC I b Two-machine equivalent representation Fig.. Two-machine power system model with SVC figure,generators G, G,..., G n arestronglyconnected i.e., with small reactances and form one area, say Area G /09/$ AACC 6

2 ,whilegenerators G, G,..., G n formadifferentarea, namely Area, which is assumed to be connected weakly tothemachinesinarea.inotherwords,thereactanceof the lossless transmission line contained in between buses andisassumedtobesignificantlyhighercomparedtothe individual reactances internal to each area. Each machine is assumed to be connected to its individual terminal bus through a transformer having reactance x Tki, k =,, i =,..., n. A SVC is installed at Bus 3, located between busesandnotnecessarilyatthemidpoint. The two-machine interarea aggregate model of the above system is shown in Figure b. This reduced system consists oftwoaggregatedgenerators G and G withaggregatedinertias H and H representingeachcoherentarea,connected to Buses and through equivalentthevenin transformer reactances x T and x T,respectively.Thevariablereactance provided by the compensator at Bus 3 is denoted as x c. ThevoltagephasorsatBuses, and 3aregivenas Ṽ = V θ, Ṽ = V θ, Ṽ c = V c θ c V θ denotes the polar representation V ε jθ. The transmission line between Buses and is assumed to be lossless, with a line reactance x e = x e + x e. Following a classical model representation, each generator is modeled asavoltagesource E i δ i, i =,, δ i representsthe stateofthe i th generatorand E i isconstant,inserieswith a direct-axis transient reactance x di, i =,, respectively. The reactances connecting the generator internal voltages to Busesandarethusgivenby x i = x Ti + x di, i =,. We define the notations σ = x + x e, σ = x + x e. The dynamic model of the two-machine system in Figure, neglecting damping, is given by[5] δ = Ω ω, H ω = P m P e 3, δ = δ δ, ω = ω ω with δ i, ω i i =, being the angle and speed of the i th machine respectively, H = H H /H +H istheequivalentinertia, P m = H P m H P m /H + H with P mi beingthemechanicalpower input to the i th machine, and P e is the effective electrical power exchange between the two generators. The constant Ωistheconversionfactorfrompuspeedtorad/s.TheSVC connectedtobus3regulatesthebusvoltagemagnitude V c to afixedsetpointvoltage V r,andtherebyprovidesintermediate voltage support. For perfect regulation, P e = E E x eq sinδ eq 4 for a symmetrical network x eq = σ = σ and δ eq = δ δ /.Foranasymmetricalnetworkthesetwo quantities are given as x eq = σ + σ and δ = δ δ satisfying the equation E σ sinδ θ c = E σ sinθ c δ. 5 In our subsequent analysis we will refer to the equivalent angular difference between the generators as simply δ. We next assume that measurement devicesphasor measurement units are located at Buses, and 3, i.e., high sampling-rate dynamic measurements of the voltage phasors atthesebusesaswellasthecurrentphasorsatdifferentparts of the system are available over time. Therefore, considering the two-area system described above, the identification problemthatwe want to solve can bestated as: Given the measured time-synchronized phasor variables V t, θ t, θ t, V t, θ t, θ t, V c t, θ c t, I t, θ I t, I t, and θ I tthatexhibitafewcyclesofinterareaoscillations, and assuming that E and E are some constant values, compute x e, x e, E, δ t, E, δ t, x, x, H, and H tocompletelycharacterizethedynamicbehaviorofthe two-machine reduced system given by equation3. Two ofthesequantities,namely x e and x e canreadily becomputedfrom Ṽ, Ṽ,and Ṽc,andthecurrents Ĩ and Ĩ atanyfixedpointoftime,usingacohm slaw: jx e = Ṽ Ṽc/Ĩ, jx e = Ṽc Ṽ/Ĩ. 6 Inaddition,if x and x areknown,thenthemachineinternal voltagescanbecomputedfromthebusvoltagesandtheline currents.thustheproblemreducesto theestimationof x and x,aswellastheinertias H and H.Inthefollowing sections we develop techniques to extrapolate these quantities for the two types of SVC feedback mechanisms listed in Section. III. COMBINED SVC VOLTAGE AND CURRENT FEEDBACK Considering a proportional controller for voltage regulation at Bus 3, if the SVC voltage and current signals are fed back, then the susceptance B = /x c evolves in time according to the closed-loop control Ḃ = B τ + k V r V c k I Ĩc I co τ V r isasetpointvoltageforbus3, k istheregulation gain, k I isthecurrentfeedbackgain, I co isthesteady-state current,and τ is the SVC time constant. Typically τ is in therangeof 50to 300msec[7],whichismuchfasterthan the time constants for the voltage variations in the system. Therefore,7 can be approximated as the static equation Using equation8 reduces to 7 B = k V r V c k I Ĩc I co 8 Ĩc = B V c, I co = B V r 9 B = k V r V c + k V c V r 0 with k = k k I.Withoutlossofgenerality,weassume δ = 0sothat δ = δ.tofindthefunctionalrelationshipbetween 7

3 B and δ,weproceedasfollows.forthenetworkinfigure b,afterafewcalculationsitcanbeshownthat V c = σ E + σ E cosδ + σ E sinδ σ + σ Bσ σ which gives = V c σ + σ Bσ σ σ E + σ E + σ σ E E cosδ ψδ. Weconsiderthepositivesquarerooti.e. ψδ > 0because both V c > 0and σ +σ Bσ σ > 0.Using0,equation reduces to the quadratic equation k σ + σ + k σ σ V c + σ + σ k V r k σ σ V r k ψδv c k V r ψδ = 0 3 ψδisdefinedin.since V c > 0,thesolutionis given as, V c δ = + 3 = k σ σ V r+k ψδ σ +σ k V r = + = σ +σ k V r k σ σ V r k ψδ = 4k σ σ +k σ +σ k V rψδ 3 = k σ σ +k σ +σ. Hence, from0 the susceptance can be written as 4 Bδ = k V r V c δ + k V c δ. 5 Considering the right half of the network in Figure b, the voltage phasor at any point between Bus 3 and Generator, at a reactance of jx from the Generator node can be written as Ṽ = [ σ E a + a χ + a σ [ +j a σ ] χ E sinδ a = x/σ [0, ],and ] χ E cosδ 6 χδ σ + σ Bδσ σ. 7 The voltage magnitude at this point is given by equation 8, Σ r isindependentof δ.linearizing3and8 aboutanequilibrium δ 0, ω 0, V ss a,asmallperturbation in the voltage magnitude at the given point can, therefore, be written as V a, δ 0 = Ja, δ 0 δ 9 with the Jacobian function defined as Ja, δ 0 = Ṽ = δ V ss a Υ ra, χ, χ, δ 0 0 δ=δ0 thefunction Υ r,,, isasinwithallpowers of χδand χ δcomputedat δ = δ 0,and V ss a isthepredisturbance equilibrium voltage at the given point. Denoting V n a, δ 0 := V a, δ 0 V ss a,itfollowsfrom9and 0 that V n a, δ 0 = Υ r a, χ, χ, δ 0 δ. Assuming the reactance to be distributed uniformly along thetransferpath,thevariable a canbeequivalentlytreated as a dimensionless spatial variable. Equation, therefore, indicates how the productof the two voltages on the LHS varies spatially between Generator and Bus 3. Similarly, forthelefthalfofthenetworkbetweenbus3andgenerator,itcanbeshownthatatanypointatareactance jxaway frombus3wehave V n a, δ 0 = Υ l a, χ, χ, δ 0 δ 3 a = x/σ [0, ], and the function Υ l,,, is as in 4 with all powers of χδ and χ δ computed at δ = δ 0. Next,ifmeasurementunitsarelocatedatBus,and3, then it implies that, following a disturbance in the system, the V n valuesatthesethreepointsonthetransferpath,atany fixed instant of time, are available from the measurements. We note that Bus correspondsto a = x /σ, Bus 3 to a = or a = 0,andBusto a = x e /σ.wedenote thesethreeconstantsas a, a3 and a,respectively,withthe superscripts indicating the bus number. Then, from and 3wecanwrite V n a, δ 0 V n a 3, δ 0 V n a, δ 0 V n a 3, δ 0 = Υ la, χ, χ, δ 0 Υ l a 3, χ, χ, δ 0 5 = Υ la, χ, χ, δ 0 Υ l a 3, χ, χ, δ 0. 6 Since the LHS of 5 and 6 are known from the bus measurements, the two unknown parameters x and x, contained in the RHS, can be easily solved for by using standard nonlinear equation solvers. Figurecomparesthevariationof V n followingand 4 from one end of the transfer path to the other, for the additional SVC current feedback case with that due to SVCbusvoltagefeedbackonlyasin[].Weuse k = 0 for the voltage feedback case, and k = k I = 0 for the current feedback case. It can be seen that for these equal values of feedback gains, the oscillations in voltage reduce significantly when the SVC current is fed back. Moreover, more efficient setpoint regulation is achieved at the point of voltage support. The values of the other parameters used for thesimulationaresameasthosein[]. A. Estimation of Machine Inertias The IME algorithm enables us to estimate the equivalent inertia constant of the system given by H = H H H + H 7 8

4 Ṽ Σ = r + E σ a χ + E a a σ χ + σ E a χ + E E a σ cosδ χ χ Υ r a, χ, χ, δ 0 = Eσ a χ 3 E σ a a χ χ E σa χ χ 3 σ a + σ χ +E E σ a a χ sinδ 0 + χ cosδ 0 χ sinδ 0 + χχ cosδ 0 χ a σ χ 4 8 Υ l a, χ, χ, δ 0 = Eσ a χ χ 3 E σ a a χ χ E σ a χ χ 3 χ sinδ 0 + χχ cosδ 0 χ sinδ 0 + χ cosδ 0 +E E σ a a σ χ 4 a χ 4 Jacobian Curve Fit SVC Bus Voltage Feedback SVC Voltage and Current Feedback Normalized Reactance r Fig.. Comparison of voltage distribution between SVC bus voltage feedback k = 0, solid line and additional SVC current feedback k = 0, k I = 0,dashedline from the expression for the swing frequency, obtained from thelinearizationoftheswingmodel3about δ 0 [0] E V c Ω cosδ 0 θ c0 ω s = 8 Hx e + x δ 0 and θ c0 arethepre-disturbanceequilibriumvalues of δ and θ c. The swing frequency ω s can be computed fromthemeasuredvoltageoscillations, θ c0 isavailablefrom the measured Bus 3 voltage, while all other parameters are calculable once x and x are estimated by the method discussed in the foregoing section. In order to calculate H, H we need a second equation involving these two parameters. As described in[], this second equation can be derived from the law of conversation of angular momentum for the two-machine system, and is given as H H = ω ω 9 ω and ω aretherotorspeedsofthetwomachines. As these two speeds are not known, we extrapolate them fromthemeasuredfrequenciesatbusesand.thebasic methodology to achieve this is to consider the voltage phasor Ṽ atanypointinanypartofthetransferpathandexpressit intermsof E δ, E δ,andthereactance xwithrespect to some chosen reference point, calculate the phasor angle θ = tan ImṼ /ReṼ andcomputethetimederivative of θ as a function of x, ω and ω. Hence, considering the two measured frequencies at Buses and, one can readily solve for these two speeds using the values of the other parameters obtained from the reactance extrapolation method.wecanshowthatforthecaseofcombinedvoltage and current feedback from the SVC, the expression for the frequencyatanypointonleftandrighthalvesofthetransfer path are as follows: For any point between Generator and Bus 3 at a reactance jx away from Generator : ν r a = ϑ r δ, δ ω + ϑ r δ, δ ω p + p + p p cosδ δ 30 a = x/σ,andtheexpressionsfor ϑ r, ϑ r, p and p as functions of a, δ, δ are given in Appendix. For any point between Bus 3 and Generator at a reactance jxawayfrombus3: ν l a = ϑ l δ, δ ω + ϑ l δ, δ ω m + m + m m cosδ δ 3 a = x/σ,andtheexpressionsfor ϑ l, ϑ l, m and m as functions of a, δ, δ are given in Appendix. Substituing a = x /σ and a = x e /σ in the above expressionsand using the measuredvaluesof ν r and ν l at busesand,respectively,wecansolvefor ω and ω,and hence,for H and H using7-9. B. Simulation Results In this section, we apply the IME algorithmto the twomachine system in Figure operating with a power transfer of300mwfromareatoarea.thedisturbanceappliedis a three-phase fault at Bus 3 cleared after 0.05 second without 9

5 any line switching.the machinedata can be foundin []. Hereweconsider k = 0pu/pua 5%droop, k = 5pu/pu weakcurrentfeedback, V r = puand τ = 0msec. The voltage magnitudes at the three buses are shown in Figure3.Choosingthepeakofthesecondcycleasafixed time instant, we get V n = 0.0, V n = 0.035, V 3n = Using the IME algorithm 5-6, we obtain x = Voltage Magnitude pu Bus Bus Bus Time sec Fig. 3. Voltage oscillations at three buses in the two-machine system pu, x = pu as the actual values of these parameters are x = 0.34 pu, x = 0.39 pu. Also theestimates for H, H areobtainedas 6.39 and 9.6pu while the actual values are 6.5 and 9.5 pu, respectively. IV. DAMPING IMPROVEMENT WITH LINE CURRENT FEEDBACK In this section, we assume that the SVC in the twomachine system in Figure is employed for providing additional damping to the interarea oscillations, by supplementary line current feedback for voltage regulation there will be an additional washout filter to eliminate the feedback effect in steady-state. Therefore, the susceptance is given as B = k 3 Ĩ 33 k 3 is a constant gain and Ĩ is the line current flowing between Bus 3 and Bus. Alternatively, Ĩ can also be considered for feedback. If additionally, the SVC is also expected to achieve steady-state voltage regulation, then there will be an additional voltage error feedback term ontherhsof33,similartoasinsection3. Wefirstderiveanexpressionfor Ĩintermsoftheangular difference δ. From network equations, it is straightforward toshowthat cosθ c = σ E + σ E cosδ ψδ ψδ 34 ψδ = σ E + σ E + σ σ E E cosδ as definedin.next,wenotethat I := Ĩ = σ V c + E V ce ψδ. 35 so that, using33 and35, the equivalent of equation3 forthiscase,canbewrittenas σ + σ V c + σ k 3 V c V c + E V ce ψδ = ψδ which reduces to the quartic equation α V 4 c + α V 3 c + α 3V c + α 4V c + α 5 = 0 36 α = σ k 3, α = σ k 3 E ψδ 37 α 3 = σ + σ + σ k 3 E 38 α 4 = σ + σ ψδ, α 5 = ψδ. 39 The solution is given by Ferrari s formula for quartic equation as V c δ = α 4α + ± s β ± t 3β + β 3 ± s β 4 β 40 the subscripts s and t denote dependent and independent signs respectively, and the construction of the coefficients β, β, β 3 and β 4 in termsof α i i =,...,5 is shown in Appendix. For typical values of the system parameters,thesignsmaybechosenas s = and t = +. The subsequent derivations for the normalized voltage at any pointonthetransferpathoneithersideofthesvcbusare similartothoseinsection3,i.e., V n a = Υ l a, χ, χ, δ 0 δ, 4 V n a = Υ r a, χ, χ, δ 0 δ 4 χδ = σ + σ + σ k 3 V c δ + E V cδe ψδ, 43 the functions Υ r and Υ l are as in and 4, and the expressionfor χ δcanbederivedas χ δ = σ k 3 σ I δ V c δv cδ E Vc δ ψ δ + V c δ ψδ 44 with ψ δand V c δbeingthederivativesof ψδand V c δ in 34 and 40 respectively, with respect to δ. It should be remembered that equations 4-4 are considered for δ = δ 0. From 4-44, the method of solving for x and x followsinthesamewayasinsection3. Jacobian Curve Fit Supplementary Line Current Feedback SVC Voltage and Current Feedback Normalized Reactance r Fig. 4. Comparison of voltage distribution between SVC voltage and current feedback k = 0, k I = 0, solid line and supplementary line currentfeedbackk 3 = 0,dashedline 0

6 Figure 4 compares the variation of V n for the case of combined SVC voltage and current feedbackas in Section 3 with that due to the supplementary line current feedback. We use k = k I = 0 for the SVC currentfeedbackcase, and k 3 = 0forthesupplementaryfeedback.Itcanbeseen that for approximately the same amount of voltage regulation at Bus 3, the voltage oscillations at each point of the transfer path are significantly damped by the supplementary feedback. This testifies to the fact that supplementary control can be a useful tool for improving interarea damping, as used in[7]. If additional voltage feedback is used for voltage regulationthenthedashedcurvewilldipmoreat r =,the point of voltage support. The values of the other parameters usedforthesimulationaresameasthosein[]. V. CONCLUSION In this paper our main objective has been to study how voltage oscillations at any point in a radial two-area power system with intermediate SVC control, vary as a function of the electrical distance of that point from a common reference following a disturbance in the system. This has enabled us to estimate the aggregated model parameters for each area, using dynamic measurements of voltage, currents and frequencies available from specific parts of the system, so that a reduced order two-machine interarea equivalent for the system can be constructed. This reduced order model cannowbeusedforcontroldesignfordampingthetransient interarea swings in the system following any disturbance. VI. APPENDIX Wefirstdefine V c δ, δ tobethesolution4with ψδ in replacedby ψδ δ, and define Bδ, δ as in 5. We next define S r δ, δ = p ÃE σ sinδ + E σ sinδ + δ Then p ÃE σ sinδ + E σ sinδ + δ Ã = a σ σ E E sinδ δ k ψδ δ +k V c+k k V r V c p = E a + a σ χδ,δ, p = a σ E χδ,δ. 45 ϑ rδ,δ = p πe σ cosδ δ +p πe σ Sr λδ,δ ϑ rδ,δ = p E a +p E a cosδ δ + p πe σ cosδ δ +p πe σ +Sr λδ,δ λδ,δ = χδ,δ +k V cδ,δ 46 πδ,δ = a +k V cδ,δ χδ,δ. 47 Fortheleftsideofthenetwork,wecanderivethat ϑ l δ,δ = m E a cosδ δ +m E a + m πe σ +m πe σ cosδ δ S l λδ,δ ϑ l δ,δ = m πe σ +m πe σ cosδ δ +S l λδ,δ m = σ a E, m χδ,δ =E a + σ a χδ,δ 48 πδ,δ = a +k V cδ,δ χδ,δ 49 andtheexpressionfor S l isexactlythesameasthatof S r butwith Ãreplacedby Ā Ā = a σ σ E E sinδ δ k + k V c ψδ δ +k k V r V c. 50 The symmetry of the Jacobian functions on either side of thetransferpathwithrespecttothespatialvariables a and a canbereadilyseenfromtheexpressionsfor ϑ ri and ϑ li, i =,.Also,fromtheaboveequationsand30,3itcan beverifiedthat ν r 0 = ω, ν l = ω and ν r = ν l 0. VII. APPENDIX Let α,.., α 5 be as in Then, by Ferrari s formula, the coefficients β,.., β 4 in the solution 40 of the quartic equation36 are given as follows: β = 3α 8α + α 3, β 4 = α3 α 8α 3 α α 3 α + α 4 5 α µ = 3α4 56α 4 + α 3α 6α 3 α α 4 4α + α 5 5 α µ = β µ, µ 3 = β β µ 3 µ 4 = µ 3 + β µ µ3 7, µ 5 = 3 µ 4 54 β 3 = 5β 6 µ 5 + µ 3µ 5 µ β = β + β 3 56 REFERENCES [] A. Chakrabortty and J. H. Chow, Interarea Model Estimation for Radial Power System Transfer Paths with Voltage Support using Synchronized Phasor Measurements, in the Proceedings of IEEE PES Society General Meeting, Pittsburgh, PA, July 008. [] A. Chakrabortty, Estimation, Analysis and Control Methods for Large-scale Electric Power Systems using Synchronized Phasor Measurements, PhD thesis, Rensselaer Polytechnic Institute, Troy, NY, August 008. [3] J. H. Chow, G. Peponides, P. V. Kokotovic, B. Avramovic, and J. R. Winkelman, Time-Scale Modeling of Dynamic Networks with Applications to Power Systems, Springer-Verlag, New York, 98. [4] R. A. Date and J. H. Chow, Aggregation Properties of Linearized Two-Time-Scale Power Networks, IEEE Transactions on Circuits and Systems, 387, July 99. [5] A. Fouad and V. Vittal, Power System Transient Stability Analysis using the Transient Energy Function Method, Prentice Hall, 996. [6] N. Hingorani and L. Gyugyi, Understanding FACTS, IEEE Press, NJ, 000. [7] E. V. Larsen and J. H. Chow, SVC Control Design Concepts for System Dynamic Performance, Invited paper for IEEE Special Symposium on Application of Static VAR Systems for System Dynamic Performance, 987. [8] R. W. de Mello, R. Podmore, and K. N. Stanton, Coherency-based dynamic equivalents: Applications in transient stability studies, in Proceedings of the PICA Conference, pp. 33, 975. [9] M. Parniani and M. R. Iravani, Optimal Robust Control Design of Static VAr Compensators, IEE Proceedings on Generation, Transmission&Distribution,Vol.45,No.3,pp ,May998. [0] G. Rogers, Power System Oscillations, Kluwer Academic Publishers, 999.