The Incorporation of a Discrete, Dynamic LTC Transformer Model in a Dynamic Power Flow Algorithm

Transcription

1 Downoaded from orbit.dtu.dk on: Dec 6, 08 The Incorporation of a Discrete, Dynamic LTC Transformer Mode in a Dynamic Power Fow Agorithm Garcia-Vae, Rodrigo; Acha, Enrique Pubished in: The Internationa Association of Science and Technoogy for Deveopment Pubication date: 007 Link back to DTU Orbit Citation APA: Garcia-Vae, R., & Acha, E The Incorporation of a Discrete, Dynamic LTC Transformer Mode in a Dynamic Power Fow Agorithm. In The Internationa Association of Science and Technoogy for Deveopment Pama de Maorca, Spain: Acta Press. Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-making activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta If you beieve that this document breaches copyright pease contact us providing detais, and we wi remove access to the work immediatey and investigate your caim.

2 THE INCORPORATION OF A DISCRETE, DYNAMIC LTC TRANSFORMER MODEL IN A DYNAMIC POWER FLOW ALGORITHM Rodrigo Garcia-Vae Department of Eectronics and Eectrica Engineering University of Gasgow Gasgow, UK emai: r.garcia@eec.ga.ac.uk Enrique Acha Department of Eectronics and Eectrica Engineering University of Gasgow Gasgow, UK emai: e.acha@eec.ga.ac.uk ABSTRACT This paper reports on the deveopment of a dynamic LTC transformer mode suitabe for a dynamic power fow agorithm where the generators dynamic equations are soved simutaneousy and in conjunction with the network agebraic equations using Newton s method. The method may be used for ong-term power systems dynamic assessment, and in addition to the oad tap changing transformer mode, a representation of the generator boier is aso incuded. These are sow acting devices which are known to impact on system votage coapse phenomena; events which may evove over periods of time asting severa tens of minutes. Two test systems are presented to show the versatiity of the simuation too, incuding a trip cascading event eading up to a wide-area votage coapse. Where appropriate, resut comparisons with the output of a conventiona transient stabiity program have been carried out. KEY WORDS Dynamic power fows, votage stabiity, dynamic stabiity, votage coapse, LTC transformer, oad shedding. Introduction Society s appetite for eectricity continues unabated and with many transmission expansion programmes experiencing deays, existing transmission assets are being overstretched. Moreover, increased transmission system oadings and wide area disturbance propagation brought about by widespread interconnectivity, are caing for more accurate assessments of reactive power margins. On the main, power systems engineers have reied on the use of two kinds of appication programs for buk power systems assessments: power fows and transient stabiity. The former has been used to assess system votage performance and active and reactive power fows and osses; a this from a snapshot of time perspective. Static power fows have been suitaby extended to dea with votage coapse assessment, with resuts normay provided in the form of P-V nose curves on the power-votage pane. The approach yieds a rather cear picture as to where the point of votage coapse ies, as cacuated by the static power fow soution. Nevertheess, it has been argued that the static approach incorporates no information about the dynamic behaviour of votage coapse and that suitabe eements of power systems dynamics shoud be incuded in the formuation. On the other hand, transient stabiity agorithms have traditionay been used to assess power systems dynamic phenomena from tens of miiseconds up to severa seconds. These appication programs incude detaied modes of synchronous generators with their excitation systems, turbines and governors; as we as dynamic modes of oads and a wide range of power eectronics-based converters. The votage coapse phenomena is said to be dynamic in nature and yet when it presents itsef, it normay does it in a manner that is rather sow when compared to the dynamic time frame of traditiona transient stabiity programs. It is not uncommon for a votage coapse event to evove over a period of about 30 minutes; a time frame we beyond the intended scope of transient stabiity agorithms and modes. Sow acting devices such as the time-dependent, discrete nature of oad tap-changing transformer and boiers are not normay accounted for in transient stabiity programs and yet these sow acting devices together with the system oad characteristic, have been reported [] as having a direct bearing on the extent of a votage coapse event. Commensurate with industry s interest in achieving more reaistic dynamic reactive power margins assessments, an existing soution approach [] based on fu time domain simuations is deveoped further in this paper. It is a unifying framework where the power fows representation of the network is combined with the dynamic modes of the power system to enabe combined soutions at pre-defined discrete time steps using the Newton-Raphson method. Such an approach is, in essence, a dynamic power fow method; a method that takes fu account of a dynamic components in the power system, some of which may be non-inear and discontinuous in nature. However, to enabe comprehensive assessments of the many issues surrounding reactive power margins assessments and votage coapse issues in today s power systems, fuer modes of power pant components are required than those presented in []. Among these is the deveopment of a dynamic LTC transformer with discrete taps. Where appropriate, the resuts reported in this paper are compared with the output of PSAT [3], a widey used transient stabiity simuation package within the academic community.

3 Dynamic power fows The evoution of power fows in time, foowing an event, panned or unpanned, that aters a perceived steady-state of the network, may be tracked by suitaby combining the dynamic equations of synchronous generators, oads and contros with the equations associated with a norma power fow soution: 0 = fx, y ẏ = gy, x where y and x are vectors of integrabe and non-integrabe agebraic variabes. Generay, f and g are non-inear vector functions and the non-integrabe network variabes cannot be eiminated agebraicay. In such a situation, equations and must be soved simutaneousy as a function of time [4]; where the possibiity exists of using a unified framework to sove for y and x. The set comprises network equations and the stator equations of synchronous machine, transformed into the network reference frame. Set comprises the differentia equations describing the synchronous generators and their contros.. Synchronous generator modeing Three-phase synchronous generators represent the main source of eectricity generation in interconnected power systems, where a key aspect of system operation is to maintain stabe operation by running a synchronous generators at or near synchronous speed. However, power system disturbances, sowy-acting interactions between contros and emergency operations in the network may ead one or more generators in the network deviating from their intended synchronous speed. To a greater or esser extent, a power systems dynamic probems invoving synchronous generators require dynamic representation of their rotors, which may be in the form of their swing equations: ω = πf0 H [P m P e D ω ω 0 ] 3 δ = ω ω o 4 where H is the inertia constant, MW s/mva, P m is the mechanica input power, p.u., P e is the eectrica output power, p.u., D is the damping coefficient, s/rad, ω is the synchronous speed, rad/s, f 0 is the system operating frequency, Hz and δ is the oad ange, rad. Assuming baanced operation and equipment, the set of equations that describe the dynamic behaviour of a saient-poe synchronous generator, in the dq pane, are: T d0 E d = E f + X d X di d E q 5 E q = E d X q X qi q 6 T q0 T q0 E T d0 E d = E d + X q X q I q E d 7 q = E q + X d X d I d E q 8 E q = V q + R a I q + X d X I d 9 E d = V d + R a I d X q X I q 0 E q = V q + R ai q + X d X I d E d = V d + R ai d X q X I q where T d0, T q0, T d0, T q0 are open-circuit transient and sub-transient time constants in d and q axis, respectivey. E d, E q, E d, E q, E d, E q, E d, E q are the interna transient and sub-transient fux votages in d and q axis, and their time derivatives, respectivey. E f is the excitation interna votage. V d, V q are generators termina votages in d and q axis, respectivey. I d, I q, are the generators currents in d and q axis. X d, X q, X d, X q are transient and subtransient reactances in d and q axis, respectivey, X d, X q, are synchronous reactances in d and q axis, respectivey, X is the eakage reactance, R a is the armature resistance.. Network modeing The basic equations representing the static part of the network are the active and reactive power injections at each network bus, say k: P k =e k G km e m B km f m + f k Q k =f k e k m=,n m=,n m=,n m=,n G km f m + B km e m G km e m B km f m G km f m + B km e m 3 4 where P k and Q k are the active and reactive powers injected at bus k, e k and f k are the votage s rea and imaginary parts at bus k, G km and B km are the rea and imaginary parts of the noda admittance reating buses k and m. It shoud be noted that k = m gives rise to the sef admittance of bus k. Each pant component of the power network is represented by its own form of noda admittance eements G kk, B kk, G km and B km..3 Power transformer mode Power transformers are essentias pant components of the power system. In addition to those that perform the basic functions of transforming an AC suppy votage into one or more different AC votages and providing eectrica insuation between the suppy votage and the user s equipment, there are transformers that are provided with an on-oad tap changing mechanism to reguate votage magnitude at a specified point of the network or to contro the amount of active power that fows through the transformer. They are known as tap-changing and phase-shifting transformers, respectivey. The equivaent circuit representation of a tap-changing transformer of impedance Z, with the tap mechanism T k ocated on the primary side, and connected between buses k and m, is shown in Fig..

4 The transfer admittance matrix for this equivaent circuit is: Ykk Y km Y mk Y mm = Z T k Z T k Z Tk Z 5 As indicated in Fig. the transformer tap is discrete and updated according to the foowing reation: Vhigh V ow T k = V ow + T pos 6 T steps The tap s dynamic is represented by the foowing differentia Equation: T k = e k H + f k V 0 7 T where H T is the tap s servomotor inertia, e k and f k are the rea and imaginary parts of the votage at the reguated bus, V 0 is the target votage magnitude. Vk T k Z Tk Z T T Z k k Figure. Tap-changing transformer equivaent circuit. Vk T pos V high Vow Vm Figure. Transformer discrete tap. 3 Numerica Soution Technique The set of agebraic and differentia equations that describe each pant component of the power network are interdependent and it has been reported that their simutaneous cacuation provides for a more stabe numerica soution []. To this end, the system differentia equations are transformed into agebraic equations and appended to the existing set of agebraic equations for a unified soution. One way of ensuring a numericay stabe transformation is to use the impicit trapezoida method, a technique known for giving reasonaby accurate resuts even when reativey arge integration time steps are seected [5]. The first step in the appication of the trapezoida method is to express the system differentia equations in the form of Equation. To exempify the procedure, the method is appied to dynamic equations 3 and 4: ω t ω t t = t [ πf0 Pm P et D ω t ω 0 H + πf0 Pm P et t D ] ω t t ω 0 8 H δ t δ t t = t δt + δ t t 9 Substituting in Equation 8 the expression of eectrica power P e, where the effects of saiency and fux decay are considered, but assuming constant P m, we have, ω t ω t t = t [ πf 0 H Pm +E dt R ai dt + X q X I qt I dt +E qt R a I qt + X d X I dt I qt Pm D πf 0 ω t ω 0 + H +E dt t R ai dt t + X q X I qt t I dt t +E qt t R a I qt= t + X d X I dt t I qt t D ω t t ω 0 ] 0 Notice that in this Equation ony the transient effects have been represented, as opposed to sub-transient ones, to keep the expression at a manageabe eve. Re-arranging terms, eads to a more compact expression, where F ωt = ω t + πf 0 H F ω = F ωt + F ωt t + C ω = 0 F δ = F δt + F δt t + C δ = 0 Pet + Dω t F ωt t = ω t t + πf 0 H C ω = tπf 0 H P m + Dω 0 Likewise, expression 9 is re-arranged, F δt = δ t t ω t Pet t + Dω t t F δt t = δ t t t ω t t C δ = πf 0 t Equations 3 and 4, describing the active and reactive powers injected in the network buses, are augmented to incorporate expicity the active and reactive powers contributed by a generators in the network. Hence, P k + P egenbus Q k + Q egenbus 3 for k=,...,nbus and =,...,ngen, where genbus is an array of generator-connected buses.

5 Equations 3-4 for oad buses and Equations - for generator buses form the necessary set with which to carry out dynamic power fow soutions of a power system. One option to achieve this is to sove the equation set, which is non-inear, using the Newton-Raphson method. Hence, the foowing inearised equation, around a base point, provides the computing engine with which to carry out the soution by iteration: P Q F ω F δ F E d F E q F E d F E q = J J + J J e f ω δ E d E q E d E q 4 Where J is a matrix of first order partia derivatives of the active and reactive power, P and Q, with respect to noda votages, e,f. The active and reactive powers contributed by the generators are taken into consideration in the Jacobian term, J. J is a matrix of partia derivatives of generators discretised functions of the form and, with respect to generator state variabes ω, δ, E d, E q, E d and E q. J is a matrix of first order partia derivatives of active and reactive power at generator buses with respect to generator state variabes. J is a matrix of first order partia derivatives of generator discretised functions of the form and, with respect to noda votages e,f. It shoud be remarked that in Equation 4 the mechanica power P m contributed by the boier-turbine-governor set has been taken to remain constant, for the sake of simpifying the equation. In practice, however, the boierturbine-governor set is an important dynamic eement in power systems ong term dynamics and the inearised Equation 4 may be suitaby expanded to accommodate the boier-turbine-governor representation. 4 Resuts 4. Vaidation To vaidate the soution technique, a time domain simuation has been carried out and compared with PSAT [3]. The test system used is the nine-bus, three-machine power system depicted at Fig. 3 [6]. The scenario considered for this study is a oad change at bus 8. A oad of 0MW is added at t = m, using a 0 p.u. resistive impedance. The simuation is run for five seconds. It shoud be noted from Fig. 4 and Fig. 5, that the system response at Bus, is more damped than those at Buses and 3. This is due to the fact that these two buses are coser to the perturbation point. Fig. 6 presents the machines ange responses. It is noted that the anguar differences of the machines are sma, thus, the system wi remain stabe after the disturbance. The simuation resuts provided by Bus votage, pu TRANF 8 TRANF 3 G G 3 Load 7 Load j pu j pu j pu G TRANF 9 Load Figure 3. 3 Machines 9 buses power system time, sec 4 3 V Bus V Bus V Bus 3 Figure 4. Bus votage response to a oad change V Bus V Bus V Bus time s Figure 5. PSAT output for bus votage response to a oad change. the dynamic power fow simuation agree on we with the response provided by PSAT, Fig Test Case To demonstrate the appicabiity and versatiity of the dynamic power fow computer program, the oad connected at bus 8 is increased by 50%. A oads are represented by their exponentia oad mode [7]. The simuation period is 5 minutes. Fig. 7 represents the system with a mechanica power

6 Ange deviation differences, rad δ δ δ 3 δ Figure 6. δ i δ response to the oad change. P m constant, dash ine, whereas the continuous ine represents the mechanica power associated with the boierturbine-governor set mode. Mechanica power, pu Figure 7. Synchronous generators mechanica power. 4.3 Test Case To assess the performance of the Dynamic Load Tap- Changer DLTC mode, a separate test case is carried out. A oad increment in bus 8 is used as the source of the perturbation. Fig. 8 shows the DLTC performance foowing the perturbation. Dynamic tap-changers and 3 are coser to the perturbation point and their action is more pronounced than dynamic tap-changer which is farther away; it is noted that after an initia drop, it returns to a vaue coser to its pre-disturbance vaue. Fig. 9 gives the votage profies at buses where the oad tap changers are connected to. When the oad tap changer is active, ON, given by the continuous ine, and when the oad tap changer is inactive, OFF, given by the dashed ine. In this test case, the dynamic oad tap changer pays a very important roe in improving the votage profie at these specific points of the system. However, this may be at the expenses of a temporary surge in reactive power consumption from the network, as shown in Fig Test Case 3 To show the wider appicabiity of the proposed approach, the New Engand system [8] depicted in Fig., is used. The system has been modified to incude one tap-changing transformer between buses 40 and 3. At first, transmission ines which connect bus 7 with buses 7, 6 and 8 are tripped; as a consequence bus 7 incuding its oad, is isoated from the system, producing a votage drop at the majority of nodes. Next, the transmission ine connecting buses 3 and 8 is taken out of service, causing that node 8 and its oad become separated from the system, bringing a recovery of system votages. This is foowed by a tripping of the transmission ine connecting buses 3 to 4; an event that triggers a votage coapse of the entire system. Fig. shows the votage magnitudes in the remaining 37 buses of the system. Fig. 3 shows the discrete response of the tap-changer under abrupt votage changes. It can be seen that this device works in its pre-specified working range, which is from 0.90 to.0. The tap changer performance can be observed in Fig. 3. Fig. 4 shows the votage profie at the bus where the votage is controed by the tap-changer. As shown, the tap-changer transformer is abe to contro the votage around its specified vaue, p.u., even when the system is undergoing unfavourabe conditions. 5 Concusions This paper presents a unified framework for dynamic simuations, where the appication of the trapezoida rue and Newton-Raphson technique are appied to sove the combined system. The trapezoida rue is appied to discretise the synchronous machines variabe and the dynamic equations corresponding to the DLTC. A detaied mathematica modeing for the synchronous generator is presented. The soution technique is vaidated, obtaining acceptabe accuracy resuts. The New Engand power system has been tested and subjected to this approach getting good resuts. Tap performance, pu Dynamic Load Tap-Changer DLTC perfor- Figure 8. mance DLTC DLTC DLTC

7 V Bus 4 V Bus 7 V Bus Bus votage magnitude V i, pu Figure 9. Votage magnitude at Buses 4, 7 and 9, tap changer ON and tap changer OFF Figure. Bus votages response Tap position, pu.04.0 Reactive power, pu DLTC 3 DLTC DLTC Figure 3. Dynamic oad tap changer performance Figure 0. Reactive power consumption of the Dynamic Load Tap-Changer DLTC. N N30 8 N37 N6 N5 N N8 N7 N7 N8 N38 N4 9 N9 Bus votage magnitude V 3, pu N3 N4 N5 N4 N6 N 6 N35 N Figure 4. Votage magnitude at Bus 3. N39 N9 N5 N7 N3 N8 N6 N N0 N3 N Figure. One-ine New Engand test system diagram. References [] IEEE Power Engineering Society, Votage Stabiity Assessment: Concepts, Practices and Toos", Fina document, August 00. [] M. Rafian, M. J. H. Stering, and M. R. Irving, Reatime power system simuation, IEE Proceedings, vo. 34, no. 3, pp. 09 3, May 987. [3] F. Miano, An open source power system anaysis toobox, IEEE Transaction, Power Systems, vo. 0, 3 N3 N0 5 N9 N34 4 N33 7 N3 N36 pp , August 005. [4] C. W. Gear, Simutaneous numerica soution of differentia-agebraic equations, IEEE Transactions Circuit Theory, vo. 8, pp , January 97. [5] J. Machowski, J. W. Biaek, and J. R. Bumby, Power System Dynamic and Stabiity. John Wiey & Sons, Inc., 997. [6] P. M. Anderson and A. A. Fouad, Power System Contro and Stabiity. IEEE Press, 997. [7] IEEE Task Force on Load Representation, Bibiography on oad modes for power fow and dynamic performance simuation, IEEE Transaction on Power Systems, vo. 0, pp , February 995. [8] J. H. Chow, Time-Scae Modeing of Dynamic Networks with Appications to Power Systems. Springer- Verag, 98.