Electric Power System Transient Stability Analysis Methods

Transcription

1 1 Electric Power Syste Transient Stability Analysis Methods João Pedro de Carvalho Mateus, IST Abstract In this paper are presented the state of the art of Electric Power Syste transient stability analysis ethods the results of a hybrid ethod ipleentation. There are presented several ethods also a coparison between the. The ipleented hybrid ethod uses indexes to lower the coputation tie uses a reduced odel equivalent, based on the identification of critical achines, which are responsible for the loss of synchronis. Index Ters - Critical clearing tie, Hybrid ethods, Power syste transient stability I. INTRODUCTION In the last years, due to the spread of electric generation facilities econoic factors, Electric Power Systes operate ore closer to their liits [1]. Thus, ore than before, it s of crucial iportance the existence of ethods to assess the syste stability. There are two kinds of stability probles: voltage stability transient stability [2]. This paper addresses the transient stability. This paper is organized as follows. In Section II are presented the odels used to describe the network coponents the network itself. Section III is dedicated to present the state of the art of Electric Power Syste transient stability analysis ethods. The ipleented hybrid ethod is presented in Section IV. In Section V are presented the results of coputational tests to which the ethod was subjected. In the last section, Section VI, are presented guide lines for future works. II. ELECTRIC POWER SYSTEM MODELLING In order to study Electric Power Syste transient stability the odels to describe their coponents should be defined. The coponents are defined using the classical odel, which is valid to tie periods up to 2 seconds. [2] It s also iportant to define how the disturbances are siulated. A. Synchronous Generators Electrically, the synchronous generators are described by a transient reactance X di next to an electrootive force (ef) E i [2]. So, the i th generator are characterized by the E i = V i + jx di I i (1) being V i the voltage at the generator terinals I i the current supplied by the generator. The rotor dynaics are described using the swing 2H i d 2 δ i ω 0 dt 2 + D dδ i i dt = P i P ei (2) where H i is the inertia constant, δ i is the rotor angle, D i is the daping-torque coefficient, ω 0 is the synchronous speed, P i is the echanical power P ei is the real power [2]. The rotor angle tie derivative can be expressed as dδ i dt = ω i ω 0 (3) where ω i is the rotor speed. Usually, the variables that describe rotor dynaics, δ i ω i, are described using as reference the Centre Of Inertia (COI). The COI is defined as δ 0 = 1 M M i δ i (4) T where M T is the total coefficient of inertia, given by M T = M i M i is the coefficient of inertia of the i th generator, which is given by (5) M i = 2H i ω 0. (6) With this changes, the rotor dynaics is described by dθ i dt = ω i M i dω i dt = f i(θ i ) where θ i, ω i f i (θ i ) are the new COI variables being, respectively, the rotor angle, the speed the accelerating power of the i th generator. The COI variables are obtained using θ i = δ i δ 0, (8) (7)

2 2 ω i = dδ i dt dδ 0 dt, (9) f i (θ i ) = P i P ei M i M T P COI, (10) P COI = (P i P ei ). (11) B. Transission Lines The transission lines are characterized by the equivalent π- odel, thus they are described by the series resistance R L, the series reactance X L the shunt adittance Y T [3]. C. Transforers Three different types of transforers are considered: the two-winding transforer, the variable transforer the phase shifting transforer [2]. - two-winding transforer The two-winding transforer is described by the total series resistance R T the total series reactance X T, or by the total series ipedance Z T, given by Z T = R T + X T. (12) The agnetization current is neglected. - variable transforer The variable transforer are odelled by an ideal transforer with transforation ratio next to the total series ipedance Z T. - phase shifting transforer The phase shifting transforer is siilar to the variable transforer but the transforation ratio is coplex. So this transforer are described by an ideal transforer with coplex transforation ratio next to the total series ipedance Z T. D. Loads All the loads are odelled using a constant adittance odel [3]. So, a load with real power P ck reactive power Q ck, connected to a bus with voltage agnitude V k, nubered k, is represented by a constant adittance of value Y ck : Y ck = G ck + jb ck = P ck jq ck V k 2. (13) E. Network The network is characterized by an reduced adittance atrix Y red, which is obtained by adding the constant adittances of odelling the loads the transient reactance of each generator [2]. This atrix is obtained fro the [I] [0] = [Y ] [Y n ] [Y n ] [Y nn ] ] [E (14) [V] where [E ] is the ef vector, [I] is the current supplied by the generators vector [V] is the bus voltages vector. After soe algebra, the Y red atrix is defined as Y red = [Y ] [Y n ][Y nn ] 1 [Y n ]. (15) The real power supplied by each generator is obtained using the P ei = Re{E i I i } (16) which becoes, oitting the algebra: P ei = E 2 i G ii + E i E j G ij cos δ i δ j j=1 j i + B ij sin δ i δ j (17) where G ij B ij are, respectively, the real iaginary of the ij positions of adittance atrix that describes the network. F. Disturbances The studied disturbances are only three-pole short-circuits with contact to ground near the buses. In the prefault period, the syste is in steady state. The disturbance occurs at 0,1 s of the siulation (t def = 0,1 s) it s eliinated at the clearing tie t cl. During the fault period, the disturbance is siulated by adding a large agnitude fault adittance Y def to the adittance atrix. The disturbance is cleared spontaneously or by turning off one line, which ust be reflected in the adittance atrix [4]. III. ELECTRIC POWER SYSTEM TRANSIENT STABILITY ANALYSIS METHODS There are several ethods to study the transient stability of Electric Power Systes. These ethods can be gathered in four groups: Nuerical Integration Methods, Direct Methods, Hybrid Methods Artificial Intelligence Techniques. A. Nuerical Integration Methods An Electric Power Syste can be described by two sets of s: a differential set related to the generators an algebraic set related to the others coponents [3]. The way these ethods study the transient stability of the syste is solving, in the tie, the two sets of s entioned above. These ethods offers very good odelling capabilities. B. Direct Methods These ethods are also called Energy Function Methods, because they are based in coparisons of energy values [5]. More precisely, these ethods calculates the energy value at the clearing tie the critical energy value. If the energy value at the clearing tie is higher than de critical energy value the syste is unstable, otherwise the syste is stable.

3 3 These ethods only require to solve the s during the fault period, which leads to lower coputation ties. C. Hybrid Methods The Hybrid Methods gather the advantages fro the Nuerical Integration Methods with the advantages fro the Direct Methods [3]. Fro the first they get the odelling capabilities, whereas fro the second they get the fast analysis capabilities. D. Artificial Intelligence Techniques These are the newer approach to assess the transient stability of Electric Power Systes [3]. Unlike the previous ethods, which are all deterinistic, these kind on ethods are probabilistic. They are characterized by being necessary to do a lot of siulations before being ready to use. However, when in use, they provide a very fast analysis. E. Coparison between Methods Between all the ethods presented here, the best results are obtained with the Hybrid Methods with the Artificial Intelligence Techniques. Due to the siulations that the Artificial Intelligence Techniques require before being ready to use, the Hybrid Methods take advantage when it s necessary to choose one [3]. IV. IMPLEMENTED HYBRID METHOD The developed hybrid ethod ai is to deterine the critical clearing tie for specific disturbances, which are defined by the user. As entioned above, the disturbances are triphasic short-circuits near the buses, that are cleared spontaneously or by turning off one line. The ethod is developed in atlab. A. Data Input All the data is provided to the progra by way of Excel files. It s necessary to provide data about all the network coponents about the disturbances. The progra only needs to know the directory where the Excel files are stored, because after loading the files it s autonoous [4]. B. Prefault Values Coputation The step of prefault values coputation includes the solving of a power flow, the odelling of the loads by constant adittances the coputation of initial values of ef s echanical power of the generators [4]. The power flow is solved using the Newton-Rapshon ethod, which enables to have the voltage agnitude phase for all the network buses. The odelling of the loads by constant adittances has been explained above. The agnitude of ef s the value of echanical power stays constant trough the siulations, due to the use of classical odel. C. Critical Tie Cycle This step is the ore coplex of the developed hybrid ethod. It s started with the nuerical solving of the s that describe the syste. The solving is stopped before the total siulation tie due to the use of stability instability indexes. When the nuerical solving is stopped, it s tie to find the critical achines cluster. To find that, it s necessary to use two different criteria, which choice is based in the previously referred indexes. After that the syste is reduced to an equivalent achine connected to an infinite bus. This reduced odel is studied using the equal area criterion, fro which the transient stability argin is calculated. Due to the lack of all the values of power curve of equivalent achine, it s necessary to odel that curve to estiate the reaining values. Based on the calculated values of transient stability argin, a new critical tie estiate is ade. - stability instability indexes To stop the nuerical solving of the s it s necessary to resort to two indexes: one of instability (IDCS) one of estability (IDE) [3]. The first one is coputed with IDCS = f i 2 (18) a relative iniu in this index corresponds to an unstable situation. The second one is coputed using IDE = ω i θ i θ i cl (19) where θ i cl is the rotor angle at clearing tie, referred to COI. When this index suffers a signal change, fro positive to negative, the situation is stated to be stable. The two indexes are coputed at each step of nuerical solving in the post-fault period, when one of the indexes presents the characteristic behaviour, the solving is stopped the situation is declared stable if it was the IDE index to give the stopping order, or unstable if it was the IDCS index [4]. - critical achine cluster identification This identification is ade resorting to two different criteria, depending on situation s stability or instability, at an optial instant of nuerical solving which is found trough an index, the IDTO, which is coputed with the IDTO = f i ω i. (20) When this index, that is calculated in each step of the post-fault period, reaches an signal change, that oent atches the optial instant [3]. The criteria, fro the two available, to find the critical cluster, is choose based in the situation s stability or instability: if the situation is stable it s used an index based in the rotor angles variation; if the situation is unstable, it s used a ethod called Critical Machines Ranking (CMR) [4]. The first index is coputed, for each generator, using

4 4 t obs IA COIi = θ i (t + t) θ i (t) (21) t=t cl where t is the integration step. The indexes to all the generators are then sorted in descending order. After that, it s coputed the bigger difference between two consecutive values. The generators above the biggest difference are the critical cluster, the others are the reaining achines [3]. The ethod used to find the critical cluster, when the situation is unstable, begins with the sort in descending order of all achines rotor angle δ i [6]. Then, one proceeds as follows: 1. pick i th achine in the decreasing order list fro top to botto in order, which copose subset C; the rest is belong to the reaining achine syste, the subset R, where i = 1,2,, 1, being the nuber of generators; 2. calculate for the two systes the COI angle the difference between the; 3. repeat 1 2, until the top n-1 achines in the decreasing order list have been chosen. The two subsets with the largest difference between COI angles are the classification wanted. The COI angles for the subsets C R are coputed with the s δ C = 1 M M k δ k, (22) C M C = M k, (23) ω C (t) = 1 M M k ω k (t), (27) C δ R (t) = 1 M M j δ j (t) (28) R ω R (t) = 1 M M j ω j (t). (29) R After the reduction of each group to an equivalent achine, the equivalent achine that describes the syste can be obtained. The rotor angle δ eq speed ω eq are obtained using the s δ eq (t) = δ C (t) δ R (t) (30) ω eq (t) = ω C (t) ω R (t). (31) The total inertia coefficient is obtained using M T = M C + M R (32) whereas the equivalent inertia coefficient is coputed using M eq = M CM R M C + M R. (33) δ R = 1 M R M j δ j (24) To fully define the equivalent achine it s only issing the s to copute the equivalent echanical electrical power. The equivalent echanical power is calculated using M R = M j. (25) - syste reduction The syste reduction is done in two phases: first, the critical cluster are reduced to one equivalent achine, happening the sae with the reaining achines; second, the two equivalent achines are transfored in one single achine [3]. As stated above, C refer to critical cluster R to reaining achines. Keeping this notation, the rotor angle speed for each subset are coputed using the expressions δ C (t) = 1 M M k δ k (t), (26) C P eq (t) = 1 M M R P (t) M k C P (t) (34) j T whereas the equivalent electrical power is calculated using P e eq (t) = 1 M M R P (t) e M k C P (t) e. (35) j T - transient stability argin coputation It s based on transient stability argin values that the critical tie estiations are ade. The value of transient stability argin η is easily calculated using

5 5 δu eq η = P e eq P eq dδ eq. (36) δ0 eq 0 u where δ eq is the initial rotor angle value δ eq is the rotor angle, bigger than δ 0 eq, for which the echanical power equals the electrical power. Apparently the process to calculate the transient stability argin sees to be easy. However, due to the stop of nuerical solving before the total siulation tie, the power curve is not fully available. So, it s necessary to fit the available data in order to predict the issing values. This fitting is ade using two different ethods: a trigonoetric fitting a polynoial fitting. In the beginning it s used the trigonoetric one, until the variation between two consecutive estiates is lower than a tolerance ε 1. After that it s used the polynoial one until the variation between two consecutive estiates is lower than a tolerance ε 2, tie when the process stops. The trigonoetric fitting tries to odel the data by the P e eq δ eq = P ax e eq sin δ eq (37) ax where P e eq is the axiu electrical power the equivalent generator can supply. The polynoial fitting tries to odel the data by the P e eq δ eq = c 1 δ 2 eq + c 2 δ eq + c 3 (38) where c 1, c 2 e c 3 are constants. The use of two different fitting rises fro the fact that the trigonoetric one is strong even with a bad initial critical tie estiate but it s less accurate, whereas the polynoial one is weak with a bad initial critical tie estiate but it s ore accurate finding the correct critical tie. - critical tie estiation As entioned, the critical tie estiates are based on the transient stability argin calculated values. In fact, it s is aditted that there is a linear relation between the clearing tie the transient stability argin. So, starting fro the available transient stability argin values, it s possible to estiate a new ore near to the final solution critical tie, regarding that a null transient stability argin atches the wanted critical tie. The process is easy to explain. Starting fro the current (i) previous (i 1) iteration values of clearing tie transient stability argin it s find the line that relates that values: η(t cl ) = η t cl + b η (39) where η is the slope obtained by η = ηi η i 1 i i 1 t cl t (40) cl b η is the y-intercept of the line, obtained using (39). With the line available, setting η(t cl ) = 0, the new critical tie estiate is ade [4]. D. Output Data As entioned earlier, the progra ai is to copute the critical clearing tie for disturbances defined by the user, so the coputed value of critical clearing tie is available at the progra exit. The nuber of necessary iterations to copute the critical clearing tie value is also available at the progra exit. This values are stored in an Excel file located in the sae directory of the network data Excel files [4]. E. Code files The code needed to ipleent the hybrid ethod is spread by different code files [4]. The kernel of the progra is in a file called TC. In this file it s all the code necessary to ipleent the code except: the code to the power flow, which is in TC_TE ; the code to find the critical achine cluster to copute the reduced syste, which is in TC_MC ; the code to copute the transient stability argin, which is in TC_ME. V. COMPUTATIONAL RESULTS The developed hybrid ethod was subjected to soe coputational tests, in order to assess its reliability. The tests were done in the CIGRE test network, which has 7 generators, 10 buses 13 lines [3]. A. Coputational Tests To assess the ethod reliability were siulated 29 disturbances, which were all three-pole short-circuit with contact to ground near the buses. - results They were tested 29 different disturbances. The results show that for 23 disturbances the results were correct. For this results, the axiu absolute difference between the critical tie obtained with nuerical integration ethods the critical tie obtained with the hybrid ethod is 3 s, or 0,62%. - error analysis As said above, 23 of 29 disturbances were correctly studied so, it reains 6 disturbances to analyse. The probles identified in these 6 disturbances can be gathered in 3 groups: 1. critical achines cluster identification: the criteria used to identify the critical cluster don t offer unequivocal results; 2. power curve fitting isatch: the ethods used to fit the equivalent achine power curve are not able to do it correctly, which causes the prograe to get locked in an infinite loop; 3. initial critical tie estiation: the hybrid ethod

6 6 only converges if it receives a ore accurate first critical tie estiate. - possible corrections For the tree error groups detected there are possible solutions. 1. critical achines cluster identification: the solution is to use other criteria which, together with those already used, offer unequivocal results; 2. power curve fitting isatch: the solution ay be to use flags, which are turned on when the progra are locked, what causes the siulation to be stopped; 3. initial critical tie estiation: the solution can be to try to find an initial critical tie estiate using ethods like the extended equal area criterion. B. Exaple This exaple consist in the study of one disturbance. More precisely, it was a disturbance happening next to bus 2, which was eliinated by turning off line 1. As referred before, the disturbance occurs 0,1 s after the siulation start. The progra was started with an initial estiate of t cl = 0,6 s. In Fig. 1 is the rotor angle tie evolution. The tie that siulations lasts it s bigger than the required by the indexes, because to better see the evolution, one chose to let the siulation run. As can be seen, the achine 1 are critical, the others achines belong to the reaining achines groups, which will be confired by the critical cluster identifying criteria. It s also observable that the siulation atches an unstable situation, which will be confired by the stability instability indexes. As the situation is unstable, the criteria to identify the critical achines cluster is the CMR. The results of applying this ethod are available in Table I. As can be seen the results confir that the only critical achine is the achine 1 all the others belong to the reaining achines group. Table I Results of applying CMR Machines Rotor angles δ i [rad] Difference [rad] 1 7,012 6, ,700 2, ,699 2, ,666 1, ,628 1, ,618 1, ,600 0,000 After identifying the critical achines cluster, it s tie to copute the reduced equivalent syste, which rotor angle tie evolution can be seen in Fig. 3. This figure enables to confir that the situation is unstable. achine 1 achine 1 reaining achines reaining achines Fig. 3 - Rotor angle tie evolution of reduced equivalent syste The transient stability argin is coputed using the power curve of the equivalent achine, which can be seen in Fig. 4. It s also observable the trigonoetric fitting, which is the fitting used here because the estiate it s still far fro the final value. It s interesting that the supplied values of electrical power are negative, due to the expressions used to get the equivalent achine. Fig. 1 - Rotor angle tie evolution In Fig. 2 is the IDCS IDE indexes tie evolution. As can be seen, the IDCS index reaches a iniu, which atches an unstable situation. It can also be seen that the siulation is stopped after 0,7 s. Fig. 4 - Power curve of equivalent achine (full) the trigonoetric fitting (dashed) Fig. 2 - IDCS IDE indexes tie evolution The coputed value for transient stability argin is η 0 = 6,7809, whose polarity is consistent with the fact that the situation is unstable. As there available only one value of transient stability argin, it s not possible to do a new estiate so, the clearing tie used in the next siulation is

7 7 coputed by t 1 cr = 0,9 t 0 cr = 0,54 s, which leads to a transient stability argin of η 1 = With this two values of transient stability argin respective clearing ties, it s possibly to do a new estiate of critical tie, which is t 2 cr = Fro now on, it's only repeating the sae steps, so there is no need to show the. In Table II, it can be seen the values obtained in each iteration until the difference it s lower than a tolerance ε 1 = 0,01 s. Iteration Table II Results of iterations with the trigonoetric fitting Critical tie i estiate t cr Transient stability argin η i Difference t cr - 0,600-6, ,540-5,249 0, ,334 9,293 0, ,466-1,252 0, ,450 0,160 0, ,452-0,002 When the difference it s lower than the tolerance ε 1, it s tie to change to the polynoial fitting. With the polynoial fitting the first iteration uses the last critical tie estiate, i.e., t 5 cr = 0,452 s. As with the previous fitting, after the first iteration it s not possible to do a new critical tie estiate so, the clearing tie used in the next siulation it s obtained by t 6 cr = 0,98 t 5 cr = 0,443 s. In Table III, it can be seen the values obtained in each iteration using the polynoial fitting, until the difference is lower than a tolerance ε 2 = 0,001 s. reduced odel is studied using the equal area criteria, being coputed the transient stability argin fro which the critical tie is estiated. Although the ethod presents soe wrong results, it was presented several iportant concepts started a path to the ipleentation of this kind of ethods. B. Future Works As referred in the paper, the developed ethods presents soe liitations, specifically it was said that the critical achines cluster, the power curve fitting the initial critical tie estiate still need soe work on the. So a future work can be to keep the developing of this ethods trying to eliinate the liitations. Another option to future works is trying to insert new functionalities in the ethod, like disturbances far fro the bus, ore precise odels to describe the coponents, etc. APPENDIX The structure of the network used to test the ethods is presented below. Table III Results of iterations with the polynoial fitting Iteration Critical tie i estiate t cr Transient stability argin η i Difference t cr 5 0,452-0, ,443 0,149 0, ,445 0,010 0, ,445-0,000 When the difference is lower than the tolerance ε the siulation it s stopped. The critical tie was found in 7 iterations. To find the actual critical tie it s necessary to subtract the tie when the disturbance occurs t def, so the critical tie is equal to t cr = 0,445 t def = 0,345 s. VI. CONCLUSIONS AND FUTURE WORKS A. Conclusions The paper presents a hybrid ethod to assess the transient stability of Electric Power Systes. This ethod is able to find the critical tie for disturbances specified by the user. The ethod gathers the Nuerical Integration Methods with the Direct Methods. Due to the use of indexes the ethod is able to stop the siulation before the total tie. After the siulation is stopped, the critical achines cluster is identified a reduced equivalent odel of the syste is found, which consists of a achine connected to an infinite bus. This Fig. 5 - Diagra of CIGRE test network The reaining data about this network can be find in [4]. ACKNOWLEDGMENT The author would like to thank to all the people that had a role in the writing of this paper. REFERENCES [1] Kundur, P. Power Syste Stability Control. USA : McGraw-Hill, Inc., [2] Sucena Paiva, J. P. Redes de Energia Eléctrica: ua análise sistéica. 2ª edição. Lisboa : IST Press, 2007 [3] Machado Ferreira, C. M. Análise da Estabilidade Transitória de Sisteas Eléctricos de Energia utilizo Forulações Híbridas. Porto : Departaento de Engenharia Electrotécnica e de Coputadores, Faculdade de Engenharia da Universidade do Porto, Dissertação de Doutoraento, [4] Carvalho Mateus, J. P. Métodos de Análise da Estabilidade Transitória de Sisteas de Energia Eléctrica. Lisboa : Instituto Superior Técnico, Dissertação de Mestrado, [5] Vittal, Vijay. Direct Stability Methods. [book author] Leonard L. Grigsby. Electric Power Engineering Hbook. Second Edition. Power Syste Stability Control. Boca Raton, USA : CRC Press - Taylor & Francis Group, [6] Chan, K. W., Zhou, Q. e Chung, T. S. Transient Stability Margin Assessent for Large Power Syste using Tie Doain Siulation Based Hybrid Equal Area Criterion Method. Proceedings of the 5th International Conference on Advances in Power Syste Control, Operation Manageent. October 2000