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A New Method for Obtaining Critical Clearing Time for Transient Stability Naoto Yorino, Member, IEEE, Ardyono Priyadi, Student Member, IEEE, Hironori Kakui, and Mitsuhiro Takeshita Abstract—This paper proposes a new formulation for transient stability analysis for electric power systems. Different from existing methods, a minimization problem is formulated for obtaining critical clearing time (CCT) for transient stability. The method is based on the computation of a trajectory on the stability boundary, which is referred to as critical trajectory in this paper. The critical trajectory is defined as the trajectory that starts from a point on a fault-on trajectory at CCT and reaches a critical point of losing synchronism. The new proposal includes a modified trapezoidal formulation for numerical integration, the critical conditions for synchronism, and the unified minimization formulation. It will be demonstrated that the solution of the minimization problem successfully provides the exact CCT that agrees with the conventional numerical simulation method. Index Terms—Critical clearing time, critical trajectory method, electric power system, transient stability analysis. Fig. 1. Trajectories in a phase plane for a single machine to infinite bus system with damping.

I. INTRODUCTION

T

RANSIENT stability analysis plays an important role for maintaining security of power system operation. The analysis is mainly performed through numerical simulations, where numerical integration is carried out step by step from an initial value to obtain dynamic response to disturbances. In general, such a numerical simulation method is effective since it can easily take into account various dynamic models for complex power systems as well as various time sequences of events. Furthermore, the method is useful in analyzing various kinds of complex nonlinear phenomena such as in [1]–[3]. However, the numerical simulation is usually time consuming, and therefore, it is not necessarily suited for real time stability assessment. An alternative approach, called transient energy function methods [4]–[16], assesses system stability based on the transient energy. Those methods provide fast and efficient stability assessment for a number of disturbances. Although they are practically useful, a common disadvantage is concerned with the accuracy of stability judgment. A major limitation is that they cannot deal with detailed models for power systems since the transient energy functions are available only for limited types of power system models. Another problem is that the most of the methods require the evaluation of critical energy, which affects considerably the accuracy of stability assessment. The critical energy is not necessarily easily calculated. Manuscript received June 24, 2009; revised December 18, 2009. First published February 25, 2010; current version published July 21, 2010. Paper no. TPWRS-00481-2009. N. Yorino and A. Priyadi are with the Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan. H. Kakui and M. Takeshita are with the Chugoku Electric Power Co., Inc., Hiroshima-shi 730-8701, Japan. Digital Object Identifier 10.1109/TPWRS.2009.2040003

This paper proposes a new method for transient stability analysis. In order to describe the proposed method, typical dynamic behaviors of a power system are given in Fig. 1, where a single machine case is presented as an example. Three kinds of trajectories are given in phase plane starting at different points on a fault-on trajectory “1”. Trajectory “2” is for a stable case where the fault is cleared early enough and it oscillates around a stable equilibrium point. Trajectory “4” corresponds to an unstable case, where the fault clearing is too late. Trajectory “3” corresponds to a critical case for stability and is referred to as the critical trajectory in this paper. In a specific single machine case, the critical trajectory reaches an unstable equilibrium point (UEP) as is seen from Fig. 1. The critical trajectory is defined as the trajectory that starts from a point on a fault-on trajectory and reaches a critical point that satisfies a set of conditions of losing synchronism. The conditions will be proposed in this paper. The critical point agrees with UEP for a single machine system as in Fig. 1, although this is not the case for general multimachine systems. It is generally difficult to compute the critical trajectory by means of conventional numerical simulations. In this paper, the problem is formulated as a minimization problem for computing the critical trajectory based on preliminary examinations in [17]–[21]. Critical conditions with CCT for transient stability are directly computed as the solution of the minimization problem. II. PROBLEM FORMULATION A. Definitions Transient stability problem for an event disturbance may be expressed as follows: Initially, a power system is operating at a stable operating point, say , when a fault occurs at time

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. Then, the system is governed by the fault-on dynamics as follows: during the fault (1) . where The solution curve of (1) is called fault-on trajectory and is expressed in this paper by (2) . where The fault is cleared at time . The system is governed by the post-fault dynamics expressed by the following nonlinear equation: (3)

Fig. 2. Concept of the proposed method.

Thus, the time duration is replaced with the distance as follows:

The solution curves of (3) are called post-fault trajectory, represented by (4) is a point on the fault-on trajectory Note that initial point at time , fault clearing time:

(8) Equation (8) is substituted into (6) to obtain the following form: (9)

(5) B. Modified Trapezoidal Formulation In this section, we propose a modified trapezoidal form. Letting a solution of (3) at time be denoted as , the following equation holds using the conventional trapezoidal formula: (6) where

In this paper, superscript is used for state transition number with respect to time. As is stated in the Introduction, we pay attention to the critical trajectory, where a system fault is cleared at CCT and then the state variables converge to a critical point as stated before. In some specific case, the critical point agrees with an UEP and the trajectory reaches the UEP with infinite time. Fig. 2 shows the critical trajectory, where two boundary points, and , represent the initial point at CCT and the critical point. A difficulty in obtaining the critical trajectory is that infinite time may be taken when to reach UEP. An example is that a ball is moving towards the top of hill (UEP), whose kinetic energy is equal to the potential energy at UEP. The ball cannot reach the UEP within a finite time since the speed tends to zero as it becomes close to the UEP. For such computation, infinite time steps are required. To avoid the problem, we have developed a new method for numerical integration as follows. First, the distance between the two points in (6) is defined as (7)

By the above equation, the numerical integration with respect to time is transformed into that with distance. This transformation makes it possible to represent the critical trajectory by finite points with a same distance as shown in Fig. 2. C. Critical Condition for Synchronism As is given in Fig. 1, the critical trajectory converges to UEP for a single machine infinite bus system. However, this is not the case in general. In this section, we propose critical condition for synchronism that suffices in general multimachine systems. It is known in a single machine case that the synchronizing force or , where and disappears when respectively are the synchronizing torque and power, and is the rotor angle. A natural extension of the condition for multimachine case may be written based on singularity condition of synchronizing force coefficient matrix as follows: (10) where is the eigenvector corresponding to zero , and is the eigenvalue of matrix number of generators. We further intuitively assume a condition that the eigenvector must agree with change direction of . That : is, the following equation holds with a scalar (11) It will be assumed that the above conditions (10) and (11) hold at a point (the end point) on the critical trajectory. All the variables in (11) will be treated as decision variables in the minimization problem in the next section.

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Although it is not the complete proof of the stability condition for dynamic system, the equations represent the stationary conditions for the synchronizing torque or power, or as follows: (12) Since is basically a function of the rotor angles of generators, the following equation holds: (13) The above equation implies that (11) and (12) are equivalent to each other under condition (10). Thus, the newly proposed conditions are to capture the stationary conditions for or caused by the singularity of synchronizing torque/power matrix. The conditions have been derived intuitively by the author and will be used in the paper without complete proof for instability. The conditions have successfully worked so far with no exceptions. D. Problem Formulation Based on the above discussion, the problem for obtaining the critical condition for transient stability for system (3) is formulated as follows:

additionally as in (18), then solves the redundant equations as a are propminimization problem so that the individual errors erly distributed. The solution of the problem, (14)–(18), is interpreted as folto , represent the critlows. The set of points, ical trajectory, where is automatically determined when the number of integration steps, , is specified; CCT and the critical point are, respectively, obtained as and at the solution. Note that the proposed method is exact without major approximations in the formulation, and that is an important parameter that affects the accuracy and computation time since is roughly proportional to the size of the proposed minimization problem. III. NUMERICAL EXAMINATIONS A. Power System Model In this section, numerical examinations are carried out using multimachine power system represented by the Xd’ generator model, where each generator is represented by two dimensional differential equations. The center of inertia (COI) swing equation in [4] is used for both the conventional numerical simulation and the proposed method as follows: (19) (20)

(14)

where

where (15) (16) with boundary conditions (17) (18) where is a square weighting matrix with positive diagonal does not affect the convergence terms. Since the selection of nor accuracy of proposed method, identity matrix will be used for all the simulations in the latter section. After the minfor imization of (14), becomes ideally zero, where the proposed trapezoidal equation of (9) hold to connect all point to , in Fig. 2. Equation (17), a boundary condition for the initial point, expresses a fault-on trajectory as a function of fault clearing time, . Equation (18) is the other boundary , sub-vector of , satisfies the critcondition, where ical conditions (10) and (11). It is noted that the latter boundary condition is additional compared with the conventional numerical integration formulated as an initial value problem. In such is accumulated as a conventional method, numerical error increases so that a final point in general has a considerable error. On the other hand, the proposed method specifies the final point

B. Simulation Method for Exact Solutions In order to confirm the effectiveness of the proposed method, we have carried out numerical examinations using five test systems as follows: • three-machine nine-bus system (Anderson and Fouad in [22]); • four-machine nine-bus system (Modified Anderson and Fouad system depicted in Fig. 3); • six-machine 30-bus system (IEEE 30); • seven-machine 57-bus system (IEEE 57); • 30-machine 115-bus system (IEE Japan West 30). It is assumed that every transmission line consists of double parallel circuits, and that a three phase fault occurs at a point very close to a bus on one of the parallel lines. After a while, the fault is cleared by opening the faulted line. The 4th-order Runge–Kutta method is used for numerical integration with time step of 0.001 [s]. First, the fault-on trajectory as a function is obtained numerically, which is stored as with a specified is selected as an iniof time, . Then, tial condition to simulate the dynamic behavior to judge the stability of the system. This process is repeated by setting different

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TABLE III CCTS OF IEEE SIX-MACHINE 30-BUS SYSTEM

Fig. 3. Modified Anderson and Fouad four-machine nine-bus system.

TABLE I CCTS OF ANDERSON AND FOUAD THREE-MACHINE NINE-BUS SYSTEM

TABLE IV CCTS OF IEEE SEVEN-MACHINE 57-BUS SYSTEM

TABLE II CCTS OF MODIFIED ANDERSON AND FOUAD FOUR-MACHINE NINE-BUS SYSTEM

value of . The binary search method is used to estimate a critical value of , that is CCT. The obtained results are given by Tables I–V, where the fault location and CCTs are given. For example, in Table I, the expression of “0.342–0.343” implies that the system is stable with clearing time of 0.342 [s] but unstable with 0.343 [s] and that exact value of CCT exists between 0.342 and 0.343 [s]. The CPU time for the simulation method implies the amount of CPU time to perform ten times of the conventional simulations to obtain the corresponding accuracy of the CCT given by the proposed method. C. Proposed Method The proposed method is performed as in the following procedure.

using the 1) The fault-on trajectory is obtained as conventional numerical simulation method and is approximated as a cubic spline data interpolation to define (17). 2) Equations (19)–(20) are used to define (16). 3) The least square minimization problem, (14)–(18), is solved using the Newton Raphson (NR) method with as a convergence criterion to obtain CCT. The CCT for the test systems obtained by the proposed method are listed in Tables I–V, where the number of iterations, computation time (CPU time) is also shown. The CPU time for . It is confirmed the proposed method is for in those tables that the CCTs obtained by the proposed method are exact enough compared with the conventional numerical simulation method. It is understood that the proposed method is also numerically robust enough to obtain the exact CCT without major approximation, where no computational difficulties exist. It is also observed that the CPU time for the proposed method is faster compared to the conventional simulation method. The method can save significant time in offline stability studies, and may be applied for online use in the future. As is mentioned in the Introduction, the transient energy function methods are alternative methods to obtain CCT. In order to

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TABLE V CCTS OF IEE JAPAN WEST 30-MACHINE 115-BUS SYSTEM

Fig. 5. Critical, unstable, and stable waveforms of rotor angle of generator 1 for IEEJ 30-machine 115-bus system for fault at point A.

Fig. 6. Critically stable rotor angle curves for Critical Time (CT) = 0:173 [s] for IEEJ 30-machine 115-bus system for fault at point A.

Fig. 4. Angular velocity ! and rotor angle curves of the generators 1–10 for IEE Japan West 30-machine 115-bus system for fault at point D.

compare the proposed method with the transient energy function method, we chose the BCU Shadowing Method [12] for comparison. The results are listed in Table V for 30-machine system. It is observed that the proposed method provides accurate CCTs, while this is not the case for the Shadowing method. Fig. 4 shows the critical trajectories obtained by the proposed method, together with stable and unstable trajectories (1 and 2)

given by the conventional numerical simulations. In this figure, the ordinate is angular velocity in [rad/s] and the abscissa is rotor angle in [rad] for each generator. It is observed that the critical trajectory lies almost between the stable and unstable trajectories, implying the validity of the proposed method. Figs. 5–7 show the waveforms of rotor angles of generator , where generator 30 is used as angle reference to observe the phenomena clearly. All those figures are for fault A in the 30-machine system. Fig. 5 focuses on generator 1 showing the critical waveform obtained by the proposed method, with the stable and unstable waveforms obtained by the conventional simulations. It is seen that the proposed method provides the accurate critical waveform as well as the CCT. Figs. 6 and 7, respectively, show the stable and unstable waveforms of generators 1, 6, 11, 12, 18, 22, 24, and 29 obtained by the numerical simulations with the critical waveforms obtained by the proposed method in common. As the system becomes lager, the unstable phenomena tend to become complicated, but the proposed method successfully computes the critical condition.

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YORINO et al.: A NEW METHOD FOR OBTAINING CRITICAL CLEARING TIME FOR TRANSIENT STABILITY

Fig. 7. Critically unstable rotor angle curves for 30-machine 115-bus system for fault at point A.

CT = 0 174 [s] for IEEJ :

Fig. 8. Errors in CCT [s] related to convergence criterion for 30-machine system Maxjdx j.

It should be noted that parameter is used for every test system. This setting usually provides accurate CCT with negligible error. Figs. 8–10, respectively, show the errors in CCT [s], the number of iterations, the CPU time for different setting of the convergence criterion of the NR method. It is understood that the obtained CCTs are accurate enough even for , resulting in the reduced iterations as well as computation time. Some comment is added for the objective (14): all ’s are regarded as numerical errors in the equations as mentioned before, implying that the objective value should be zero at the solution. The objective values were around 0.008 for all the cases even for 30-machine system. Note that an Intel® Core™ 2 Duo E8500 with 3 GB of RAM is used for the numerical examinations. IV. CONCLUSION This paper proposes a new formulation for transient stability analysis as a minimization problem for electric power systems. Different from conventional simulation methods, the formulation is not based on an initial value problem but based on

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Fig. 9. Average number of iterations related to convergence criterion for 30-machine system Maxjdx j.

Fig. 10. Average CPU time [s] related to convergence criterion for 30-machine system Maxjdx j.

boundary value problem to directly obtain a critical condition for stability such as a CCT. The method computes the critical trajectory that represents a critical case for stability. It is confirmed that the proposed method can provide the exact CCTs, consistent with the conventional numerical simulations. The result is quite important since no such methods have existed so far to compute the exact CCT without major errors. Another important point is that the proposed method is numerically robust for detecting CCTs for various patterns of complicated instability phenomena. It is expected that the method is able to be applied to larger systems. At present, the proposed method cannot analyze multiple swing instabilities, which should be studied in the future. Although the proposed method can deal with various types of power system models at least in theory, only the Xd’ generator model is used to confirm the validity of the new formulation in this paper. Since the Xd’ generator model is not good enough for transient stability studies, a further study is necessary in order to take into account more detailed generator models together with various types of controllers in the future.

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REFERENCES [1] IEEE Power Engineering Society, Inter-Area Oscillations in Power Systems, System Dynamic Performance Subcommittee Special Publication, 95TP101, 1995. [2] IEEE Power Engineering Society, Voltage Stability Assessment: Concepts, Practices and Tools, Power System Stability Subcommittee Special Publication, SP101PSS, 2003. [3] N. Yorino, H. Sasaki, Y. Tamura, and R. Yokoyama, “A generalized analysis method of auto-parametric resonances in power systems,” IEEE Trans. Power Syst., vol. 4, no. 3, pp. 1057–1064, Aug. 1989. [4] T. Athey, R. Podmore, and S. Virmani, “A practical method for direct analysis of transient stability,” IEEE Trans. Power App. Syst., vol. PAS-98, pp. 573–584, 1979.. [5] N. Kakimoto, Y. Ohsawa, and M. Hayashi, “Transient stability analysis of electric power system via Luré type Lyapunov function,” Trans. IEE Jpn., vol. 98-E, no. 5/6, pp. 63–79, 1978. [6] G. A. Maria, C. Tang, and J. Kim, “Hybrid transient stability analysis,” IEEE Trans. Power Syst., vol. 5, no. 2, pp. 384–393, May 1990. [7] Y. Xue, L. Wehenkel, R. Belhomme, P. Rous-Seaux, M. Pavella, E. Euxibie, B. Heilbronn, and J. F. Lesigne, “Extended equal area criterion revised,” IEEE Trans. Power Syst., vol. 7, no. 3, pp. 1012–1022, Aug. 1992. [8] Y. Mansour, E. Vaahedi, A. Y. Chang, B. R. Corns, B. W. Garrett, K. Demaree, T. Athey, and K. Cheung, “B. C. Hydro’s on-line transient stability assessment (TSA): Model development, analysis, and postprocessing,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 241–253, Feb. 1995. [9] G. D. Irisarri, G. C. Ejebe, and J. G. Waight, “Efficient solution for equilibrium points in transient energy function analysis,” IEEE Trans. Power Syst., vol. 9, no. 2, pp. 693–699, May 1994. [10] H. D. Chiang, F. Wu, and P. Varaiya, “A BCU method for direct analysis of power system transient stability,” IEEE Trans. Power Syst., vol. 9, no. 3, pp. 1194–1208, Aug. 1994. [11] H. D. Chiang, C. C. Chu, and G. Cauley, “Direct stability analysis of electric power systems using energy functions: Theory, applications, and perspective,” Proc. IEEE, vol. 83, no. 11, pp. 1497–1529, Nov. 1995. [12] R. T. Treinen, V. Vittal, and W. Kliemann, “An improved technique to determine the controlling unstable equilibrium point in a power system,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 43, no. 4, pp. 313–323, Apr. 1996. [13] Y. Kataoka, Y. Tada, H. Okamoto, and R. Tanabe, “Improvement of search efficiency of unstable equilibrium for transient stability assessment,” Nat. Conv. Rec., IEE Jpn., Power Syst., pp. 1349–1350, 1999, in Japanese. [14] A. Llamas, J. De La Ree Lopez, L. Mili, A. G. Phadke, and J. S. Thorp, “Clarifications of the BCU method for transient stability analysis,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 210–219, Feb. 1995. [15] F. Paganini and B. C. Lesieutre, “Generic properties, one-parameter deformations, and the BCU method,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 46, no. 6, pp. 760–763, Jun. 1999. [16] L. F. C. Alberto and N. G. Bretas, “Application of melnikov’s method for computing heteroclinic orbits in a classical SMIB power system model,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 47, no. 7, pp. 1085–1089, Jul. 2000. [17] N. Yorino, T. Saito, H. Q. Li, Y. Kamei, and H. Sasaki, “A new method for transient stability analysis,” in IEE Jpn. Papers of Technical Meeting on Power System Technology, 2003, no. PE-03-83, PSE-03-94, pp. 57–62, in Japanese. [18] N. Yorino, Y. Kamei, and Y. Zoka, “A new method for transient stability assessment based on critical trajectory,” in Proc. 15th Power System Computation Conf. (PSCC), Paper ID: 20-3, session 10, paper #4, pp. 1–6, 2005. [19] N. Yorino, A. Priyadi, Y. Zoka, H. Yasuda, and H. Kakui, “A novel method for transient stability analysis as boundary value problem,” in Proc. Int. Conf. Electrical Engineering (ICEE), Okinawa, Japan, Jul. 2008, pp. 1–6.

[20] A. Priyadi, N. Yorino, M. Eghbal, Y. Zoka, Y. Sasaki, H. Yasuda, and H. Kakui, “Transient stability assessment as boundary value problem,” in Proc. IEEE 8th Annu. Electrical Power and Energy Conf., Vancouver, BC, Canada, Oct. 2008, pp. 1–6. [21] N. Yorino, A. Priyadi, Y. Zoka, Y. Sasaki, M. Tanaka, H. Yasuda, and H. Kakui, “Transient stability analysis as boundary value problem,” in Proc. XI Symp. Specialists in Electric Operational and Expansion Planning (XI SEPOPE), Belem, Brazil, Mar. 2009, paper #32, pp. 1–11. [22] P. M. Anderson and A. A. Fouad, Power System Control and Stability. Ames, IA: The Iowa State Univ. Press, 1977, vol. 1. Naoto Yorino (M’90) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Waseda University, Tokyo, Japan, in 1981, 1983, and 1987, respectively. He is a Professor in the Graduate School of Engineering, Hiroshima University, Hiroshima, Japan. He was with Fuji Electric Co. Ltd., Japan from 1983 to 1984. He was a Visiting Professor at McGill University, Montreal, QC, Canada, from 1991 to 1992. Dr. Yorino is a member of the IEE of Japan, CIGRE, iREP, and ESCJ.

Ardyono Priyadi (S’07) received the Sarjana Teknik degree (equivalent to the B.S. degree) in electrical engineering from Institute Technology of Sepuluh Nopember (ITS), Surabaya, Indonesia, in 1997 and the M.Eng. degree in electrical engineering from Hiroshima University, Hiroshima, Japan in 2008. He is pursuing the Ph.D. degree in the transient stability area at the Department of Artificial Complex Systems Engineering, Hiroshima University. He worked as a Lecturer in ITS starting from 1997 and is currently on study-leave. His research interests are power system dynamic and control, transient stability, and application of the AI techniques in power system.

Hironori Kakui received the B.S. degree in electrical engineering from Waseda University, Tokyo, Japan, in 1994. He joined the Chugoku Electric Power Co., Inc. in 1994. Presently, he has been engaged in power system engineering such as the investigation of power system stability enhancement. Mr. Kakui is a member of the IEE of Japan.

Mitsuhiro Takeshita received the B.S. and M.S degrees from Kyushu University, Fukuoka, Japan, in 1992 and 1994, respectively. He joined the Chugoku Electric Power Co., Inc. in 1994. Presently, he has been engaged in power system engineering. Mr. Takeshita is a member of the IEE of Japan.

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010

A New Method for Obtaining Critical Clearing Time for Transient Stability Naoto Yorino, Member, IEEE, Ardyono Priyadi, Student Member, IEEE, Hironori Kakui, and Mitsuhiro Takeshita Abstract—This paper proposes a new formulation for transient stability analysis for electric power systems. Different from existing methods, a minimization problem is formulated for obtaining critical clearing time (CCT) for transient stability. The method is based on the computation of a trajectory on the stability boundary, which is referred to as critical trajectory in this paper. The critical trajectory is defined as the trajectory that starts from a point on a fault-on trajectory at CCT and reaches a critical point of losing synchronism. The new proposal includes a modified trapezoidal formulation for numerical integration, the critical conditions for synchronism, and the unified minimization formulation. It will be demonstrated that the solution of the minimization problem successfully provides the exact CCT that agrees with the conventional numerical simulation method. Index Terms—Critical clearing time, critical trajectory method, electric power system, transient stability analysis. Fig. 1. Trajectories in a phase plane for a single machine to infinite bus system with damping.

I. INTRODUCTION

T

RANSIENT stability analysis plays an important role for maintaining security of power system operation. The analysis is mainly performed through numerical simulations, where numerical integration is carried out step by step from an initial value to obtain dynamic response to disturbances. In general, such a numerical simulation method is effective since it can easily take into account various dynamic models for complex power systems as well as various time sequences of events. Furthermore, the method is useful in analyzing various kinds of complex nonlinear phenomena such as in [1]–[3]. However, the numerical simulation is usually time consuming, and therefore, it is not necessarily suited for real time stability assessment. An alternative approach, called transient energy function methods [4]–[16], assesses system stability based on the transient energy. Those methods provide fast and efficient stability assessment for a number of disturbances. Although they are practically useful, a common disadvantage is concerned with the accuracy of stability judgment. A major limitation is that they cannot deal with detailed models for power systems since the transient energy functions are available only for limited types of power system models. Another problem is that the most of the methods require the evaluation of critical energy, which affects considerably the accuracy of stability assessment. The critical energy is not necessarily easily calculated. Manuscript received June 24, 2009; revised December 18, 2009. First published February 25, 2010; current version published July 21, 2010. Paper no. TPWRS-00481-2009. N. Yorino and A. Priyadi are with the Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan. H. Kakui and M. Takeshita are with the Chugoku Electric Power Co., Inc., Hiroshima-shi 730-8701, Japan. Digital Object Identifier 10.1109/TPWRS.2009.2040003

This paper proposes a new method for transient stability analysis. In order to describe the proposed method, typical dynamic behaviors of a power system are given in Fig. 1, where a single machine case is presented as an example. Three kinds of trajectories are given in phase plane starting at different points on a fault-on trajectory “1”. Trajectory “2” is for a stable case where the fault is cleared early enough and it oscillates around a stable equilibrium point. Trajectory “4” corresponds to an unstable case, where the fault clearing is too late. Trajectory “3” corresponds to a critical case for stability and is referred to as the critical trajectory in this paper. In a specific single machine case, the critical trajectory reaches an unstable equilibrium point (UEP) as is seen from Fig. 1. The critical trajectory is defined as the trajectory that starts from a point on a fault-on trajectory and reaches a critical point that satisfies a set of conditions of losing synchronism. The conditions will be proposed in this paper. The critical point agrees with UEP for a single machine system as in Fig. 1, although this is not the case for general multimachine systems. It is generally difficult to compute the critical trajectory by means of conventional numerical simulations. In this paper, the problem is formulated as a minimization problem for computing the critical trajectory based on preliminary examinations in [17]–[21]. Critical conditions with CCT for transient stability are directly computed as the solution of the minimization problem. II. PROBLEM FORMULATION A. Definitions Transient stability problem for an event disturbance may be expressed as follows: Initially, a power system is operating at a stable operating point, say , when a fault occurs at time

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. Then, the system is governed by the fault-on dynamics as follows: during the fault (1) . where The solution curve of (1) is called fault-on trajectory and is expressed in this paper by (2) . where The fault is cleared at time . The system is governed by the post-fault dynamics expressed by the following nonlinear equation: (3)

Fig. 2. Concept of the proposed method.

Thus, the time duration is replaced with the distance as follows:

The solution curves of (3) are called post-fault trajectory, represented by (4) is a point on the fault-on trajectory Note that initial point at time , fault clearing time:

(8) Equation (8) is substituted into (6) to obtain the following form: (9)

(5) B. Modified Trapezoidal Formulation In this section, we propose a modified trapezoidal form. Letting a solution of (3) at time be denoted as , the following equation holds using the conventional trapezoidal formula: (6) where

In this paper, superscript is used for state transition number with respect to time. As is stated in the Introduction, we pay attention to the critical trajectory, where a system fault is cleared at CCT and then the state variables converge to a critical point as stated before. In some specific case, the critical point agrees with an UEP and the trajectory reaches the UEP with infinite time. Fig. 2 shows the critical trajectory, where two boundary points, and , represent the initial point at CCT and the critical point. A difficulty in obtaining the critical trajectory is that infinite time may be taken when to reach UEP. An example is that a ball is moving towards the top of hill (UEP), whose kinetic energy is equal to the potential energy at UEP. The ball cannot reach the UEP within a finite time since the speed tends to zero as it becomes close to the UEP. For such computation, infinite time steps are required. To avoid the problem, we have developed a new method for numerical integration as follows. First, the distance between the two points in (6) is defined as (7)

By the above equation, the numerical integration with respect to time is transformed into that with distance. This transformation makes it possible to represent the critical trajectory by finite points with a same distance as shown in Fig. 2. C. Critical Condition for Synchronism As is given in Fig. 1, the critical trajectory converges to UEP for a single machine infinite bus system. However, this is not the case in general. In this section, we propose critical condition for synchronism that suffices in general multimachine systems. It is known in a single machine case that the synchronizing force or , where and disappears when respectively are the synchronizing torque and power, and is the rotor angle. A natural extension of the condition for multimachine case may be written based on singularity condition of synchronizing force coefficient matrix as follows: (10) where is the eigenvector corresponding to zero , and is the eigenvalue of matrix number of generators. We further intuitively assume a condition that the eigenvector must agree with change direction of . That : is, the following equation holds with a scalar (11) It will be assumed that the above conditions (10) and (11) hold at a point (the end point) on the critical trajectory. All the variables in (11) will be treated as decision variables in the minimization problem in the next section.

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Although it is not the complete proof of the stability condition for dynamic system, the equations represent the stationary conditions for the synchronizing torque or power, or as follows: (12) Since is basically a function of the rotor angles of generators, the following equation holds: (13) The above equation implies that (11) and (12) are equivalent to each other under condition (10). Thus, the newly proposed conditions are to capture the stationary conditions for or caused by the singularity of synchronizing torque/power matrix. The conditions have been derived intuitively by the author and will be used in the paper without complete proof for instability. The conditions have successfully worked so far with no exceptions. D. Problem Formulation Based on the above discussion, the problem for obtaining the critical condition for transient stability for system (3) is formulated as follows:

additionally as in (18), then solves the redundant equations as a are propminimization problem so that the individual errors erly distributed. The solution of the problem, (14)–(18), is interpreted as folto , represent the critlows. The set of points, ical trajectory, where is automatically determined when the number of integration steps, , is specified; CCT and the critical point are, respectively, obtained as and at the solution. Note that the proposed method is exact without major approximations in the formulation, and that is an important parameter that affects the accuracy and computation time since is roughly proportional to the size of the proposed minimization problem. III. NUMERICAL EXAMINATIONS A. Power System Model In this section, numerical examinations are carried out using multimachine power system represented by the Xd’ generator model, where each generator is represented by two dimensional differential equations. The center of inertia (COI) swing equation in [4] is used for both the conventional numerical simulation and the proposed method as follows: (19) (20)

(14)

where

where (15) (16) with boundary conditions (17) (18) where is a square weighting matrix with positive diagonal does not affect the convergence terms. Since the selection of nor accuracy of proposed method, identity matrix will be used for all the simulations in the latter section. After the minfor imization of (14), becomes ideally zero, where the proposed trapezoidal equation of (9) hold to connect all point to , in Fig. 2. Equation (17), a boundary condition for the initial point, expresses a fault-on trajectory as a function of fault clearing time, . Equation (18) is the other boundary , sub-vector of , satisfies the critcondition, where ical conditions (10) and (11). It is noted that the latter boundary condition is additional compared with the conventional numerical integration formulated as an initial value problem. In such is accumulated as a conventional method, numerical error increases so that a final point in general has a considerable error. On the other hand, the proposed method specifies the final point

B. Simulation Method for Exact Solutions In order to confirm the effectiveness of the proposed method, we have carried out numerical examinations using five test systems as follows: • three-machine nine-bus system (Anderson and Fouad in [22]); • four-machine nine-bus system (Modified Anderson and Fouad system depicted in Fig. 3); • six-machine 30-bus system (IEEE 30); • seven-machine 57-bus system (IEEE 57); • 30-machine 115-bus system (IEE Japan West 30). It is assumed that every transmission line consists of double parallel circuits, and that a three phase fault occurs at a point very close to a bus on one of the parallel lines. After a while, the fault is cleared by opening the faulted line. The 4th-order Runge–Kutta method is used for numerical integration with time step of 0.001 [s]. First, the fault-on trajectory as a function is obtained numerically, which is stored as with a specified is selected as an iniof time, . Then, tial condition to simulate the dynamic behavior to judge the stability of the system. This process is repeated by setting different

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TABLE III CCTS OF IEEE SIX-MACHINE 30-BUS SYSTEM

Fig. 3. Modified Anderson and Fouad four-machine nine-bus system.

TABLE I CCTS OF ANDERSON AND FOUAD THREE-MACHINE NINE-BUS SYSTEM

TABLE IV CCTS OF IEEE SEVEN-MACHINE 57-BUS SYSTEM

TABLE II CCTS OF MODIFIED ANDERSON AND FOUAD FOUR-MACHINE NINE-BUS SYSTEM

value of . The binary search method is used to estimate a critical value of , that is CCT. The obtained results are given by Tables I–V, where the fault location and CCTs are given. For example, in Table I, the expression of “0.342–0.343” implies that the system is stable with clearing time of 0.342 [s] but unstable with 0.343 [s] and that exact value of CCT exists between 0.342 and 0.343 [s]. The CPU time for the simulation method implies the amount of CPU time to perform ten times of the conventional simulations to obtain the corresponding accuracy of the CCT given by the proposed method. C. Proposed Method The proposed method is performed as in the following procedure.

using the 1) The fault-on trajectory is obtained as conventional numerical simulation method and is approximated as a cubic spline data interpolation to define (17). 2) Equations (19)–(20) are used to define (16). 3) The least square minimization problem, (14)–(18), is solved using the Newton Raphson (NR) method with as a convergence criterion to obtain CCT. The CCT for the test systems obtained by the proposed method are listed in Tables I–V, where the number of iterations, computation time (CPU time) is also shown. The CPU time for . It is confirmed the proposed method is for in those tables that the CCTs obtained by the proposed method are exact enough compared with the conventional numerical simulation method. It is understood that the proposed method is also numerically robust enough to obtain the exact CCT without major approximation, where no computational difficulties exist. It is also observed that the CPU time for the proposed method is faster compared to the conventional simulation method. The method can save significant time in offline stability studies, and may be applied for online use in the future. As is mentioned in the Introduction, the transient energy function methods are alternative methods to obtain CCT. In order to

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TABLE V CCTS OF IEE JAPAN WEST 30-MACHINE 115-BUS SYSTEM

Fig. 5. Critical, unstable, and stable waveforms of rotor angle of generator 1 for IEEJ 30-machine 115-bus system for fault at point A.

Fig. 6. Critically stable rotor angle curves for Critical Time (CT) = 0:173 [s] for IEEJ 30-machine 115-bus system for fault at point A.

Fig. 4. Angular velocity ! and rotor angle curves of the generators 1–10 for IEE Japan West 30-machine 115-bus system for fault at point D.

compare the proposed method with the transient energy function method, we chose the BCU Shadowing Method [12] for comparison. The results are listed in Table V for 30-machine system. It is observed that the proposed method provides accurate CCTs, while this is not the case for the Shadowing method. Fig. 4 shows the critical trajectories obtained by the proposed method, together with stable and unstable trajectories (1 and 2)

given by the conventional numerical simulations. In this figure, the ordinate is angular velocity in [rad/s] and the abscissa is rotor angle in [rad] for each generator. It is observed that the critical trajectory lies almost between the stable and unstable trajectories, implying the validity of the proposed method. Figs. 5–7 show the waveforms of rotor angles of generator , where generator 30 is used as angle reference to observe the phenomena clearly. All those figures are for fault A in the 30-machine system. Fig. 5 focuses on generator 1 showing the critical waveform obtained by the proposed method, with the stable and unstable waveforms obtained by the conventional simulations. It is seen that the proposed method provides the accurate critical waveform as well as the CCT. Figs. 6 and 7, respectively, show the stable and unstable waveforms of generators 1, 6, 11, 12, 18, 22, 24, and 29 obtained by the numerical simulations with the critical waveforms obtained by the proposed method in common. As the system becomes lager, the unstable phenomena tend to become complicated, but the proposed method successfully computes the critical condition.

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YORINO et al.: A NEW METHOD FOR OBTAINING CRITICAL CLEARING TIME FOR TRANSIENT STABILITY

Fig. 7. Critically unstable rotor angle curves for 30-machine 115-bus system for fault at point A.

CT = 0 174 [s] for IEEJ :

Fig. 8. Errors in CCT [s] related to convergence criterion for 30-machine system Maxjdx j.

It should be noted that parameter is used for every test system. This setting usually provides accurate CCT with negligible error. Figs. 8–10, respectively, show the errors in CCT [s], the number of iterations, the CPU time for different setting of the convergence criterion of the NR method. It is understood that the obtained CCTs are accurate enough even for , resulting in the reduced iterations as well as computation time. Some comment is added for the objective (14): all ’s are regarded as numerical errors in the equations as mentioned before, implying that the objective value should be zero at the solution. The objective values were around 0.008 for all the cases even for 30-machine system. Note that an Intel® Core™ 2 Duo E8500 with 3 GB of RAM is used for the numerical examinations. IV. CONCLUSION This paper proposes a new formulation for transient stability analysis as a minimization problem for electric power systems. Different from conventional simulation methods, the formulation is not based on an initial value problem but based on

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Fig. 9. Average number of iterations related to convergence criterion for 30-machine system Maxjdx j.

Fig. 10. Average CPU time [s] related to convergence criterion for 30-machine system Maxjdx j.

boundary value problem to directly obtain a critical condition for stability such as a CCT. The method computes the critical trajectory that represents a critical case for stability. It is confirmed that the proposed method can provide the exact CCTs, consistent with the conventional numerical simulations. The result is quite important since no such methods have existed so far to compute the exact CCT without major errors. Another important point is that the proposed method is numerically robust for detecting CCTs for various patterns of complicated instability phenomena. It is expected that the method is able to be applied to larger systems. At present, the proposed method cannot analyze multiple swing instabilities, which should be studied in the future. Although the proposed method can deal with various types of power system models at least in theory, only the Xd’ generator model is used to confirm the validity of the new formulation in this paper. Since the Xd’ generator model is not good enough for transient stability studies, a further study is necessary in order to take into account more detailed generator models together with various types of controllers in the future.

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REFERENCES [1] IEEE Power Engineering Society, Inter-Area Oscillations in Power Systems, System Dynamic Performance Subcommittee Special Publication, 95TP101, 1995. [2] IEEE Power Engineering Society, Voltage Stability Assessment: Concepts, Practices and Tools, Power System Stability Subcommittee Special Publication, SP101PSS, 2003. [3] N. Yorino, H. Sasaki, Y. Tamura, and R. Yokoyama, “A generalized analysis method of auto-parametric resonances in power systems,” IEEE Trans. Power Syst., vol. 4, no. 3, pp. 1057–1064, Aug. 1989. [4] T. Athey, R. Podmore, and S. Virmani, “A practical method for direct analysis of transient stability,” IEEE Trans. Power App. Syst., vol. PAS-98, pp. 573–584, 1979.. [5] N. Kakimoto, Y. Ohsawa, and M. Hayashi, “Transient stability analysis of electric power system via Luré type Lyapunov function,” Trans. IEE Jpn., vol. 98-E, no. 5/6, pp. 63–79, 1978. [6] G. A. Maria, C. Tang, and J. Kim, “Hybrid transient stability analysis,” IEEE Trans. Power Syst., vol. 5, no. 2, pp. 384–393, May 1990. [7] Y. Xue, L. Wehenkel, R. Belhomme, P. Rous-Seaux, M. Pavella, E. Euxibie, B. Heilbronn, and J. F. Lesigne, “Extended equal area criterion revised,” IEEE Trans. Power Syst., vol. 7, no. 3, pp. 1012–1022, Aug. 1992. [8] Y. Mansour, E. Vaahedi, A. Y. Chang, B. R. Corns, B. W. Garrett, K. Demaree, T. Athey, and K. Cheung, “B. C. Hydro’s on-line transient stability assessment (TSA): Model development, analysis, and postprocessing,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 241–253, Feb. 1995. [9] G. D. Irisarri, G. C. Ejebe, and J. G. Waight, “Efficient solution for equilibrium points in transient energy function analysis,” IEEE Trans. Power Syst., vol. 9, no. 2, pp. 693–699, May 1994. [10] H. D. Chiang, F. Wu, and P. Varaiya, “A BCU method for direct analysis of power system transient stability,” IEEE Trans. Power Syst., vol. 9, no. 3, pp. 1194–1208, Aug. 1994. [11] H. D. Chiang, C. C. Chu, and G. Cauley, “Direct stability analysis of electric power systems using energy functions: Theory, applications, and perspective,” Proc. IEEE, vol. 83, no. 11, pp. 1497–1529, Nov. 1995. [12] R. T. Treinen, V. Vittal, and W. Kliemann, “An improved technique to determine the controlling unstable equilibrium point in a power system,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 43, no. 4, pp. 313–323, Apr. 1996. [13] Y. Kataoka, Y. Tada, H. Okamoto, and R. Tanabe, “Improvement of search efficiency of unstable equilibrium for transient stability assessment,” Nat. Conv. Rec., IEE Jpn., Power Syst., pp. 1349–1350, 1999, in Japanese. [14] A. Llamas, J. De La Ree Lopez, L. Mili, A. G. Phadke, and J. S. Thorp, “Clarifications of the BCU method for transient stability analysis,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 210–219, Feb. 1995. [15] F. Paganini and B. C. Lesieutre, “Generic properties, one-parameter deformations, and the BCU method,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 46, no. 6, pp. 760–763, Jun. 1999. [16] L. F. C. Alberto and N. G. Bretas, “Application of melnikov’s method for computing heteroclinic orbits in a classical SMIB power system model,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 47, no. 7, pp. 1085–1089, Jul. 2000. [17] N. Yorino, T. Saito, H. Q. Li, Y. Kamei, and H. Sasaki, “A new method for transient stability analysis,” in IEE Jpn. Papers of Technical Meeting on Power System Technology, 2003, no. PE-03-83, PSE-03-94, pp. 57–62, in Japanese. [18] N. Yorino, Y. Kamei, and Y. Zoka, “A new method for transient stability assessment based on critical trajectory,” in Proc. 15th Power System Computation Conf. (PSCC), Paper ID: 20-3, session 10, paper #4, pp. 1–6, 2005. [19] N. Yorino, A. Priyadi, Y. Zoka, H. Yasuda, and H. Kakui, “A novel method for transient stability analysis as boundary value problem,” in Proc. Int. Conf. Electrical Engineering (ICEE), Okinawa, Japan, Jul. 2008, pp. 1–6.

[20] A. Priyadi, N. Yorino, M. Eghbal, Y. Zoka, Y. Sasaki, H. Yasuda, and H. Kakui, “Transient stability assessment as boundary value problem,” in Proc. IEEE 8th Annu. Electrical Power and Energy Conf., Vancouver, BC, Canada, Oct. 2008, pp. 1–6. [21] N. Yorino, A. Priyadi, Y. Zoka, Y. Sasaki, M. Tanaka, H. Yasuda, and H. Kakui, “Transient stability analysis as boundary value problem,” in Proc. XI Symp. Specialists in Electric Operational and Expansion Planning (XI SEPOPE), Belem, Brazil, Mar. 2009, paper #32, pp. 1–11. [22] P. M. Anderson and A. A. Fouad, Power System Control and Stability. Ames, IA: The Iowa State Univ. Press, 1977, vol. 1. Naoto Yorino (M’90) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Waseda University, Tokyo, Japan, in 1981, 1983, and 1987, respectively. He is a Professor in the Graduate School of Engineering, Hiroshima University, Hiroshima, Japan. He was with Fuji Electric Co. Ltd., Japan from 1983 to 1984. He was a Visiting Professor at McGill University, Montreal, QC, Canada, from 1991 to 1992. Dr. Yorino is a member of the IEE of Japan, CIGRE, iREP, and ESCJ.

Ardyono Priyadi (S’07) received the Sarjana Teknik degree (equivalent to the B.S. degree) in electrical engineering from Institute Technology of Sepuluh Nopember (ITS), Surabaya, Indonesia, in 1997 and the M.Eng. degree in electrical engineering from Hiroshima University, Hiroshima, Japan in 2008. He is pursuing the Ph.D. degree in the transient stability area at the Department of Artificial Complex Systems Engineering, Hiroshima University. He worked as a Lecturer in ITS starting from 1997 and is currently on study-leave. His research interests are power system dynamic and control, transient stability, and application of the AI techniques in power system.

Hironori Kakui received the B.S. degree in electrical engineering from Waseda University, Tokyo, Japan, in 1994. He joined the Chugoku Electric Power Co., Inc. in 1994. Presently, he has been engaged in power system engineering such as the investigation of power system stability enhancement. Mr. Kakui is a member of the IEE of Japan.

Mitsuhiro Takeshita received the B.S. and M.S degrees from Kyushu University, Fukuoka, Japan, in 1992 and 1994, respectively. He joined the Chugoku Electric Power Co., Inc. in 1994. Presently, he has been engaged in power system engineering. Mr. Takeshita is a member of the IEE of Japan.

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