Phase-shifting transformers in a structure-preserving energy function

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1 Electric Power Systems Research 74 (2005) Phase-shifting transformers in a structure-preserving energy function A. Gabrijel a,, B. Mihalic b a Eletro-Slovenija, d.o.o., Hajdrihova 2, 1000 Ljubljana, Slovenia b University of Ljubljana, Faculty of Electrical Engineering, Tržaša 25, 1000 Ljubljana, Slovenia Received 15 June 2004; accepted 8 December 2004 Abstract With some exceptions, direct methods have so far not been properly able to assess the transient stability of general power systems with phase-shifting transformers (PSTs) because there were no energy functions that would include their action in a post-fault situation. In this paper, we propose two additional potential-energy function parts for the two most common PSTs that can be added to any existing structurepreserving energy functions (SPEF). In this way, modularity is achieved and other FACTS devices can easily be taen into consideration. Using the proposed energy functions, it is possible to apply direct methods to any system that consists of an arbitrary number of phase-angle regulators (PAR) and quadrature-boosting transformers (QBT). These new energy functions assume a constant controllable parameter; however, we extended the use of the presented energy functions by a simple procedure that maes the direct method able to consider any parameter change in a post-fault system. The proposed energy-function extensions were tested in a longitudinal system, because only in such a system will direct methods and a simulation give the same results. Another demonstration was made with an IEEE nine-bus test system using a potential-energy boundarysurface (PEBS) direct method Elsevier B.V. All rights reserved. Keywords: Phase-shifting transformers; Direct method; Lyapunov function 1. Introduction Many power blacouts in the recent past can be attributed to the deregulation of the power sector which in general did not bring the anticipated dramatic fall in the prices charged to consumers quite the opposite in some cases. In the past, power systems were built with a certain safety margin; this margin is now slowly decreasing and many people believe that power systems are more vulnerable than they were in the past. One of the reasons for this is that deregulation has not clearly answered the questions that relate to the construction of new facilities, and as a result, cross-border trades are increased while systems are often pushed to their limits, where even a relatively small disturbance can lead to widespread blacouts. One of the ey issues in the reliable operation of power systems is the preservation of Corresponding author. Tel.: address: uros.gabrijel@eles.si (A. Gabrijel). stability after large disturbances. This problem is referred to as the transient stability of electric power systems. Since a system s nonlinear differential algebraic equations (DAEs) must not be linearized in the case of a large disturbance, the assessment of transient stability is one of the most technically and computationally demanding tass. The problems associated with transient-stability assessment have long been the subject of extensive research [1,2]. In spite of this, however, there is still no universal answer. Instead, there are several approaches, each with their pros and cons. Conventionally, transient stability is assessed by a digital simulation repetition until certain critical clearing times (CCTs) are estimated. The CCT is the maximum duration of a fault that still allows the system to preserve its stability. This approach can, however, be very time consuming, since it requires numerous simulation scenarios of the fault occurrences. Nevertheless, this method provides very accurate results because there are almost no limitations on the modeling. Another drawbac is that it can only provide a simple yes or /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.epsr

2 324 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) no answer to the question of stability, and cannot provide information about stability margins. Clearly, it is not possible to use this method for real-time operation, which is required for congestion management and calculations of available transfer capabilities. Alternative options are to use the so-called direct methods that are founded on the Lyapunov stability theory. In addition to information on a system s stability, these methods can also provide a stability-margin estimation and various stability indices that help to identify critical operation points. Direct methods use the energy function of a post-fault system and the trajectory of a fault-on system to estimate CCTs. The computation time is considerably less than that required for the conventional time-domain (T-D) method. However, this gain does not come without costs. Direct methods often suffer from simplifications of the models and uncertainties in the evaluation of the critical energy. Other hybrid approaches combining T-D simulations with an energy-function evaluation are also possible [3]. It has long been nown that FACTS devices can help to damp power-system oscillations and improve transient stability [4]. This technology is still very expensive, however, and in practice it is only applied to avoid barriers relating to new transmission-line constructions, where it can help to fully utilize the existing transfer lines (up to their thermal limits). The exceptions are phase-shifting transformers (PSTs), which are very common and are used for power-flow control and redirections. There are two possible constructions: the first is to use a tap changer with mechanical switches; the second substitutes the mechanical tap-changer with thyristors, and is called a static phase shifter (SPS). If continuous control is required, a pulse-width modulation (PWM) voltageor current-source converter can be applied. No matter how FACTS devices act during the pre-fault, fault and post-fault periods their actions have to be considered when the transient stability is assessed. In our opinion, the easiest and most transparent way is to use a structurepreserving energy function [5,6] in which every element is represented by its own part. This principle is also used for a static var compensator (SVC) which energy function is presented in [7]. Alternatively [8], presents the energy functions and control laws of thyristor-controlled phase-shifting transformers (TCPSTs) in a reduced single machine-infinite bus (SMIB) system. This approach is good as far as the reduction of a power system to an SMIB can be performed without losing the information about the FACTS location. However, their use is limited to longitudinal systems and cases of an interconnection. Another drawbac is that they can only handle one FACTS device, and the use of a more sophisticated generator and load models is not possible, at least not without a system-trajectory approximation technique, such as the one needed for an extended equal-area criterion. Here, we tae a step forward by proposing new energy functions for two types of PSTs that enable direct methods to properly estimate the CCTs of a general electric power system with any number of PSTs installed. These energy functions can also be used if PSTs are controlled in order to Fig. 1. Basic PST modeling. augment the transient stability. In other words, the proposed energy functions can accommodate changes in the PST s controllable parameter. Generally, energy functions based on a structure-preserving framewor benefit from modularity. In other words, if a system contains several different inds of FACTS devices they can all be considered in the SPEF by means of proper FACTS energy functions, e.g., SVCs or SSSCs [9]. The new energy functions were validated on a longitudinal and an IEEE nine-bus test system. For this purpose we used a direct method called potential-energy boundarysurface (PEBS) [1], which says that the time instant when the total energy of the system along the fault-on trajectory equals the maximum of the potential energy along the same fault-on trajectory acts as a good CCT estimation. 2. Modeling 2.1. Models of PSTs For the reader s convenience we will briefly describe the basic PST features that are relevant for our derivations (more can be found in [4,10]). PSTs have the ability to introduce a phase shift between terminal voltage phasors more or less independently of the throughput current. If losses are neglected, then PSTs do not produce or consume active and reactive power [4]. As mentioned in the introduction, there are several basic structures possible. However, two types are commonly used, i.e., the phase-angle regulator (PAR), which can separate a phase from its terminal voltage phasors without changing their magnitude; and the quadrature-boosting transformer (QBT), which has a fixed injected voltage phase in terms of the input-voltage phasor (β =90 ). A possible conventional PST construction consists of a series and parallel transformer coupled with a mechanical tap changer (Fig. 1a). An equivalent model is presented in Fig. 1b. A corresponding phasor diagram for the case of a QBT application is in Fig. 1c, while Fig. 1d represents the operation of a PAR.

3 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) Fig. 2. PST in a longitudinal system. A longitudinal transmission system with a PST inserted into a transmission path with losses neglected (Fig. 2) isdescribed with the following equations: U - PST1 = U - 1 I - 1jX 1 (1) U - PST2 = U - PST1e jα 1 (2) cos α I - 2 = I - 1e jα cos α = U - PST2 U - 2 (3) jx 2 From expressions (1) to (3), the following PST transmission characteristic can be derived using U - PST1I -1 = U - PST2I -2 [4]: P QBT = P 1 = P 2 = U 1 U 2 (X 1 / cos(α)) + X 2 cos(α) sin(δ + α) (4) P PAR = P 2 = P 1 = U 1U 2 sin(δ + α) (5) X From expressions (4) and (5), it is evident that a QBT is a non-symmetrical device. This is because the transferred power is dependent on both the orientation and the location of the QBT; these parameters do not influence the transmitted power in the case of a PAR. Consider now a PST with its series-transformer reactance inserted between nodes i and j, as shown in Fig. 3. Omitting the reactance X 1 in Fig. 2 and in Eqs. (4) and (5), we can give the following mathematical expressions for the reactive and real power flows that are valid for Fig. 3, which shows the PST in orientation 1: Fig. 3. PST model with a transformer reactance. Fig. 4. Structure-preserving model with classical generators. QBT case: P ij = P ji = U iu j X s cos α sin(θ ij + α) (6) ( ) U i Ui Q ij = X s cos α cos α U j cos(θ ij + α) (7) ) Q ji = U j X s PAR case: ( U j U i cos α cos(θ ij + α) P ij = P ji = U iu j X s sin(θ ij + α) (9) Q ij = U i X s (U i U j cos(θ ij + α)) (10) Q ji = U j (U j U i cos(θ ij + α)) (11) X s where X S represents the PST s series-transformer reactance, while θ ij = θ i θ j. If we consider an alternative orientation of the QBT, it is necessary to mae the following replacements U i U j, α α in θ ij θ ji in formulae (6) (8) The system model In order to understand the structure of the newly developed SPEF, which also considers PSTs, we provide a brief explanation of the electric power system model that is used as a basis for SPEF development. A classical, structure-preserving model of a power system with m generators and N networ nodes is presented in Fig. 4 [11]. We augment the networ-admittance matrix Y N with m generators constant voltages E - i behind the transient reactances X di. ( ) 1 y = Diag (12) - jx di [ ] y- 0 Y - aug = Y - N (8) (13)

4 326 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) We now neglect the transmission-line resistances and rewrite the simplified, augmented admittance matrix as Y - = [jb ij ]. We have adopted the nomenclature of the N-bus and m- generators power system from [11]: δ is a vector consisting of m rotor angles, ω is a vector consisting of m rotor velocities, U N is a vector consisting of N bus-voltage magnitudes, θ is a vector consisting of N bus-voltage angles. Introducing new vectors that comprise all the phase angles and all the voltage magnitudes, we obtain: φ = [δ T, θ T ] T = [φ 1 = δ 1,...,φ m = δ m,φ m+1 = θ m+1,...,φ = θ ] T U = [U 1 = E 1,...,U m = E m,u m+1,...,u ] T (14) Using a center of inertia (COI) reference frame, defined by δ COI = 1 M T ω COI = 1 M T m M i δ i ; M T = m m M i M i ω i ; ω COI = 1 M T (15) m M i ω i where M i is a generators inertia coefficient, we can rewrite the state variables as: [ ] φ i = φ i δ COI (16) φ i = φ i ω COI The tilde above a variable symbolizes a COI. Based on this nomenclature the following system equations can be expressed in terms of the COI [12]: φ i = ω i, (17) j=1 M i ω i = P mi B ij U i U j sin φ ij M i P COI ; M T i = 1,...,m (18) P i (U i ) = B ij U i U j sin φ ij ; j=1 i = m + 1,...,m+ N (19) Q i (U i ) = j=1 B ij U i U j cos φ ij ; i = m + 1,...,m+ N (20) where P mi is the mechanical power input of the ith generator, which is assumed to be constant during the transients; P i (U i ) and Q i (U i ) represent voltage-dependent real and reactive power loads at the ith node and: P COI = M COI ω COI = 3. Energy functions m P mi i=m+1 P i (U i ) (21) QBTs are widely used in Europe; however, in the USA, PARs are more common. Almost all these conventional PSTs are used for power sharing among parallel lines. Since they are equipped with mechanical switches they cannot offer the dynamic performance of expensive, static PSTs, which employ semiconductor switches, i.e., thyristors. Therefore, their tap-changer is set for a desired steady-state power-flow regulation and does not alter during the transients. Assuming a constant tap change, we have derived the energy functions in a structure-preserving framewor for a PAR and a QBT. These energy functions can be used as extra modules for any of the existing structure-preserving energy functions (SPEF) that represent a post-fault electric power system, e.g., ones from [12] or [13]. Consider now a QBT, modeled as in Fig. 3, inserted into a system between the nodes and l. We can modify the corresponding power-flow Eqs. (6) (8) so they match the new configuration: P l = P l = U U l X s cos α sin( φ l + α) (22) ( ) U U Q l = X s cos α cos α U l cos( φ l + α) (23) Q l = U ( l U l U ) X s cos α cos( φ l + α) (24) A SPEF can be constructed as a first integral of the system equations in the COI reference frame; at this point we have reproduced the SPEF from [12]: V ( ω, φ, U) = V ( ω) + V p1 ( φ, U) + V p2 ( φ) + K (25) where φ = [ δ T, θ T ] T ; K is an arbitrary constant, usually chosen so that it places the origin of (25) at zero; V ( ω), the inetic energy. The rest of the total system energy (25) is the potential energy, where V p2 ( φ) represents the potential energy of the system loads (real power only), and V p1 ( φ, V) stands for V p1 ( φ, U) = 1 2 m P mi φ i + j=1 i=m+1 Q(U i ) dv i U i B ij U i U j cos φ ij (26)

5 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) The energy-function construction procedure for a QBT follows the one for a system without FACTS devices from [12]: Considering φ = [ δ T, θ T ] T, we multiply the active power transfers (P l and P l ) from (22) by the time derivatives φ and φ l, consecutively, and then sum them together: P l φ l = U U l X s cos α sin( φ l + α) φ l (27) where φ l = φ φ l. Then, similarly, we divide the terms (19) and (20) by U and U l, and then multiply them by U and U l, respectively. ( U cos α U l cos( φ l + α) ) Q l U U = (28) U X s cos α Q l U l = U ( l U l U ) U l X s cos α cos( φ l + α) (29) The two terms obtained are added to (27), and the result is P l φ l + Q l U + Q l U l = 1 U U l X s [ U U l cos α sin( φ l + α) φ l + U U cos 2 α + U l U l U U l + U U l cos α ] cos( φ l + α) (30) By intuition we can see that if α is considered constant, the right-hand side of (30) can be expressed alternatively as: ( ( )) d 1 U 2 dt 2X s cos 2 α + U2 l 2U U l cos α cos( φ l + α) (31) The first integral of (31) is an extension to the existing SPEF (25) in the case of a single QBT inserted into the system between the nodes and l. Hence, a complete SPEF in the case of more than one QBT can be expressed as: V ( ω, φ, U, α) = V ( ω) + V p1 ( φ, U) + V p2 ( φ) g + V QBT ( φ, U,α i ) + K (32) where g is the number of QBTs in a system, α = [ α1,...,α g ] T and V QBTi ( φ, U,α i ) = 1 2X s ( U 2 cos 2 α i + U 2 l 2U U l cos α i cos( φ l + α i ) ) (33) Note that the indices and l no longer represent just one pair of nodes. They are fictitious and should be altered in accordance with the QBT s connection points. In the case of a reversed QBT orientation, a change of indices is necessary, as explained in Section 2.1. The rest of the energy function (32) is determined as in (25). The QBT s energy function (33) can be introduced into other SPEFs, which allows detailed models of synchronous machines and loads to be taen into account, e.g., [13]. If, for instance, the controllable parameter α of a single QBT, after the fault is cleared, is not constant, rather it changes its value a few times by means of a var-tap switch, it is also possible to consider this change with a modified PEBS direct method, as described below. The calculation is done numerically and always straightforward along the post-fault system trajectory. It is similar to a numerical integration, and in fact becomes one if a continuous control of α is adopted. By moving along the fault-on system trajectory we can distinguish several sections in which the controllable parameter α is constant. We denote the number of sections by w. We now calculate, based on the trajectory, the corresponding value of V QBT at each time instance in a particular section z by using this formula: V QBT ( φ, U, α) w = [V z QBT ( φ, U, α z ) V z QBT 0 ( φ, U, α z )] (34) z=1 V z QBT ( φ, U,α z ) = 1 2X s ( U 2 cos 2 α z + U 2 l 2U U l cos α z cos( φ l + α z ) ) (35) α = [α 1,...,α w ] T (36) where V z QBT ( φ, U,α z )isasin(33), while only one QBT is present. V z QBT 0 ( φ, U,α z ) is a constant that is different in each section, defined by the V z QBT ( φ, U,α z ) as the first value of V QBT in each section z, and VQBT 1 0 ( φ, U,α 1 ) = 0. In order to correctly apply the energy function (34) we have to now when the change of α will occur. In the case of a SMIB and two possible α values, α = ±α max, this instance is set when the generator s angle equals π/2. A similar method was already considered in [8], although in this case it was for the SMIB energy functions. The same energy-function derivation procedure used for a QBT can be performed for a PAR inserted between nodes and l. Hence, the corresponding energy function is defined as: V ( ω, φ, U, α) = V ( ω) + V p1 ( φ, U) + V p2 ( φ) h + V PARi ( φ, U,α i ) + K (37)

6 328 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) where h is the number of PARs in a system, α = [α 1,...,α h ] T and V PARi ( φ, U,α i ) = 1 2X s (U 2 + U2 l 2U U l cos( φ l + α i )) (38) Applying the same principle used for a QBT to a PAR, the energy function in the case of a controllable-parameter change can be determined as: Fig. 5. Longitudinal test system. V PAR ( φ, U, α) w = [V z PAR ( φ, U,α z ) V z PAR 0 ( φ, U,α z )] (39) z=1 V z PAR ( φ, U,α z ) = 1 2X s (U 2 + U2 l 2U U l cos( φ l + α z )) (40) α = [α 1,...,α w ] T (41) where similarly, V z PAR 0 ( φ, U,α z ) is a constant defined by the value V z PAR ( φ, U,α z ) in the first calculation step after the change of α. The value of V z PAR 0 ( φ, U,α z ) in the first section is equal to zero. It should be noted at this stage since this is a very important point that if an instantaneous change in a PST controllable parameter occurs, all the system parameters, except the generators rotor angles and speeds experience an instantaneous change, i.e., the voltages and the angles of the system. Therefore, the potential parts of the energy function V p1 ( φ, V) and V p2 ( φ) also need to be subjected to the same procedure as V PAR and V QBT in (34) and (39). If the change in the controllable parameters is continuous then the potential-energy parts have to be put through numerical integration. For all the presented energy functions it can be shown that they satisfy all the conditions for a Lyapunov function. In fact, it can be done in the same way as it was in [12]. 4. Numerical examples A SMIB test system is a very simple test that shows if an energy function properly represents a FACTS s effect on transient stability. Namely, the results of a direct method must match the simulation results, if the energy function is good assuming that a generator is represented by a classical model. This is due to the fact that the critical energy of an SMIB system is uniformly given. Therefore, we have first checed the proposed energy functions on a longitudinal test system presented in Fig. 5, and only then on an IEEE nine bus test system (Fig. 6). For the purposes of the time-domain simulations the PSTs were modeled by means Fig. 6. An SSSC in an IEEE nine-bus test system. of the mathematical expressions (6) (11) in both test systems. The longitudinal system is the same as in [9], consisting of a classical generator connected to the infinite bus through two sections of parallel 500-V power lines. A PST of 265 MVA is placed in the middle of the system, comprising a series transformer with a short-circuit voltage u = 3.75%. The pre-fault and fault-on PST s angle α is set to 0 and its tap change is introduced only with the short-circuit elimination (instantaneously). Lossless lines are sectioned in 100- m Π segments and lumped together. The generator is presented as a classical model with an initial voltage 1 p.u. at 30. Additional data that were used can be found in the Appendix A. The CCT for the fault near BUS1 was obtained using a step-by-step method, and directly with the use of the proposed energy functions (33) and (38). The results for both PST types and different angles α are presented in Table 1. Table 1 CCTs obtained in a longitudinal system α ( ) PAR QBT (orientation 1) Simulation Direct method Simulation Direct method CCT (ms)

7 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) Table 2 CCTs using PSTs with α change at δ = π/2 α ( ) PAR QBT (orientation 1) Simulation Direct method Simulation Direct method CCT (ms) ± ± ± ± We made an assumption that the fault is cleared without linetripping. The active power load was modelled as a constant admittance and evaluated numerically for the purposes of the direct method. As one can see, the results are within the range of a time step, and hence they prove that the corresponding energy functions are appropriate. If, for instance, we were to use energy function (25), which does not tae PSTs into consideration, the first result obtained by a direct method in Table 1 would be 31 ms smaller, which represents an error of 21%. By applying a simple, yet efficient, transient augmentation we can increase the CCT. This can be done by changing the angle α during the first-swing angle propagation. The results in Table 2 show the effect of changing the α sign from positive to negative at a generator angle δ = π/2. Again, we can see that the results are within the range of a time step, and hence they prove that the energy functions (34) and (39) are appropriate. We can also see the improvement in transient stability, as the resulting CCTs in Table 2 are higher than those in Table 1. It is also interesting that the results from the QBT do not differ from those of the PAR, which implies that both devices have a similar effect for small α ranges. This also corresponds with findings [4] and [10], that near the electrical middle of a longitudinal system a PAR and a QBT have a similar effect on the transmission characteristics and that the QBT s orientation plays only a minor role. The second test system was an IEEE nine-bus machine system [14], in which we inserted a PAR and a QBT simultaneously, as shown in Fig. 6. Again, the generators were presented with the classical model, the loads were modeled as constant admittances and the lines were considered lossless. Table 3 CCTs obtained in an IEEE nine bus system β ( ) Simulation Direct method CCT (ms) PSTs, rated at 100 MW and u = 3.75%, are connected and functional all the time. They share the same constant α value. In multi-machine systems, the critical energy is not uniformly given, but only estimated. This leads to the deviations of the results of the PEBS direct method presented in Table 3. Nevertheless, the results are good and prove the usefulness of the proposed energy functions. If we were to use, for instance, the energy function of a system without PAR and QBT energy functions, the first result in Table 3 obtained by a direct method would be 145 ms, which is a very conservative and misleading result. By applying the proposed PSTs energy functions in a direct method over a simplified networ their correctness is proved but the true essence lies in application of real power systems where direct methods have to cope with power system modelled and simulated in detail. As any other approximation to the reality direct methods have a certain degree of compromise to deal with and their downside is usually the accuracy. Proposed energy functions were not applied in real-lie environment because many direct method approaches [1] are possible and their comparison is out of scope. Some of them are discussed in [15,16]. Currently, also hybrid approaches are very popular, because they combine the complementary characteristics of time-domain simulations and direct methods accuracy and speed. 5. Conclusions So far, with some exceptions, direct methods were not able to properly assess the transient stability of general power systems with PSTs because there were no energy functions available that would include their action in a post-fault situation. In this paper we propose two additional potential-energy function parts for the two most common PSTs that can be added to any existing structure-preserving energy functions. In this way, modularity is achieved and other FACTS devices can easily be taen into consideration. Using the proposed energy functions it is possible to apply direct methods to any system that consists of an arbitrary number and type of PSTs, which are either PAR or QBT. These new energy functions assume constant PST s controllable parameter; however, we extended the use of the presented energy functions by a simple procedure (modified PEBS) that maes the direct method possible for considering any parameter change in a post-fault system. The proposed energy-function extensions were tested in a longitudinal system, because only in such a system do direct methods and the simulation give the same results. A further demonstration was made with an IEEE nine-bus test system using a potential-energy boundary-surface direct method. A true challenge lies in comparison of existing direct methods and energy functions of modern power systems meshed with FACTS devices and HVDCs.

8 330 A. Gabrijel, B. Mihalic / Electric Power Systems Research 74 (2005) Appendix A Longitudinal test system data Generator data Lines data P n = 1500 MVA x d =45 x = 0.33 /m T m = 6.6 s c = 12 nf/m References [1] M.A. Pai, Energy Function Analysis for Power System Stability, Kluwer Academic Publishers, [2] M. Pavella, P.G. Murthy, Transient Stability of Power Systems, Wiley, [3] M. Pavella, D. Ernst, D. Ruiz-Vega, Transient Stability of Power Systems. A Unified Approach to Assessment and Control, Kluwer Academic Publishers, [4] Y.H. Song, A.T. Johns (Eds.), Flexible ac Transmission Systems (FACTS), IEE Power and Energy Series, vol. 30, IEE, Corp., London, [5] A. Bergen, D.J. Hill, A structure preserving model for power system stability analysis, IEEE Trans. Power Apparatus Syst. PAS-100 (1) (1981). [6] N.A. Tsolas, A. Arapostathis, P.P. Varaiya, A structure preserving energy function for power system transient stability analysis, IEEE Trans. Circuits Syst. Cas-32 (10) (1985). [7] I.A. Hisens, D.J. Hill, Incorporation of SVCs into energy function methods, Trans. Power Syst. 7 (1) (1992) [8] R. Mihalič, U. Gabrijel, Transient stability assessment of systems comprising phase-shifting FACTS devices by direct methods, Int. J. Electrical Power Energy Syst. 26 (6) (2004). [9] R. Mihalič, U. Gabrijel, A structure-preserving energy function for a static series synchronous compensator, IEEE Trans. Power Syst., submitted for publication. [10] R. Mihalič, P. Žuno, Phase-shifting transformer with fixed phase between terminal voltage and voltage boost tool for transient stability margin enhancement, IEE Proc. Gen. Transm. Distrib. 142 (3) (1995). [11] P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, Prentice Hall, [12] Th. Van Cutsem, M. Ribbens-Pavella, Structure preserving direct methods for transient stability analysis of power systems, in: Proceedings of 24th Conference on Decision and Control, Ft. Lauderdale, FL, December, 1985, pp [13] K.R. Padiyar, K.K. Ghosh, Direct stability evaluation of power systems with detailed generator models using structure-preserving energy functions, Int. J. Electrical Power Energy Syst. 11 (1) (1989) [14] P.M. Anderson, A.A. Fouad, Power system control and stability, IEEE Press Power Systems Engineering Series, Revised Printing, [15] M. Ribbens-Pavella, F.J. Evans, Direct methods for studying dynamics of large-scale electric power systems a survey, Automatica 21 (1) (1985) [16] B. Toumi, R. Dhifaoui, Th. Van Cutsem, M. Ribbens-Pavella, Fast transient stability assessment revisited, IEEE Trans. Power Syst. PWRS-1 (2) (1986).