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Energy function for an interline power-flow controller
Electric Power Systems Research 79 (2009) 945–952
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Energy function for an interline power-flow controller V. Azbe ∗ , R. Mihalic Faculty of Electrical Engineering, University of Ljubljana, Trzaska 25, 1000 Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 10 March 2008 Received in revised form 8 July 2008 Accepted 16 December 2008 Keywords: FACTS devices IPFC Lyapunov energy function Power system control Power system stability
a b s t r a c t An IPFC may be applied for steady-state power-flow and voltage control as well as for mastering dynamic phenomena like transient-stability margin enhancement, oscillation damping, etc. For these tasks the Lyapunov energy-function approach is frequently used as a convenient way to control or analyze the electric-power system (EPS). The basis for the implementation of such an approach is to know the energy function of the EPS. Currently, this is not possible for the EPSs that include IPFCs, because the alreadyknown energy functions that proved to be suitable for an EPS do not include such a device. Therefore, in this paper, energy functions that consider the IPFC’s action in the form of a supplement to the alreadyknown structure-preserving energy functions were constructed. They are based on a structure-preserving frame and can be applied for an arbitrary number of IPFCs, which may consist of an arbitrary number of series branches. The developed energy functions were applied for a transient-stability assessment using the Lyapunov direct method, and they proved to be adequate. © 2008 Elsevier B.V. All rights reserved.
1. Introduction With the increased importance of online dynamic security assessment the Lyapunov direct method might be applied to avoid the time-consuming repetition of solving a system’s nonlinear differential equations in a step-by-step manner. However, to apply this direct method one has to have the proper Lyapunov function for the electric-power system (EPS). Although many different Lyapunov functions have been constructed for a system without FACTS devices, the function obtained by integrating a system’s swing equations and thus representing the sum of the kinetic and potential energies has provided the best results [1]. With the introduction of FACTS devices to the EPS these Lyapunov energy functions have to be supplemented in order to assure the proper use of direct methods. Furthermore, these energy functions can be applied in control strategies for transient-stability enhancement, oscillation damping, etc. In this paper the Lyapunov energy functions for an EPS with IPFCs were constructed. They are based on a structure-preserving frame that allows more realistic representations of the power-system components, especially the behavior of the loads. In a structurepreserving energy function (SPEF) a FACTS device is considered as an additional term to the SPEF for the EPS without FACTS devices. Therefore, various FACTS devices can be simultaneously considered in a SPEF at different points of the EPS.
A FACTS device with two or more power converters that simultaneously provides a series compensation for two or more lines was firstly described by Gyugyi et al. [2] and referred to as an “Interline Power Flow Controller (IPFC)”. Some authors, e.g. ref. [3], use this term for a combination of multiple series and at least one parallel branch. However, this kind of FACTS device, that includes parallel compensation, authors, e.g., in refs. [4,5], denote as a generalized unified power-flow controller (GUPFC). After 1999 a number of articles were published [2–11] describing various applications of an IPFC. In refs. [6–9] control schemes for steady-state power-flow control are proposed. In refs. [10,11] the effectiveness of an IPFC is compared with other multiconverter FACTS devices. Calculations based on the energy function that consider the IPFC’s dynamic behavior—for example, a transientstability assessment or control strategies for a transient-stability improvement or oscillation damping—have not been proposed so far because the energy function for an EPS has not been known. The energy function presented in this paper was applied for a transientstability assessment using the Lyapunov direct method. The paper is organized as follows: Section 2 describes the basic characteristics of the IPFC’s operation and the injection model. Section 3 describes the construction of a new Lyapunov energy function for the EPS comprising IPFCs. Section 4 presents the numerical examples of the Lyapunov direct method for a transient-stability assessment that can be used as a verification of adequacy for an IPFC’s energy function. Section 5 draws the conclusions. 2. Operating characteristics of the IPFC
∗ Corresponding author. Tel.: +386 1 4768 415; fax: +386 1 4768 289. E-mail address: [email protected] (V. Azbe). 0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.12.003
For the readers’ convenience the basic IPFC features relevant to our derivations, which are summarized from refs. [2,6–9], are
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Fig. 3. Single IPFC’s branch control area in the P–Q diagram. Fig. 1. Model of an IPFC consisting of n-series branches.
briefly described first. Later in this section some new aspects of IPFCs are described that are important when the limitations of such a device are taken into consideration. It should be pointed out that an IPFC is not an all-powerful device and there are some limitations (i.e., operation areas that cannot be achieved and depend on the IPFC’s operating point) that have to be considered either in the simulation or the control definition. 2.1. General consideration An IPFC is a FACTS device that compensates two or more lines simultaneously; into each compensated line it inserts a series voltage by applying a voltage-source converter (VSC). All VSCs are connected to a DC link, which includes a condenser. In this way active power can be exchanged between the compensated lines. Fig. 1 shows a model of an IPFC for n compensated lines. The functionality of an IPFC can be described by a single line with one IPFC branch included. Fig. 2 shows (a) the functional scheme and (b) the phasor diagram of such a line. The reactance XTRS is the sum of the reactances of the IPFC’s series transformer and the reactance of a line. The system is assumed to be lossless. It should be noted that the magnitude of the series-injected voltage UT is independent of the magnitude of the bus voltage Ui , as it is generated by a VSC. Therefore, the control parameters of each IPFC’s branch are the magnitude and the angle of the series-injected voltage, i.e., UT
and ϕT . Of course, by changing these two parameters other system parameters could be controlled, such as Ui and Pi or Pi and Qj , etc. The reactive power QT , according to Fig. 2, is provided by a VSC, while PT represents the active-power injection of a series transformer that has to be provided via a DC circuit from other branches. In reality, in an IPFC consisting of n branches, 2n − 1 parameters can be independently controlled, as one parameter has to provide the active-power balance of a device. In the phasor diagram in Fig. 2(b) the lines inside the circle represent “voltage-compensation lines” [2], i.e., the lines of constant active-power injection PT . If the voltage UT is controlled so as to keep its end on this line, the active power PT injected by the VSC is constant. It can be proved that these lines are parallel to the line connecting the ends of the voltages Ui and Uj . The circle in Fig. 2(b) represents the control area of the IPFC’s series branch that is limited by the maximum voltage magnitude UTmax . It is possible to translate this control area into a P–Q diagram. Like in the phasor diagram, the control area is a circle with parallel lines of constant active power injection PT , as presented in Fig. 3. The line with PT = 0 is oriented toward the origin of the P–Q diagram. Depending on the angle ij the center of this circle moves along on the ij -curve. The proof that the control area is a circle is presented later in this section. The expressions for the active and reactive powers as functions of the system and the IPFC’s controllable parameters for a single compensated line according to Fig. 2 can be obtained after some algebraic calculations. They are as follows: Pi = Qi = Pj = Qi =
Ui · Uj · sin(ij ) + Ui · UT · sin(ϕT ) XTRS Ui2 · Ui · Uj · cos(ij ) + Ui · UT · sin(ϕT ) XTRS Ui · Uj · sin(ij ) + Uj · UT · sin(ij + ϕT ) XTRS −Uj2 · Ui · Uj · cos(ij ) + Uj · UT · cos(ij + ϕT ) XTRS
PT = − QT =
UT · (Ui · sin(ϕT ) − Uj · sin(ij + ϕT )) XTRS
UT · (UT + Ui · cos(ϕT1 ) − Uj · cos(ij + ϕT )) XTRS
(1)
(2) (3)
(4) (5) (6)
Let us denote the number of series branches of an IPFC by n. The sum of active powers injected into all the series branches must be (when losses are neglected) equal to zero. n
PTi = 0
(7)
i=1
Fig. 2. Single compensated line: (a) functional scheme; (b) phasor diagram.
This restriction requires additional care with the IPFC’s control actions. The consequences of this power-balance requirement are presented in the following Section 2.2.
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In order to describe the control area of the IPFC’s series branch in a P–Q diagram, let us denote the general equations for the circle in the Cartesian “x–y” coordinate system: x = x1 + r · cos(˛)
(8)
y = y1 + r · sin(˛)
(9)
The point (x1 ,y1 ) is the origin of the circle and r is the radius of the circle. Considering (3) and (4) it is obvious that Pj and Qj represent equations of a circle in the P–Q diagram with the following parameters: x1 = y1 = r=
Ui · Uj · sin(ij )
(10)
XTRS −Uj2 + Ui · Uj · cos(ij )
(11)
XTRS
Uj · UT
˛=
(12)
XTRS − (ij + ϕT ) 2
(13)
2.2. Consequences of the power-balance requirement The control parameters UT and ϕT of a branch that compensates the active power injections of other IPFC’s branches can be calculated from (1) to (6). After a few algebraic transformations we can denote UT and ϕT as: PT · XTRS UT = − Ui · sin(ϕT ) − Uj · sin(ij + ϕT )
ϕT = 2 · arctg
−UT · (Ui − Uj · cos(ij )) ±
(14) √ A
PT · XTRS + UT · Uj · sin(ij )
(15)
where A in (15) stands for: 2 A = UT2 · (Ui2 + Uj2 − 2 · Ui · Uj · cos(ij )) − PT2 · XTRS
(16)
and PT is the injected active power necessary to meet (7). In order to be able to set the system’s and the IPFC’s operation point according to (1)–(6), A in (15) has to be positive (it is under a square root)—otherwise no solution is possible. Consequently, according to (16), the active power PT is limited by: PT ≤
UT · U XTRS
(17)
where U denotes the magnitude of the voltage difference between the phasors Ui and Uj . Therefore, if the angle ij is too small, the IPFC’s branch cannot compensate the required active power PT . This restriction is presented in Fig. 4, which presents the control area of an IPFC’s branch in the P– ij diagram when constant PT is considered, and is valid for voltage magnitudes Ui = Uj = 1 p.u., UT = 0.25 p.u. and PT = 0.1 p.u. The p.u. basis for the active power P in Fig. 4 represents the maximum of the transmission characteristic without an IPFC (UT is set to 0). In Fig. 4, the curves PTMAX and PTMIN represent the maximum and minimum active power PT that can be retrieved from the IPFC’s branch, while the curves PjMAX and PjMiN represent the maximum and minimum active power Pj that can be transferred between the nodes i and j. The dashed curve Pj0 represents the active power Pj when the IPFC’s branch is inactive, i.e., UT is set to 0. As can be seen from Fig. 4, the control area is defined only in the area where PTMAX is higher than the required PT . In the area of small angles ij where PTMAX is less than PT there is a prohibited area where the IPFC’s branch cannot provide the required PT . Consequently, the control of many branches of the IPFC operating at various angles ij should consider the limitations of the maximum possible PT in
Fig. 4. Control area of an IPFC’s branch in a P– ij diagram when constant PT is considered.
the branches that are responsible for maintaining the active power balance between all the IPFC’s branches. It is obvious that at the point Pj = Qj = ij = 0 it is not possible to transmit real power, PT , in or out of the IPFC’s branch and consequently it acts as an SSSC, as described in ref. [12]. From observing the IPFC’s branch control area in a P–Q diagram for various angles ij it is clear that the distance between the parallel lines of constant active-power injection PT that represents the same increment of PT is not constant, but decreases with the increasing angle ij , as presented in Fig. 5. 2.3. Injection model of an IPFC The functionality of the IPFC can be represented by two VSCs that compensate two lines. For further calculations let us denote the parameters of each branch with an additional index, i.e., 1 for the first branch and 2 for the second branch. The voltages at the line ends Ui1 and Ui2 are connected to a common node, and thus they are equal and can be denoted as Ui , while the voltages Uj1 and Uj2 are different. In order to be able to construct the Lyapunov structurepreserving energy function of an IPFC, an IPFC should be represented by an injection model. The injection model for each branch of an IPFC is similar to the general SSSC injection model presented in ref. [13]. By connecting both branches to a common DC circuit, an IPFC can be represented by an injection model, shown in Fig. 6. Since the control parameters of an IPFC are the magnitude and the angle of the series injected voltage in each branch, i.e., UT1 , ϕT1 , UT2 and ϕT2 —although they are not independent, as (7) has to be fulfilled—the active and the reactive power injections of the equivalent injection model according to Fig. 6 can be determined. These power injections of the injection model are the basis for the development of the energy function of an IPFC. Psi1 =
Ui UT1 sin(ϕT1 ) XTRS1
(18)
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3.1. Construction of the Lyapunov energy function The energy function on the structure-preserving frame for the EPS without FACTS devices was developed in the 1980s. More details about its construction can be found in refs. [14–16]. For further use let us denote it as Vwithout FACTS . The energy function for an IPFC is constructed as an extension to the Vwithout FACTS , which is presented in refs. [14]. The energy function Vwithout FACTS was obtained as the first integral of the modified swing equations. In order to develop the energy function for an IPFC that can be added to Vwithout FACTS , it should be constructed in the same way: as the first integral of the sum of power injections (18)–(25) that are modified in the same way as the swing equations in ref. [14]. This means that the power injections (18)–(25) should be modified as follows: • the active-power injections (18)–(21) are multiplied with the time derivative of the angle of the bus voltage, • the reactive power injections (22)–(25) are divided by the magnitude of the bus voltage and multiplied by the time derivative of the magnitude of the bus voltage. Since no active power is added to the system by an IPFC, the sum of the active-power injections at the beginning of the IPFC branches is equal to the sum of the active-power injections at their end. For an IPFC with two branches this means:
Fig. 5. P–Q diagram for various angles ij .
Psi1 + Psi2 = −(Psj1 + Psj2 ) Psi2 =
Ui UT2 sin(ϕT2 ) XTRS2
Psj1 = − Psj2 = − Qsi1 = Qsi2
Uj1 UT1 XTRS1 Uj2 UT2 XTRS2
sin(ij1 + ϕT1 ) sin(ij2 + ϕT2 )
U UT2 = i cos(ϕT2 ) XTRS2
Qsj1 = − Qsj2 = −
XTRS1 Uj2 UT2 XTRS2
(20)
By multiplying the active-power injections (18)–(21) by the time derivative of the angle of the bus voltage and considering (26), the following expressions are obtained (the time derivative of a variable is denoted as a point above the variable):
(Psi1 + Psi2 ) · ˙ i =
Ui UT1 cos(ϕT1 ) XTRS1
Uj1 UT1
(19)
Uj1 UT1 XTRS1
(21)
Psj1 · ˙ j1 = − (23) Psj2 · ˙ j2 = (24)
cos(ij2 + ϕT2 )
(25)
3. Energy function for an IPFC The energy function for an IPFC is constructed for two series branches and is then expanded for an arbitrary number of branches.
sin(ij1 + ϕT1 )+
Uj2 UT2 XTRS2
sin(ij2 + ϕT2 ) · ˙ i (27)
(22)
cos(ij1 + ϕT1 )
(26)
Uj1 UT1 XTRS1
Uj2 UT2 XTRS2
sin(ij1 + ϕT1 ) · ˙ j1
(28)
sin(ij2 + ϕT2 ) · ˙ j2
(29)
Further reactive-power injections (22)–(25) are divided by the magnitude of the bus voltage and multiplied by the time derivative of the magnitude of the bus voltage: U˙ UT1 U˙ UT2 Qsi1 + Qsi2 U˙ i = i cos(ϕT1 ) + i cos(ϕT2 ) Ui XTRS1 XTRS2 Qsj1 Uj1 Qsj2 Uj2
U˙ j1 = − U˙ j2 = −
U˙ j1 UT1 XTRS1 U˙ j2 UT2 XTRS2
(30)
cos(ij1 + ϕT1 )
(31)
cos(ij2 + ϕT2 )
(32)
The sum of (27)–(32) can be rewritten as: Uj1 · UT1 XTRS1 + −
Fig. 6. Injection model of an IPFC with two series branches.
· sin(ij1 + ϕT1 ) · ˙ ij1 +
Uj2 · UT2 XTRS2
· sin(ij2 + ϕT2 ) · ˙ ij2
U˙ j1 · UT1 U˙ i · UT1 U˙ · UT2 cos(ϕT1 ) + i cos(ϕT2 ) − · cos(ij1 + ϕT1 ) XTRS1 XTRS2 XTRS1 U˙ j2 · UT2 XTRS2
· cos(ij2 + ϕT2 )
(33)
The construction procedure from ref. [14] leads us to (33). The crucial step in the energy-function construction is how to obtain an
V. Azbe, R. Mihalic / Electric Power Systems Research 79 (2009) 945–952
analytical solution of the first integral of (33). There is no uniform procedure to find this integral and it can be—for any kind of application concepts and formulation procedures of deriving an energy function as a first integral—obtained only with some intuition, as already reported in ref. [14], where it was also stressed that there is no uniform procedure that exists to obtain a Lyapunov energy function. To find the first integral of (33), i.e., to obtain a function for which the time derivative is known, we can use the fact that alreadyknown energy functions for some of the FACTS devices are very closely related to the reactive powers injected into the system by these devices [16–22]. The derivations of the energy functions for CSC, SVC, SSSC, STATCOM, PST and UPFC have shown that for devices that from the system’s point of view operate as controllable voltage or current sources, the energy function is equal to the total sum of the reactive-power injections. In contrast, for devices that from the system’s point of view operate as controllable reactances, the energy function is equal to half of the sum of the reactive powers injected into the system by these devices. So we denote the sum of the IPFC’s reactive-power injections as: Qinj = Qsi1 + Qsi2 + Qsj1 + Qsj2
+ − −
U · UT1 Ui · U˙ T1 cos(ϕT1 ) − i sin(ϕT1 ) · ϕ˙ T1 XTRS1 XTRS1
Ui · U˙ T2 U · UT2 cos(ϕT2 ) − i sin(ϕT2 ) · ϕ˙ T2 XTRS2 XTRS2 Uj1 · U˙ T1 XTRS1 Uj2 · U˙ T2 XTRS2
cos(ij1 + ϕT1 ) + cos(ij2 + ϕT2 ) +
(35) can be denoted as:
VIPFC = Qinj +
Uj2 · UT2 Ui · UT2 sin(ϕT2 ) − sin(ij2 + ϕT2 ) dϕT2 XTRS2 XTRS2 (36)
The expression in square brackets of (36) is equal to the active power PT2 and consequently we can rewrite it as:
VIPFC = Qinj +
PT2 dϕT2
(37)
Eq. (37) is the final form of an energy function for an IPFC when the controllable parameters are constant. The integral part of (37) cannot be analytically solved. Although (37) cannot be proved to be a true Lyapunov function (it is path dependent), it can be successfully used in the Lyapunov direct method for transient-stability assessment or the optimum IPFC parameters in the Lyapunov sense can be calculated. For the purposes of the Lyapunov direct method the integral part of (37) is calculated numerically in the same way as, e.g., active loads.
(34)
Now let us find the time derivative of this sum Qinj , and compare it with (33). All the parts of (33) are included in this derivative. The first integral of (33), which represents the energy function for an IPFC, can now be rewritten as: VIPFC = Qinj −
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Uj1 · UT1 XTRS1
Uj2 · UT2 XTRS2
sin(ij1 + ϕT1 ) · ϕ˙ T1
sin(ij2 + ϕT 2 ) · ϕ˙ T2 dt (35)
In order to be able to solve the integral part of (35), the time behavior of the control parameters has to be known, i.e., the control strategy of the IPFC has to be known. While it is obvious that this integral cannot be solved for every type of control strategy, let us try to solve it for two simple control strategies: (a) for constant control parameters and (b) for a constant active power PT between the two series branches. 3.1.1. The energy function for constant control parameters If the rating of the IPFC is considered, then the magnitude UT cannot exceed its maximum possible value UTmax . On the other hand, it can be proved that the “optimum” ϕT does not change a great deal when the first-swing stability is considered and the results—i.e., the critical clearing times—are similar if the constant ϕT is considered. Of course, this statement could be made only after the optimum control strategy has been implemented. In addition, it has been found that the optimum strategy requires the maximum possible UT , i.e., UTmax . So the natural conclusion is to consider the control parameters as being constant. Although this strategy is not optimal, it can be used when sectional-constant parameters are applied, as described in refs. [16]. In this way the control strategy becomes nearly optimized in the Lyapunov sense. Let us, for the further development of an energy function (35), decide that the angle ϕT2 depends on three other parameters—i.e., UT1 , ϕT1 and UT2 , in order to fulfill (7). As UT1 , ϕT1 and UT2 are considered constant (and ϕT2 not) their time derivatives equal 0, and
3.1.2. The energy function for the constant active power PT between the two series branches If the energy function without an integral part is derived and can be proved to be a true Lyapunov function, the voltage magnitudes UT1 and UT2 as well as the active-power exchange between the IPFC branches (i.e., PT ) should be considered constant. While this approach can be justified for UT , as explained in the previous paragraph, constant PT cannot be considered a kind of optimum control. However, this can be overcome by applying sectional constant control, which is based on the assumption that during electromechanical transients the system parameters (voltage magnitudes and angles, rotor angles) do not change abruptly. Furthermore, according to findings described in Section 2.2, when searching for optimum parameters we should avoid the prohibited area presented in Fig. 4. As the derivations of the system dynamics are calculated numerically, it is important that the starting-point of the search does not fall in this prohibited area, because in this case the calculation fails. Because PT defines the border of the prohibited area, its setting makes it possible to choose good starting points when searching for other optimum controllable parameters for the IPFC. As the active-power transfer PT between the two series branches as well as the voltage magnitudes UT1 and UT2 are considered time independent, (35) can be rewritten as:
VIPFC = Qinj +
+
Uj1 · UT1 Ui · UT1 sin(ϕT1 ) − sin(ij1 + ϕT1 ) dϕT1 XTRS1 XTRS1
Uj2 · UT2 Ui · UT2 sin(ϕT2 ) − sin(ij2 + ϕT2 ) dϕT2 XTRS2 XTRS2 (38)
The expressions in the square brackets of (38) are equal to the active power PT1 and PT2 , respectively. Because this active power is assumed to be constant, (38) can be rewritten as: VIPFC = Qinj + PT1 · ϕT1 + PT2 · ϕT2 = Qinj + PT1 · (ϕT1 − ϕT2 )
(39)
Eq. (39) is a true Lyapunov energy function for an IPFC when the active power PT between the two series branches and the controllable parameters UT1 and UT2 are constant. In this case the integral part of (35) was analytically solved. Its adequacy is verified in Section 4. It is interesting to note that the consideration of the activepower exchange between the series branches of an IPFC in the
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Fig. 7. SMIB test system with an IPFC with two series branches.
energy-function construction is the same as the consideration of the active loads presented in ref. [14]. According to this reference, the energy function for the active loads, i.e., the integral, can be analytically solved only if the active loads are considered as a constant active power. In any other case, the integral has to be calculated numerically or an approximate expression (which does not satisfy the Lyapunov conditions) has to be used. The obtained energy functions (37) and (39) act like an extension to any structure-preserving energy function, e.g., Vwithout FACTS . The energy function for the power system with an IPFC is, therefore: V = Vwithout
FACTS
+ VIPFC
(40)
Obviously, the energy function VIPFC can be extended to deal with any number of IPFCs installed in the system. 3.2. Generalization for an arbitrary number of series branches Let n be the number of series branches and let UTk and ϕTk be the voltage magnitude and the voltage angle of the kth branch. The energy function (37) can be generalized for any number of series branches of an IPFC. Considering the parameters UTk and ϕTk to be constant in all branches except the power-balancing branch, where only the injected voltage magnitude UTk is constant, on the basis of (35) it is easy to show that the energy function of such an IPFC is equal to (37). However, the sum of all the reactive power injections Qinj in (37) should include the reactive power injections of all the series branches:
n
Qinj =
[Qsik + Qsjk ]
(41)
k=1
and PT2 and ϕT2 denote the injected-power parameters of the power-balancing branch. In the same way the energy function (39) can be generalized. Since the active power injected by a VSC and the voltage magnitude of a VSC in each series branch are constant, on the basis of (35) and (38) it is easy to show that the generalization of (39) leads to (42): VIPFC = Qinj +
n
PTk · ϕTk
(42)
k=1
Because the critical energy in a SMIB system is uniformly given, the CCTs estimated by the Lyapunov direct method should be the same as that obtained using time-domain simulations and, consequently, the adequacy of the energy functions (37) and (39) can be verified. However, exactly the same system model has to be used in both the simulation and the direct method. For this reason we applied the Mathematica computer program. We are aware that a mathematical tool is not the most appropriate for a simulation of electric-power system dynamics, and the use of, e.g., PSCAD, EMTP or the Netomac program, would have been much easier to implement. However, in order to ensure that the system modeling is absolutely identical in the simulation and when applying direct methods we chose Mathematica. For the purposes of time-domain simulations an IPFC was modeled by means of the mathematical expressions (18)–(25). Additionally, the Lyapunov direct method for transient-stability assessment that uses the IPFC’s energy function was applied on the IEEE 9-bus 3-machine test system. In a multi-machine system the system fault and the post-fault trajectory are no longer uniformly given, and direct methods do not give exactly the same results as the simulation. Such a model cannot serve for validation purposes. However, it can serve to demonstrate the Lyapunov direct method applied to a multi-machine system comprising IPFCs. 4.1. SMIB test system The SMIB test system is presented in Fig. 7. The data for this system can be found in ref. [16]. An IPFC consists of two series branches, each rated at 132.5 MVA and connected to the system through a series transformer with a short-circuit voltage uk = 3.75%. The disturbance is a three-phase short-circuit near BUS1, see Fig. 7, and is assumed to be eventually cleared, i.e., the system’s post-fault configuration is identical to the pre-fault one. The pre-fault and fault-on values of the IPFC’s controllable parameters are set to 0. The CCTs were obtained by repeating the time-domain simulations and applying the direct method with the use of the newly proposed IPFC energy function. In the first numerical example we apply the IPFC energy function (37) that is valid for the constant controllable parameters UT1 , UT2 and ϕT1 . The results for various UT1 and UT2 limits and for various constant values of ϕT1 are presented in Table 1. As one can see, the results are identical, and hence it can be concluded that
4. Verification of adequacy for an IPFC’s energy function In order to verify the adequacy of the derived energy functions (37) and (39), they were applied in the Lyapunov direct method for transient-stability assessment, i.e., for the determination of the critical clearing times (CCTs), on the single-machine infinite-bus (SMIB) test system. For the CCT estimation we used the potentialenergy boundary surface (PEBS) method [15], in which the CCT is the time instant when the total energy of the system along the faulton trajectory equals the maximum of the potential energy along the same fault-on trajectory.
Table 1 CCTs obtained in a SMIB test system including an IPFC with the constant controllable parameters UT1 , UT2 and ϕT1 . UT1 [p.u.]
0 0.1 0.15 0.2 0.25
UT2 [p.u.]
0 0.1 0.15 0.2 0.25
ϕT1 [◦ ]
– 100 100 100 100
CCT [ms] Simulation
Direct method
139 151 156 160 164
139 151 156 160 164
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Table 2 CCTs obtained in a SMIB test system including an IPFC with the constant controllable parameters UT1 , UT2 and PT . UT1 [p.u.]
0.0 0.15 0.15 0.15 0.3 0.3 0.3
UT2 [p.u.]
0.0 0.15 0.15 0.15 0.3 0.3 0.3
PT [p.u.]
0 −0.05 0 0.075 −0.1 0 0.1
CCT [ms] Simulation
Direct method
139 154 156 138 165 168 164
139 154 156 138 165 168 164
Table 3 CCTs obtained in an IEEE 9-bus test system including an IPFC with the constant controllable parameters UT1 , UT2 and ϕT1 . UT1 [p.u.]
UT2 [p.u.]
ϕT1 [◦ ]
CCT [ms] Simulation
Simulation
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0.1 0.1 0.1 0.2 0.2 0.2
– −60 −45 −10 −45 −20 −10
242 245 246 245 248 249 248
244 245 246 246 248 249 249
the energy function (37) is appropriate for such a control strategy. The CCTs gathered in Table 1 are those obtained with the constant controllable parameter ϕT1 set to a value that gives a maximum transient-stability improvement, i.e., maximum CCTs. In the second numerical example of the SMIB test system we apply the IPFC energy function (39) that is valid for the constant controllable parameters UT1 , UT2 and PT . The results are presented in Table 2. Again, the results are identical, and in this way they validate the proposed IPFC energy function (39). 4.2. IEEE 9-bus 3-machine test system In order to demonstrate the application of the newly proposed energy functions (37) and (39) for an IPFC in a multi-machine system, CCTs were obtained using the Lyapunov direct method for the IEEE 9-bus 3-machine test system. Because the trajectory of the machine rotors and hence the critical energy in a multi-machine system is not uniformly given, the CCTs estimated by the Lyapunov direct method are in general different from that obtained using time-domain simulations. However, as can be seen from the results later in this section (Tables 3 and 4), the Lyapunov direct method that uses the newly proposed energy functions (37) and (39) provides a good estimation of the CCTs. The data for the presented system can be found in ref. [23]. The generators were represented by a classical model, the loads were modeled as constant admittances and the lines were considered to be lossless. The disturbance is a three-phase short-circuit near Table 4 CCTs obtained in an IEEE 9-bus test system including an IPFC with the constant controllable parameters UT1 , UT2 and PT . UT1 [p.u.]
UT2 [p.u.]
PT [◦ ]
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0 0.07 −0.07 0 0.1 −0.17
CCT [ms] Simulation
Simulation
242 246 245 246 249 248 249
244 246 245 247 249 247 249
Fig. 8. An IEEE 9-bus test system with an IPFC.
bus 7. An IPFC is included in the system as shown in Fig. 8. Both the IPFC’s branches are rated at 100 MVA and uk = 3.75%. The prefault and fault-on values of the IPFC’s controllable parameters are set to 0. Like in a SMIB test system, the CCTs were obtained using time-domain simulations, and directly with the use of the newly proposed IPFC energy function. In the first numerical example of the IEEE 9-bus test system we applied IPFC energy function (37) that is valid for the constant controllable parameters UT1 , UT2 and ϕT1 . According to the selected UT1 , UT2 and ϕT1 , two values of the angle ϕT2 fulfill the requirement for active power compensation between the IPFC’s branches: the one that injects the inductive reactive power and the one that injects the capacitive reactive power. In this numerical example we considered the injection of capacitive reactive power in the second branch that gives a better transient stability improvement. The results for various UT1 and UT2 limits and for various constant values of ϕT1 are presented in Table 3. The CCTs obtained with the direct method deviate from the reference (time-domain simulation) by no more than 2 ms. In the second numerical example on the IEEE 9-bus test system the IPFC energy function (39) that is valid for constant controllable parameters UT1 , UT2 and PT was applied. According to the selected UT1 , UT2 and PT , two values of the angle ϕT1 and two values of the angle ϕT2 fulfill the requirement for active power compensation between the IPFC’s branches—inductive or capacitive reactive power can be injected into each of the branches. Again, for a better transient-stability improvement the injection of capacitive reactive power into each branch was considered. The results are presented in Table 4. The CCTs obtained with the direct method deviate from the reference by no more than 1 ms, which is equal to the integration time step. Both multi-machine numerical examples show that the Lyapunov direct method using the proposed IPFC’s energy function can be successfully applied in transient-stability assessment. However, with constant controllable parameters it is hard to fulfill the requirement for active power compensation between the IPFC’s two branches for the period of the first swing. As we showed in Section 2, the active power injected into a series branch depends on the voltage angles at the connection points of that branch. Because those angles experience large changes for the period of the first swing, the controllable parameter ϕT1 or PT can only fulfill the IPFC’s power
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compensation between the two branches for the entire period of the first swing in a narrow range. 5. Conclusions An IPFC can be considered the most powerful device for dynamic power flow control in EPS as it is able to control active and/or reactive power flow in each line of a chosen network node. In this sense it may be applied for steady-state power-flow and voltage control, transient-stability margin enhancement, oscillation damping, etc. In order to be able to use a Lyapunov energy-function approach for these tasks, the energy function was constructed for a system that includes IPFCs. In our work the structure-preserving approach has been applied. Another important issue described in this paper is to clearly define the possible operating areas and the limitations of an IPFC. Theoretical considerations have been described that show where such operational limitations are, so now areas with impossible operational states can be avoided when numerical calculations are carried out. In order to verify the adequacy of the derived energy functions, they were applied in the Lyapunov direct method for transientstability assessment, i.e., for the determination of the CCTs, on the SMIB test system. In order to demonstrate the application in a multi-machine system, CCTs were obtained using the Lyapunov direct method for the IEEE 9-bus 3-machine test system. The results validate the proposed IPFC energy functions. References [1] P. Sauer, M.A. Pai, Power System Dynamics and Stability, Prentice Hall, 1998. [2] L. Gyugyi, K.K. Sen, C.D. Schauder, The interline power flow controller concept: a new approach to power flow management in transmission systems, IEEE Trans. Power Deliv. 14 (1999) 1115–1123. [3] S. Mishra, P.K. Dash, P.K. Hota, M. Tripathy, Genetically optimized Neuro-fuzzy IPFC for damping modal oscillations of power system, IEEE Trans. Power Syst. 17 (2002) 1140–1147. [4] B. Fardanesh, B. Spherling, E. Uzunovic, S. Zelingher, Multi-converter FACTS devices: the generalized unified power flow controller (GUPFC), in: IEEE Power Engineering Society Summer Meeting, vol. 2, 2000, pp. 1020–1025. [5] Z. Xiao-Ping, E. Handschin, M. Yao, Modeling of the generalized unified power flow controller (GUPFC) in a nonlinear interior point OPF, IEEE Trans. Power Syst. 16 (2001) 367–373. [6] J. Chen, T.T. Lie, D.M. Vilathgamuwa, Basic control of interline power flow controller, in: IEEE Power Engineering Society Winter Meeting, vol. 1, 2002, pp. 521–525. [7] D. Menniti, A. Pinnarelli, N. Sorrentino, A fuzzy logic controller for interline power flow controller model implemented by ATP-EMTP, in: Proceedings of International Conference on Power System Technology, PowerCon 2002, vol. 3, 2002, pp. 1898–1903. [8] S. Teerathana, A. Yokoyama, An optimal power flow control method of power system using interline power flow controller (IPFC), in: Proceedings of IEEE Region 10 Conference, TENCON 2004, 2004, pp. 343–346.
[9] X. Wei, J.H. Chow, B. Fardanesh, A.A. Edris, A dispatch strategy for an interline power flow controller operating at rated capacity, in: Proceedings of IEEE/PES Power Systems Conference and Exposition 2004, 2004, pp. 1459–1466. [10] B. Fardanesh, Optimal utilization, sizing, and steady-state performance comparison of multiconverter VSC-based FACTS controllers, IEEE Trans. Power Deliv. 19 (2004) 1321–1327. [11] B. Fardanesh, A. Schuff, Dynamic studies of the NYS transmission system with the Marcy CSC in the UPFC and IPFC configurations, in: Proceedings of IEEE/PES Transmission and Distribution Conference and Exposition 2003, vol. 3, 2003, pp. 1126–1130. [12] R. Mihalic, Power flow control with controllable reactive series elements, IEE Proceedings—Generation, Transmission and Distribution 145 (1998) 493–498. [13] M. Noroozian, L. Angquist, G. Ingestrom, Series compensation, in: Y.H. Song, A.T. Johns (Eds.), Flexible ac Transmission Systems (FACTS), IEE, London, U.K., 1999, pp. 199–242. [14] 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, 1985, pp. 70–77. [15] M.A. Pai, Energy Function Analysis For Power System Stability, Kluwer Academic Publishers, 1989. [16] V. Azbe, U. Gabrijel, R. Mihalic, D. Povh, The energy function of a general multimachine system with a unified power flow controller, IEEE Trans. Power Syst. 20 (2005) 1478–1485. [17] U. Gabrijel, R. Mihalic, Direct methods for transient stability assessment in power systems comprising controllable series devices, IEEE Trans. Power Syst. 17 (2002) 1116–1122. [18] U. Gabrijel, R. Mihalic, Transient stability enhancement and assessment of systems comprising controllable series devices, in: Proceedings of 2nd IASTED International Conference on Power and Energy Systems, 2002, pp. 251–258. [19] R. Mihalic, U. Gabrijel, Transient stability assessment of systems comprising phase-shifting FACTS devices by direct methods, International Journal of Electrical Power & Energy Systems 26 (2004) 445–453. [20] R. Mihalic, U. Gabrijel, A structure-preserving energy function for a Static Series Synchronous Compensator, IEEE Trans. Power Syst. 19 (2004) 1501–1507. [21] R. Mihalic, P. Zˇ unko, D. Povh, Improvement of transient stability using unified power flow controller, IEEE Trans. Power Deliv. 11 (1996) 485–492. [22] I.A. Hiskens, D.J. Hill, Incorporation of SVCs into energy function methods, Trans. Power Syst. 7 (1992) 133–140. [23] P.M. Anderson, A.A. Fouad, Power System Control and Stability, IEEE Press Power Systems Engineering Series, Revised Printing, 1999. Valentin Azbe received his B.Sc., M.Sc. and Dr.Sc. degrees from The University of Ljubljana, Slovenia, in 1996, 2003, and 2005, respectively. After receiving his diploma he worked with IBE, consulting engineers, Slovenia, as a project manager in the Department for Overhead-Lines design. In 2000 he joined the Department of Power Systems and Devices at the Faculty of Electrical Engineering, The University of Ljubljana, where he has since worked as a junior researcher. In 2005 he became a Teaching Assistant. His areas of interest include system analysis, FACTS devices, power-system protection and DC power-system analysis. Rafael Mihalic received the Dipl. Eng., M.Sc. and Dr.Sc. degrees from The University of Ljubljana, Ljubljana, Slovenia, in 1986, 1989, and 1993, respectively. He became a Teaching Assistant in the Department of Power Systems and Devices, Faculty for Electrical and Computer Engineering, The University of Ljubljana, in 1986. Between 1988 and 1991, he was a member of the Siemens Power Transmission and Distribution Group, Erlangen, Germany. Since 2005 he has been a Professor at the University of Ljubljana. His areas of interest include system analysis and FACTS devices. Prof. Mihalic is a member of Cigre (Paris, France).
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Energy function for an interline power-flow controller V. Azbe ∗ , R. Mihalic Faculty of Electrical Engineering, University of Ljubljana, Trzaska 25, 1000 Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 10 March 2008 Received in revised form 8 July 2008 Accepted 16 December 2008 Keywords: FACTS devices IPFC Lyapunov energy function Power system control Power system stability
a b s t r a c t An IPFC may be applied for steady-state power-flow and voltage control as well as for mastering dynamic phenomena like transient-stability margin enhancement, oscillation damping, etc. For these tasks the Lyapunov energy-function approach is frequently used as a convenient way to control or analyze the electric-power system (EPS). The basis for the implementation of such an approach is to know the energy function of the EPS. Currently, this is not possible for the EPSs that include IPFCs, because the alreadyknown energy functions that proved to be suitable for an EPS do not include such a device. Therefore, in this paper, energy functions that consider the IPFC’s action in the form of a supplement to the alreadyknown structure-preserving energy functions were constructed. They are based on a structure-preserving frame and can be applied for an arbitrary number of IPFCs, which may consist of an arbitrary number of series branches. The developed energy functions were applied for a transient-stability assessment using the Lyapunov direct method, and they proved to be adequate. © 2008 Elsevier B.V. All rights reserved.
1. Introduction With the increased importance of online dynamic security assessment the Lyapunov direct method might be applied to avoid the time-consuming repetition of solving a system’s nonlinear differential equations in a step-by-step manner. However, to apply this direct method one has to have the proper Lyapunov function for the electric-power system (EPS). Although many different Lyapunov functions have been constructed for a system without FACTS devices, the function obtained by integrating a system’s swing equations and thus representing the sum of the kinetic and potential energies has provided the best results [1]. With the introduction of FACTS devices to the EPS these Lyapunov energy functions have to be supplemented in order to assure the proper use of direct methods. Furthermore, these energy functions can be applied in control strategies for transient-stability enhancement, oscillation damping, etc. In this paper the Lyapunov energy functions for an EPS with IPFCs were constructed. They are based on a structure-preserving frame that allows more realistic representations of the power-system components, especially the behavior of the loads. In a structurepreserving energy function (SPEF) a FACTS device is considered as an additional term to the SPEF for the EPS without FACTS devices. Therefore, various FACTS devices can be simultaneously considered in a SPEF at different points of the EPS.
A FACTS device with two or more power converters that simultaneously provides a series compensation for two or more lines was firstly described by Gyugyi et al. [2] and referred to as an “Interline Power Flow Controller (IPFC)”. Some authors, e.g. ref. [3], use this term for a combination of multiple series and at least one parallel branch. However, this kind of FACTS device, that includes parallel compensation, authors, e.g., in refs. [4,5], denote as a generalized unified power-flow controller (GUPFC). After 1999 a number of articles were published [2–11] describing various applications of an IPFC. In refs. [6–9] control schemes for steady-state power-flow control are proposed. In refs. [10,11] the effectiveness of an IPFC is compared with other multiconverter FACTS devices. Calculations based on the energy function that consider the IPFC’s dynamic behavior—for example, a transientstability assessment or control strategies for a transient-stability improvement or oscillation damping—have not been proposed so far because the energy function for an EPS has not been known. The energy function presented in this paper was applied for a transientstability assessment using the Lyapunov direct method. The paper is organized as follows: Section 2 describes the basic characteristics of the IPFC’s operation and the injection model. Section 3 describes the construction of a new Lyapunov energy function for the EPS comprising IPFCs. Section 4 presents the numerical examples of the Lyapunov direct method for a transient-stability assessment that can be used as a verification of adequacy for an IPFC’s energy function. Section 5 draws the conclusions. 2. Operating characteristics of the IPFC
∗ Corresponding author. Tel.: +386 1 4768 415; fax: +386 1 4768 289. E-mail address: [email protected] (V. Azbe). 0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.12.003
For the readers’ convenience the basic IPFC features relevant to our derivations, which are summarized from refs. [2,6–9], are
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Fig. 3. Single IPFC’s branch control area in the P–Q diagram. Fig. 1. Model of an IPFC consisting of n-series branches.
briefly described first. Later in this section some new aspects of IPFCs are described that are important when the limitations of such a device are taken into consideration. It should be pointed out that an IPFC is not an all-powerful device and there are some limitations (i.e., operation areas that cannot be achieved and depend on the IPFC’s operating point) that have to be considered either in the simulation or the control definition. 2.1. General consideration An IPFC is a FACTS device that compensates two or more lines simultaneously; into each compensated line it inserts a series voltage by applying a voltage-source converter (VSC). All VSCs are connected to a DC link, which includes a condenser. In this way active power can be exchanged between the compensated lines. Fig. 1 shows a model of an IPFC for n compensated lines. The functionality of an IPFC can be described by a single line with one IPFC branch included. Fig. 2 shows (a) the functional scheme and (b) the phasor diagram of such a line. The reactance XTRS is the sum of the reactances of the IPFC’s series transformer and the reactance of a line. The system is assumed to be lossless. It should be noted that the magnitude of the series-injected voltage UT is independent of the magnitude of the bus voltage Ui , as it is generated by a VSC. Therefore, the control parameters of each IPFC’s branch are the magnitude and the angle of the series-injected voltage, i.e., UT
and ϕT . Of course, by changing these two parameters other system parameters could be controlled, such as Ui and Pi or Pi and Qj , etc. The reactive power QT , according to Fig. 2, is provided by a VSC, while PT represents the active-power injection of a series transformer that has to be provided via a DC circuit from other branches. In reality, in an IPFC consisting of n branches, 2n − 1 parameters can be independently controlled, as one parameter has to provide the active-power balance of a device. In the phasor diagram in Fig. 2(b) the lines inside the circle represent “voltage-compensation lines” [2], i.e., the lines of constant active-power injection PT . If the voltage UT is controlled so as to keep its end on this line, the active power PT injected by the VSC is constant. It can be proved that these lines are parallel to the line connecting the ends of the voltages Ui and Uj . The circle in Fig. 2(b) represents the control area of the IPFC’s series branch that is limited by the maximum voltage magnitude UTmax . It is possible to translate this control area into a P–Q diagram. Like in the phasor diagram, the control area is a circle with parallel lines of constant active power injection PT , as presented in Fig. 3. The line with PT = 0 is oriented toward the origin of the P–Q diagram. Depending on the angle ij the center of this circle moves along on the ij -curve. The proof that the control area is a circle is presented later in this section. The expressions for the active and reactive powers as functions of the system and the IPFC’s controllable parameters for a single compensated line according to Fig. 2 can be obtained after some algebraic calculations. They are as follows: Pi = Qi = Pj = Qi =
Ui · Uj · sin(ij ) + Ui · UT · sin(ϕT ) XTRS Ui2 · Ui · Uj · cos(ij ) + Ui · UT · sin(ϕT ) XTRS Ui · Uj · sin(ij ) + Uj · UT · sin(ij + ϕT ) XTRS −Uj2 · Ui · Uj · cos(ij ) + Uj · UT · cos(ij + ϕT ) XTRS
PT = − QT =
UT · (Ui · sin(ϕT ) − Uj · sin(ij + ϕT )) XTRS
UT · (UT + Ui · cos(ϕT1 ) − Uj · cos(ij + ϕT )) XTRS
(1)
(2) (3)
(4) (5) (6)
Let us denote the number of series branches of an IPFC by n. The sum of active powers injected into all the series branches must be (when losses are neglected) equal to zero. n
PTi = 0
(7)
i=1
Fig. 2. Single compensated line: (a) functional scheme; (b) phasor diagram.
This restriction requires additional care with the IPFC’s control actions. The consequences of this power-balance requirement are presented in the following Section 2.2.
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In order to describe the control area of the IPFC’s series branch in a P–Q diagram, let us denote the general equations for the circle in the Cartesian “x–y” coordinate system: x = x1 + r · cos(˛)
(8)
y = y1 + r · sin(˛)
(9)
The point (x1 ,y1 ) is the origin of the circle and r is the radius of the circle. Considering (3) and (4) it is obvious that Pj and Qj represent equations of a circle in the P–Q diagram with the following parameters: x1 = y1 = r=
Ui · Uj · sin(ij )
(10)
XTRS −Uj2 + Ui · Uj · cos(ij )
(11)
XTRS
Uj · UT
˛=
(12)
XTRS − (ij + ϕT ) 2
(13)
2.2. Consequences of the power-balance requirement The control parameters UT and ϕT of a branch that compensates the active power injections of other IPFC’s branches can be calculated from (1) to (6). After a few algebraic transformations we can denote UT and ϕT as: PT · XTRS UT = − Ui · sin(ϕT ) − Uj · sin(ij + ϕT )
ϕT = 2 · arctg
−UT · (Ui − Uj · cos(ij )) ±
(14) √ A
PT · XTRS + UT · Uj · sin(ij )
(15)
where A in (15) stands for: 2 A = UT2 · (Ui2 + Uj2 − 2 · Ui · Uj · cos(ij )) − PT2 · XTRS
(16)
and PT is the injected active power necessary to meet (7). In order to be able to set the system’s and the IPFC’s operation point according to (1)–(6), A in (15) has to be positive (it is under a square root)—otherwise no solution is possible. Consequently, according to (16), the active power PT is limited by: PT ≤
UT · U XTRS
(17)
where U denotes the magnitude of the voltage difference between the phasors Ui and Uj . Therefore, if the angle ij is too small, the IPFC’s branch cannot compensate the required active power PT . This restriction is presented in Fig. 4, which presents the control area of an IPFC’s branch in the P– ij diagram when constant PT is considered, and is valid for voltage magnitudes Ui = Uj = 1 p.u., UT = 0.25 p.u. and PT = 0.1 p.u. The p.u. basis for the active power P in Fig. 4 represents the maximum of the transmission characteristic without an IPFC (UT is set to 0). In Fig. 4, the curves PTMAX and PTMIN represent the maximum and minimum active power PT that can be retrieved from the IPFC’s branch, while the curves PjMAX and PjMiN represent the maximum and minimum active power Pj that can be transferred between the nodes i and j. The dashed curve Pj0 represents the active power Pj when the IPFC’s branch is inactive, i.e., UT is set to 0. As can be seen from Fig. 4, the control area is defined only in the area where PTMAX is higher than the required PT . In the area of small angles ij where PTMAX is less than PT there is a prohibited area where the IPFC’s branch cannot provide the required PT . Consequently, the control of many branches of the IPFC operating at various angles ij should consider the limitations of the maximum possible PT in
Fig. 4. Control area of an IPFC’s branch in a P– ij diagram when constant PT is considered.
the branches that are responsible for maintaining the active power balance between all the IPFC’s branches. It is obvious that at the point Pj = Qj = ij = 0 it is not possible to transmit real power, PT , in or out of the IPFC’s branch and consequently it acts as an SSSC, as described in ref. [12]. From observing the IPFC’s branch control area in a P–Q diagram for various angles ij it is clear that the distance between the parallel lines of constant active-power injection PT that represents the same increment of PT is not constant, but decreases with the increasing angle ij , as presented in Fig. 5. 2.3. Injection model of an IPFC The functionality of the IPFC can be represented by two VSCs that compensate two lines. For further calculations let us denote the parameters of each branch with an additional index, i.e., 1 for the first branch and 2 for the second branch. The voltages at the line ends Ui1 and Ui2 are connected to a common node, and thus they are equal and can be denoted as Ui , while the voltages Uj1 and Uj2 are different. In order to be able to construct the Lyapunov structurepreserving energy function of an IPFC, an IPFC should be represented by an injection model. The injection model for each branch of an IPFC is similar to the general SSSC injection model presented in ref. [13]. By connecting both branches to a common DC circuit, an IPFC can be represented by an injection model, shown in Fig. 6. Since the control parameters of an IPFC are the magnitude and the angle of the series injected voltage in each branch, i.e., UT1 , ϕT1 , UT2 and ϕT2 —although they are not independent, as (7) has to be fulfilled—the active and the reactive power injections of the equivalent injection model according to Fig. 6 can be determined. These power injections of the injection model are the basis for the development of the energy function of an IPFC. Psi1 =
Ui UT1 sin(ϕT1 ) XTRS1
(18)
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3.1. Construction of the Lyapunov energy function The energy function on the structure-preserving frame for the EPS without FACTS devices was developed in the 1980s. More details about its construction can be found in refs. [14–16]. For further use let us denote it as Vwithout FACTS . The energy function for an IPFC is constructed as an extension to the Vwithout FACTS , which is presented in refs. [14]. The energy function Vwithout FACTS was obtained as the first integral of the modified swing equations. In order to develop the energy function for an IPFC that can be added to Vwithout FACTS , it should be constructed in the same way: as the first integral of the sum of power injections (18)–(25) that are modified in the same way as the swing equations in ref. [14]. This means that the power injections (18)–(25) should be modified as follows: • the active-power injections (18)–(21) are multiplied with the time derivative of the angle of the bus voltage, • the reactive power injections (22)–(25) are divided by the magnitude of the bus voltage and multiplied by the time derivative of the magnitude of the bus voltage. Since no active power is added to the system by an IPFC, the sum of the active-power injections at the beginning of the IPFC branches is equal to the sum of the active-power injections at their end. For an IPFC with two branches this means:
Fig. 5. P–Q diagram for various angles ij .
Psi1 + Psi2 = −(Psj1 + Psj2 ) Psi2 =
Ui UT2 sin(ϕT2 ) XTRS2
Psj1 = − Psj2 = − Qsi1 = Qsi2
Uj1 UT1 XTRS1 Uj2 UT2 XTRS2
sin(ij1 + ϕT1 ) sin(ij2 + ϕT2 )
U UT2 = i cos(ϕT2 ) XTRS2
Qsj1 = − Qsj2 = −
XTRS1 Uj2 UT2 XTRS2
(20)
By multiplying the active-power injections (18)–(21) by the time derivative of the angle of the bus voltage and considering (26), the following expressions are obtained (the time derivative of a variable is denoted as a point above the variable):
(Psi1 + Psi2 ) · ˙ i =
Ui UT1 cos(ϕT1 ) XTRS1
Uj1 UT1
(19)
Uj1 UT1 XTRS1
(21)
Psj1 · ˙ j1 = − (23) Psj2 · ˙ j2 = (24)
cos(ij2 + ϕT2 )
(25)
3. Energy function for an IPFC The energy function for an IPFC is constructed for two series branches and is then expanded for an arbitrary number of branches.
sin(ij1 + ϕT1 )+
Uj2 UT2 XTRS2
sin(ij2 + ϕT2 ) · ˙ i (27)
(22)
cos(ij1 + ϕT1 )
(26)
Uj1 UT1 XTRS1
Uj2 UT2 XTRS2
sin(ij1 + ϕT1 ) · ˙ j1
(28)
sin(ij2 + ϕT2 ) · ˙ j2
(29)
Further reactive-power injections (22)–(25) are divided by the magnitude of the bus voltage and multiplied by the time derivative of the magnitude of the bus voltage: U˙ UT1 U˙ UT2 Qsi1 + Qsi2 U˙ i = i cos(ϕT1 ) + i cos(ϕT2 ) Ui XTRS1 XTRS2 Qsj1 Uj1 Qsj2 Uj2
U˙ j1 = − U˙ j2 = −
U˙ j1 UT1 XTRS1 U˙ j2 UT2 XTRS2
(30)
cos(ij1 + ϕT1 )
(31)
cos(ij2 + ϕT2 )
(32)
The sum of (27)–(32) can be rewritten as: Uj1 · UT1 XTRS1 + −
Fig. 6. Injection model of an IPFC with two series branches.
· sin(ij1 + ϕT1 ) · ˙ ij1 +
Uj2 · UT2 XTRS2
· sin(ij2 + ϕT2 ) · ˙ ij2
U˙ j1 · UT1 U˙ i · UT1 U˙ · UT2 cos(ϕT1 ) + i cos(ϕT2 ) − · cos(ij1 + ϕT1 ) XTRS1 XTRS2 XTRS1 U˙ j2 · UT2 XTRS2
· cos(ij2 + ϕT2 )
(33)
The construction procedure from ref. [14] leads us to (33). The crucial step in the energy-function construction is how to obtain an
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analytical solution of the first integral of (33). There is no uniform procedure to find this integral and it can be—for any kind of application concepts and formulation procedures of deriving an energy function as a first integral—obtained only with some intuition, as already reported in ref. [14], where it was also stressed that there is no uniform procedure that exists to obtain a Lyapunov energy function. To find the first integral of (33), i.e., to obtain a function for which the time derivative is known, we can use the fact that alreadyknown energy functions for some of the FACTS devices are very closely related to the reactive powers injected into the system by these devices [16–22]. The derivations of the energy functions for CSC, SVC, SSSC, STATCOM, PST and UPFC have shown that for devices that from the system’s point of view operate as controllable voltage or current sources, the energy function is equal to the total sum of the reactive-power injections. In contrast, for devices that from the system’s point of view operate as controllable reactances, the energy function is equal to half of the sum of the reactive powers injected into the system by these devices. So we denote the sum of the IPFC’s reactive-power injections as: Qinj = Qsi1 + Qsi2 + Qsj1 + Qsj2
+ − −
U · UT1 Ui · U˙ T1 cos(ϕT1 ) − i sin(ϕT1 ) · ϕ˙ T1 XTRS1 XTRS1
Ui · U˙ T2 U · UT2 cos(ϕT2 ) − i sin(ϕT2 ) · ϕ˙ T2 XTRS2 XTRS2 Uj1 · U˙ T1 XTRS1 Uj2 · U˙ T2 XTRS2
cos(ij1 + ϕT1 ) + cos(ij2 + ϕT2 ) +
(35) can be denoted as:
VIPFC = Qinj +
Uj2 · UT2 Ui · UT2 sin(ϕT2 ) − sin(ij2 + ϕT2 ) dϕT2 XTRS2 XTRS2 (36)
The expression in square brackets of (36) is equal to the active power PT2 and consequently we can rewrite it as:
VIPFC = Qinj +
PT2 dϕT2
(37)
Eq. (37) is the final form of an energy function for an IPFC when the controllable parameters are constant. The integral part of (37) cannot be analytically solved. Although (37) cannot be proved to be a true Lyapunov function (it is path dependent), it can be successfully used in the Lyapunov direct method for transient-stability assessment or the optimum IPFC parameters in the Lyapunov sense can be calculated. For the purposes of the Lyapunov direct method the integral part of (37) is calculated numerically in the same way as, e.g., active loads.
(34)
Now let us find the time derivative of this sum Qinj , and compare it with (33). All the parts of (33) are included in this derivative. The first integral of (33), which represents the energy function for an IPFC, can now be rewritten as: VIPFC = Qinj −
949
Uj1 · UT1 XTRS1
Uj2 · UT2 XTRS2
sin(ij1 + ϕT1 ) · ϕ˙ T1
sin(ij2 + ϕT 2 ) · ϕ˙ T2 dt (35)
In order to be able to solve the integral part of (35), the time behavior of the control parameters has to be known, i.e., the control strategy of the IPFC has to be known. While it is obvious that this integral cannot be solved for every type of control strategy, let us try to solve it for two simple control strategies: (a) for constant control parameters and (b) for a constant active power PT between the two series branches. 3.1.1. The energy function for constant control parameters If the rating of the IPFC is considered, then the magnitude UT cannot exceed its maximum possible value UTmax . On the other hand, it can be proved that the “optimum” ϕT does not change a great deal when the first-swing stability is considered and the results—i.e., the critical clearing times—are similar if the constant ϕT is considered. Of course, this statement could be made only after the optimum control strategy has been implemented. In addition, it has been found that the optimum strategy requires the maximum possible UT , i.e., UTmax . So the natural conclusion is to consider the control parameters as being constant. Although this strategy is not optimal, it can be used when sectional-constant parameters are applied, as described in refs. [16]. In this way the control strategy becomes nearly optimized in the Lyapunov sense. Let us, for the further development of an energy function (35), decide that the angle ϕT2 depends on three other parameters—i.e., UT1 , ϕT1 and UT2 , in order to fulfill (7). As UT1 , ϕT1 and UT2 are considered constant (and ϕT2 not) their time derivatives equal 0, and
3.1.2. The energy function for the constant active power PT between the two series branches If the energy function without an integral part is derived and can be proved to be a true Lyapunov function, the voltage magnitudes UT1 and UT2 as well as the active-power exchange between the IPFC branches (i.e., PT ) should be considered constant. While this approach can be justified for UT , as explained in the previous paragraph, constant PT cannot be considered a kind of optimum control. However, this can be overcome by applying sectional constant control, which is based on the assumption that during electromechanical transients the system parameters (voltage magnitudes and angles, rotor angles) do not change abruptly. Furthermore, according to findings described in Section 2.2, when searching for optimum parameters we should avoid the prohibited area presented in Fig. 4. As the derivations of the system dynamics are calculated numerically, it is important that the starting-point of the search does not fall in this prohibited area, because in this case the calculation fails. Because PT defines the border of the prohibited area, its setting makes it possible to choose good starting points when searching for other optimum controllable parameters for the IPFC. As the active-power transfer PT between the two series branches as well as the voltage magnitudes UT1 and UT2 are considered time independent, (35) can be rewritten as:
VIPFC = Qinj +
+
Uj1 · UT1 Ui · UT1 sin(ϕT1 ) − sin(ij1 + ϕT1 ) dϕT1 XTRS1 XTRS1
Uj2 · UT2 Ui · UT2 sin(ϕT2 ) − sin(ij2 + ϕT2 ) dϕT2 XTRS2 XTRS2 (38)
The expressions in the square brackets of (38) are equal to the active power PT1 and PT2 , respectively. Because this active power is assumed to be constant, (38) can be rewritten as: VIPFC = Qinj + PT1 · ϕT1 + PT2 · ϕT2 = Qinj + PT1 · (ϕT1 − ϕT2 )
(39)
Eq. (39) is a true Lyapunov energy function for an IPFC when the active power PT between the two series branches and the controllable parameters UT1 and UT2 are constant. In this case the integral part of (35) was analytically solved. Its adequacy is verified in Section 4. It is interesting to note that the consideration of the activepower exchange between the series branches of an IPFC in the
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Fig. 7. SMIB test system with an IPFC with two series branches.
energy-function construction is the same as the consideration of the active loads presented in ref. [14]. According to this reference, the energy function for the active loads, i.e., the integral, can be analytically solved only if the active loads are considered as a constant active power. In any other case, the integral has to be calculated numerically or an approximate expression (which does not satisfy the Lyapunov conditions) has to be used. The obtained energy functions (37) and (39) act like an extension to any structure-preserving energy function, e.g., Vwithout FACTS . The energy function for the power system with an IPFC is, therefore: V = Vwithout
FACTS
+ VIPFC
(40)
Obviously, the energy function VIPFC can be extended to deal with any number of IPFCs installed in the system. 3.2. Generalization for an arbitrary number of series branches Let n be the number of series branches and let UTk and ϕTk be the voltage magnitude and the voltage angle of the kth branch. The energy function (37) can be generalized for any number of series branches of an IPFC. Considering the parameters UTk and ϕTk to be constant in all branches except the power-balancing branch, where only the injected voltage magnitude UTk is constant, on the basis of (35) it is easy to show that the energy function of such an IPFC is equal to (37). However, the sum of all the reactive power injections Qinj in (37) should include the reactive power injections of all the series branches:
n
Qinj =
[Qsik + Qsjk ]
(41)
k=1
and PT2 and ϕT2 denote the injected-power parameters of the power-balancing branch. In the same way the energy function (39) can be generalized. Since the active power injected by a VSC and the voltage magnitude of a VSC in each series branch are constant, on the basis of (35) and (38) it is easy to show that the generalization of (39) leads to (42): VIPFC = Qinj +
n
PTk · ϕTk
(42)
k=1
Because the critical energy in a SMIB system is uniformly given, the CCTs estimated by the Lyapunov direct method should be the same as that obtained using time-domain simulations and, consequently, the adequacy of the energy functions (37) and (39) can be verified. However, exactly the same system model has to be used in both the simulation and the direct method. For this reason we applied the Mathematica computer program. We are aware that a mathematical tool is not the most appropriate for a simulation of electric-power system dynamics, and the use of, e.g., PSCAD, EMTP or the Netomac program, would have been much easier to implement. However, in order to ensure that the system modeling is absolutely identical in the simulation and when applying direct methods we chose Mathematica. For the purposes of time-domain simulations an IPFC was modeled by means of the mathematical expressions (18)–(25). Additionally, the Lyapunov direct method for transient-stability assessment that uses the IPFC’s energy function was applied on the IEEE 9-bus 3-machine test system. In a multi-machine system the system fault and the post-fault trajectory are no longer uniformly given, and direct methods do not give exactly the same results as the simulation. Such a model cannot serve for validation purposes. However, it can serve to demonstrate the Lyapunov direct method applied to a multi-machine system comprising IPFCs. 4.1. SMIB test system The SMIB test system is presented in Fig. 7. The data for this system can be found in ref. [16]. An IPFC consists of two series branches, each rated at 132.5 MVA and connected to the system through a series transformer with a short-circuit voltage uk = 3.75%. The disturbance is a three-phase short-circuit near BUS1, see Fig. 7, and is assumed to be eventually cleared, i.e., the system’s post-fault configuration is identical to the pre-fault one. The pre-fault and fault-on values of the IPFC’s controllable parameters are set to 0. The CCTs were obtained by repeating the time-domain simulations and applying the direct method with the use of the newly proposed IPFC energy function. In the first numerical example we apply the IPFC energy function (37) that is valid for the constant controllable parameters UT1 , UT2 and ϕT1 . The results for various UT1 and UT2 limits and for various constant values of ϕT1 are presented in Table 1. As one can see, the results are identical, and hence it can be concluded that
4. Verification of adequacy for an IPFC’s energy function In order to verify the adequacy of the derived energy functions (37) and (39), they were applied in the Lyapunov direct method for transient-stability assessment, i.e., for the determination of the critical clearing times (CCTs), on the single-machine infinite-bus (SMIB) test system. For the CCT estimation we used the potentialenergy boundary surface (PEBS) method [15], in which the CCT is the time instant when the total energy of the system along the faulton trajectory equals the maximum of the potential energy along the same fault-on trajectory.
Table 1 CCTs obtained in a SMIB test system including an IPFC with the constant controllable parameters UT1 , UT2 and ϕT1 . UT1 [p.u.]
0 0.1 0.15 0.2 0.25
UT2 [p.u.]
0 0.1 0.15 0.2 0.25
ϕT1 [◦ ]
– 100 100 100 100
CCT [ms] Simulation
Direct method
139 151 156 160 164
139 151 156 160 164
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Table 2 CCTs obtained in a SMIB test system including an IPFC with the constant controllable parameters UT1 , UT2 and PT . UT1 [p.u.]
0.0 0.15 0.15 0.15 0.3 0.3 0.3
UT2 [p.u.]
0.0 0.15 0.15 0.15 0.3 0.3 0.3
PT [p.u.]
0 −0.05 0 0.075 −0.1 0 0.1
CCT [ms] Simulation
Direct method
139 154 156 138 165 168 164
139 154 156 138 165 168 164
Table 3 CCTs obtained in an IEEE 9-bus test system including an IPFC with the constant controllable parameters UT1 , UT2 and ϕT1 . UT1 [p.u.]
UT2 [p.u.]
ϕT1 [◦ ]
CCT [ms] Simulation
Simulation
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0.1 0.1 0.1 0.2 0.2 0.2
– −60 −45 −10 −45 −20 −10
242 245 246 245 248 249 248
244 245 246 246 248 249 249
the energy function (37) is appropriate for such a control strategy. The CCTs gathered in Table 1 are those obtained with the constant controllable parameter ϕT1 set to a value that gives a maximum transient-stability improvement, i.e., maximum CCTs. In the second numerical example of the SMIB test system we apply the IPFC energy function (39) that is valid for the constant controllable parameters UT1 , UT2 and PT . The results are presented in Table 2. Again, the results are identical, and in this way they validate the proposed IPFC energy function (39). 4.2. IEEE 9-bus 3-machine test system In order to demonstrate the application of the newly proposed energy functions (37) and (39) for an IPFC in a multi-machine system, CCTs were obtained using the Lyapunov direct method for the IEEE 9-bus 3-machine test system. Because the trajectory of the machine rotors and hence the critical energy in a multi-machine system is not uniformly given, the CCTs estimated by the Lyapunov direct method are in general different from that obtained using time-domain simulations. However, as can be seen from the results later in this section (Tables 3 and 4), the Lyapunov direct method that uses the newly proposed energy functions (37) and (39) provides a good estimation of the CCTs. The data for the presented system can be found in ref. [23]. The generators were represented by a classical model, the loads were modeled as constant admittances and the lines were considered to be lossless. The disturbance is a three-phase short-circuit near Table 4 CCTs obtained in an IEEE 9-bus test system including an IPFC with the constant controllable parameters UT1 , UT2 and PT . UT1 [p.u.]
UT2 [p.u.]
PT [◦ ]
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0.1 0.1 0.1 0.2 0.2 0.2
0 0 0.07 −0.07 0 0.1 −0.17
CCT [ms] Simulation
Simulation
242 246 245 246 249 248 249
244 246 245 247 249 247 249
Fig. 8. An IEEE 9-bus test system with an IPFC.
bus 7. An IPFC is included in the system as shown in Fig. 8. Both the IPFC’s branches are rated at 100 MVA and uk = 3.75%. The prefault and fault-on values of the IPFC’s controllable parameters are set to 0. Like in a SMIB test system, the CCTs were obtained using time-domain simulations, and directly with the use of the newly proposed IPFC energy function. In the first numerical example of the IEEE 9-bus test system we applied IPFC energy function (37) that is valid for the constant controllable parameters UT1 , UT2 and ϕT1 . According to the selected UT1 , UT2 and ϕT1 , two values of the angle ϕT2 fulfill the requirement for active power compensation between the IPFC’s branches: the one that injects the inductive reactive power and the one that injects the capacitive reactive power. In this numerical example we considered the injection of capacitive reactive power in the second branch that gives a better transient stability improvement. The results for various UT1 and UT2 limits and for various constant values of ϕT1 are presented in Table 3. The CCTs obtained with the direct method deviate from the reference (time-domain simulation) by no more than 2 ms. In the second numerical example on the IEEE 9-bus test system the IPFC energy function (39) that is valid for constant controllable parameters UT1 , UT2 and PT was applied. According to the selected UT1 , UT2 and PT , two values of the angle ϕT1 and two values of the angle ϕT2 fulfill the requirement for active power compensation between the IPFC’s branches—inductive or capacitive reactive power can be injected into each of the branches. Again, for a better transient-stability improvement the injection of capacitive reactive power into each branch was considered. The results are presented in Table 4. The CCTs obtained with the direct method deviate from the reference by no more than 1 ms, which is equal to the integration time step. Both multi-machine numerical examples show that the Lyapunov direct method using the proposed IPFC’s energy function can be successfully applied in transient-stability assessment. However, with constant controllable parameters it is hard to fulfill the requirement for active power compensation between the IPFC’s two branches for the period of the first swing. As we showed in Section 2, the active power injected into a series branch depends on the voltage angles at the connection points of that branch. Because those angles experience large changes for the period of the first swing, the controllable parameter ϕT1 or PT can only fulfill the IPFC’s power
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compensation between the two branches for the entire period of the first swing in a narrow range. 5. Conclusions An IPFC can be considered the most powerful device for dynamic power flow control in EPS as it is able to control active and/or reactive power flow in each line of a chosen network node. In this sense it may be applied for steady-state power-flow and voltage control, transient-stability margin enhancement, oscillation damping, etc. In order to be able to use a Lyapunov energy-function approach for these tasks, the energy function was constructed for a system that includes IPFCs. In our work the structure-preserving approach has been applied. Another important issue described in this paper is to clearly define the possible operating areas and the limitations of an IPFC. Theoretical considerations have been described that show where such operational limitations are, so now areas with impossible operational states can be avoided when numerical calculations are carried out. In order to verify the adequacy of the derived energy functions, they were applied in the Lyapunov direct method for transientstability assessment, i.e., for the determination of the CCTs, on the SMIB test system. In order to demonstrate the application in a multi-machine system, CCTs were obtained using the Lyapunov direct method for the IEEE 9-bus 3-machine test system. The results validate the proposed IPFC energy functions. References [1] P. Sauer, M.A. Pai, Power System Dynamics and Stability, Prentice Hall, 1998. [2] L. Gyugyi, K.K. Sen, C.D. Schauder, The interline power flow controller concept: a new approach to power flow management in transmission systems, IEEE Trans. Power Deliv. 14 (1999) 1115–1123. [3] S. Mishra, P.K. Dash, P.K. Hota, M. Tripathy, Genetically optimized Neuro-fuzzy IPFC for damping modal oscillations of power system, IEEE Trans. Power Syst. 17 (2002) 1140–1147. [4] B. Fardanesh, B. Spherling, E. Uzunovic, S. Zelingher, Multi-converter FACTS devices: the generalized unified power flow controller (GUPFC), in: IEEE Power Engineering Society Summer Meeting, vol. 2, 2000, pp. 1020–1025. [5] Z. Xiao-Ping, E. Handschin, M. Yao, Modeling of the generalized unified power flow controller (GUPFC) in a nonlinear interior point OPF, IEEE Trans. Power Syst. 16 (2001) 367–373. [6] J. Chen, T.T. Lie, D.M. Vilathgamuwa, Basic control of interline power flow controller, in: IEEE Power Engineering Society Winter Meeting, vol. 1, 2002, pp. 521–525. [7] D. Menniti, A. Pinnarelli, N. Sorrentino, A fuzzy logic controller for interline power flow controller model implemented by ATP-EMTP, in: Proceedings of International Conference on Power System Technology, PowerCon 2002, vol. 3, 2002, pp. 1898–1903. [8] S. Teerathana, A. Yokoyama, An optimal power flow control method of power system using interline power flow controller (IPFC), in: Proceedings of IEEE Region 10 Conference, TENCON 2004, 2004, pp. 343–346.
[9] X. Wei, J.H. Chow, B. Fardanesh, A.A. Edris, A dispatch strategy for an interline power flow controller operating at rated capacity, in: Proceedings of IEEE/PES Power Systems Conference and Exposition 2004, 2004, pp. 1459–1466. [10] B. Fardanesh, Optimal utilization, sizing, and steady-state performance comparison of multiconverter VSC-based FACTS controllers, IEEE Trans. Power Deliv. 19 (2004) 1321–1327. [11] B. Fardanesh, A. Schuff, Dynamic studies of the NYS transmission system with the Marcy CSC in the UPFC and IPFC configurations, in: Proceedings of IEEE/PES Transmission and Distribution Conference and Exposition 2003, vol. 3, 2003, pp. 1126–1130. [12] R. Mihalic, Power flow control with controllable reactive series elements, IEE Proceedings—Generation, Transmission and Distribution 145 (1998) 493–498. [13] M. Noroozian, L. Angquist, G. Ingestrom, Series compensation, in: Y.H. Song, A.T. Johns (Eds.), Flexible ac Transmission Systems (FACTS), IEE, London, U.K., 1999, pp. 199–242. [14] 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, 1985, pp. 70–77. [15] M.A. Pai, Energy Function Analysis For Power System Stability, Kluwer Academic Publishers, 1989. [16] V. Azbe, U. Gabrijel, R. Mihalic, D. Povh, The energy function of a general multimachine system with a unified power flow controller, IEEE Trans. Power Syst. 20 (2005) 1478–1485. [17] U. Gabrijel, R. Mihalic, Direct methods for transient stability assessment in power systems comprising controllable series devices, IEEE Trans. Power Syst. 17 (2002) 1116–1122. [18] U. Gabrijel, R. Mihalic, Transient stability enhancement and assessment of systems comprising controllable series devices, in: Proceedings of 2nd IASTED International Conference on Power and Energy Systems, 2002, pp. 251–258. [19] R. Mihalic, U. Gabrijel, Transient stability assessment of systems comprising phase-shifting FACTS devices by direct methods, International Journal of Electrical Power & Energy Systems 26 (2004) 445–453. [20] R. Mihalic, U. Gabrijel, A structure-preserving energy function for a Static Series Synchronous Compensator, IEEE Trans. Power Syst. 19 (2004) 1501–1507. [21] R. Mihalic, P. Zˇ unko, D. Povh, Improvement of transient stability using unified power flow controller, IEEE Trans. Power Deliv. 11 (1996) 485–492. [22] I.A. Hiskens, D.J. Hill, Incorporation of SVCs into energy function methods, Trans. Power Syst. 7 (1992) 133–140. [23] P.M. Anderson, A.A. Fouad, Power System Control and Stability, IEEE Press Power Systems Engineering Series, Revised Printing, 1999. Valentin Azbe received his B.Sc., M.Sc. and Dr.Sc. degrees from The University of Ljubljana, Slovenia, in 1996, 2003, and 2005, respectively. After receiving his diploma he worked with IBE, consulting engineers, Slovenia, as a project manager in the Department for Overhead-Lines design. In 2000 he joined the Department of Power Systems and Devices at the Faculty of Electrical Engineering, The University of Ljubljana, where he has since worked as a junior researcher. In 2005 he became a Teaching Assistant. His areas of interest include system analysis, FACTS devices, power-system protection and DC power-system analysis. Rafael Mihalic received the Dipl. Eng., M.Sc. and Dr.Sc. degrees from The University of Ljubljana, Ljubljana, Slovenia, in 1986, 1989, and 1993, respectively. He became a Teaching Assistant in the Department of Power Systems and Devices, Faculty for Electrical and Computer Engineering, The University of Ljubljana, in 1986. Between 1988 and 1991, he was a member of the Siemens Power Transmission and Distribution Group, Erlangen, Germany. Since 2005 he has been a Professor at the University of Ljubljana. His areas of interest include system analysis and FACTS devices. Prof. Mihalic is a member of Cigre (Paris, France).