Power system damping from energy function analysis implemented by voltage-source-converter stations

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Electric Power Systems Research 78 (2008) 1353–1360

Power system damping from energy function analysis implemented by voltage-source-converter stations Si-Ye Ruan a , Guo-Jie Li a,∗ , Boon-Teck Ooi b , Yuan-Zhang Sun a a

State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing, 100084, PR China b Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada Received 4 April 2007; received in revised form 25 August 2007; accepted 11 December 2007 Available online 19 February 2008

Abstract Because weakly damped system oscillations can endanger the secure operation of power systems, this paper is a study on how damping can be increased. As the power system is nonlinear, an energy function approach, patterned after the direct method of Lyapunov function, is used in the analysis. The analysis develops an energy function W and shows that a real power term (proportional to local frequency) and/or a reactive power term (proportional to the line voltage differentiated with respect to time) increase the rate of diminution of the energy function W. This implies effective damping of power swings by such power signals and is verified by simulations of a multi-machine system model. The damping signals are introduced by voltage-source-converter (VSC) stations. © 2007 Elsevier B.V. All rights reserved. Keywords: VSC; Real and reactive power modulations; Energy function

1. Introduction Large interconnected power systems often suffer from weakly damped power swings. Such lack of damping occurs particularly in areas connected by weak tie-lines, systems with longitudinal structures and generators connected by long lines to the rest of the system. To enhance the damping of power system oscillations, HVDC and FACTS controllers have been used [1–4]. This paper focuses on VSC stations (the inverters/rectifiers of VSC–HVDC and of distributed generation/renewable energy sources) [5–9], because on top of the steady-state real and reactive power deliveries, they allow real power modulation [7], reactive power modulation [8] and a combination of real and reactive power modulations [9] to be applied for dynamic performance enhancement. So far, many studies have been based on small signal analysis of a simple model, which has certain limitations. Firstly, the linearized model is valid only in the vicinity of the chosen operating point. Then, power systems are large and complex. Therefore one needs a methodology that can deal with large

∗

Corresponding author. Tel.: +86 10 62795771; fax: +86 10 62772469. E-mail address: [email protected] (G.-J. Li).

0378-7796/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2007.12.003

nonlinear systems. To achieve this goal, this paper adopts the energy function method, patterned after the direct method of Lyapunov. Traditionally, Lyapunov theory deals with a dynamical system without inputs. Recently, it is also applied in feedback design by making the Lyapunov derivative negative when choosing the control [10]. Such ideas have been made precise with the introduction of the concept of a Control Lyapunov Functions for systems with control input, which has been successfully applied to FACTS controllers [11,12,16]. Developing on [11,12,16], this paper shows that real power term proportional to frequency and/or a reactive power term proportional to time rate of voltage differentiation, increase the rate of diminution of the energy function. Implicitly, the damping is increased. The theoretical conclusions are verified by simulations in a multi-area power system using VSC stations to implement the damping strategies. The outline of the paper is as follows: In Section 2, a general power system is modeled for energy function stability analysis. The analysis identifies the type of real and reactive power modulations in the network buses which increase the rate of diminution of the energy function and therefore increase power system damping. Further, it describes how the aforesaid modulations are carried out by VSC stations. Simulations and the analysis of results are presented in Section 3. Conclusions are drawn in Section 4.

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2. System model and the control strategy In the interest of arriving at the contributions of this paper quickly, the detailed derivations have been moved to Appendix A. Appendix A contains the standard techniques used by the energy function method but are nevertheless necessary for this paper to be self sufficient. This section is a sketch of how the present status of the research is reached and against this, the theoretical contribution of this paper can be appreciated. Following in the footsteps of [1], the equations are referenced with respect to the center of inertia (COI). 2.1. Generators In the power system in Fig. 1, there are n generation sources, each represented by Vi ∠ φi (i = 1. . .n). The symbols Vi (i = 1. . .n) represent constant emfs behind transient reactances xi and φi are the mechanical rotor angle. The dynamic equation of motion of each generator (i = 1. . .n) is [13]: φ˙ i = ωi

(1)

Mi ω˙ i = Pmi − Pei − Di ωi −

Mi PCOI MT

Fig. 2. Simplified model of VSC station.

(2)

where Mi represents moment of inertia, the turbine power is Pmi , the generated power is Pei = Bij Vi Vj sin(φI –φj ) the subscript j = i + n being the bus to which it is connected and the power n associated with the center of inertia (COI) is PCOI = i=1 (Pmi − Pei ). 2.2. Network equations

PFi =

Bij Vi Vj sin φij

2n+m 

Bij Vi Vj cos φij

(3a)

(3b)

j=1

where φij = φI –φj . Complex power balance at the ith bus requires; {Pi (ωi ) + jQi (Vi )} + (PFi + jQFi )} = 0.0

There are N = n + m buses, numbered i = n + 1, n + 2,. . .2n, 2n + 1,. . .2n + m, each bus having voltage Vi ∠ φi . At each bus i, the load Pi (ωi ) + jQi (Vi ) is, in general, functions of frequency ωi and voltage Vi . Usually, Pi (ωi ) are assumed to be constant. Emanating from the ith bus are connections to the other buses and the generators which take a total complex power PFi + jQFi . It is assumed that the line resistances can be neglected. 2n+m 

QFi = −

(4)

2.3. Energy function method Following [13], the energy function W is defined as: W = W1 + W2

(5)

where W1 is a positive definite function based on the kinetic energy of the generators: n

j=1

W1 =

1 Mi ωi2 2

(6)

i=1

and W2 , which addresses the complex powers at the n + m buses, consists of 6 components, and as Appendix A elaborates.

Fig. 1. Model of power system.

2.3.1. Status of energy function research The end-point of existing research is the proof of negative definiteness in dW/dt. It is pertinent to draw attention to the fact that in the chain rule differentiation, when W2 is differentiated with respect to φi , it has a dφi /dt term and likewise when differentiated with respect to Vi , it has dVi /dt. Although provision has been made that for the load Pi (ωi ) + jQi (Vi ) at each bus i, to be functions of frequency ωi and voltage Vi , in all previous work

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of active loads,       dW dW1 dW2 dW2 = + + dt dt ωi dt φi dt Vi =−

n 

Di ωi2 −

i=1

g+h 

Pi φ˙ i −

i=g+1

g+h 

Qi

i=g+1

V˙ i Vi

(8)

Two power modulations are proposed: Real power modulation: Pi = kPi φ˙ i

Fig. 3. VSC station based on 2-level topology.

the buses are connected to constant loads.       n  dW dW2 dW2 dW1 + + = − Di ωi2 = dt dt ωi dt φi dt Vi

(7)

i=1

which is a negative definite function Thus in the domain where W can be shown to be positive, the system is stable. The viscous damping coefficients in the generator shaft Di are the only dissipative term because resistances have been neglected in the model.

i = g + 1, . . . , g + h

Reactive power modulation:   V˙ i i = g + 1, . . . , g + h Qi = kQi Vi

(9)

(10)

Substituting (9) and (10), Eq. (8) becomes  2 g+h g+h n    dW V˙ i kPi φ˙ i2 − kQi = − Di ωi2 − dt Vi i=1

i=g+1

(11)

i=g+1

2.3.2. Increasing damping by complex loads If the loads in the some of the buses, (i = g + 1,. . .g + h), [g + 1,. . . g + h] ⊂ [n + 1,. . .2n + m], are not constant, but are capable of injecting perturbation complex power Pi + jQi . This perturbation complex power does not affect the energy function W. But it will affect dW/dt. Since PFi (ωi ) + jQFi (Vi ) + Pi (ωi ) + jQi (Vi ) = −(Pi + jQi ), dW/dt can be made more negative and thereby increase damping. More details can be found in Appendix A. With the inclusion

Fig. 4. A 4-machine power system.

Fig. 5. Frequency deviations when no additional damping is adopted. (a) Deviations of rotor angles in individual machines. (b) Deviations of inter-area and local frequency oscillations.

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another load. The difference is that when the station operates as an inverter/rectifier, the polarity of Pvi is −ve/+ve. In the simulations carried out in this paper, Qvi = 0.0. The second function is to apply perturbation complex power, i.e. Pvi + jQvi for the purpose of damping system oscillations. The damping power only affects the interconnected system in the transient state while it is equal to zero in the steady state. In practice, because the VSC stations have the main function of delivering steady-state complex power, Pvi + jQvi is usually a limited amount left over by the MVA rating and is required for the short duration of damping a transient. As is well known, time differentiation such as in φ˙ i and ˙ (Vi /Vi ) can be easily obtained at each bus. Applying local feedback, the strategies adopted are: for real power modulation Pi = PVi = kpi (ωi ), for reactive power modulation Qi = QVi = kQi (V˙ i /Vi ) [4]. The control of the VSC station [5] is based on operating the two-level topology of the voltage-source converter of Fig. 3 under Sinusoidal Pulse Width Modulation (SPWM). The research is carried out by simulations using an industry-standard software, EMTDC-PSCAD.

Fig. 6. Frequency deviations when real and reactive power modulations are adopted. (a) Pure real power modulation (PVi + jQVi = 8.0 + j0.0 MVA for each VSC station; kP1 = 13, kP2 = 12.4, kP3 = 20, kQ1 = kQ2 = kQ3 = 0). (b) Pure reactive power modulation (PVi + jQVi = 0.0 + j8.0 MVA for each VSC station; kP1 = kP2 = kP3 = 0, kQ1 = 451, kQ2 = 451, kQ3 = 475).

which is a more negative definite function than (7). The bigger the damping coefficients kPi and kQi are, the more negative is the value of dW/dt. Thus, the system returns to the equilibrium operating point faster. An active complex load (Pi + Pi ) + j(Qi + Qi ) could be realized by the combination of VSC station (Pvi + Pvi ) + j(Qvi + Qvi ) and the normal load PLi + jQLi , where Pi + jQi = (Pvi + PLi ) + j(Qvi + QLi ) and Pi + jQi = Pvi + jQvi . Fig. 2 is the diagrammatic representation of a VSC station with the single-line ac connected to a bus at one end and the other end to “DC System”. The DC System may be: (i) the dc line of a point-to-point VSC–HVDC, (ii) the dc bus of multi-terminal VSC–HVDC. Fig. 2 can also represent the power injection of photo-voltaics, turbine-generator outputs of: run-of the river hydro, gas turbines, wind turbines [6]. These distributed/renewable generation sources unavoidably require conversion of variable frequencies to dc and Fig. 2 represent the inverter end which changes dc to supply frequency. It is assumed that the number of VSC stations in the interconnected system is h, which are installed at nodes g + 1,. . .g + h. The function of a VSC station can be divided into two: First part is a constant complex power, i.e. Pvi + jQvi , just like

Fig. 7. Frequency deviations when damping power rating of real and reactive power modulation is increased. (a) Pure real power modulation (PVi + jQVi = 16.0 + j0.0 MVA for each VSC station, kP1 = 29, kP2 = 27 kP3 = 43, kQ1 = kQ2 = kQ3 = 0). (b) Pure reactive power modulation (PVi + jQVi = 0.0 + j16.0 MVA for each VSC station kP1 = kP2 = kP3 = 0, kQ1 = 923, kQ2 = 923, kQ3 = 990).

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Fig. 8. MVA Rating of VSC station |Simax |=|PVi + jQVi |.

3. Simulation results and analysis The results reached in Section 2 are demonstrated in simulation tests using the 4-machine system of Fig. 4. EMTDC/PSCAD [14], a widely accepted software by industry, is used to run the simulation. Appendix B lists the parameters of the system. The

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system is divided into Area A with generators G1 and G2 and Area B with generators G3 and G4, the 2 areas being joined by Bus 7. Three VSC stations (VSC1, VSC2 and VSC3), installed at Bus 5, 6 and 8, modulate their perturbation real and reactive powers to increase damping. Note that kPi and kQi (i = 1, 2 and 3) are their real and reactive power modulation coefficients. In the steady-state, each VSC station injects 80 MW into the system. Power system disturbances are initiated by three-phase faults of 80 ms duration at Bus 7. The response oscillations are classified by the local-area mode frequencies ωlocal and the inter-area mode frequency ωinter . The local area mode oscillation in Area A is due to the speed difference ω12 between G1 and G2 and in area B, ω34 , the speed difference between G3 and G4. Their frequencies are in the 2 Hz range. The oscillation between two areas, at frequency ωinter = [(M1 ω1 + M2 ω2 )/(M1 + M2 )] − [(M3 ω3 + M4 ω4 )/(M3 + M4 )], where [(M1 ω1 + M2 ω2 )/(M1 + M2 )] is assumed to be the ‘COI frequency’ in AreaA while [(M3 ω3 + M4 ω4 )/(M3 + M4 )] is that in AreaB. ωinter is at a lower frequency of about 1 Hz. The waveforms in Fig. 5 provide the base for comparison. The damping modulations of the VSC stations are deactivated. The simulations show that inter-area oscillation intrudes and beats with the local oscillations. The deviations of rotor angles in individual machines are provided as well (ϕ1 , ϕ2 ,

Fig. 9. Frequency deviations under three cases. (a) Case 1(kP1 = 13, kP2 = 12.4, kP3 = 20, kQ1 = kQ2 = kQ3 = 0). (b) Case 2(kP1 = 7.6, kP2 = 6.5, kP3 = 13, kQ1 = 1132, kQ2 = 1140, kQ3 = 1189). (c) Case 3, (kP1 = kP2 = kP3 = 0, kQ1 = 2243, kQ2 = 2257, kQ3 = 2361).

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specifically to the local modes, leaving the VSC stations to damp the inter-area mode. 3.1. Influence of damping power rating In the study of this paper, the total load in the system is 1200 MW, with each of the 3 VSC stations injecting 80 MW. In the test of Fig. 6, the peaks modulated powers are 8 MW and or 8 MVAR, which represent 10% of the 80 MW. As power system equipment are frequently operated at 10% below their rating, the modulation power can be found in the VSC stations. The total damping power introduced is only 2% of the total system load and this is only for the brief transient. A previous study [15] also estimated the power required for damping to be around 2% of the total load power. As the damping transient is short, a short duration overload may be tolerated. Fig. 7 shows the effect of doubling the modulation powers to 16 MW and 16 MVAR or 20% of the 80 MW. Fig. 7(a) shows that the inter-area oscillation is damped within 3 s by real power modulation. However, the damping by reactive power modulation remains poor as shown in Fig. 7(b). 3.2. Mixed damping strategy

Fig. 10. Frequency deviations when installed locations of VSC–HVDC systems are getting closer to generators. (a) Pure real power modulation (PVi + jQVi = 8 + j0 MVA for each VSC station, kP1 = 24, kP2 = 17, kP3 = 29, kQ1 = kQ2 = kQ3 = 0). (b) Pure reactive power modulation (PVi + jQVi = 0 + j8 MVA for each VSC station; kP1 = kP2 = kP3 = 0, kQ1 = 525, kQ2 = 525, kQ3 = 550).

ϕ3 and ϕ4 are the deviations of rotor angles in G1, G2, G3 and G4). The frequencies of inter-area oscillation and the frequencies of the local area oscillations have very little damping when Di = 0. Note that the effects of the amorissseur or damper windings could contribute to the damping torque which have been neglected in the deduction as being very small. Fig. 6(a and b) show the capability of VSC stations to damp oscillations. Note that the movement trends of machine angles in these two cases are similar as that in Fig. 5(a). Thus, they are not given any more for the limitation of the paper. System responses in Fig. 6(a) are for real power modulation alone. The responses to reactive power modulation alone are displayed in Fig. 6(b). Comparing them with Fig. 5, the conclusion is that the inter-area oscillation is significantly damped, more so by real power modulation than by reactive power modulation. On the other hand, the damping of the local oscillation is poor, which is consistent with the result in [17]. More studies will be required to find out whether the VSC stations can simultaneously damp all the modes. As generators are frequently equipped with Power System Stabilizers (PSS), their damping capability can be tuned

Up to this point, by using the same amount of real power modulation PVi and reactive power modulation QVi as the basis for comparison, reactive power has been shown to be less effective in damping. However, more reactive power for modulation can be drawn from a VSC station than real power for modulation. In the examples above, consider the constant output power PVi + jQVi = 80.0 + j0.0 MVA to be drawn from a VSC station which is rated at 88.0 MVA. Fig. 8 shows a circle of radius |Simax | = 88 MVA representing the magnitude of the MVA rating. The components of the complex modulating power SVi = PVi + jQVi are allowed to satisfy |Simax | = [(PVi + PVi )2 + (QVi + QVi )2 ]1/2 = 88.0, as shown in Fig. 8. It can be shown that the magnitudes of real power and reactive power for modulation are constrained by  80

80

P = |88 cos ϕ − 80| Vi

QVi = |88 sin ϕ|

, where − a cos

88

≤ ϕ ≤ a cos

88

Three cases are considered: pure real power modulation, mixed modulation and pure reactive power modulation, which are presented as Case 1, 2 and 3, respectively. Case 1: PVi + jQVi = 8.0 + j0.0 MVA for each VSC station Case 2: PVi + jQVi = 6.0 + j18.6 MVA for each VSC station Case 3: PVi + jQVi = 0.0 + j 36.6 MVA for each VSC station System responses in three cases are presented in Fig. 9. The system responses with pure real and reactive power modulations, which are Case 1 and 3, are shown in Fig. 9(a and c) respectively. The damping in Case 3 is better than that in Case1. This result does not contradict Fig. 6 which is based on the same amount of damping power. This is because 4–5 times more reactive power is drawn from the VSC station for the same MVA rating.

S.-Y. Ruan et al. / Electric Power Systems Research 78 (2008) 1353–1360

System response with mixed modulation, which is Case 2, is demonstrated in Fig. 9(b). It offers the best damping. This is a very useful result. It means that more damping can be obtained with the same MVA rating by utilizing the mixture of real and reactive power modulations. 3.3. Influence of locations of VSC stations The locations of the installations influence the damping capability. Damping is improved when the VSC stations are located closer to the generators. For example, the improved damping shown in Fig. 10 is for the case when VSC stations are installed at bus 1, 2 and 4, which are closer to the generators. Apart from the relocation, the system conditions are unaltered. The magnitude of the real modulating power is 8 MW, the same as for Fig. 6(a), but the local damping ability is even better than for 16 MW in Fig. 7(a). Comparably, reactive power modulation is not so sensitive to the change of location. 4. Conclusion This paper has advanced the Energy Function Method (patterned after the Direct Method of Liapunov) one step forward. This consists of proving mathematically the conditions for increasing the damping of a stable power system. Proof of concept has been carried out by simulations using EMTP/PSCAD, an industry standard simulation software. In the simulations, VSC stations implement the damping strategies using the real power term proportional to frequency and/or a reactive power term proportional to time rate of voltage differentiation, The simulation research further estimates that the method is feasible because: (i) the size of the perturbation complex powers needed for damping is about 10% of the MVA of the VSC station which implements the damping strategy; (ii) in total the perturbation complex powers required is around 2% of the size of the total load of the system. In the foreseeable future, distributed/renewable power sources will supply 20% of the load and with the 10% reserve margin of the VSC stations, there will be the 2% needed to increase damping. Some detail conclusions are: • Real power modulation is more effective than reactive power modulation. • Damping increases with the magnitude of the damping modulation. • A combination of real and reactive power modulations has been found to yield the best damping. • The damping from real power modulation becomes stronger when the VSC stations are located closer to the generators in the power system. • In the 2-area test system, simulations reveal that the inter-area oscillation is damped significantly but not the local oscillation. Further work will be needed to damp the local modes as well. Presently, local oscillations are within the frequency range of Power System Stabilizers (PSS) to damp out and the worry is what to do with the very low frequency interarea oscillations. Thus, the capability to damp out inter-area

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oscillation is a promising direction of research to follow up. Acknowledgements This work was supported in part by Beijing Natural Science Foundation (3073021) and National Natural Science Foundation of China (50377017). Appendix A The energy function [13] is defined as: W = W 1 + W2 + C

(A.1)

W1 and W2 are kinetic and potential energy, respectively. C is a constant such that at the post fault stable equilibrium point the energy function is zero. n

1 Mi ωi2 2

W1 =

i=1

The subscript i in W2 has a different meaning W2 =

6 

W2i

i=1 n  W21 = − Pmi φi , i=1

W22 =

2n+m 

Pi φi ,

i=n+1

W23 =

2n+m 



i=n+1

W24 = −

Qi dVi Vi

2n+m 1  Bii Vi2 , 2 i=n+1

n  2n  Bij Vi Vj cos φij , W25 = − i=1 j=i+n 2n+m−1  2n+m 

W26 = −

Bij Vi Vj cos φij

i=n+1 j=i+1

It can be verified that the time differentiations of the energy function components are:     n  dW1 Mi (A.2) = ωi Pmi − Pei − Di ωi − PCOI dt ω˜ i MT i=1



dW21 dW25 + dt dt

 φi (i=1...n)

=−

n  i=1

ωi (Pmi − Pei )

(A.3)

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dW22 dW25 dW26 + + dt dt dt

φi (i=n+1...2n+m)

=

2n+m 

(PFi + Pi )φ˙ i

i=n+1

(A.4) 

dW23 dW24 dW25 dW26 + + + dt dt dt dt =

2n+m 

(QFi + Qi )

i=n+1

 Vi (i=n+1...2n+m)

V˙ i Vi

(A.5)

If the loads in the some of the buses are not constant, but active and capable of injecting perturbation complex power Pi + jQi (i = g + 1,. . .g + h), [g + 1,. . .g + h] ⊂ [n + 1,. . . 2n + m]. This perturbation complex power doesn’t alter energy function W. But it will affect dW/dt. Since (PFi (ωi ) + Pi (ωi )) + j(QFi (Vi ) + jQi (Vi )) = −(Pi + jQi ), dW/dt is affected in (A.4) and (A.5). Thus,       dW2 dW2 dW1 dW + + = dt dt ωi dt φi dt Vi =

n 

Di ωi2 −

i=1

g+h 

Pi φ˙ i −

i=g+1

g+h 

Qi

i=g+1

V˙ i Vi

(A.6)

Selecting the appropriate real and reactive power modulations, dW/dt can be made more negative. Two power modulations are proposed: Real power modulation: Pi = kPi φ˙ i

i = g + 1, . . . , g + h

Reactive power modulation:   V˙ i Qi = kQi i = g + 1, . . . , g + h Vi

(A.7)

(A.8)

Substituting (A.7) and (A.8), Eq. (A.6) becomes  2 g+h g+h n    V˙ i dW 2 2 ˙ = − Di ωi − kPi φi − kQi dt Vi i=1

i=g+1

(A.9)

i=g+1

Appendix B 1) Generators (G1–G4) M = 4s, SB = 500 MVA, VB = 13.8 kV, Ra = 0.02 p.u., xd = 1.05 p.u., xp = 0.12 p.u., D = 0 The output power of generators: PG1 = 310 MW; PG2 = 300 MW; PG3 = 200 MW, PG4 = 150 MW

2) Transmission Lines l = 0.001 H/km L1—60 km; L2—80 km; L3—40 km; L4—10 km; L5—60 km; L6—40 km; L7—20 km Two parallel tie-lines: L8—160 km; L9—160 km; 3) Transformers (T1–T4) 13.8 kV/115 kV, SB = 500 MVA xT = 0.1 p.u. 4) Loads SL1 —200 + j0 MVA; SL2 —250 + j0 MVA; SL3 —150 + j0 MVA SL5 —150 + j0 MVA; SL4 —250 + j0 MVA; SL6 —200 + j0 MVA References [1] P. Kundur, Power system stability and control, McGraw-Hill, 1994. [2] K.P. Padiyar, R.K. Varma, Damping torque analysis of static var system controllers, IEEE Trans. Power Syst. 6 (2) (1991) 458–465. [3] H.F. Wang, F.J. Swift, M. Li, A unified model for the analysis of FACTS devices in damping power system oscillations. II. Multi-machine power systems, IEEE Trans. Power Deliv. 13 (4) (1998) 1355–1362. [4] T. Smed, G. Andersson, Utilizing HVDC to damp power oscillations, IEEE Trans. Power Deliv. 8 (2) (1993) 620–623. [5] G.J. Li, S.Y. Ruan, L. Peng, Y.Z. Sun, X. Li, A novel nonlinear control for stability improvement in HVDC light system, in: IEEE Power Engineering Society General Meeting 2005, June, 2005, pp. 837–845. [6] W.X. Lu, B.T. Ooi, Optimal acquisition and aggregation of offshore wind power by multi-terminal Voltage-Source HVDC, IEEE Trans. Power Deliv. 18 (1) (2003) 201–206. [7] F.A.R. Jowder, B.T. Ooi, VSC-HVDC station with SSSC characteristics, IEEE Trans. Power Electron. 19 (4) (2004) 1053–1059. [8] W.X. Lu, B.T. Ooi, Simultaneous inter-area decoupling and local area damping by voltage source HVDC, in: IEEE Power Engineering Society Meeting, February, 2001, pp. 1079–1084. [9] Rotor Angle Stability, Available: http://www.abb.com, accessed Dec. 2006. [10] R.A. Freeman, P.V. Kokotovic, Robust Nonlinear Control Design, Birkhauser, 1996. [11] M. Noroozian, M. Ghandhari, G. Andersson, J. Gronquist, I. Hiskens, A robust control strategy for shunt and series reactive compensators to damp electromechanical oscillations, IEEE Trans. Power Deliv. 16 (4) (2001) 812–817. [12] M. Ghandhari, G. Andersson, I.A. Hiskens, Control lyapunov functions for controllable series devices, IEEE Trans. Power Syst. 16 (4) (2001) 689– 694. [13] M.A. Pai, Energy Function Analysis for Power System Stability, Kluwer Academic Publishers, 1989. [14] Manitoba HVDC Research Centre, ‘PSCAD/ EMTDC user’s manual’, Available: http://www.hvdc.ca, accessed Dec. 2006. [15] DeLeon, F. and Ooi, B.T., ‘Damping Power System Oscillations by Unidirectional Control of Alternative Generation Plants’, IEEE Power Engineering Society Meeting, 2001, Paper 0-7803-6672-7/01. [16] M. Noroozian, et al., Improving power system dynamics by series connected FACTS devices, IEEE Trans. Power Deliv. 12 (4) (1997) 1635– 1641. [17] M. Ghandhari, G. Andersson, Damping of inter-area and local modes by the use of controllable components, IEEE Trans. Power Deliv. 10 (4) (1995) 2007–2012.