Simultaneous Stabilization of Power System Using UPFC-Based ...

Electric Power Components and Systems, 34:941–959, 2006 Copyright © Taylor & Francis Group, LLC ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325000600596635

Simultaneous Stabilization of Power System Using UPFC-Based Controllers ALI T. AL-AWAMI Electrical Engineering Department King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia

Y. L. ABDEL-MAGID Electrical Engineering Program Petroleum Institute Abu Dhabi, United Arab Emirates

M. A. ABIDO Electrical Engineering Department King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia This article studies the use of robust UPFC-based stabilizers to damp low frequency oscillations. The potential of the UPFC-based stabilizers to enhance the dynamic stability is evaluated by singular value decomposition. Particle swarm optimization technique is used to optimize the parameters of each stabilizer, first individually, then concurrently. To ensure the robustness of the proposed stabilizers, the design process considers a wide range of operating conditions. The effectiveness of the proposed controllers is verified through several linear and nonlinear analysis techniques. These techniques prove that the coordinated design of UPFC-based stabilizers is superior over any of the individual designs. Keywords FACTS devices, UPFC, particle swarm optimization, simultaneous stabilization, power system stability

1.

Introduction

As power demand grows rapidly and expansion in transmission and generation is restricted, power systems are today much more loaded than before. This causes the power systems to be operated near their stability limits. In addition, interconnection between Manuscript received in final form on 24 January 2006. The authors would like to acknowledge King Fahd University of Petroleum and Minerals for the support of this work under Project # KFUPM EE/Power/282. Address correspondence to Dr. M. A. Abido, Electrical Engineering Dept., KFUPM Box 183, Dhahran 31261, Saudi Arabia. E-mail: [email protected]

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remotely located power systems gives rise to low frequency oscillations in the range of 0.1–3.0 Hz. If not well damped, these oscillations may keep growing in magnitude, resulting in loss of synchronism. Power system stabilizers (PSSs) have been used in the last few decades to serve the purpose of enhancing power system damping to low frequency oscillations. PSSs have proved to be efficient in performing their assigned tasks. However, they may adversely affect voltage profile and may not be able to suppress oscillations resulting from severe disturbances, especially those that may occur at the generator terminals. A wide spectrum of PSS tuning approaches has been proposed. These approaches have included pole placement [1], damping torque concepts [2], H∞ [3], variable structure [4], and the different optimization and artificial intelligence techniques [5–7]. FACTS devices have shown very promising results when used to improve power system steady-state performance. Because of the extremely fast control action associated with FACTS-device operations, they have been potential candidates for utilization in power system damping enhancement. A unified power flow controller (UPFC) is the most versatile device in the FACTS concept. It has the ability to adjust the three control parameters, i.e., the bus voltage, transmission line reactance, and phase angle between two buses, either simultaneously or independently. A UPFC performs this through the control of the in-phase voltage, quadrature voltage, and shunt compensation. Until now, not much research has been dedicated to the analysis and control of UPFCs. Several trials have been reported in the literature to model a UPFC for steadystate and transient studies. Based on Nabavi-Iravani model [8], Wang developed two UPFC models [9, 10] that have been linearized and incorporated into the Heffron-Phillips model [11]. A number of control schemes have been suggested to perform the oscillation-damping task. Huang et al. [12] attempted to design a conventional fixed-parameter lead-lag controller for a UPFC installed in the tie line of a two-area system to damp the interarea mode of oscillation. Mok et al. [13] considered the design of an adaptive fuzzy logic controller for the same purpose. Dash et al. [14] suggested the use of a radial basis function NN for a UPFC to enhance system damping performance. Robust control schemes, such as H∞ and singular value analysis, have also been explored [15, 16]. To avoid pole-zero cancellation associated with the H∞ approach, the structured singular value analysis have been utilized in [17] to select the parameters of the UPFC controller to have the robust stability against model uncertainties. In this work, singular value decomposition (SVD) is used to select the control signals that are most suitable for damping the electromechanical (EM) mode oscillations. This is done as SVD analysis can be readily used to estimate the EM mode controllability of the different UPFC-based stabilizers. A weakly connected system equipped with a UPFC is used in this study. The problem of designing the damping stabilizers is formulated as an optimization problem to be solved using PSO. The aim of the optimization is to search for the optimum controller parameter settings that minimize the maximum damping factor of the system complex modes. To ensure the robustness of the proposed stabilizers, the design process simultaneously takes into account several loading conditions with and without system parameter uncertainties, a process called simultaneous stabilization. In addition, a coordinated scheme of the UPFC-based stabilizers is presented. Damping torque coefficient, eigenvalue and nonlinear simulation analyses are used to assess the effectiveness of the proposed stabilizers to damp low frequency oscillations.

Simultaneous Stabilization of Power System

2. 2.1.

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Problem Statement Nonlinear Model of Power System with UPFC

Figure 1 shows a SMIB system equipped with a UPFC. The UPFC consists of an excitation transformer (ET), a boosting transformer (BT), two three-phase GTO based voltage source converters (VSCs), and DC link capacitors. The four input control signals to the UPFC are mE , mB , δE , and δB , where mE is the excitation amplitude modulation ratio, mB is the boosting amplitude modulation ratio, δE is the excitation phase angle, and δB is the boosting phase angle. By applying Park’s transformation and neglecting the resistance and transients of the ET and BT transformers, the UPFC can be modeled as [8–10]:  m cos δ v  E E dc


   2 vEtd 0 −xE iEd  = + (1)  iEq vEtq xE 0 mE sin δE vdc  2  m cos δ v  B B dc


   2 vBtd 0 −xB iBd  = + (2)  m sin δ v  vBtq xB 0 iBq B B dc 2



iEd

iBd 3mB 3mE cos δE sin δE cos δE sin δB v˙dc = + (3) iEq iBq 4Cdc 4Cdc where vEt , iE , vBt , and iB are the excitation voltage, excitation current, boosting voltage, and boosting current, respectively; Cdc and vdc are the DC link capacitance and voltage, respectively.

Figure 1. SMIB power system equipped with UPFC.

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A. T. Al-Awami et al. The nonlinear model of the SMIB system of Figure 1 is δ˙ = ωb (ω − 1)

(4)

ω˙ = (Pm − Pe − D(ω − 1))/M

(5)

 E˙ q = [−Eq + Ef d + (xd − xd )id ]/Tdo

(6)

E˙ f d = (KA (Vref − v + uPSS ) − Ef d )/TA

(7)

where Pm and Pe are the input and output power, respectively. Pe = vd id + vq iq ,

v = (vd2 + vq2 )1/2 , id = iEd + iBd ,

vd = xq iq ,

vq = Eq − xd id ,

iq = iEq + iBq

(8)

M and D the inertia constant and damping coefficient, respectively; ωb the synchronous speed; δ and ω the rotor angle and speed, respectively; Eq , Ef d , and v the generator  the open circuit field time constant; internal, field and terminal voltages, respectively; Tdo  xd , xd , and xq the d-axis reactance, d-axis transient reactance, and q-axis reactance, respectively; KA and TA the exciter gain and time constant, respectively; Vref the reference voltage; and uPSS the PSS control signal. Also from Figure 1, we have v = j xtE i t + v Et v Et = v Bt + j xBV i B + v b

(9) (10)

where it and vb , are the armature current and infinite bus voltage, respectively; vEt , vBt , and iB the ET voltage, BT voltage, and BT current, respectively. From (9) we have vd + j vq = xq (iEq + iBq ) + j [Eq − xd (iEd + iBd )] = j xtE (iEd + iBd + j iEq + j iBq ) + vEtd + vEtq

(11)

And from (10) we have vEtd + j vEtq = vBtd + j vBtq + j xBV iBd − xBV iBq + vb sin δ + j vb cos δ

(12)

From (1)–(2) and (11)–(12) we can find  xBB  mE sin δE vdc xBd xdE mB sin δB vdc vb cos δ + Eq − + xd 2xd xd 2  mE cos δE vdc xBq xqE mB cos δB vdc vb sin δ + = − 2xq xq 2  xE  mE sin δE vdc xdE xdt mB sin δB vdc vb cos δ + = Eq + − xd 2xd xd 2  mE cos δE vdc xqE xqt mB cos δB vdc vb sin δ + =− + 2 2xq xq

iEd =

(13)

iEq

(14)

iBd iBq

(15) (16)

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where xq = (xq + xtE + xE )(xB + xBV ) + xe (xq + xtE ), xBq = xB + xBV + xq + xtE xqt = xq + xtE + xE , xd

xqE = xq + xtE

= (xd + xtE + xE )(xB + xBV ) + xE (xd + xtE ), xBd = xB + xBV + xd + xtE xdt = xd + xtE + xE ,

xdE = xd + xtE

xBB = xB + xBV

(17)

(18)

(19)

(20) (21)

xE and xB are the ET and BT reactances, respectively. 2.2.

Linearized Model of Power System with UPFC

The nonlinear dynamic equations can be linearized around a given operating point to have the linear model given below: !δ˙ = ωb !ω

(22)

!ω˙ = (!Pm − !Pe − D!ω)/M

(23)

 !E˙ q = [−!Eq + !Ef d + (xd − xd )!id ]/Tdo

!E˙ f d = [−!Ef d + KA (!Vref − !v + !upss ]/TA !v˙dc = K7 !δ + K8 !Eq − K9 !vdc + Kce !mE + Kcδe !δE + Kcb !mB + Kcδb !δB

(24) (25)

(26)

where !Pe = K1 !δ + K2 !Eq + Kpd !vdc + Kpe !mE + Kpδe !δE + Kpb !mB + Kpδb !δB !Eq = K4 !δ + K3 !Eq + Kqd !vdc + Kqe !mE + Kqδe !δE + Kqb !mB + Kqδb !δB !v = K5 !δ + K6 !Eq + Kvd !vdc + Kve !mE + Kvδe !δE + Kvb !mB + Kvδb !δB K1 –K9 , Kpu , Kqu , and Kvu are linearization constants.

(27)

(28)

(29)

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A. T. Al-Awami et al. In state-space representation, the power system can be modeled as x˙ = Ax + Bu

(30)

where the state vector x and control vector u are

T x = !δ !ω !Eq !Ef d !vdc u = !upss 

!mE ωb

0

 K1   −  M   K4  −  A=  Tdo    K A K5 −  TA 

   0     0 B=     KA  T  A 0

0

0

T

K2 M

0

0

−

K3  Tdo

1  Tdo

−

K A K6 TA K8

0 0

−

(32)      Kqd   −   Tdo    KA Kvd   − TA   −

1 TA

0

0

Kpe M

−

Kpδe M

−

Kpb M

−

Kqe  Tdo

−

Kqδe  Tdo

−

Kqb  Tdo

KA Kve TA Kce

−

KA Kvδe TA Kcδe

Kpd M

(33)

−K9

0

−

−



0

−

0

0

!δb

D M

−

K7 

!mb

!δE

(31)

−

KA Kvb TA Kcb

0



     Kqδb   −   Tdo    KA Kvδb   TA   −

Kpδb M

(34)

Kcδb

The linearized dynamic model of the state-space representation is shown in Figure 2. Only one UPFC control input is considered in this figure. 2.3.

UPFC-Based Stabilizers

The UPFC damping stabilizers are the widely used lead-lag scheme of the structure, shown in Figure 3, where u can be mE , mB , or δB . In order to maintain the power balance between the series and shunt converters, a DC voltage regulator must be incorporated. The DC voltage is controlled through modulating the phase angle of the ET voltage, δE . Therefore, the δE damping controller to be considered is that shown in Figure 4, where the DC voltage regulator is a PIcontroller. 2.4.

Objective Function and Stabilizer Design

In this study, several loading conditions, including nominal, light, heavy, and leading power factor without and with system parameter uncertainties, are considered simultaneously to ensure the robustness of the proposed stabilizers. To select the best stabilizer

Simultaneous Stabilization of Power System

Figure 2. Modified Heffron-Phillips linearized model.

Figure 3. UPFC with lead-lag controller.

Figure 4. UPFC with lead-lag controller and DC voltage regulator.

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parameters that enhance the most power system transient performance, the problem is formulated so as to optimize a selected objective function J subject to some inequality constraints, which are the maximum and minimum limits of each controller gain K and time constants T1 –T4 . In this work, J = max{σi }

(35)

where σi is the damping factor corresponding to the least damped complex mode of the ith loading condition. Hence, the design problem can be formulated as: minimize J Subject to

K min ≤ K ≤ K max T1min ≤ T1 ≤ T1max T2min ≤ T2 ≤ T2max T3min ≤ T3 ≤ T3max T4min ≤ T4 ≤ T4max

The proposed approach employs PSO to search for the optimum parameter settings of the given controllers.

3.

Controllability Measure

To measure the controllability of the EM mode by a given input (control signal), the singular value decomposition (SVD) is employed. Mathematically, if G is an m × n complex matrix, then there exist unitary matrices W and V with dimensions of m × m and n × n, respectively, such that G=W V

(36)

where
=

1

0

 0 , 0

1

= diag(σ1 , . . . , σr )

(37)

with σ1 ≥ · · · ≥ σr ≥ 0 where r = min{m, n} and σ1 , . . . , σr are the singular values of G. The minimum singular value σr represents the distance of the matrix G from all the matrices with a rank of r − 1. This property can be used to quantify modal controllability [18]. The matrix B can be written as B = [b1 b2 b3 b4 b5 ] where bi is a column vector corresponding to the ith input. The minimum singular value, σmin , of the matrix [λI − A bi ] indicates the capability of the ith input to control the mode associated with the eigenvalue λ. Actually, the higher the σmin , the higher the controllability of this mode by the input considered. As such, the controllability of the EM mode can be examined with all inputs in order to identify the most effective one to control the mode.

Simultaneous Stabilization of Power System

4.

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Particle Swarm Optimizer

A novel population based optimization approach, called particle swarm optimization (PSO) approach, was introduced first in [19]. This new approach features many advantages; it is simple, fast and can be coded in few lines. Also, its storage requirement is minimal. Moreover, this approach is advantageous over evolutionary and genetic algorithms in many ways. First, PSO has memory. That is, every particle remembers its best solution (local best) as well as the group best solution (global best). Another advantage of PSO is that the initial population of the PSO is maintained, and so there is no need for applying operators to the population, a process that is time and memory storage consuming [19–21]. PSO starts with a population of random solutions “particles” in a D-dimension space. The ith particle is represented by Xi = (xi1 , xi2 , . . . , xiD ). Each particle keeps track of its coordinates in hyperspace, which are associated with the fittest solution it has achieved so far. The value of the fitness for particle i (pbest) is also stored as Pi = (pi1 , pi2 , . . . , piD ). The global version of the PSO keeps track of the overall best value (gbest), and its location, obtained thus far by any particle in the population [20, 21]. PSO consists of, at each step, changing the velocity of each particle toward its pbest and gbest according to Eq. (38). The velocity of particle i is represented as Vi = (vi1 , vi2 , . . . , viD ). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and gbest. The position of the ith particle is then updated according to Eq. (39) [20, 21]. vid = w ∗ vid + c1 ∗ rand() ∗ (pid − xid ) + c2 ∗ Rand() ∗ (pgd − xid )

(38)

xid = xid + vid

(39)

where pid = pbest

and

pgd = gbest

(40)

The PSO algorithm can be described as follows: Step 1: Initialize an array of particles with random positions and their associated velocities to satisfy the inequality constraints. Step 2: Check for the satisfaction of the equality constraints and modify the solution if required. Step 3: Evaluate the fitness function of each particle. Step 4: Compare the current value of the fitness function with the particles’ previous best value (pbest). If the current fitness value is less, then assign the current fitness value to pbest and assign the current coordinates (positions) to pbestx. Step 5: Determine the current global minimum fitness value among the current positions. Step 6: Compare the current global minimum with the previous global minimum (gbest). If the current global minimum is better than gbest, then assign the current global minimum to gbest and assign the current coordinates (positions) to gbestx. Step 7: Change the velocities according to Eq. (38). Step 8: Move each particle to the new position according to Eq. (39) and return to Step 2. Step 9: Repeat Step 2–Step 8 until a stopping criterion is satisfied or the maximum number of iterations is reached.

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Simulation Results Controllability Measure

First, singular value decomposition is employed to measure the controllability of the EM mode from each of the UPFC input signals: mE , δE , mB , and δB . The minimum singular value, σmin , is estimated over a wide range of operating conditions. For SVD analysis, Pe ranges from 0.05 to 1.4 p.u. and Qe = [0, 0.4]. At each loading condition, the system model is linearized, the EM mode is identified, and the SVD-based controllability measure is implemented. For comparison purposes, the minimum singular value for the four inputs at Qe = 0.0 p.u. and Qe = 0.4 p.u. are shown in Figures 5 and 6, respectively. From these figures, it is evident that δE and mB are more powerful than δB and mE . Hence, they are the UPFC signals to be considered throughout this study. It can be also concluded that: • δE capability of controlling the EM mode is almost independent of loading. • The controllability of δE is generally higher than those of the other stabilizers, especially for low and moderate loading. • At very low loading, all the stabilizers, except the mB -based stabilizer at nominal loading and the δE -based stabilizer, suffer from poor controllability. 5.2.

Stabilizer Design

The objective is to design robust δE - and mB -based stabilizers. Both individual and coordinated designs are to be considered. To ensure the stabilizer effectiveness over a wide range of operating conditions, the design process takes into account several loading conditions including nominal, light, heavy, and leading Pf conditions. These conditions are considered without and with system parameter uncertainties, such as machine inertia, line impedance, and field time constant. The total number of operating conditions considered during the design process is 16, as given in Table 1. First, PSO is used to optimize the parameters of each stabilizer individually. The objective function is to minimize the maximum damping factors of all the complex eigenvalues associated with the 16 operating points simultaneously. Then, PSO is used to search for the optimum parameters of both controllers simultaneously. The final settings of the optimized parameters for the proposed stabilizers are given in Table 2. For reference, Table 3 lists the open-loop eigenvalues associated with the EM modes

Table 1 Loading conditions and parameter uncertainties considered in the design stage Loading condition (Pe , Qe ) p.u.

Parameter uncertainties

Nominal (1.0, 0.015) Light (0.3, 0.100) Heavy (1.1, 0.100) Leading Pf (0.7, −0.30)

No parameter uncertainties 30% increase of line reactance X 25% decrease of machine inertia M  30% decrease of field time constant Tdo

Simultaneous Stabilization of Power System

Figure 5. Minimum singular value with all stabilizers at Qe = 0.0.

Figure 6. Minimum singular value with all stabilizers at Qe = 0.4.

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A. T. Al-Awami et al. Table 2 The optimal parameter settings of the individual and coordinated designs Individual

K T1 T2 T3 T4

Coordinated

δE

mB

δE

mB

−100.00 5.0000 1.0379 0.0643 1.5452

96.862 4.9980 2.5728 0.1203 0.0100

−66.1817 1.5345 1.6110 4.4207 3.9503

100 5.0000 3.0948 5.0000 3.3213

of all the 16 operating points considered in the robust design process. It is evident that all these modes are unstable. To evaluate the performances of the proposed stabilizers, damping torque coefficient analysis, eigenvalue analysis, and nonlinear time-domain simulations are carried out. It is worth mentioning that all the time domain simulations are carried out using the nonlinear power system model. The system data is given in the appendix. 5.3.

Damping Torque Coefficient

To evaluate the effectiveness of the proposed stabilizers, the damping torque coefficient (Kd ) has been estimated with those stabilizers when designed individually. In addition, the damping torque coefficient of the coordinated design is also estimated. Figures 7 and 8 show Kd versus the loading variations with the different schemes at Q = 0.0 and Q = 0.4 p.u., respectively. These figures address the highly effective coordinated design, as compared with δE and mB individual designs, over the different loading conditions. 5.4.

Eigenvalue Analysis and Time-Domain Simulations

The system EM modes with the proposed UPFC-based controllers when applied at the four loading conditions—nominal, light, heavy, and leading Pf—are given in Table 4.

Table 3 Open-loop eigenvalues associated with the EM modes of all the 16 points considered in the robust design process

Loading

No parameter uncertainties

30% increase of line reactance X

25% decrease of machine inertia M

30% decrease of field time  constant Tdo

Nominal (N) Light (L) Heavy (H) Leading Pf (P)

1.5033 ± 5.3328i 1.3952 ± 5.0825i 1.4138 ± 5.0066i 1.4502 ± 5.3584i

1.4191 ± 4.9900i 1.3254 ± 4.7414i 1.2553 ± 4.5295i 1.4092 ± 5.0888i

1.8047 ± 5.9408i 1.6730 ± 5.6676i 1.7027 ± 5.5775i 1.7461 ± 5.9774i

1.5034 ± 5.4025i 1.3951 ± 5.0913i 1.4038 ± 5.0846i 1.4498 ± 5.3952i

Simultaneous Stabilization of Power System

Figure 7. Damping coefficient with load variations, Q = 0.0 p.u.

Figure 8. Damping coefficient with load variations, Q = 0.4 p.u.

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A. T. Al-Awami et al. Table 4 System eigenvalues with all the stabilizers at different loading conditions

Loading

δE

mB

δE and mB

N L H P

−3.5239 ± 5.3275i −2.9370 ± 5.6541i −1.9204 ± 1.7604i −1.8267 ± 5.4732i

−3.9100 ± 12.720i −3.7100 ± 12.190i −3.5600 ± 13.120i −3.5300 ± 2.6100i

−7.5110 ± 10.648i −5.8186 ± 11.047i −0.6485 ± 0.3310i −5.8657 ± 6.4690i

It is evident that, using the proposed stabilizers design, the damping factors of the EM mode eigenvalues are greatly enhanced. It is also clear that the coordinated design outperforms the two individual designs in shifting the system electromechanical mode to the left of the s-plane. Moreover, the nonlinear time-domain simulations are carried out at the nominal and light loading conditions specified previously. The speed deviations, DC voltage, and electrical power for a 6-cycle three-phase fault at nominal and light loading conditions are shown in Figures 9–14, respectively. The simulation results obtained indicate a clear enhancement of the proposed coordinated δE -mB design over both the individual designs. This enhancement can be easily recognized from the sound reduction in overshoot and settling time of the speed, electrical power and DC voltage responses.

Figure 9. Speed response to 6-cycle fault disturbance for nominal loading.

Simultaneous Stabilization of Power System

Figure 10. Speed response to 6-cycle fault disturbance for light loading.

Figure 11. UPFC DC voltage response to 6-cycle fault disturbance for nominal loading.

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Figure 12. UPFC DC voltage response to 6-cycle fault disturbance for light loading.

Figure 13. Electrical power response to 6-cycle fault disturbance for nominal loading.

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Figure 14. Electrical power response to 6-cycle fault disturbance for light loading.

The control efforts of the coordinated design are also greatly reduced as compared with the control efforts of the two individual designs, but are not shown here for space limitations.

6.

Conclusions

In this article, SVD has been employed to evaluate the EM mode controllability to the UPFC control signals. An optimization technique has been proposed to design the UPFCbased stabilizers individually. PSO has been utilized to search for the optimal controller parameter settings that optimize an eigenvalue-based objective function. To guarantee the robustness of the proposed controllers, the design process is carried out considering a wide range of operating conditions. A similar approach has been used to design the UPFCbased stabilizers in a coordinated manner. To compare the effectiveness of the proposed stabilizer schemes, damping torque coefficient analysis, eigenvalue analysis, and nonlinear time domain simulation analysis have been carried out. The different analysis results have proved that the coordinated scheme of the UPFC-based stabilizers outperforms the individual schemes.

7.

Appendix

The test system parameters are:  = 5.044, freq = 60, Machine: xd = 1, xq = 0.6, xd = 0.3, D = 0, M = 8.0, Tdo v = 1.05;

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A. T. Al-Awami et al.

Exciter: KA = 50, TA = 0.05, Ef d_ max = 7.3, Ef d_ min = −7.3; PSS: Tw = 5; Ti_ min = 0.01; Ti_ max = 5; i = 1, 2, 3, 4; |uPSS | ≤ 0.2; |BSVC | ≤ 0.4; |XTCSC | < 0.5; |2TCPS | < 15 deg; DC voltage regulator: kdp = −10, kdi = 0; Transmission line: X = 0.7; UPFC: xE = 0.1, xB = 0.1, Ks = 1, Ts = 0.05, Cdc = 3, Vdc = 2.

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