# Optimal Coordinated Design of Multiple Damping Controllers Based on PSS and UPFC Device to Improve Dynamic Stability in the Power System

Authors: A. N. Hussain; F. Malek; M. A. Rashid; L. Mohamed; N. A. Mohd Affendi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Unified Power Flow Controller (UPFC) device is applied to control power flow in transmission lines. Supplementary damping controller can be installed on any control channel of the UPFC inputs to implement the task of Power Oscillation Damping (POD) controller. In this paper, we have presented the simultaneous coordinated design of the multiple damping controllers between Power System Stabilizer (PSS) and UPFC-based POD or between different multiple UPFC-based POD controllers without PSS in a single-machine infinite-bus power system in order to identify the design that provided the most effective damping performance. The parameters of the damping controllers are optimized utilizing a Chaotic Particle Swarm Optimization (CPSO) algorithm based on eigenvalue objective function. The simulation results show that the coordinated design of the multiple damping controllers has high ability in damping oscillations compared to the individual damping controllers. Furthermore, the coordinated design of UPFC-based POD controllers demonstrates the superiority over the coordinated design of PSS and UPFC-based POD controllers for enhancing greatly the stability of the power system.


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## Body

## 1. Introduction

When large power systems are interconnected through weak tie lines, Low Frequency Oscillations (LFO) in the range of 0.1–3 Hz are observed. These oscillations may sustain and grow to cause system separation if no adequate damping is available [1].The installation of a Power System Stabilizer (PSS) appears as a simple and inexpensive technique for many years to produce an amount of damping torque through the injection of a supplementary stabilizing signal at a voltage reference input of an Automatic Voltage Regulator (AVR), which has increased the stability of the power system. However, their performance deteriorated when the system operating conditions varied widely [2–4]. Hence, in order to operate power systems effectively without lessening the system safety and quality of supply, even in the case of emergency conditions such as loss of transmission lines and/or generating units, new control strategies need to be applied. Flexible AC Transmission System (FACTS) device is an effective technology that has appeared as an alternative way in controlling power flow and damping power system oscillations. Although the damping ratio of  FACTS-based stabilizer controllers is not often their main function, but its ability to increase power system damping characteristics has been acknowledged [5, 6].Unified Power Flow Controller (UPFC) is one of the most important FACTS device families. Its primary function is to control and optimize the real and reactive power flow in a given line, voltage, and current at the UPFC bus [7, 8]. This is achieved by regulating the controllable parameters of the transmission system such as line impedance, phase angle, and voltage magnitude. Beside these primary functions, it can also provide appropriate damping effect to the tie interconnected modern power systems oscillation through its supplementary controller and increase system stability [9].To improve general power system performance, possible interactions between PSSs and FACTS-damping controllers are considered, but uncoordinated local design of PSS and FACT-damping controller may cause destabilizing interaction on the damping of system oscillations [10]. Therefore, coordinated design of PSSs and FACTS-based stabilizers is a necessity both to make use of the benefits of multiple stabilizers, which will enhance the system stability and prevent the drawbacks accompanied with their operation.Several researches have been carried out for the coordination between PSSs and FACTS damping controllers. Some of these researches were depending on the complex nonlinear simulation while the others based on the linearized model for power system [11–17]. One approach for achieving the required performance is to make the coordinated design of the controller as a constrained optimization problem. The constraints include limits of controller parameters. The most traditional optimization methods used some deterministic rules to move from one point to another in the decision hyperspace. Therefore, probability of getting stuck at a local optimum is the main problem with these methods [18].Recently, Particle Swarm Optimization (PSO) technique has appeared as a useful tool for engineering global optimization. PSO is a population-based stochastic optimization method, which employs the swarm intelligence produced by the cooperation and rivalry between the particles in a swarm. PSO is unlike the other evolutionary algorithms having many advantages. The major advantages are as follows: it is fast and simple, does need to apply operators such as GA algorithm, and is easy to be implemented [19, 20]. The simple PSO greatly depends on its parameters, and it is not guaranteed to be global convergent. In order to improve the global searching ability and prevent a slide into the premature convergence to local minima, PSO and chaotic sequence techniques are combined to form a Chaotic Particle Swarm Optimization (CPSO) technique.In this paper, we present the results of our comprehensive comparison and assessment of the damping function of multiple damping stabilizers using different coordinated designs in order to identify the design that provided the most effective damping performance. The two alternative designs we evaluated are listed below:(i)
coordinated design between PSS and any one out of the four input control channels of the series and shunt structure of UPFC device because any control loop can superimpose a supplementary damping controller to implement the required damping,(ii)
coordinated design between any two out of the four input control channels of the series and shunt structure of UPFC device as a multiple damping controller without using PSS.The parameters of the damping controllers for individual and coordinated design are optimized utilizing CPSO technique based on eigenvalue objective function. The simulation results of the individual damping controllers show the best damping effects resulting from using the POD controllersδE and mB are higher than PSS response. In addition, the coordinated designs (mB&δE), (mE&δE), (δE&δB), and (PSS&δE) are able to damp the LFO effectively and can accomplish this task faster and with lower overshoot in comparison to their individual control responses or the best individual controls δE and mB.


## 2. Mathematical Model of Power System for SMIB Equipped with UPFC

Figure1 shows a single-machine infinite bus (SMIB) equipped with a UPFC. The synchronous generator is supplying electric power to the infinite bus through a transmission line and UPFC. The UPFC consists of two three-phase GTO based voltage source converters (VSCs) coupled through a common dc terminal. VSC-1 is connected in shunt with the line through an excitation transformer (ET), and VSC-2 is inserted in series with the transmission line through a boosting transformer (BT). The dc voltage for both converters is provided by a common capacitor bank [21]. Among the four input control signals to the UPFC, mE and δE represent the amplitude modulation ratio and phase angle signals of shunt converter VSC-1; mB and δB represent the amplitude modulation ratio and phase angle signals of converter VSC-2. These parameters are considered as UPFC control inputs to provide synchronized power compensation in series line without external voltage source [4, 22].Figure 1
SMIB power system equipped with UPFC.The structure of the UPFC controller is shown in Figure2, where N can be mE, mB, δE, and δB. In order to maintain the power flow 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 is to be considered as shown in Figure 3, where the dc voltage regulator is a PI-controller [23].Figure 2
UPFC with damping controller.Figure 3
UFPC with dc voltage regulator and damping controller.


### 2.1. Nonlinear Dynamic Model of UPFC

Referring to Figure1, the nonlinear differential equations describe the dynamic behavior of the UPFC, the three-phase excitation current, three-phase boosting current, and dc link voltage which can be written as follows [24]:
(1)[diEadtdiEbdtdiEcdt]=[-rElE000-rElE000-rElE][iEaiEbiEc]-mEvdc2lE[cos(ωt+δE)cos(ωt+δE-120°)cos(ωt+δE+120°)]+[1lE0001lE0001lE][vEtavEtbvEtc],[diBadtdiBbdtdiBcdt]=[-rBlB000-rBlB000-rBlB][iBaiBbiBc]-mBvdc2lB[cos(ωt+δB)cos(ωt+δB-120°)cos(ωt+δB+120°)]+[1lB0001lB0001lB][vBtavBtbvBtc],dvdcdt=mE2Cdc[cos(ωt+δE)cos(ωt+δE-120°)cos(ωt+δE+120°)]T[iEaiEbiEc]+mB2Cdc[cos(ωt+δB)cos(ωt+δB-120°)cos(ωt+δB+120°)]T[iBaiBbiBc],
where vEt, iE, rE, and lE are the excitation transformer voltage, current, resistance, and inductance, respectively; vBt, iB, rB, and lB are the boosting voltage, current, resistance, and inductance respectively; Cdc, vdc are the  dc link capacitance and voltage respectively.By applying Park’s transformation and ignoring the resistance and transient of the UPFC transformers, equations (1) become as follows [25]:
(2)[vEtdvEtq]=[0-xExE0][iEdiEq]+[vdcmEcosδE2vdcmEsinδE2],[vBtdvBtq]=[0-xBxB0][iBdiBq]+[vdcmBcosδB2vdcmBsinδB2],(3)dvdcdt=3mE4Cdc[cosδEsinδE][iEdiEq]+3mB4Cdc[cosδBsinδB][iBdiBq],
where xE is the ET reactance and xB is the BT reactance. From Figure 1, we can have
(4)V-t=jxtEI-+V-Et,V-Et=V-Bt+jxBvI-B+V-b,(5)I-=I-E+I-B,
where I is the armature current, Vt is the generator terminal voltage, Vb is the infinite-bus voltage, xtE and xBv are the reactances of a transmission line. The terminal voltage and the armature current of the generator can be expressed in terms of the d-axis and q-axis components:
(6)vt=vtd+jvtq(6′)vtd=xqiq;vtq=Eq′-x`did(7)i=id+iq(7′)id=iEd+iBd;iq=iEq+iBq,
where x`d is the d-axis transient reactance, xq is the q-axis reactance, and Eq′ is the transient generator internal voltage. Substituting (6′) and (7′) into (6) gives
(8)vtd+jvtq=xq(iEq+iBq)+j[Eq′-x`d(iEd+iBd)].
Equation (4) can be expressed in the d-q axis reference frame as:
(9)vtd+jvtq=jxtE(iEd+jiEq+iBd+jiBq)+vEtd+jvEtq,vEtd+jvEtq=vBtd+jvBtq+jxBviBd-xBviBq+vbsinδ+jvbcosδ.
From (2), (8), and (9), we have obtained the current injection equations of UPFC:
(10)iEd=xBBxdΣEq′-mEsinδEvdcxBd2xdΣ+xdExdΣ(Vbcosδ+mBsinδBvdc2),iEq=mEcosδEvdcxBq2xqΣ-xqExqΣ(Vbsinδ+mBcosδBvdc2),iBd=xExdΣEq′+mEsinδEvdcxdE2xdΣ-xdtxdΣ(Vbcosδ+mBsinδBvdc2),iBq=-mEcosδEvdcxqE2xqΣ+xqtxqΣ(Vbsinδ+mBcosδBvdc2),
where
(11)xBB=xB+xBv,xdΣ=(x`d+xtE+xE)(xB+xBv)+xE(x`d+xtE)xBd=xB+xBv+x`d+xtE,xdE=x`d+xtE,xBq=xB+xBv+xq+xtExqΣ=(xq+xtE+xE)(xB+xBv)+xE(xq+xtE),xdt=x`d+xtE+xExqt=xq+xtE+xE,xqE=xq+xtE,
where xd is the d-axis reactance.


### 2.2. Nonlinear Model of the Power System

The nonlinear dynamic equations of the SMIB system shown in Figure1 are described by [26]:
(12)δ˙=ωb(ω-1),ω˙=[Pm-Pe-D(ω-1)]M,E˙q′=[Efd-(xd-x`d)id-Eq′]Tdo′,
where ωb is the synchronous speed, δ and ω are the angle and speed of the rotor, respectively, Pm and Pe are the input mechanical and output electrical powers of the generator, respectively, M and D are the machine inertia constant and damping coefficient, respectively, Efd is the generator field voltage, and Tdo′ is the open-circuit field time constant.The excitation system is represented by a first-order model (IEEE type—ST1) [27]:
(13)E˙fd=[Ka(Vref-vt)-Efd]Ta,
where Ka and Ta are the gain and time constant of excitation system, respectively; vref is the reference voltage.The output power of the generator can be expressed in terms of thed-axis and q-axis components of the armature current and terminal voltage as
(14)Pe=vtdid+vtqiq.


### 2.3. Linearized Model of a Power System Equipped with UPFC

Linear dynamic model of the power system is obtained by linearizing the nonlinear equations (2)–(14) around nominal operating point. The linearized model of the power system as shown in Figure 1 is given as follows:(15)Δδ˙=ωbΔω,Δω˙=(ΔPm-ΔPe-DΔω)M,ΔE˙q′=[-ΔEq′+ΔEfd-(xd-x`d)Δid]Tdo′,(16)ΔE˙fd=(-KaΔvt-ΔEfd)Ta,
where
(17)ΔPe=K1Δδ+K2ΔEq′+KpcΔvdc+KpeΔmE+KpδeΔδE+KpbΔmB+KpδbΔδB,ΔEq′=K4Δδ+K3ΔEq′+KqcΔvdc+KqeΔmE+KqδeΔδE+KqbΔmB+KqδbΔδB,Δvt=K5Δδ+K6ΔEq′+KvcΔvdc+KveΔmE+KvδeΔδE+KvbΔmB+KvδbΔδB,(18)Δv˙dc=K7Δδ+K8ΔEq′-K9Δvdc+KceΔmE+KcδeΔδE+KcbΔmB+KcδbΔδB,
where the linearization constants K1–K9, Kpc, Kpe, Kpδe, Kpb, Kpδb, Kqc, Kqe, Kqδe, Kqb, Kqδb, Kvc, Kve, Kvδe, Kvb, Kvδb, Kce, Kcδe, Kcb, and Kcδb are functions of the system parameters and the initial operating conditions.Referring to Figure2, the UPFC dynamic models of ΔmE, ΔmB, and ΔδB are described by the following state equations:
(19)Δm˙E=1Ts1(-ΔmE+KS1ΔUmE),Δm˙B=1TS3(-ΔmB+KS3ΔUmB),Δδ˙B=1TS4(-ΔδB+KS4ΔUδB).In addition, from Figure3, the UPFC dynamic model of a dc voltage regulator is described by the following state equations:
(20)Δδ˙E=1TS2(KS2ΔXdc-ΔδE+KS2ΔUδE),(21)ΔX˙dc=-KdPK7Δδ-KdPK8ΔEq′-(Kdi+KdPK9)Δvdc-KdPKceΔmE-KdPKcδeΔδE-KdPKcbΔmB-KdPKcδbΔδB,
where KdP and Kdi are the proportional and integral gains of the dc voltage regulator controller, respectively; (KS1–KS4) and (TS1–TS4) are the gains and times constants of the main control loops for UPFC, respectively.Equation (15) describes the model of the machine. Equations (16) and (18) represent the models for the exciter and the UPFC dc voltage regulators link, respectively. Equations (19) and (20) represent the control action of the main control loops of the UPFC with damping controller, and (21) represents the control action of PI dc voltage regulator. In state-space representation, these equations can be arranged in compact form as
(22)ΔX˙=AΔX+BΔU,
where the state vector ΔX and control vector ΔU are(23)ΔX=[ΔδΔωΔE`qΔEfdΔVdcΔXdcΔmEΔδEΔmBΔδB]T,ΔU=[ΔUmEΔUδEΔUmBΔUδB]T.The structure of the matricesA and B is(24)A=[0ωb00000000-K1M-DM-K2M0-KpdM0-KpeM-KpδeM-KpbM-KpδbM-K4T`do0-K3T`do1T`do-KqdT`do0-KqeT`do-KqδeT`do-KqbT`do-KqδbT`do-K5KaTa0-K6KaTa-1Ta-KvdKaTa0-KveKaTa-KvδeKaTa-KvbKaTa-KvδbKaTaK70K80-K90KceKcδeKcbKcδb-KdPK70-KdPK80(Kdi+KdPK9)0-KdPKce-KdPKcδe-KdPKcb-KdPKcδb000000-1TS100000000KS2TS20-1TS20000000000-1TS30000000000-1TS4],B=[000000000000000000000000KS1TS10000KS2TS20000KS3TS30000KS4TS4].The modified Phillips-Heffron transfer function model including UPFC has 28 constants; on the other hand, the Phillips-Heffron model has only 6 constants as shown in Figure4. This figure shows that the damping torque provided by the UPFC device consists of two parts. The first part is applied directly to the electromechanical oscillation loop of the generator. Its sensitivity is mainly measured by coefficients Kpc, Kpe, Kpδe, Kpb, and Kpδb, which is called the direct-damping torque. The second part consists of coefficients Kvc, Kve, Kvδe, Kvb, and Kvδb, which is applied through the field channel of the generator. Its sensitivity is related to the deviation of field voltage, which is referred to as the indirect-damping torque.Figure 4
Modified Phillips-Heffron transfer function  model of SMIB system with UPFC.


### 2.4. Eigenvalue Analysis without Stabilizer

For nominal operating condition, the dynamic behavior of the system is recognized through the eigenvalues of the system matrixA. By solving the system characteristic equation |λI-A|=0, the eigenvalues of the system are computed which are given below:
(25)λ1=-19.752,λ2=-19.826,λ3=0.48394+3.4508i,λ4=0.48394-3.4508i,λ5=-1.7178,λ6=-0.13878+0.30433i,λ7=-0.13878-0.30433i,λ8=-20,λ9=-20,λ10=-20.It is clearly seen from eigenvalues of the matrixA that the system is unstable and needs a supplementary stabilizer for stability.


## 2.1. Nonlinear Dynamic Model of UPFC

Referring to Figure1, the nonlinear differential equations describe the dynamic behavior of the UPFC, the three-phase excitation current, three-phase boosting current, and dc link voltage which can be written as follows [24]:
(1)[diEadtdiEbdtdiEcdt]=[-rElE000-rElE000-rElE][iEaiEbiEc]-mEvdc2lE[cos(ωt+δE)cos(ωt+δE-120°)cos(ωt+δE+120°)]+[1lE0001lE0001lE][vEtavEtbvEtc],[diBadtdiBbdtdiBcdt]=[-rBlB000-rBlB000-rBlB][iBaiBbiBc]-mBvdc2lB[cos(ωt+δB)cos(ωt+δB-120°)cos(ωt+δB+120°)]+[1lB0001lB0001lB][vBtavBtbvBtc],dvdcdt=mE2Cdc[cos(ωt+δE)cos(ωt+δE-120°)cos(ωt+δE+120°)]T[iEaiEbiEc]+mB2Cdc[cos(ωt+δB)cos(ωt+δB-120°)cos(ωt+δB+120°)]T[iBaiBbiBc],
where vEt, iE, rE, and lE are the excitation transformer voltage, current, resistance, and inductance, respectively; vBt, iB, rB, and lB are the boosting voltage, current, resistance, and inductance respectively; Cdc, vdc are the  dc link capacitance and voltage respectively.By applying Park’s transformation and ignoring the resistance and transient of the UPFC transformers, equations (1) become as follows [25]:
(2)[vEtdvEtq]=[0-xExE0][iEdiEq]+[vdcmEcosδE2vdcmEsinδE2],[vBtdvBtq]=[0-xBxB0][iBdiBq]+[vdcmBcosδB2vdcmBsinδB2],(3)dvdcdt=3mE4Cdc[cosδEsinδE][iEdiEq]+3mB4Cdc[cosδBsinδB][iBdiBq],
where xE is the ET reactance and xB is the BT reactance. From Figure 1, we can have
(4)V-t=jxtEI-+V-Et,V-Et=V-Bt+jxBvI-B+V-b,(5)I-=I-E+I-B,
where I is the armature current, Vt is the generator terminal voltage, Vb is the infinite-bus voltage, xtE and xBv are the reactances of a transmission line. The terminal voltage and the armature current of the generator can be expressed in terms of the d-axis and q-axis components:
(6)vt=vtd+jvtq(6′)vtd=xqiq;vtq=Eq′-x`did(7)i=id+iq(7′)id=iEd+iBd;iq=iEq+iBq,
where x`d is the d-axis transient reactance, xq is the q-axis reactance, and Eq′ is the transient generator internal voltage. Substituting (6′) and (7′) into (6) gives
(8)vtd+jvtq=xq(iEq+iBq)+j[Eq′-x`d(iEd+iBd)].
Equation (4) can be expressed in the d-q axis reference frame as:
(9)vtd+jvtq=jxtE(iEd+jiEq+iBd+jiBq)+vEtd+jvEtq,vEtd+jvEtq=vBtd+jvBtq+jxBviBd-xBviBq+vbsinδ+jvbcosδ.
From (2), (8), and (9), we have obtained the current injection equations of UPFC:
(10)iEd=xBBxdΣEq′-mEsinδEvdcxBd2xdΣ+xdExdΣ(Vbcosδ+mBsinδBvdc2),iEq=mEcosδEvdcxBq2xqΣ-xqExqΣ(Vbsinδ+mBcosδBvdc2),iBd=xExdΣEq′+mEsinδEvdcxdE2xdΣ-xdtxdΣ(Vbcosδ+mBsinδBvdc2),iBq=-mEcosδEvdcxqE2xqΣ+xqtxqΣ(Vbsinδ+mBcosδBvdc2),
where
(11)xBB=xB+xBv,xdΣ=(x`d+xtE+xE)(xB+xBv)+xE(x`d+xtE)xBd=xB+xBv+x`d+xtE,xdE=x`d+xtE,xBq=xB+xBv+xq+xtExqΣ=(xq+xtE+xE)(xB+xBv)+xE(xq+xtE),xdt=x`d+xtE+xExqt=xq+xtE+xE,xqE=xq+xtE,
where xd is the d-axis reactance.


## 2.2. Nonlinear Model of the Power System

The nonlinear dynamic equations of the SMIB system shown in Figure1 are described by [26]:
(12)δ˙=ωb(ω-1),ω˙=[Pm-Pe-D(ω-1)]M,E˙q′=[Efd-(xd-x`d)id-Eq′]Tdo′,
where ωb is the synchronous speed, δ and ω are the angle and speed of the rotor, respectively, Pm and Pe are the input mechanical and output electrical powers of the generator, respectively, M and D are the machine inertia constant and damping coefficient, respectively, Efd is the generator field voltage, and Tdo′ is the open-circuit field time constant.The excitation system is represented by a first-order model (IEEE type—ST1) [27]:
(13)E˙fd=[Ka(Vref-vt)-Efd]Ta,
where Ka and Ta are the gain and time constant of excitation system, respectively; vref is the reference voltage.The output power of the generator can be expressed in terms of thed-axis and q-axis components of the armature current and terminal voltage as
(14)Pe=vtdid+vtqiq.


## 2.3. Linearized Model of a Power System Equipped with UPFC

Linear dynamic model of the power system is obtained by linearizing the nonlinear equations (2)–(14) around nominal operating point. The linearized model of the power system as shown in Figure 1 is given as follows:(15)Δδ˙=ωbΔω,Δω˙=(ΔPm-ΔPe-DΔω)M,ΔE˙q′=[-ΔEq′+ΔEfd-(xd-x`d)Δid]Tdo′,(16)ΔE˙fd=(-KaΔvt-ΔEfd)Ta,
where
(17)ΔPe=K1Δδ+K2ΔEq′+KpcΔvdc+KpeΔmE+KpδeΔδE+KpbΔmB+KpδbΔδB,ΔEq′=K4Δδ+K3ΔEq′+KqcΔvdc+KqeΔmE+KqδeΔδE+KqbΔmB+KqδbΔδB,Δvt=K5Δδ+K6ΔEq′+KvcΔvdc+KveΔmE+KvδeΔδE+KvbΔmB+KvδbΔδB,(18)Δv˙dc=K7Δδ+K8ΔEq′-K9Δvdc+KceΔmE+KcδeΔδE+KcbΔmB+KcδbΔδB,
where the linearization constants K1–K9, Kpc, Kpe, Kpδe, Kpb, Kpδb, Kqc, Kqe, Kqδe, Kqb, Kqδb, Kvc, Kve, Kvδe, Kvb, Kvδb, Kce, Kcδe, Kcb, and Kcδb are functions of the system parameters and the initial operating conditions.Referring to Figure2, the UPFC dynamic models of ΔmE, ΔmB, and ΔδB are described by the following state equations:
(19)Δm˙E=1Ts1(-ΔmE+KS1ΔUmE),Δm˙B=1TS3(-ΔmB+KS3ΔUmB),Δδ˙B=1TS4(-ΔδB+KS4ΔUδB).In addition, from Figure3, the UPFC dynamic model of a dc voltage regulator is described by the following state equations:
(20)Δδ˙E=1TS2(KS2ΔXdc-ΔδE+KS2ΔUδE),(21)ΔX˙dc=-KdPK7Δδ-KdPK8ΔEq′-(Kdi+KdPK9)Δvdc-KdPKceΔmE-KdPKcδeΔδE-KdPKcbΔmB-KdPKcδbΔδB,
where KdP and Kdi are the proportional and integral gains of the dc voltage regulator controller, respectively; (KS1–KS4) and (TS1–TS4) are the gains and times constants of the main control loops for UPFC, respectively.Equation (15) describes the model of the machine. Equations (16) and (18) represent the models for the exciter and the UPFC dc voltage regulators link, respectively. Equations (19) and (20) represent the control action of the main control loops of the UPFC with damping controller, and (21) represents the control action of PI dc voltage regulator. In state-space representation, these equations can be arranged in compact form as
(22)ΔX˙=AΔX+BΔU,
where the state vector ΔX and control vector ΔU are(23)ΔX=[ΔδΔωΔE`qΔEfdΔVdcΔXdcΔmEΔδEΔmBΔδB]T,ΔU=[ΔUmEΔUδEΔUmBΔUδB]T.The structure of the matricesA and B is(24)A=[0ωb00000000-K1M-DM-K2M0-KpdM0-KpeM-KpδeM-KpbM-KpδbM-K4T`do0-K3T`do1T`do-KqdT`do0-KqeT`do-KqδeT`do-KqbT`do-KqδbT`do-K5KaTa0-K6KaTa-1Ta-KvdKaTa0-KveKaTa-KvδeKaTa-KvbKaTa-KvδbKaTaK70K80-K90KceKcδeKcbKcδb-KdPK70-KdPK80(Kdi+KdPK9)0-KdPKce-KdPKcδe-KdPKcb-KdPKcδb000000-1TS100000000KS2TS20-1TS20000000000-1TS30000000000-1TS4],B=[000000000000000000000000KS1TS10000KS2TS20000KS3TS30000KS4TS4].The modified Phillips-Heffron transfer function model including UPFC has 28 constants; on the other hand, the Phillips-Heffron model has only 6 constants as shown in Figure4. This figure shows that the damping torque provided by the UPFC device consists of two parts. The first part is applied directly to the electromechanical oscillation loop of the generator. Its sensitivity is mainly measured by coefficients Kpc, Kpe, Kpδe, Kpb, and Kpδb, which is called the direct-damping torque. The second part consists of coefficients Kvc, Kve, Kvδe, Kvb, and Kvδb, which is applied through the field channel of the generator. Its sensitivity is related to the deviation of field voltage, which is referred to as the indirect-damping torque.Figure 4
Modified Phillips-Heffron transfer function  model of SMIB system with UPFC.


## 2.4. Eigenvalue Analysis without Stabilizer

For nominal operating condition, the dynamic behavior of the system is recognized through the eigenvalues of the system matrixA. By solving the system characteristic equation |λI-A|=0, the eigenvalues of the system are computed which are given below:
(25)λ1=-19.752,λ2=-19.826,λ3=0.48394+3.4508i,λ4=0.48394-3.4508i,λ5=-1.7178,λ6=-0.13878+0.30433i,λ7=-0.13878-0.30433i,λ8=-20,λ9=-20,λ10=-20.It is clearly seen from eigenvalues of the matrixA that the system is unstable and needs a supplementary stabilizer for stability.


## 3. Structure of the Supplementary Damping Stabilizers PSS and POD Controller

In order to overcome the LFO problem, supplemental control action is applied to the generator excitation in the form of PSS or UPFC device as POD controller. The four main control parameters of the UPFC (mE, δE, mB, and δB) can be modulated in order to produce the appropriate damping torque, which is illustrated in Figure 2.The POD controller has a structure similar to that of the PSS. Figure5 shows a sample block diagram of a POD controller. The controller contains three main blocks, that is, the gain block, the washout filter block, and two blocks (lead-lag) phase compensators. It provides an electrical torque in phase with the speed deviation in order to improve damping of power system oscillations. The washout filter block acts as a high-pass filter to eliminate the dc offset of the POD output and prevent steady-state changes in the terminal voltage of the generator. From this perspective, the washout time Tω should have a value in the range of 1 to 20 seconds defined to the electromechanical oscillation modes [1]. In this study, the time constants Tω, T2, and T4 were assigned specific values of 10 s, 0.1 s, and 0.1 s, respectively, while the parameters of the POD controller such as  KN, T1, and T3 need to be determined. The speed deviation Δω used as an input signal to the POD and ΔUN is the controller output, where N=PSS,mE,δE,mB,andδB. With usage of the above dynamic damping controller, the number of matrix state variables increases from 10 to 13, by adding three state variables ΔX˙1,ΔX˙2, and ΔU˙N.Figure 5
Structure of the supplementary damping stabilizers (PSS or POD).


### 3.1. Optimal Design of the Supplementary Damping Stabilizers (PSS or POD)

The main objective of optimization technique is to improve the dynamic stability of the power system against disturbances at different loading conditions. It can be achieved by suitable tuning of damping controller parameters.The supplementary damping stabilizer (lead-lag type) can be described mathematically as(26)U(s)=G(s)Y(s),
where G(s) is the transfer function of the supplementary damping stabilizer, Y(s) is the measurement signal, and U(s) is the output signal of the supplementary damping stabilizer, which will provide additional damping by moving electromechanical mode to the left. Equation (26) can be expressed in state-space form as
(27)ΔX˙C=ACΔXC+BCΔU,
where ΔXC is the controller state vector. Equation (22) describes a linear model of the power system extracted around a certain operating point. combining (22) with (27), we obtained a closed-loop system:
(28)ΔX˙Cℓ=AcℓΔXCℓ,ΔXCℓ=[ΔXΔXC],ζi=-Real(λi)|λi|,J=min(ζi),
where ΔXCℓ  is the state vector of the closed loop system, λi is the ith eigenvalue mode of the closed loop matrix Acℓ and ζi is the damping coefficient of the ith eigenvalue. It is clear that the objective function J will identify the minimum value of the damping coefficient among all modes.The goal of optimization process is to maximizeJ in order to realize appropriate damping for all modes including electromechanical mode, by exploring the search space of admissible control parameters, which enhance the system damping characteristics, and maximum J is searched within the limited range of control parameters:
(29)KNmin≤KN≤KNmax,TNimin≤TNi≤TNimax,N=PSS,mE,δE,mB,δB,i=1,3.Typical ranges of the optimized parameters are 0.01–100 forKN and 0.001–1 for TNi.


### 3.2. Optimization Process

The problem of tuning the parameters for individual and coordinated design for multiple damping controllers, which would ensure maximum damping performance, was solved via a PSO optimization procedure that appeared to be a promising evolutionary technique for handling optimization problems. PSO is a population-based, stochastic-optimization technique that was inspired by the social behavior of flocks of birds and schools of fish [28].The advantages of PSO algorithm are that it is simple and easy to implement and it has a flexible and well-balanced mechanism to enhance the local and global exploration capabilities. Recently, it has acquired wide range of applications in solving optimization design problems featuring nonlinearity, nondifferentiability, and high dimensionality in many area search spaces [29, 30].


#### 3.2.1. Classical PSO Algorithm

In the PSO, each possible solution is represented as a particle, and each set of particles comprises a population. Each particle keeps its position in hyperspace, which is related to the fittest solution it ever experiences in a special memory calledpbest. In addition, the position related to the best value obtained so far by any particle in the population is called gbest. For each iteration of the PSO algorithm, the pbest and gbest values are updated and each particle changes its velocity toward them randomly. This concept can be expressed as [18]
(30)vik+1=wvik+c1r1(pbesti-xik)+c2r2(gbest-xik)(31)xik+1=xik+vik+1,i=1,2,…,n,
where v is the particle velocity, x is the particle position, k is the number of iterations, w is the inertia weight factor, c1 and c2 are the cognitive and asocial acceleration factors respectively, n is the number of particles, and r1 and r2 are the uniformly distributed random numbers in the range of 0 to 1. Figure 6 shows the flow chart of the PSO algorithm.Figure 6
PSO algorithm for the tuning parameters of an individual and coordinated design.


#### 3.2.2. Chaotic Particle Swarm Optimization (CPSO)

The main disadvantage of the simple PSO algorithm is that the performance of it greatly depends on its parameters and it is not guaranteed to be global  convergent. In order to improve the global searching ability and prevent a slide into the premature convergence to local minima, PSO and chaotic sequence techniques are combined to form a chaotic particle swarm optimization (CPSO) technique, which practically combines the population-based evolutionary searching ability of PSO and chaotic searching behavior. The logistic equation employed for constructing hybrid PSO is described as [31]:
(32)βk+1=μβk(1-βk),0≤β1≤1,
where μ is the control parameter with a real value between 0 to 4.  Although (32) is deterministic, it exhibits chaotic dynamics when μ= 4 and β0∉{0,0.25,0.5,0.75,1}. It exhibits the sensitive dependence on initial conditions, which is the basic characteristic of chaos. The inertia weighting function in (30) is usually evaluated utilizing the following equation:
(33)w=wmax-[(wmax-wmin)(iteritermax)],
where wmax and wmin are maximum and minimum values of w, itermax is the maximum number of iterations, and iter is the current iteration number. The new weight parameter wnew is defined by multiplying weight parameter w in (33) and logistic equation (32):
(34)wnew=w×βk+1.To improve the global searching capability of PSO, we have to introduce a new velocity update equation as follows:(35)vik+1=wnewvik+c1r1(pbesti-xik)+c2r2(gbest-xik).We have observed that the proposed new weight decreases and oscillates simultaneously for total iteration, whereas the conventional weight decreases monotonously fromwmax to wmin. The final choice of a parameter was considered to be the optimal choice: n, itermax, c1, c2, wmin, wmax, μ, and β0 are chosen as 30, 100, 2, 2, 0.3, 0.9, 4, and 0.3, respectively.


## 3.1. Optimal Design of the Supplementary Damping Stabilizers (PSS or POD)

The main objective of optimization technique is to improve the dynamic stability of the power system against disturbances at different loading conditions. It can be achieved by suitable tuning of damping controller parameters.The supplementary damping stabilizer (lead-lag type) can be described mathematically as(26)U(s)=G(s)Y(s),
where G(s) is the transfer function of the supplementary damping stabilizer, Y(s) is the measurement signal, and U(s) is the output signal of the supplementary damping stabilizer, which will provide additional damping by moving electromechanical mode to the left. Equation (26) can be expressed in state-space form as
(27)ΔX˙C=ACΔXC+BCΔU,
where ΔXC is the controller state vector. Equation (22) describes a linear model of the power system extracted around a certain operating point. combining (22) with (27), we obtained a closed-loop system:
(28)ΔX˙Cℓ=AcℓΔXCℓ,ΔXCℓ=[ΔXΔXC],ζi=-Real(λi)|λi|,J=min(ζi),
where ΔXCℓ  is the state vector of the closed loop system, λi is the ith eigenvalue mode of the closed loop matrix Acℓ and ζi is the damping coefficient of the ith eigenvalue. It is clear that the objective function J will identify the minimum value of the damping coefficient among all modes.The goal of optimization process is to maximizeJ in order to realize appropriate damping for all modes including electromechanical mode, by exploring the search space of admissible control parameters, which enhance the system damping characteristics, and maximum J is searched within the limited range of control parameters:
(29)KNmin≤KN≤KNmax,TNimin≤TNi≤TNimax,N=PSS,mE,δE,mB,δB,i=1,3.Typical ranges of the optimized parameters are 0.01–100 forKN and 0.001–1 for TNi.


## 3.2. Optimization Process

The problem of tuning the parameters for individual and coordinated design for multiple damping controllers, which would ensure maximum damping performance, was solved via a PSO optimization procedure that appeared to be a promising evolutionary technique for handling optimization problems. PSO is a population-based, stochastic-optimization technique that was inspired by the social behavior of flocks of birds and schools of fish [28].The advantages of PSO algorithm are that it is simple and easy to implement and it has a flexible and well-balanced mechanism to enhance the local and global exploration capabilities. Recently, it has acquired wide range of applications in solving optimization design problems featuring nonlinearity, nondifferentiability, and high dimensionality in many area search spaces [29, 30].


### 3.2.1. Classical PSO Algorithm

In the PSO, each possible solution is represented as a particle, and each set of particles comprises a population. Each particle keeps its position in hyperspace, which is related to the fittest solution it ever experiences in a special memory calledpbest. In addition, the position related to the best value obtained so far by any particle in the population is called gbest. For each iteration of the PSO algorithm, the pbest and gbest values are updated and each particle changes its velocity toward them randomly. This concept can be expressed as [18]
(30)vik+1=wvik+c1r1(pbesti-xik)+c2r2(gbest-xik)(31)xik+1=xik+vik+1,i=1,2,…,n,
where v is the particle velocity, x is the particle position, k is the number of iterations, w is the inertia weight factor, c1 and c2 are the cognitive and asocial acceleration factors respectively, n is the number of particles, and r1 and r2 are the uniformly distributed random numbers in the range of 0 to 1. Figure 6 shows the flow chart of the PSO algorithm.Figure 6
PSO algorithm for the tuning parameters of an individual and coordinated design.


### 3.2.2. Chaotic Particle Swarm Optimization (CPSO)

The main disadvantage of the simple PSO algorithm is that the performance of it greatly depends on its parameters and it is not guaranteed to be global  convergent. In order to improve the global searching ability and prevent a slide into the premature convergence to local minima, PSO and chaotic sequence techniques are combined to form a chaotic particle swarm optimization (CPSO) technique, which practically combines the population-based evolutionary searching ability of PSO and chaotic searching behavior. The logistic equation employed for constructing hybrid PSO is described as [31]:
(32)βk+1=μβk(1-βk),0≤β1≤1,
where μ is the control parameter with a real value between 0 to 4.  Although (32) is deterministic, it exhibits chaotic dynamics when μ= 4 and β0∉{0,0.25,0.5,0.75,1}. It exhibits the sensitive dependence on initial conditions, which is the basic characteristic of chaos. The inertia weighting function in (30) is usually evaluated utilizing the following equation:
(33)w=wmax-[(wmax-wmin)(iteritermax)],
where wmax and wmin are maximum and minimum values of w, itermax is the maximum number of iterations, and iter is the current iteration number. The new weight parameter wnew is defined by multiplying weight parameter w in (33) and logistic equation (32):
(34)wnew=w×βk+1.To improve the global searching capability of PSO, we have to introduce a new velocity update equation as follows:(35)vik+1=wnewvik+c1r1(pbesti-xik)+c2r2(gbest-xik).We have observed that the proposed new weight decreases and oscillates simultaneously for total iteration, whereas the conventional weight decreases monotonously fromwmax to wmin. The final choice of a parameter was considered to be the optimal choice: n, itermax, c1, c2, wmin, wmax, μ, and β0 are chosen as 30, 100, 2, 2, 0.3, 0.9, 4, and 0.3, respectively.


## 3.2.1. Classical PSO Algorithm

In the PSO, each possible solution is represented as a particle, and each set of particles comprises a population. Each particle keeps its position in hyperspace, which is related to the fittest solution it ever experiences in a special memory calledpbest. In addition, the position related to the best value obtained so far by any particle in the population is called gbest. For each iteration of the PSO algorithm, the pbest and gbest values are updated and each particle changes its velocity toward them randomly. This concept can be expressed as [18]
(30)vik+1=wvik+c1r1(pbesti-xik)+c2r2(gbest-xik)(31)xik+1=xik+vik+1,i=1,2,…,n,
where v is the particle velocity, x is the particle position, k is the number of iterations, w is the inertia weight factor, c1 and c2 are the cognitive and asocial acceleration factors respectively, n is the number of particles, and r1 and r2 are the uniformly distributed random numbers in the range of 0 to 1. Figure 6 shows the flow chart of the PSO algorithm.Figure 6
PSO algorithm for the tuning parameters of an individual and coordinated design.


## 3.2.2. Chaotic Particle Swarm Optimization (CPSO)

The main disadvantage of the simple PSO algorithm is that the performance of it greatly depends on its parameters and it is not guaranteed to be global  convergent. In order to improve the global searching ability and prevent a slide into the premature convergence to local minima, PSO and chaotic sequence techniques are combined to form a chaotic particle swarm optimization (CPSO) technique, which practically combines the population-based evolutionary searching ability of PSO and chaotic searching behavior. The logistic equation employed for constructing hybrid PSO is described as [31]:
(32)βk+1=μβk(1-βk),0≤β1≤1,
where μ is the control parameter with a real value between 0 to 4.  Although (32) is deterministic, it exhibits chaotic dynamics when μ= 4 and β0∉{0,0.25,0.5,0.75,1}. It exhibits the sensitive dependence on initial conditions, which is the basic characteristic of chaos. The inertia weighting function in (30) is usually evaluated utilizing the following equation:
(33)w=wmax-[(wmax-wmin)(iteritermax)],
where wmax and wmin are maximum and minimum values of w, itermax is the maximum number of iterations, and iter is the current iteration number. The new weight parameter wnew is defined by multiplying weight parameter w in (33) and logistic equation (32):
(34)wnew=w×βk+1.To improve the global searching capability of PSO, we have to introduce a new velocity update equation as follows:(35)vik+1=wnewvik+c1r1(pbesti-xik)+c2r2(gbest-xik).We have observed that the proposed new weight decreases and oscillates simultaneously for total iteration, whereas the conventional weight decreases monotonously fromwmax to wmin. The final choice of a parameter was considered to be the optimal choice: n, itermax, c1, c2, wmin, wmax, μ, and β0 are chosen as 30, 100, 2, 2, 0.3, 0.9, 4, and 0.3, respectively.


## 4. Simulation and Comparison Results

In this section, the ability of UPFC in damping system oscillation and the dynamic interactions of UPFC-POD controllers are investigated intensively. The CPSO technique has been applied to design individual and coordinated damping controllers. To evaluate the performance of the proposed simultaneous coordinated design approaches, the responses with the proposed controllers were compared with the responses of the individual design of the PSS and UPFC-POD controllers for two schemes. The resultant optimal parameters of the individual controllers and coordinated designs are given in Tables1, 2, and 3, respectively.Table 1
The optimal parameters of the individual controllers.
Individual controllers
Type of algorithm
Optimal values
K
T
1 (Sec.)
T
3 (Sec.)
PSS
CPSO
38.8433
0.2490
0.2490
PSO
12.2151
0.8227
0.8227
m
E
CPSO
26.1560
0.8812
0.0015
PSO
31.5718
0.3635
0.0344
δ
E
CPSO
11.5577
0.1960
0.0331
PSO
12.1740
0.2143
0.0028
m
B
CPSO
66.2928
0.0011
0.2369
PSO
75.4243
0.0998
0.3171
δ
B
CPSO
99.6244
0.7206
0.0017
PSO
25.4779
0.0307
0.5386Table 2
The optimal parameters of the coordinated designs between PSS and UPFC-POD controllers.
Coordinated designs
Type of algorithm
Optimal values
K
*
T
1
* (Sec.)
T
3
* (Sec.)
K
#
T
1
# (Sec.)
T
3
# (Sec.)
PSS
* & mE#
CPSO
45.5204
0.0418
0.2659
38.3455
0.0316
0.6943
PSO
44.0127
0.2628
0.0613
52.8507
0.5057
0.0072
PSS
* & δE#
CPSO
8.1683
0.2501
0.1518
10.0208
0.1993
0.0361
PSO
5.2118
0.7275
0.0301
10.9942
0.1579
0.0284
PSS
* & mB#
CPSO
57.2645
0.1963
0.1756
54.6790
0.1057
0.2037
PSO
61.6657
0.2949
0.4783
59.6813
0.3536
0.0816
PSS
* & δB#
CPSO
63.0262
0.2377
0.2371
3.9384
0.7395
0.9415
PSO
15.2002
0.9788
0.0074
0.3729
0.5007
0.5611Table 3
The optimal parameters of the coordinated designs between different UPFC-POD controllers.
Coordinated designs
Type of algorithm
Optimal values
K
*
T
1
* (Sec.)
T
3
* (Sec.)
K
#
T
1
# (Sec.)
T
3
# (Sec.)
m
E
* & δE#
CPSO
50.7479
0.6966
0.0656
8.0139
0.0513
0.0530
PSO
57.7561
0.4707
0.0895
7.9823
0.0460
0.0136
m
B
* & δB#
CPSO
97.9661
0.0014
0.0019
69.2724
0.0022
0.1816
PSO
68.0064
0.2596
0.0413
42.4356
0.2982
0.0070
m
E
* & mB#
CPSO
26.1813
0.0081
0.8276
2.3838
0.6509
0.4527
PSO
41.5012
0.9941
0.0023
0.1583
0.8581
0.0941
δ
E
* & δB#
CPSO
89.0042
0.0735
0.0087
63.4526
0.0490
0.2352
PSO
4.7214
1.0000
0.6398
11.4511
0.1435
0.7305
m
E
* & δB#
CPSO
99.9911
0.2171
0.0407
81.5964
0.0005
0.0063
PSO
17.2632
0.1916
0.8990
87.1383
0.1447
0.0056
m
B
* & δE#
CPSO
57.8685
0.0530
0.2428
10.5448
0.0462
0.0788
PSO
65.7520
0.1876
0.0403
10.7924
0.0619
0.0561


### 4.1. Scheme 1

Figures7–11 show the effect of applying the individual controllers and coordinated designs for PSS and different UPFC-POD controllers in a SMIB of speed deviation of Δω with 10% step change in mechanical input power. The system eigenvalues with the proposed individual stabilizers and coordinated designs for nominal operating condition are given in Tables 4 and 5, respectively. The first and second rows represent the electromechanical mode and their damping ratio ζEM using participation factor to identify the eigenvalue associated with electromechanical mode. It is clear that the proposed controllers stabilize the system effectively. The maximum overshoot as well as the settling time of the system responses has been compared to the capability of the simulated coordinate approaches in oscillation damping.Table 4
System eigenvalues of the individual controllers.
PSS
m
E
δ
E
m
B
δ
B
-
2
.
29
±
j
7
.
02
-
2
.
11
±
j
7
.
31
-
3
.
69
±
j
6
.
08
-
2
.
79
±
j
5
.
11
-
1.38
±
j
6.06
ζ
E
M
=
0
.
3101
ζ
E
M
=
0
.
2773
ζ
E
M
=
0
.
5188
ζ
E
M
=
0
.
4792
ζ
E
M
=
0
.
2220
-
4.18
±
j
8.27
-
5.38
±
j
10
-
3.7
±
j
5.56
-
2.24
±
j
3.13
-
1.7
±
j
4.25
-
3.78
±
j
5.55
-
3.3
±
j
7.22
-
4.38
±
j
5.07
-
5.24
±
j
7.46
-
2.09
±
j
4.95
-
24.84
-
28.96
-
4.85
±
6.37
-
25.27
-
28.52
-
19.83
-
16.58
-
23.72
19.17
-
19.71
-
0.69
-
19.87
-
19.83
-
1.93
-
16.49
-
0.33
-
2.14
-
0.26
-
0.2
-
1.98
-
20
-
0.21
-
20
-
20
-
0.29
-
20
-
20
-
20
-
20
-
20
-
20
-
20
-
20
-
20Table 5
System eigenvalues of the coordinated designs between PSS and UPFC-POD controllers.
PSS & mE
PSS & δE
PSS & mB
PSS & δB
-
2
.
85
±
j
5
.
67
-
3
.
86
±
j
5
.
32
-
1
.
23
±
j
2
.
11
-
2
.
67
±
j
6
.
39
ζ
E
M
=
0
.
45
ζ
E
M
=
0
.
5873
ζ
E
M
=
0
.
504
ζ
E
M
=
0
.
3855
-
6.03
±
j
6.11
-
4.94
±
j
5.14
-
4.45
±
j
6.91
-
2.79
±
j
5.18
-
4.49
±
j
4.58
-
3.28
±
j
5.61
-
5.91
±
j
3.73
-
4.08
±
j
7.95
-
2.7
±
j
6.73
-
4.89
±
j
7.02
-
2.48
±
j
3.62
-
20.16
±
j
7.11
-
23.87
-
10
±
j
6.97
-
19.68
±
j
4.4
-
3.33
±
j
5.58
-
19.67
-
23.33
-
19.87
-
10
±
j
4.26
-
14.16
-
19.83
-
10
-
19.713
-
10
-
16.99
-
10
-
0.2
-
10
-
0.26
-
0.2
-
20
-
0.2
-
20
-
20
-
20
-
20
-
20
-
20
-
20Figure 7
Speed variation responses for individual damping controllers (PSS,mE, δE, mB, and δB).Figure 8
Speed variation responses for individual damping controllersPSS, mE and coordinated design PSS & mE.Figure 9
Speed variation responses for individual damping controllersPSS, δB and coordinated design PSS & δB.Figure 10
Speed variation responses for individual damping controllersPSS, mB and coordinated design PSS & mB.Figure 11
Speed variation responses for individual damping controllersPSS, δE and coordinated design PSS & δE.Figure7 shows the speed deviation responses of the four UPFC-POD controllers and PSS individually. All these responses are shown in one figure for comparison purposes. It can be seen that by using the UPFC-POD controllers δE and  mB provide a robust damping characteristic higher than that of PSS and enhance the stability. However, the UPFC-POD controllers mE and δB do not perform well and have poor capabilities in damping the system oscillation.Figures8 and 9 show the improvement in overshoot and settling time of the system response while using the coordinated designs (PSS&mE) and (PSS&δB) over the individual control responses but less than the best individual controls δE and mB. Figure 10 shows that the coordinated design (PSS&mB) improves the system damping compared to their individual controls but no more than the best one δE. Figure 11 shows the best response in Scheme 1 while using the coordinated design (PSS&δE) over their individual control responses because it improved both damping parameters overshoot and settling time.


### 4.2. Scheme 2

In this scheme, the power system is considered to possess a UPFC device without PSS. Figures12–17 show the effect of applying the individual controllers and coordinated designs between different UPFC-POD controllers in a SMIB of speed deviation of Δω with 10% step change in mechanical input power. The eigenvalues of the system with the proposed controllers in scheme 2 are given in Table 6. It is clear that the proposed controllers stabilize the system effectively.Table 6
System eigenvalues of the coordinated designs between different UPFC-POD controllers.
m
E & δE
m
B & δB
m
E & mB
δ
E & δB
m
E & δB
m
B & δE
-
4
.
71
±
j
7
.
04
-
2
.
85
±
j
5
.
18
-
1
.
71
±
j
3
.
02
-
1
.
46
±
j
2
.
31
-
2
.
39
±
j
6
.
35
-
4
.
74
±
j
5
.
1
ζ
E
M
=
0
.
5561
ζ
E
M
=
0
.
482
ζ
E
M
=
0
.
4927
ζ
E
M
=
0
.
5343
ζ
E
M
=
0
.
35
ζ
E
M
=
0
.
6808
-
5.1
±
j
7.33
-
2.71
±
j
5.04
-
5.39
±
j
10.01
-
1.37
±
j
2.09
-
4.24
±
j
4.93
-
5.7
±
j
6.08
-
10.97
±
j
11.2
-
3.66
±
j
4.51
-
2
±
j
4.23
-
4.24
±
j
4.93
-
2.96
±
j
5.68
-
4.9
±
j
5
-
16.17
±
j
2.23
-
10
±
j
3.01
-
1.9
±
j
3.46
-
2.48
±
j
4.15
-
3.22
±
j
7.25
-
6.29
±
j
8.17
-
5.83
±
j
5.89
-
24.85
-
10
±
j
2.13
-
10
±
j
2.18
-
10
±
j
4.03
-
23.73
-
6.46
±
j
8.25
-
20.26
-
28.96
-
20.33
-
20.33
-
16.3
-
19.87
-
19.38
-
19.87
-
19.76
-
19.76
-
19.83
-
0.2
-
0.88
-
0.71
-
18.75
-
18.46
-
20
-
20
-
0.36
-
0.2
-
0.2
-
0.2
-
10
-
20
-
0.2
-
20
-
20
-
20
-
10
-
20
-
20
-
20
-
20
-
0.2
-
20
-
20Figure 12
Speed variation responses for individual damping controllersmB, δE and coordinated design mB & δE.Figure 13
Speed variation responses for individual damping controllersmE, δE and coordinated design mE & δE.Figure 14
Speed variation responses for individual damping controllersδB, δE and coordinated design δB & δE.Figure 15
Speed variation responses for individual damping controllersmE, mB and coordinated design mE & mB.Figure 16
Speed variation responses for individual damping controllersδB, mB and coordinated design mB & δB.Figure 17
Speed variation responses for individual damping controllersmE, δB and coordinated design mE & δB.From Figures12, 13, and 14, it is clear that the system damping resulting from using coordinated designs (mB&δE), (mE&δE), and (δE&δB) are much higher than those resulting from using the best individual controls δE and mB. Figures 15 and 16 show that the coordinated designs (mE&mB) and (mB&δB) improve the system damping compared to their individual controls but not more than the best one δE. Figure 17 shows the improvement in overshoot and settling time of the system response while using the coordinated design (mE&δB) over the individual control responses but less than the best individual controls δE and mB.


## 4.1. Scheme 1

Figures7–11 show the effect of applying the individual controllers and coordinated designs for PSS and different UPFC-POD controllers in a SMIB of speed deviation of Δω with 10% step change in mechanical input power. The system eigenvalues with the proposed individual stabilizers and coordinated designs for nominal operating condition are given in Tables 4 and 5, respectively. The first and second rows represent the electromechanical mode and their damping ratio ζEM using participation factor to identify the eigenvalue associated with electromechanical mode. It is clear that the proposed controllers stabilize the system effectively. The maximum overshoot as well as the settling time of the system responses has been compared to the capability of the simulated coordinate approaches in oscillation damping.Table 4
System eigenvalues of the individual controllers.
PSS
m
E
δ
E
m
B
δ
B
-
2
.
29
±
j
7
.
02
-
2
.
11
±
j
7
.
31
-
3
.
69
±
j
6
.
08
-
2
.
79
±
j
5
.
11
-
1.38
±
j
6.06
ζ
E
M
=
0
.
3101
ζ
E
M
=
0
.
2773
ζ
E
M
=
0
.
5188
ζ
E
M
=
0
.
4792
ζ
E
M
=
0
.
2220
-
4.18
±
j
8.27
-
5.38
±
j
10
-
3.7
±
j
5.56
-
2.24
±
j
3.13
-
1.7
±
j
4.25
-
3.78
±
j
5.55
-
3.3
±
j
7.22
-
4.38
±
j
5.07
-
5.24
±
j
7.46
-
2.09
±
j
4.95
-
24.84
-
28.96
-
4.85
±
6.37
-
25.27
-
28.52
-
19.83
-
16.58
-
23.72
19.17
-
19.71
-
0.69
-
19.87
-
19.83
-
1.93
-
16.49
-
0.33
-
2.14
-
0.26
-
0.2
-
1.98
-
20
-
0.21
-
20
-
20
-
0.29
-
20
-
20
-
20
-
20
-
20
-
20
-
20
-
20
-
20Table 5
System eigenvalues of the coordinated designs between PSS and UPFC-POD controllers.
PSS & mE
PSS & δE
PSS & mB
PSS & δB
-
2
.
85
±
j
5
.
67
-
3
.
86
±
j
5
.
32
-
1
.
23
±
j
2
.
11
-
2
.
67
±
j
6
.
39
ζ
E
M
=
0
.
45
ζ
E
M
=
0
.
5873
ζ
E
M
=
0
.
504
ζ
E
M
=
0
.
3855
-
6.03
±
j
6.11
-
4.94
±
j
5.14
-
4.45
±
j
6.91
-
2.79
±
j
5.18
-
4.49
±
j
4.58
-
3.28
±
j
5.61
-
5.91
±
j
3.73
-
4.08
±
j
7.95
-
2.7
±
j
6.73
-
4.89
±
j
7.02
-
2.48
±
j
3.62
-
20.16
±
j
7.11
-
23.87
-
10
±
j
6.97
-
19.68
±
j
4.4
-
3.33
±
j
5.58
-
19.67
-
23.33
-
19.87
-
10
±
j
4.26
-
14.16
-
19.83
-
10
-
19.713
-
10
-
16.99
-
10
-
0.2
-
10
-
0.26
-
0.2
-
20
-
0.2
-
20
-
20
-
20
-
20
-
20
-
20
-
20Figure 7
Speed variation responses for individual damping controllers (PSS,mE, δE, mB, and δB).Figure 8
Speed variation responses for individual damping controllersPSS, mE and coordinated design PSS & mE.Figure 9
Speed variation responses for individual damping controllersPSS, δB and coordinated design PSS & δB.Figure 10
Speed variation responses for individual damping controllersPSS, mB and coordinated design PSS & mB.Figure 11
Speed variation responses for individual damping controllersPSS, δE and coordinated design PSS & δE.Figure7 shows the speed deviation responses of the four UPFC-POD controllers and PSS individually. All these responses are shown in one figure for comparison purposes. It can be seen that by using the UPFC-POD controllers δE and  mB provide a robust damping characteristic higher than that of PSS and enhance the stability. However, the UPFC-POD controllers mE and δB do not perform well and have poor capabilities in damping the system oscillation.Figures8 and 9 show the improvement in overshoot and settling time of the system response while using the coordinated designs (PSS&mE) and (PSS&δB) over the individual control responses but less than the best individual controls δE and mB. Figure 10 shows that the coordinated design (PSS&mB) improves the system damping compared to their individual controls but no more than the best one δE. Figure 11 shows the best response in Scheme 1 while using the coordinated design (PSS&δE) over their individual control responses because it improved both damping parameters overshoot and settling time.


## 4.2. Scheme 2

In this scheme, the power system is considered to possess a UPFC device without PSS. Figures12–17 show the effect of applying the individual controllers and coordinated designs between different UPFC-POD controllers in a SMIB of speed deviation of Δω with 10% step change in mechanical input power. The eigenvalues of the system with the proposed controllers in scheme 2 are given in Table 6. It is clear that the proposed controllers stabilize the system effectively.Table 6
System eigenvalues of the coordinated designs between different UPFC-POD controllers.
m
E & δE
m
B & δB
m
E & mB
δ
E & δB
m
E & δB
m
B & δE
-
4
.
71
±
j
7
.
04
-
2
.
85
±
j
5
.
18
-
1
.
71
±
j
3
.
02
-
1
.
46
±
j
2
.
31
-
2
.
39
±
j
6
.
35
-
4
.
74
±
j
5
.
1
ζ
E
M
=
0
.
5561
ζ
E
M
=
0
.
482
ζ
E
M
=
0
.
4927
ζ
E
M
=
0
.
5343
ζ
E
M
=
0
.
35
ζ
E
M
=
0
.
6808
-
5.1
±
j
7.33
-
2.71
±
j
5.04
-
5.39
±
j
10.01
-
1.37
±
j
2.09
-
4.24
±
j
4.93
-
5.7
±
j
6.08
-
10.97
±
j
11.2
-
3.66
±
j
4.51
-
2
±
j
4.23
-
4.24
±
j
4.93
-
2.96
±
j
5.68
-
4.9
±
j
5
-
16.17
±
j
2.23
-
10
±
j
3.01
-
1.9
±
j
3.46
-
2.48
±
j
4.15
-
3.22
±
j
7.25
-
6.29
±
j
8.17
-
5.83
±
j
5.89
-
24.85
-
10
±
j
2.13
-
10
±
j
2.18
-
10
±
j
4.03
-
23.73
-
6.46
±
j
8.25
-
20.26
-
28.96
-
20.33
-
20.33
-
16.3
-
19.87
-
19.38
-
19.87
-
19.76
-
19.76
-
19.83
-
0.2
-
0.88
-
0.71
-
18.75
-
18.46
-
20
-
20
-
0.36
-
0.2
-
0.2
-
0.2
-
10
-
20
-
0.2
-
20
-
20
-
20
-
10
-
20
-
20
-
20
-
20
-
0.2
-
20
-
20Figure 12
Speed variation responses for individual damping controllersmB, δE and coordinated design mB & δE.Figure 13
Speed variation responses for individual damping controllersmE, δE and coordinated design mE & δE.Figure 14
Speed variation responses for individual damping controllersδB, δE and coordinated design δB & δE.Figure 15
Speed variation responses for individual damping controllersmE, mB and coordinated design mE & mB.Figure 16
Speed variation responses for individual damping controllersδB, mB and coordinated design mB & δB.Figure 17
Speed variation responses for individual damping controllersmE, δB and coordinated design mE & δB.From Figures12, 13, and 14, it is clear that the system damping resulting from using coordinated designs (mB&δE), (mE&δE), and (δE&δB) are much higher than those resulting from using the best individual controls δE and mB. Figures 15 and 16 show that the coordinated designs (mE&mB) and (mB&δB) improve the system damping compared to their individual controls but not more than the best one δE. Figure 17 shows the improvement in overshoot and settling time of the system response while using the coordinated design (mE&δB) over the individual control responses but less than the best individual controls δE and mB.


## 5. Conclusions

This paper is concerned with the damping of LFO via PSS and UPFC-POD controllers applied independently and also through the simultaneous coordinated designs in a SMIB power system. To improve the global searching ability and prevent a slide into the premature convergence to local minima, PSO and chaos theory are combined to form a CPSO. For the proposed controller design problem, a CPSO algorithm was used as the optimization technique to search for the optimal damping controller parameters in both the individual and the coordinated designs. The simulation results of the individual damping controllers showed the best damping effects resulting from usingδE and  mB as POD controllers are much higher than those resulting from using PSS, mE, and δB responses. In addition, the coordinated designs (PSS&δE) in scheme 1; (mB&δE),(mE&δE) and (δE&δB) in scheme 2 provide superior performance in comparison with the best individual controls δE and mB while the coordinated designs (PSS&mB) in scheme 1; (mE&mB), and (mB&δB) in scheme 2 improve the system damping compared to the individual controls except δE, which is the best one. Finally, the coordinated designs (PSS&mE) and (PSS&δB) in scheme 1; (mE&δB) in scheme 2 provide robust damping effects over their individual control responses but no more than the best individual controls δE and mB.



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*Source: 965282-2013-03-28.xml*