Adaptive SVC Damping Controller Design, Using ...

Adaptive SVC Damping Controller Design, Using Residue Method in a Multi-Machine System Abolfazl Jalilvand and Morteza Daviran Keshavarzi Department of Electrical Engineering, Faculty of Engineering, Zanjan University, Zanjan, Iran [email protected] & [email protected] Abstract- In this paper the design of an adaptive Power Oscillation Damping (POD) controller has been investigated for a Static Var Compensator (SVC). Simultaneous linearization of power system is carried out through a Recursive Least Squares (RLS) identification algorithm, after which a Residue Method is applied for adaptive POD parameter tuning. Proposed methodology has been simulated on a four machine test system considering current injection model for SVC. Obtained results through this approach demonstrate that this Adaptive POD gives more appropriate performance in comparison with fixed parameter controller in a wide range of operating conditions.

I.

INTRODUCTION

Occurrence of the disturbance in power system makes electromechanical oscillations and following this phenomenon fluctuations appear in system variables such as voltage, frequency and generator rotor angles. Power system stability has been recognized as an important problem for secure system operation since the 1920s. Many major blackouts caused by power system instability have illustrated the importance of this phenomenon [1]. Satisfactory damping of power oscillations is an important issue addressed when dealing with the rotor angle stability of power systems. Existing oscillatory modes in power system can be divided into two general categories, local and inter-area modes. In local mode, one generator (a group of generators) swings against the rest of the system at 1.0 to 2.0 Hz. The more complex Interarea modes of oscillations are observed over a large part of the network. It involves two coherent groups of generators swinging against each other at 1 Hz or less [2-3]. A conventional damping control design considers a single operating condition of the system. In this kind of controller the feedback is fixed and the way in which the error is processed is the same for all operating conditions [4, 5]. In case of contingencies, changed operating conditions can cause poorly damped or even unstable oscillations since the controller parameters yielding satisfactory damping for one operating condition may no longer provide sufficient damping for another one [6]. Therefore it is necessary to design a controller that can stabilize the system against various disturbances in a wide range of operating conditions. For this purpose the adaptive control scheme may be used which can modify its behavior in response to changes in the dynamics of the system and the character of the disturbances [7-8]. Since power system is a completely nonlinear system, therefore in adaptive system we need to employ an on-line

identification algorithm. In [9] Kalman filtering technique has been utilized in order to identify power system oscillations. In this paper the linear model is identified using Recursive Least Squares (RLS) estimation. Then linearized system is analyzed in state space framework and dominant oscillatory mode is recognized. Controller parameters are updated based on Residue Method. To improve the damping of oscillations in power systems, supplementary control laws are referred to as POD control [6] can be applied to existing devices. In this work, POD control has been applied to SVC. II. STATIC VAR COMPENSATOR (SVC) A basic topology of Static Var Compensator (SVC) consists of a series capacitor bank C in parallel with a thyristorcontrolled reactor L , is shown in Figure 1. Vj

C L

Figure 1. Basic SVC topology

The SVC can be seen as an adjustable susceptance which is a function of thyristors firing angle α [10]. XC 1 ⎧ [2(π − α ) + sin(2α )]⎫⎬ (1) BSVC (α ) = ⎨X L − π XC X L ⎩ ⎭ where X C and X L are the nominal reactance of capacitor bank and inductive reactance of shunt reactor respectively. The effect of SVC can be considered as a variable capacitor that is equal to a voltage source which depends on voltage V j . The current injection model of the SVC is obtained by replacing the voltage source by an equivalent current source I j injected to bus j as follows [11]:

I j = jB SVC V j Vj

Vj jB SVC

(2)

Ij

Figure 2. Current injection model of SVC

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Figure 3 indicates the adaptive SVC control block diagram where Vt is the voltage magnitude at the SVC terminal, V ref is the voltage to be maintained by SVC, ΔV is the terminal voltage deviation as the controller input and α ref is the reference firing angle to attain to desired compensation ratio. α ref

Parameter tuning

V ref +

Σ @

ΔV

Adaptive POD

+ +

Δαm ax

Σ

Δα m in

SVC

Ij

Power system

Vt

reflected in the feedback signal. The measures for those two properties are the modal controllability and observability, respectively, are defined as follows [12-13]:

Bm = Φ −1 B

(9) Cm = C Φ The mode is not controllable if the corresponding row of the matrix Bm is a zero vector, and the mode is not observable if the corresponding column of the matrix C m is a zero vector.

B. Residues Figure 3. Adaptive SVC control block diagram

Considering (3) with single input and single output (SISO) and assuming D = 0 , the open loop transfer function of the system can be described by [13]:

III. POWER SYSTEM LINEAR MODEL The total linearized system model including SVC can be represented by following equations [12-13]. Δx = AΔx + BΔu Δy = CΔx + DΔu

(3)

The eigenvalues of the system λi are determined by characteristic equation of matrix A as follows: det(λi I − A ) = 0 (4) For every eigenvalue λi , there is a right eigenvector φi which satisfies the following equation: Aφi = λiφi (5) Similarly left eigenvector ψ i corresponding to every

eigenvalue λi is obtained by (6). ψ i A = λiψ i (6) Using right and left eigenvectors modal matrices are introduced by: Φ = [φ1 φ2 " φn ] (7) Ψ = ⎡⎣ψ 1T ψ T2 " ψ Tn ⎤⎦ The right eigenvector indicates that on which system variables the mode is more observable. The left eigenvector, together with the system’s initial state, determines the amplitude of the mode. A left eigenvector carries the mode controllability information. For a particular eigenvalue λi = σ i ± jω i , the relative damping ratio is given by: −σ i ξi = × 100 (8) σ i2 + ω i2 The oscillatory modes having damping ratio less than 3% are said to be critical [12]. When designing damping controls one has to take care about margin due to uncertainties or disturbances. Hence the damping ratio of at least 5% may be the objective of the control design [6].

A. Controllability and Observability In order to modify a selected oscillatory mode by a feedback controller, the chosen input of the controller must influence the behavior of that mode and the mode must also be visible in the chosen feedback signal i.e. the behavior of that mode should be

G (s ) =

ΔY (s ) = C (sI − A )−1 B ΔU (s )

(10)

The transfer function G ( s ) can be expanded in partial fractions of the Laplace transform of y in terms of C and B matrices and the right and left eigenvectors. Then we have: n

G (s ) =

C ϕi ψ i B

n

Ri

∑ (s − λ ) = ∑ (s − λ ) i

i =1

(11)

i

i =1

The residue Ri of a particular mode i ψgives the measure of that mode’s sensitivity to a feedback. Residues are also suitable to find the feedback signal location in the power system. The largestresiduethenindicatesthemosteffectivelocationto applythefeedbackcontrol. IV. POD CONTROLLER DESIGN Figure 4 shows a closed-loop system where G ( s) represents the power system including SVC and H ( s ) indicates POD controller. yref

Σ

e

H (s )

u

y

G (s )

−

Figure 4. Closed-loop system with POD controller

Itcanbeproved[13],thatwhenthefeedbackcontrolis applied,theshiftofaneigenvaluecanbecalculatedby: Δλi = Ri H (λi ) (12) This shows that the shift of the eigenvalue caused by the controller is proportional to the magnitude of the corresponding residue. The POD controller, whose structure has been shown in Fig. 5, consists of an amplification block, a wash-out and low-pass filter and mc ψstages of lead-lag blocks (usually mc = 2 ). Input

KP

1 1 + Tm s

Tw s 1 + Tw s

Low-pass filter

Washout

1 + Tlead s 1 + Tlag s

1 + Tlead s Output 1 + Tlag s

mc stages

Figure 5. POD controller structure

The transfer function H ( s) , of the POD controller is given by [13]:

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⎛ 1 ⎞ ⎛ Tw s H (s ) = K P ⎜ ⎟⎜ ⎝ 1 + T m s ⎠ ⎝ 1 + Tw s

⎞ ⎛ 1 + T lead s ⎟⎜ ⎠ ⎜⎝ 1 +T lag s

2

⎞ ⎟ = K P H 1 (s ) (13) ⎟ ⎠

Where K p is a positive constant gain, Tm is the measurement time constant, Tw is the washout time constant and Tlead and Tlag are the lead and lag time constants, respectively. From (12) and Fig. 6, it can be clearly seen that with the same gain of the feedback loop, a larger residue will result in a larger change of the corresponding oscillatory mode. Therefore, the best feedback signal for the SVC damping controller is the one with the largest residue for the considered mode of oscillation. The same is true for the optimal location of the POD controller. Direction of Ri

Direction of Δλi = ΔK P H1 (λi ) Ri

Δλi

ϕ com

jω

λ(i 0)

( K P = ΔK P )

( K P = 0)

In Fig. 6 ϕ comp shows the compensation angle, which is necessary to move the eigenvalue direct to the left parallel with the real axis. This angle will be achieved by the lead-lag function and the parameters Tlead and Tlag , determined by:

T α c = lead = Tlag Tlag =

1 − sin(

1 + sin(

1

mc

ϕ comp mc

)

(14)

Linear Model

Controller Design

Mode Identifica tion

Figure 7. Adaptive POD Controller in general form

The identification purpose is to estimate ai and bi coefficients. The model is rewritten by: y (k ) = ϕ T (k − 1)θ (17)

(18)

The theoretical assumption is that the power system is working around a certain operating point for a certain period of time, which enables the estimated coefficient of the time varying linear model to converge to the actual values, then the recursive equations are obtained as follows: θˆ( k ) = θˆ(k − 1) + K (k )ε ( k ) ε (k ) = y (k ) − ϕ T (k − 1)θˆ(k − 1)

[

]

]

−1

(19)

where θˆ(t ) is the estimated parameters vector. To find transfer function of the system, it is more convenient

A (z −1 ) y (k ) = B (z −1 )u (k )

V. ADAPTIVE POWER SYSTEM IDENTIFICATION Fig. 7 shows the adaptive control scheme in general form. Linear model of the system is identified by RLS estimation. It is simpler to construct the linear model in discrete form [8]:

i =1

y

Power System

i.e z −1 y (k ) = y (k − 1) . Then we have:

The controller gain K p is computed as a function of the desired eigenvalue location according to (12): λ − λi (15) K p = id Ri H 1 (λi ) λid is the eigenvalue that is chosen to establish desired damping ratio i.e. 5%.

∑

u

POD Controller

to describe model employing backward time-shift operator z −1

where arg( Ri ) denotes the phase angle of the residue Ri , ω i is the frequency of the mode of oscillation in rad / sec .

m

ai y ( k − i ) +

e

[

)

ωi α c

n

r

K ( k ) = P(k − 1)ϕ (k − 1) I + ϕ T (k ) P(k − 1)ϕ (k ) P (k ) = I − K (k ) + ϕ T (k ) P( k − 1)

Tlead = α c Tlag

y (k ) = −

output) at time k .

θ T = [a1 a 2 ...a n b0 ...bm ] ϕ T ( k − 1) = [− y (k − 1) ... − y (k − n) u ( k ) ... u (k − m)]

σ

Figure 6. Shift of eigenvalues with the POD controller

ϕ comp = 180 D − arg( Ri ) ϕ comp

u (k ) is the value of the measured input variable (controller

where ϕ T and θ are the measured values and actual system parameters vectors respectively as follows:

arg( Ri )

λ(i1)

where y (k ) is the value of the measured output variable and

∑ bi u(k − i)

(16)

A (z −1 ) = 1 + a1z −1 + a2 z −2 + ... + an z − n

(20)

B (z −1 ) = b0 + b1z −1 + b 2 z −2 + ... + b m z − m To convert the estimated discrete-time model of the system to a continuous one the Tustin’s bilinear approximation is employed: T 1− s s 2 z −1 = e −sTs ≈ (21) Ts 1+ s 2 VI. THE CASE STUDY

The adaptive POD controller design approach is applied on a test system and the performance of the controller is examined. This system is shown in Fig. 8. The two area system has four machines and each machine is equipped with IEEE standard exciter controllers [12]. In order to construct a linear model, an

i =0

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identification system of order of n = 16 (and m = 15 ) is considered here. G1

1

5

7

6

9

L1

10

3

11

G3

L2

L7

2

L9

linearized model. Based on Simulated results it has been seen in some cases the conventionally tuned POD SVC, can not stabilize the power system under all admissible operating conditions whereas an adaptive system can update the controller in a wide range of operating points. The proposed algorithm works based on RLS estimation. 1.15

4

1.1

G2

G4

Area 1

1.05

Area 2

V7 [pu]

1

Figure 8. Two-area test system

0.9

0.8 0.75 0

10

2

4

6

8

10

12

14

16

18

20

Time[s]

9

Damping ratio = 5%

8 7

Figure 10. Bus #7(SVC terminal) voltage magnitude

Local modes

6

jω

Fixed POD Adaptive POD

0.85

The root locus of dominant eigenvalues of uncontrolled system is shown in Fig. 9.

1.1

5

1.05

4

Interarea mode

1

V9 [pu]

3 2 1 0 -6

0.95

-5

-4

-3

-2

-1

0

0.95 0.9 0.85

1

σ

Fixed POD Adaptive POD

0.8

Figure 9. Dominant eigenvalues of uncontrolled system

0.75 0

2

4

6

8

10

12

14

16

18

20

Time[s]

System has two critical oscillatory inter-area modes characterized with eigenvalue λ = 0.06 ± j3.94 , with low damping ratio, ξ = −0.015 . Table 1 shows the numerical results of the residue values associated with critical mode calculated using the transfer functions Δα SVC ΔVt at different buses, where ΔVt and Δα SVC denote SVC bus terminal voltage and thyristors firing angle deviation respectively. According to this table bus #7 has the largest residue and is then suitable location for the SVC.

Figure 11. Bus #9 voltage magnitude

REFERENCES [1]

[2] [3] [4]

TABLE 1. LOCATION INDICES OF SVC

Bus 7 9 6 10 11 5

Ri 0.8706 0.3956 0.0012 0.001 0 0

A three phase fault is applied in the middle of the line L1 and cleared after 6 cycles by opening the faulted line. A direct comparison between the fixed and adaptive POD for voltage oscillations at bus #7(SVC terminal) and bus #9 are shown in Figs. 10 and 11 respectively. VII. CONCLUSIONS In this paper an adaptive tuning method for SVC is presented based on residue approach. It is a self-tuning regulator in which controller parameters are updated based on identified

[5] [6] [7] [8] [9] [10] [11] [12] [13]

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