Parameter tuning of power system stabilizer using eigenvalue sensitivity

Electric Power Systems Research 81 (2011) 2171–2177

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Parameter tuning of power system stabilizer using eigenvalue sensitivity Damir Sumina a,∗ , Neven Bulic´ b , Mato Miˇskovic´ c a

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia University of Rijeka, Faculty of Engineering, Vukovarska 58, 51000 Rijeka, Croatia c Hydroelectric Power Plant Dubrovnik, Dubrovnik 20000, Croatia b

a r t i c l e

i n f o

Article history: Received 10 September 2010 Received in revised form 20 May 2011 Accepted 6 July 2011 Available online 31 July 2011 Keywords: Power system stabilizer Kalman filter Eigenvalue analysis

a b s t r a c t This paper proposes a method for PSS parameter tuning. The proposed method is based on the system identification using a Kalman filter. To eliminate the influence of inaccuracies of the mathematical model due to incomplete data of the system parameters, this paper proposes a procedure for identifying system parameters based on a Kalman filter. The identified system is the basis for determining the eigenvalue sensitivity function. PSS tuning is carried out based on the eigenvalue sensitivity analysis of changes of PSS parameters. In order to obtain the eigenvalue sensitivity function on the PSS parameter in a wider range of parameter values, it has been proposed to determine eigenvalues on the basis of the Bairstow method for solving polynomial equations. The entire procedure can be applied on-line, so the PSS structure and settings can be adaptive. Experimental tests carried out on a synchronous generator connected to an AC system demonstrate the effectiveness of the proposed method. Performance of the PSS tuned by using the proposed method was compared with the performance of the PSS tuned by using eigenvalue sensitivity. © 2011 Elsevier B.V. All rights reserved.

1. Introduction With the rising electric power demand, power systems can reach stressed conditions due to the need for operating closer to their limits of stability. In a power system, electromechanical oscillations usually follow a disturbance within the frequency range of 0.2–3 Hz [1,2]. Preventing this sustained oscillation is of great interest as the potential for an oscillation causes lower operational limits to be set on transfer levels. Oscillations can also arise during normal steady-state operation. The damping of sustained oscillations has been studied by a number of researches. To enhance power system damping, the generator is equipped with a power system stabilizer (PSS) that provides a supplementary feedback stabilizing the signal in the excitation system [3–5]. The problem of the PSS design is finding optimal parameters because the power system is nonlinear. The conventional PSS (CPSS) is usually designed with a fixed gain, with an aim to stabilize at the desired operating condition. However, the inherent non-linearity and multiple operating points of a power system degrade the performance of the CPSS fixed gain. A reasonable level of the CPSS robustness can be achieved depending on the parameter tuning methodology [6]. Numerous methods for determining PSS parameters have been proposed, including state-space/frequency domain techniques [7,8], residue compensation [9], phase compensation of a lead–lag controller [4], pole-placement for a PID-type controller [10], tabu

∗ Corresponding author. Tel. +385 1 6129784; fax: +385 1 6129705. E-mail address: [email protected] (D. Sumina). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.07.016

search technique [11], ant direction hybrid differential evolution [12], simulated annealing [13], particle swarm optimization [14], and evolutionary programming [15]. PSS designs use artificial intelligence methods, such as neural networks [16] and fuzzy logic [17]. Synergetic control theory [18] and other nonlinear control techniques [19,20] are also used in the PSS design. This paper presents a new method for PSS parameter tuning based both on the system identification using a Kalman filter and on determining the eigenvalue sensitivity function of an identified system. To eliminate the influence of inaccuracies of the mathematical model due to incomplete data of the system parameters, this paper proposes a procedure, described in Section 2, for identifying the system parameters based on the Kalman filter. The identified system is the basis for determining parameters of a PSS. The conventional way to determine PSS parameters, described in Section 3, is the use of eigenvalue sensitivity. A drawback to this method is a fixed defined desired damping (the pre-determined real part of a dominant conjugate pair of eigenvalues). However, the frequency and damping of electromechanical oscillations are not constant for all operating points within the PQ diagram. Especially in the cases near the stability border in the capacitive operating area [21], it is evident that there are relatively big changes in the amount of damping and frequency. So, the PSS parameters determined for one operating point are not optimal for other operating points. However, in Section 4 a method for determining the PSS parameter is described, which is based on the determination of the functional dependence of eigenvalues on the PSS parameters. PSS tuning is carried out based on the eigenvalue sensitivity analysis of changes of PSS parameters. The proposed method allows the selection of PSS

D. Sumina et al. / Electric Power Systems Research 81 (2011) 2171–2177

parameters in such a way that the changes in the system damping are observed depending on the PSS parameters. PSS parameters can be determined for each operating point to obtain the desired amount of system damping. In order to obtain the eigenvalue sensitivity function on the PSS parameter in a wider range of parameter values, the determination of eigenvalues based on the Bairstow method for solving polynomial equations has been proposed. The proposed procedure can be applied on-line, so the PSS structure and settings can be adaptive. The method is experimentally verified on a synchronous generator connected to an AC system. The performance of the PSS tuned by using the proposed method was compared with the performance of the PSS tuned by using the method described in Section 3. 2. System identification For the system identification a linear mathematical model with identified parameters is used. To identify the system parameters the measured data are applied. The identification process was performed using a Kalman filter [22,23]. In order to increase the efficiency of the procedure a multi-streaming algorithm is chosen, which comprises a number of input data at every step, and thus achieves a faster convergence. The linearized system, which includes a synchronous generator connected over transmission lines to an AC network, is represented as the following state-space model:



•

x = A · x + B · u y = C · x + D · u

(1)

where A is the power system state matrix, B is the input matrix, C is the output matrix, D is the feedforward matrix, x is the vector of state variables, u is the vector of control inputs, and y is the output vector. An estimation model was obtained from discrete state equations of the system given in Eq. (1): .



−1 

T A 2

x(k + 1)

=

y(k)

T T · I− A 2 2 = C · x(k) +

I−

· I−



−1



T A · x(k) 2 (2)

· B · [u(k) + u(k + 1)]



y

=

 ym

=

V 1 . . VN ı1 . . ıN P1 . . PN V1 . . VN ı1 . . ıN P1 . . PN

T

T

K(i) = P(i − 1)

d(t)

V PSS

Eigenvalues sensitivity function

Δω, ΔP

Model identification

ω

G ~

i1 i3 v12 v32 L1

D ig ita l co n tro l syste m

AC system Fig. 1. Structure of the laboratory model.

where P(i − 1) is the error covariance matrix, I is N · n × N · n identity matrix, n is the number of identified parameters,  is the vector of the unknown parameters that are identified in the identification procedure, and i is the number of iteration. The dependence of the output values on the identified values dy(t)/d(t) is determined by the numerical method by using the discretized identification model given in Eq. (2). The size of matrix dy(t)/d(t) is n × n · N. Rate of convergence is determined by the value of coefficient i . An increase in i results in a slower growth and a safer convergence. The initial value is set at i = 106 . This coefficient is reduced to the value of i = 102 at each step by multiplying the number of less than one. The increase in the identified values is determined by the Kalman gain K(i): (5)

Error covariance update is determined by the following recurrence relation: P(i + 1) = P(i) − K(i) ·

 dy(t) T d(t)

· P(i) + Q(i)

(6)

where, Q(i) is n × n disturbance covariance matrix. The initial value of elements on the square diagonal matrix Q is typically 0.5 and reduces to 0.0001 at the end of the procedure. 2.1. Identification results

(3)

T

where Vref is the generator voltage reference variation, V is the generator voltage deviation, ı is the load angle deviation, and P is the generator active power deviation. The dimension of the input data vector u is 1 × N and of the vectors y and ym 3 × N, where N is the number of the measured samples. Kalman gain K(i) is determined as follows [22]:

 dy(t) 

Vre f

(i) = (i) + K(i) · [ym (i) − y(i)]

where T is the sample time and k is the step time. The system identification using the Kalman filter is realized with the following algorithm. For the identification process the input data vector u is formed. The vector of the output data ym is obtained by measuring, while the vector of the output data y is obtained by using the discretized estimation model described in Eq. (2): u = [Vref1 .....VrefN ]

3x400 V Voltage co n tro lle r

Measurement unit

2172

· [(i) · I +

 dy(t) T d(t)

· P(i − 1) ·

 dy(t)  d(t)

−1

] (4)

Using the procedure described in this section, the system state matrix was identified (Fig. 1). A measurement and identification was obtained using a digital voltage control system. A pulse signal ±0.05 p.u. is superimposed to the voltage reference. This is the maximum allowed value of disturbance with respect to the assumed linear terms. The measured signals are generator voltage deviation  V, load angle deviation ı and active power deviation  P. Fig. 2 shows the measured and the estimated results. Also, Fig. 2(d) shows the response of the square error of the output data  during identification where  is determined as follows: ε = (ym − y)T · (ym − y)

(7)

For identification at every step N = 100 measured vectors were applied. The identification procedure has converged after 100 steps.

D. Sumina et al. / Electric Power Systems Research 81 (2011) 2171–2177

2173

By applying Eq. (10) the transfer function of a synchronous generator connected to an AC network with an included excitation control system is determined as follows: GSGR (s) =

N

(C · V)i

i=1

1 (W · B)i s − i

(11)

According to Eq. (11) the transfer function of the system is obtained as the sum of partial transfer functions:

y(s) GSGRi = u(s) N

GSGR (s) =

(12)

i=1

Transfer function GSGR (s) was obtained as a column vector with N members, where N is the order of the system presented in Eq. (1). The final transfer function of the system is obtained as the sum of transfer function matrices. Transfer function GSGR (s) is used for PSS parameters tuning. When the output signal of the stabilizer is included into the excitation system, the transfer function GSGR (s) can be rewritten as: GSGRS (s) =

GSGRS (s) 1 − GSGR (s) · GS (s)

(13)

For the transfer function of the open loop system GO = GSGR (s) · GS (s) the sensitivity of the eigenvalue i with respect to the parameter q is defined as [26]:



Fig. 2. Identification process: measured and estimated values. From top to bottom: generator voltage deviation, load angle deviation, active power deviation, square of error.

∂i ∂GS (s, q)

= GSGR (s)

∂q ∂q

Nominal data of the synchronous generator are given in Appendix A.

PSSs with a lead–lag structure of speed deviation and active power deviation input are considered in this study. The transfer function of the PSS is given as follows: G(s) =

3. PSS tuning using eigenvalue sensitivity The eigenvalue sensitivity approach to design the PSS is very appropriate concerning the available linearized mathematical model of a real system [24–26]. Several approaches to the determination of eigenvalue sensitivities of the power system linear model have been presented in the literature which differ in the formulation of the power system model. In [25], a hybrid form of the system model is presented, where the system model is expressed as in Eq. (1) and also described by its transfer function F(s, q). If the state matrix has N distinct eigenvalues and , V and W are the matrices of eigenvalues, and the right and the left eigenvectors, respectively, equations for all eigenvalues and eigenvectors can be expressed as: A·V W·A W·V

=V· =·W =1

(8)

Using Eq. (8), the system described in Eq. (1) can be linearly transformed in a way that x is replaced with z as follows: x z . z y

=V·z =W·x =  · z(t) + W · B · u(t) = C · V · z(t)

(9)

By applying the Laplace transform to Eq. (9) the transfer function of the output variables to the control input can be defined as follows [25]: y(s) −1 = C · V · (sI − ) · W · B u(s)

(10)

(14) s=i



1 sTw 1 + sTs Ks 1 + sT0 1 + sTw 1 + sT1

L (15)

where T1 , Ts , and Ks are the time constants and gain of the PSS and T0 is the transducer time constant. The time constant Tw in the washout filter is fixed at 5 s in this study and L is equal to 2. Assuming linearity for small changes and according to Eq. (14), the value of the time constant Ts is assumed and the parameter Ks can be determined as follows: ∂ Tw = 1 + Tw ∂Ks

 1 + T 1 + T  s s 1 + T1 1 + T1

(16)

where the selected eigenvalue  belongs to the dominant pole. 4. Tuning of PSS parameters by determining the function of the sensitivity of eigenvalues Another approach to determine the PSS parameter settings in relation to the procedure described in Section 3 is determining the functional dependence of the system damping on the PSS parameters in a defined range. Therefore, it is necessary to determine the functional dependence of eigenvalues on the PSS parameters expressed as: Re() = f (Ks , Ts )

(17)

Damping of electromechanical oscillations is determined by the real part of a complex conjugated pair of the eigenvalues, which shows considerable sensitivity to changes in the PSS parameters settings. Maximum damping is achieved for the case when the absolute value Re() in Eq. (17) is maximal. It is necessary to choose the range of gain Ks and time constant Ts . If these parameters change independently, a curve of the parameter settings can be determined. For the proposed procedure a mathematical model of the

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system is needed. By applying the transfer function of the stabilizer given in Eq. (15), a model of a stabilizer in the state space is derived:

⎡ ⎢ ⎢ ⎢ ⎢ BS = ⎢ ⎢ ⎢ ⎢ ⎣ 

Ks T0 Ks Ts T0 T1 Ks Ts2 T0 T 2 1 Ks Ts2 T0 T 2 0

1 T1 Ts − 2 T1 Ts − 2 T1

− 1 T1 1 T1

⎤

0 0 1 − T1 1 − T1

−

V, p.u.

⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 1 ⎦

AGRS =

[A] [BS C]

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 t, s

1.2

1.4

1.6

1.8

2

0 −0.2 −0.4

TW

−0.6 0.4 0.2 0 −0.2

1

−0.4



Without PSS

[BCS ] [AS ]

= 2 + u + v = p()b0 n−2 + b1 n−3 + · · · + bn−3  + bn−2 +bn−1 ( + u) + bn = bn−1 ( + u) + bn

p() P()

(19)

r



(20)

u v



⎡ =⎣

∂bn ∂u ∂bn−1 ∂u

∂bn ∂v ∂bn−1 ∂v

⎤−1 ⎦



−bn −bn−1

Matlab Real Time Workshop Voltage controller PSS Model Identification Eigenvalues sensitivity function

Data Aquisition Card D/A

PWM PC

if

AC system L1

M ω

(21)

In order for the polynomial p() to be the factor of the polynomial P() it is necessary that the rest of dividing r in Eq. (21) equals zero. For the initial values of u and v in Eq. (21) increments are calculated at each step in order to reduce the rest r. Increments u and v are determined as follows:

where AGRS is the state matrix of the system with a stabilizer. Matrix AGRS is applied for determining the eigenvalue sensitivity to the PSS parameter Ks and Ts settings. Eigenvalues are determined by applying the Bairstow method [27] that determines zeros of the characteristic polynomial P(), which are equal to the eigenvalues of the matrix AGRS :

A/D

PSS2

The Bairstow method is based on the iterative procedure to solve quadratic polynomial factors as follows:



P() = det(I − AGRS )

PSS1

Fig. 3. Simulation results for the system without and with PSS.

where AS is the state matrix of the stabilizer, BS is the matrix of input of the stabilizer and CS is the matrix of output of the stabilizer. Based on the mathematical model of the generator with a voltage regulator and the model of the stabilizer a model of the entire system can be determined by structuring the matrix state AGRS which consists of matrices as follows:



0

0.2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0

0.02

0 -0.01

0

(18)

1

CS = 0

0

δ, rad.

⎢ ⎢ ⎢ ⎢ AS = ⎢ ⎢ ⎢ ⎢ ⎣

⎤

1 − T0 1 Ts − T1 T0 T1 Ts Ts2 − T12 T0 T12 Ts T2 − s2 2 T1 T0 T1

0.04

ω, p.u.

⎡

0.06

T1

G ~

v12v32 i3 i1

Fig. 4. Experimental setup.

v12m

 (22)

D. Sumina et al. / Electric Power Systems Research 81 (2011) 2171–2177

2175

0

0

Ts=0.05

λ2

−0.2

−0.2

Ts=0.01 Ts=0.02 Ts=0.03

−0.4

Ts=0.03 λ3

−0.6

Damping ratio ζ

Damping ratio ζ

Ts=0.04

−0.8

−0.4

λ2 −0.6

λ3

−0.8

−1 0

0.5

1

1.5

2

−1

2.5

0

0.1

0.2

=

Ks1 Ts1

Ks2 Ts1

· ·

Ksn Tsn



[] =

11 · M1

· · ·

1n · Mn

(24)

0.94

0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

0.4 0.39 0.38

0.36 0.02 0.01 0

−0.01 −0.02

In order to apply the Bairstow method, the calculation of eigenvalues was carried out as follows: (1) With a fixed time constant Ts the eigenvalues are determined for all values of Ks starting from i = 1 to i = n. (2) For the parameter values Ks2 to Ksn the initial values of the coefficients u and v in Eq. (21) are determined as follows: u = −(1,n−1 + 1) + 2,n−1 + (un−1 − un−2 )

0.96

0.37



v = 1,n−1 · 2,n−1 + v + (vn−1 − vn−2 )

1 0.98

(23)

where the dimension of the parameter field is 2 × n. It is necessary to calculate the eigenvalues of the matrix AGRS :



1.02

Δω, p.u.

Ksi Tsi



0.5

1.04

V, p.u.

Eq. (22) is obtained by developing Eq. (21) in the Taylor series. In implementing the Bairstow method the relation (21) is determined by the recursive procedure. When one factor p() is determined, the polynomial order is reduced by two by dividing. This procedure is repeated until all the eigenvalues are determined. A disadvantage to this procedure is the dependence of convergence on the initial values of the coefficients u and v [28]. To determine the dependence of the eigenvalue matrix AGRS in Eq. (19) on the parameters and settings of the PSS, the Bairstow method was applied as follows. The field of the PSS parameters Ks and Ts is defined:



0.4

Fig. 6. Sensitivity function of eigenvalue on PSS parameters Ks and Ts ; input signal of PSS is active power deviation.

δ, rad.

Fig. 5. Sensitivity function of eigenvalue on PSS parameters Ks and Ts ; input signal of PSS is speed deviation.



0.3

Ks

Ks

(25)

where 1 and 2 are the solution of a certain quadratic polynomial p in Eq. (21) which was obtained for the previous value of parameter Ks . (3) Once the eigenvalues are determined for all values of the parameter Ks in Eq. (23) a new value of the time constant Ts is selected and the procedure is repeated for all values of the parameter Ks as described. The values of the time constant Ts are chosen starting from the value at which the stabilizer achieves a minimum contribution to the stability of the system to the value at which the system reaches the stability limit. Finally, as a result of applying the tuning method, PSS parameters are selected to obtain a minimum damping ratio. By using the initial values of the coefficients u and v in Eq. (25) a secure convergence is achieved. The procedure for determining

Without PSS

t,s

PSS1

PSS2

Fig. 7. Generator voltage, load angle and rotor speed response without and with PSS; input signal of PSS is speed deviation.

eigenvalues is achieved with a very small number of steps, and the obtained eigenvalues always occur in the same order. The proposed method has a certain disadvantage in the application to large scale power systems. With the increased system model complexity, the size and complexity of the state matrix will increase. The application to large scale power systems can become a difficult task due to the computational requirements. In this case it is necessary to apply a reduced order eigenanalysis [25]. Only low-frequency modes that are associated with the dynamics of the generator rotor will be observed. Another issue in implementing the proposed method is the limited number of the measurements in the large power system. Also, the proposed method is applied to a single machine infinite bus with the dominant electromechanical mode. Future work will be based on the application of the proposed method for identifying other modes, such as the exciter mode or the inter-area modes, and on the application of the method to large power systems.

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D. Sumina et al. / Electric Power Systems Research 81 (2011) 2171–2177

V, p.u.

1.05

1

0.95

0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

δ, rad.

0.4 0.39 0.38 0.37 0.36 1.4

P, p.u.

1.2 1 0.8 0.6 0.4

t,s Without PSS

PSS1

PSS2

Fig. 8. Generator voltage, load angle and active power response without and with PSS; input signal of PSS is active power deviation.

PSS performances are simulated for the identified system in Section 2.1. The synchronous generator operates at an active power 1 p.u. and reactive power 0.1 p.u. capacitive. Fig. 3 shows comparative simulation results for the system without and the system with a PSS for the step change of voltage reference 0.05 p.u. The input of the PSS is the signal of the active power deviation. Parameters of stabilizers PSS1 and PSS2 were selected by using the procedures described in Sections 3 and 4, respectively and given in Appendix B.

The experiment without and with the PSS in both cases is performed for the step change of voltage reference 0.05 p.u. (Figs. 7 and 8). Experimental results show that both stabilizers are satisfactorily tuned. However, the generator’s trajectory is less when PSS2 is used.

5. Experimental results

This paper proposes a method for PSS parameter tuning based on the model identification of a synchronous generator connected to an AC network using a Kalman filter. The identified system model is the basis for the PSS tuning method which uses an analysis of the eigenvalue sensitivity of the system state matrix. In this study, the conventional lead–lag structure of the PSS with rotor speed deviation and active power deviation inputs is used. The proposed procedure for PSS tuning based on the Bairstow method for determining the dependence of the eigenvalue sensitivity functions allows the determination of the PSS parameters. The method is experimentally verified in a laboratory environment. An algorithm was implemented in a digital control system and tested on a synchronous generator connected to an AC network. The obtained results show the effectiveness of the applied identification procedure, i.e. the PSS tuning procedure. With the integration of methods for identification and for the PSS tuning a high damping of electromechanical oscillations and increased system stability have been achieved, as demonstrated by the experiments on the real system.

By using the described procedure the functions of the eigenvalue sensitivity on the tuned PSS parameters are experimentally determined. From the set of eigenvalues, those that show extreme sensitivity to changes in the PSS parameters are selected. The PSS parameters are determined for two cases: first, when stabilization is achieved by introducing the speed deviation signal and, second, when stabilization is achieved by introducing the active power deviation signal. In both cases the parameters of stabilizers PSS1 and PSS2 were selected by using the procedures described in Sections 3 and 4, respectively, and given in Appendix B. The performance of such tuned PSSs was compared. The experimental setup is presented in Fig. 4. 5.1. Case 1 For a synchronous generator with an included voltage controller the eigenvalue sensitivity functions are determined for the case when stabilization is achieved by introducing the speed deviation signal ω (Fig. 5). The PSS has a transfer function with two identical lead–lag stages, i.e. L = 2 in Eq. (15).

6. Conclusion

Appendix A. The nominal data of the synchronous generator

5.2. Case 2 In this case the input of the PSS is the signal of the active power deviation P. The PSS includes one lead–lag stage, i.e. L = 1 in Eq. (15). Fig. 6 shows the function of the eigenvalue sensitivity to the PSS parameters.

Un Sn fn ωn cosϕn

400 V 4.5 kVA 50 Hz 3000 rpm 0.8

D. Sumina et al. / Electric Power Systems Research 81 (2011) 2171–2177

Appendix B. PSS parameters T0 T1 Tw

0.001 s 0.1 s 5s

PSS1 parameters Ks and Ts are obtained using eigenvalue sensitivity providing that the damping ratio is  = − 0.3: Input of the PSS1 is speed deviation Ks Ts

1.4 0.03 s

Input of the PSS1 is active power deviation Ks Ts

0.12 0.02 s

PSS2 parameters Ks and Ts are obtained using eigenvalue sensitivity functions: Input of the PSS2 is speed deviation Ks Ts

1.9 0.03 s

Input of the PSS2 is active power deviation Ks Ts

0.14 0.01 s

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