The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system

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Research article

The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system Miaoping Sun n, Xiaohong Nian, Liqiong Dai, Hua Guo College of information science and engineering, Central South University, Changsha, China

art ic l e i nf o

a b s t r a c t

Article history: Received 27 April 2016 Received in revised form 18 January 2017 Accepted 10 March 2017

In this paper, the delay-dependent wide-area dynamic output feedback controller (DOFC) with prescribed degree of stability is proposed for interconnected power system to damp inter-area low-frequency oscillations. Here, the prescribed degree of stability α is used to maintain all the poles on the left of s = − α in the s-plane. Firstly, residue approach is adopted to select inputoutput control signals and the schur balanced truncation model reduction method is utilized to obtain the reduced power system model. Secondly, based on Lyapunov stability theory and transformation operation in complex plane, the sufficient condition of asymptotic stability for closed-loop power system with prescribed degree of stability α is derived. Then, a novel method based on linear matrix inequalities (LMIs) is presented to obtain the parameters of DOFC and calculate delay margin of the closed-loop system considering the prescribed degree of stability α . Finally, case studies are carried out on the two-area four-machine system, which is controlled by classical wide-area power system stabilizer (WAPSS) in reported reference and our proposed DOFC respectively. The effectiveness and advantages of the proposed method are verified by the simulation results under different operating conditions. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Dynamic output feedback controller (DOFC) Interconnected power system Prescribed degree of stability Wide-area power system stabilizer (WAPSS) Linear matrix inequalities (LMIs)

1. Introduction With the increase of the scale and load ability of power systems, the inter-area low frequency oscillations become a serious problem and often suffer from poor system damping [1,2]. Many power systems in the world are affected by these electromechanical oscillations whose frequency varies between 0.1 and 2 Hz [3]. Traditionally, the damping of low frequency oscillations is provided by installing a power system stabilizer (PSS), which uses local measurements such as rotor speed and active power as feedback signals. Such PSSs can damp the local area modes effectively, while their effectiveness in damping inter-area modes is reduced because such modes are not observable/controllable directly from local signals of the generators. With the wide application of synchronized phase measurement unit (PMU) in power system, the wide-area measurement system (WAMS) technology has enabled the use of measured information from remote location for enhancing transient stability [4–7]. The availability of remote signals can overcome the aforementioned shortcoming of lacking observability and provide flexibility to n

Corresponding author. E-mail address: [email protected] (M. Sun).

damp a special critical inter-area oscillation mode of power system [8]. Although wide-area PSS provides a great potential to improve the damping of inter-area oscillation modes, the delay caused by the transmission of remote signals will degrade the damping performance, or even cause instability of the closed-loop system [9,10]. Therefore, the influence of time delay must be fully taken into consideration in the controller design. In the published works about the time delay issue of power system, three different strategies are often adopted. The first is to design controllers without considering time delay [5,11]. The second strategy is to make use of Pade approximation method, which can approximate time delay during model linearization [12–14]. It's obvious that the accuracy of Pade approximation changes with the change of order number and the higher the order number is, the higher the accuracy is. However, the amount of calculation will greatly increase. The third is to employ some robust control methods, which can keep the system stable within a certain delay [6,9,15–17]. Though the fore-mentioned controllers can ensure the stability of power systems, it may produce weakly damped inter-area oscillation mode, which is not acceptable in damping control of power system due to the larger oscillation amplitude and longer fluctuation time. The idea of prescribed degree of stability is proposed in theory to deal with weakly oscillation mode [18]. This concept is applied

http://dx.doi.org/10.1016/j.isatra.2017.03.003 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

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to power system to damp low-frequency oscillations in [19,20], where [19] designs a nonlinear decentralized feedback controller and [20] presents a WAPSS design under the condition of given damping factor and required signal transmission delay. The literature [20], above mentioned [6,9] and [21] are dedicated to obtain the parameters of WAPSS by lead-lag compensation method, but this method is no more effective because the strong coupling between the local oscillation modes and the inter-area modes would make the tuning of PSSs for damping all modes nearly impossible and the only adjusted gain would make the control effect very limited [22,23]. The WAPSS described in the form of transfer function is a dynamic output feedback controller by nature. Recently, the DOFC design has received particular research interests owing to the partially known state information and inaccurate measurement information in practical application [24,25]. However, in many cases, the design methods of state feedback controller cannot be easily applied to those of dynamic output feedback controller, which makes output feedback control generally more difficult and involved [26–29]. Without considering the time delay of closed-loop systems, the continuous and discrete DOFC are respectively designed in [26] and [27,28]. Though [29] presents a novel DOFC design and acquires its parameter matrices for discrete-time Takagi-Sugenno fuzzy system with time-varying delays, it is still very difficult for continuous timedelay system to obtain these parameter matrices. Motivated by the above investigation, the delay-dependent wide-area DOFC with prescribed degree of stability in this paper is developed to improve the damping inter-area oscillation, where the prescribed degree of stability guarantees all the poles of closed-loop power system on the left of s = − α in the s-plane. At first, the modal analysis of the linear model for power system excluding wide-area damping controller is applied to find out the low-frequency oscillation modes and identify the critical interarea mode. Next, residue approach is utilized to select the most efficient input-output control signals. Then, the delay margin based on prescribed degree of stability and the parameters of DOFC are obtained by solving the derived LMIs. Finally, based on the full-order model of two-area four-machine power system, simulation studies are undertaken to verify the effectiveness and demonstrate the advantages of the proposed method. The main contribution of the paper can be summarized as follows. 1) The prescribed degree of stability realized by transformation operation in complex plane is introduced to deal with weakly damped inter-area oscillation modes. 2) The delay-dependent sufficient conditions of asymptotic stability for closed-loop power system with prescribed degree of stability are derived. 3) A novel method based on LMI is presented to solve parameters of dynamic output feedback controller and calculate the delay margin with prescribed degree of stability. Notations. The superscript “T”, “H” and “
1” stand for matrix transposition, conjugate transpose and inverse, respectively; Rn denotes the n-dimensional Euclidean space; the notation P > 0 means that P is real symmetric and positive definite; I and 0 represent the identity matrix and zero matrix, respectively.

2. Problem formulation 2.1. Modal analysis and selection of wide-area signals The nonlinear dynamic model of power system is usually described by a set of differential-algebraic equations. The whole power system excluding the local PSS and wide-area damping controller can be linearized at an equilibrium point as follows

⎧ ẋ (t ) = A x (t ) + B u (t ) 0 0 0 0 0 ⎨ ⎪ ⎩ y0 (t ) = C0x 0(t ) ⎪

(1)

where x0(t ) ∈ Rn × 1, u0(t ) ∈ Rm × 1 and y0 (t ) ∈ Rp × 1 are the state, input and output vectors, respectively. A0 ∈ Rn × n , B0 ∈ Rn × m and C0 ∈ Rp × n are the state, input and output matrices, respectively. An eigen analysis of matrix A0 produces the distinct eigenvalues λk (assumed distinct for k¼ 1,…,n) and corresponding maT trices of the right and left eigenvectors E = ⎡⎣ e1T e2T ⋯ enT ⎤⎦ and F = ⎡⎣ f f ⋯ f ⎤⎦, respectively. 1 2

n

The transfer function of interconnected system associated with (1) is expressed by −1

H (s ) = C0( sIn − A 0 ) B0 =

n

∑ k=1

where Rk ∈ C and

p×m

Rk s − λi

is the residue matrix associated with mode

Rk = Cek fkH B

(2)

λk (3)

For i¼1,…,p and j¼1,…,m, the element rk(i, j ) of matrix Rk are given by

rk(i, j ) = Ciek fkH Bj

(4)

If the maximal value rk(i, j ) of the residues associated with mode k is obtained, then control input uj(t) and feedback signal yi(t) are the most efficient signals to damp mode k. 2.2. Modeling of power system with time delay For a large-scale power system, the order of the linearized model is comparatively high, which makes the design of a controller difficult or even infeasible. Moreover, the analysis of low frequency oscillation does not require the full-order model, as fast dynamic modes are not considered in this case. Hence, a model reduction method is always used to reduce the order of the whole power system. In this paper, the Schur balanced truncation model reduction method is used to find the reduced-order system and its details can be found in [30]. The reduced-order model of the openloop power system excluding the wide-area damping controller can be linearized at an equilibrium point as follows

⎧ ẋ(t ) = A1x(t ) + B1u(t ) ⎨ ⎩ y(t ) = C1x(t ) ⎪

(5)

where x(t), u(t) and y(t) are the state, input and output vectors, respectively. A1, B1 and C1 are the state, input and output matrices of the reduced-order power system. The structure of the closed-loop power system, which includes the reduced-order power system, wide-area damping controller and transmission delay d of the wide-area signal, is shown in Fig. 1. The wide-area damping controller adopted in the form of dynamic output feedback controller is shown in Fig. 1 as

⎧ ẋ (t ) = A x (t ) + B u (t ) W W W W W ⎨ ⎪ ⎩ yW (t ) = CW xW (t ) + DW uW (t ) ⎪

(6)

where xW(t), uW(t), yW(t) are the state, input and output vectors of DOFC, AW, BW, CW and DW are the state, input, output and transmission matrices of DOFC to be determined. The connection between the open-loop power system and WADC is represented as

⎧ u(t ) = y (t ) W ⎨ ⎩ uW (t ) = y(t − d) ⎪

⎪

(7)

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

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V (z(t )) = zT (t )Pz(t ) + 0

+

t

∫t − d zT (s)Qz(s)ds

t

∫−d ∫t + θ zṪ (s)Rz(̇ s)dsdθ

(12)

where P, Q, R are symmetric positive-definite matrices to be determined. Taking the time derivative of V (z (t )) along the trajectory of (10) is given by Fig. 1. The structure of reduced-order power system with wide-area damping controller.

V̇ (z(t )) = 2zT (t )Pz(̇ t ) + zT (t )Qz(t ) − zT (t − d)Qz(t − d) + dz Ṫ (t )Rz(̇ t ) −

t

∫t − d zṪ (s)Rz(̇ s)ds

(13)

Hence, the augmented closed-loop power system can be represented as

For any appropriately dimensioned matrices N and M, the following equations hold according to the Leibniz-Newton formula

ξ (̇ t ) = Aξ ξ (t ) + Adξ ξ (t − d)

⎧ t ⎤ ⎡ ⎪ 2ς T (t )N ⎢ z(t ) − z(t − d) − z(̇ s )ds⎥ = 0 ⎦ ⎣ t−d ⎪ ⎪ T ⎨ 2ς (t )M ⎡⎣ A z z(t ) + Adz z(t − d) − z(̇ t )⎤⎦ = 0 ⎪ t ⎪ T −1 T ς T (t )NR−1NT ς (t )ds = 0 ⎪ dς (t )NR N ς (t ) − ⎩ t−d

(8)

T ⎡ B D C 0⎤ ⎡ A BC ⎤ T where, ξ(t ) = ⎡⎣ xT (t ) xW (t )⎤⎦ , Aξ = ⎣⎢ 01 A1 W ⎦⎥, Adξ = ⎣⎢ 1B WC 1 0 ⎦⎥. W 1 W In the sequel, the definition of stability degree is introduced.

Definition. If a system is with prescribed degree of stability α, then this system is asymptotically stable and the real part of all the eigenvalues is less than −α .

3. Main results

(9)

where α > 0 is represented as the prescribed degree of stability. The transformed system can be represented as

z(̇ t ) = A z z(t ) + Adz z(t − d)

(10)

where Az = αI + Aξ , Adz = Adξ edα . Remark 1. It can be easily seen that the real part of all the eigenvalues of system (8) is α less than those of system (10). Thus, if the transformed system (10) is asymptotically stable, the augmented closed-loop system (8) is asymptotically stable with prescribed degree of stability α, which can achieve the desired damping effect. Theorem 1. For given scalars d, α, AW, BW, CW and DW, the system (8) is asymptotically stable with prescribed degree of stability α, if there exist symmetric positive-definite matrices P, Q, R and any appropriately dimensioned matrices N, M such that the following LMI is satisfied:

⎡ Ω dN ⎤ ⎢ T ⎥<0 ⎣ dN −dR ⎦

(11)

where T T T T T ¯ Ω = WPT PW P + WQ 1QWQ 1 − WQ 2QWQ 2 + dWR RWR + FWF + WF F , 0P I 00 WP = ⎡⎣ 0 0 I ⎤⎦, P¯ = ⎡⎣ P 0 ⎤⎦, WQ1 = ⎡⎣ I 0 0⎤⎦, WQ 2 = ⎡⎣ 0 I 0⎤⎦, WR = ⎡⎣ 0 0 I ⎤⎦, ⎡ I −I 0 ⎤ F = ⎣⎡ N M ⎦⎤, WF = ⎣⎢ A A −I ⎦⎥. z dz Proof. Choose the following Lyapunov-Krasovskii functional

∫

(14)

T where ς(t ) = ⎡⎣ zT (t ) zT (t − d ) z Ṫ (t )⎤⎦ . The following inequality can be obtained by substituting (14) into (13)

V̇ (z(t )) = ς T (t )Ως (t ) + dς T (t )NR−1NT ς (t )

To obtain the expected stability degree, let image function of transformation operation be

z(t ) = ξ (t )e αt

∫

t

−

∫t − d ⎡⎣ zṪ (s)R + ς T (t )N⎤⎦R−1⎡⎣ Rz(̇ s) + NT ς(t )⎤⎦ds

(

)

≤ ς T (t ) Ω + dNR−1NT ς (t )

(15)

According to Schur complement theorem, it is obvious that LMI (11) is equivalent to Ω + dNR−1NT < 0 and Ω + dNR−1NT < 0 guarantees the negativeness of V̇ (z (t )), which immediately indicates that the modified system (10) is asymptotically stable. Thus, the augmented closed-loop system (8) is asymptotically stable with prescribed degree of stability α. The proof of Theorem 1 is completed. In the sequel, the method to solve parameter matrices of DOFC will be discussed. Theorem 2. For given scalars d, α, θ1, … , θ6 , ε1, … , ε6 , the system (8)

is asymptotically stable with prescribed degree of stability α, if there ^ ^ ^ exist symmetric positive-definite matrices P , Q , R , invertible matrix ^ ^ ^ ^ ^ X1, X2 and any appropriately dimensioned matrices N , A , B , C , D , such that the following LMI is satisfied:

⎡ ^ ^⎤ ⎢ Ω dN ⎥ < 0 T ⎢ ^ ⎥ ⎣ dN −dR^ ⎦

(16)

Furthermore, if the previous conditions are satisfied, the WADC para^ ^ −1 ^ −1 −1 meters are given by AW = AX2−1 − αI, BW = BX 1 C1 , CW = CX2 , DW = ^ T ^ T ^ T^ ˜ DX1−1C1−1, where Ω^ = WpT PW P + WQ 1QWQ 1 − WQ 2QWQ 2 + dWR RWR+ ⎡ 0 P^ ⎤ ^ ^^ ^ T ^T FWF + WF F , Y = diag{X1, X2}, T = diag{Y , Y , Y}, P = YT PY , P˜ = ⎢ ^ 0 ⎥ , ⎣P ⎦ ^ ^ ⎡ ^ ^⎤ ^ ^ T T T Q = Y QY , R = Y RY , F = ⎣⎢ N M ⎦⎥ , N = T NY ,

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

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Fig. 2. Four-machine two-area power system.

dimension of LMI (16) is 8m × 8m.

T ^ ⎡ θ1I θ2I θ3I θ4I θ5I θ6I ⎤⎥ M=⎢ , ⎣ ε1I ε2I ε3I ε4I ε5I ε6I ⎦

⎡ −I I 0 ⎢ 0 I 0 ⎢ ^ ^ WF = ⎢ ^ (A1 + αI )X1 B1C e dα B1D ⎢ ^ ^ ⎢ 0 A e dα B ⎣

0 −I 0 0

0 0 ⎤ ⎥ 0 0 ⎥ . −X1 0 ⎥ ⎥ ⎥ 0 −X2 ⎦

Proof. Suppose there exist invertible matrix X1, X2 and pre- and post-multiplying (11) with diag{T T , YT } and diag{T , Y}, respectively, then we can obtain

⎡ T T ΩT dTT NY ⎤ ⎢ ⎥<0 ⎣ dY T NT T −dY T RY ⎦

4. Four-machine two-area power system simulation

(17)

After a series of calculations, we have

⎧ ⎡ Y T PY ⎤ ¯ T = WT ⎢ 0 ⎪ T T WPT PW ⎥WP P P T ⎪ ⎣ Y PY 0 ⎦ ⎪ ⎪ T T ⎨ T WQ 1QWQ 1T = WQT 1Y T QYWQ 1 ⎪ ⎪ T T WQT 2QWQ 2T = WQT 2Y T QYWQ 2 ⎪ T T T T ⎪ ⎩ T WR RWRT = WR Y RYWR

(18)

T T FWF T = ⎡⎣ T T NY T T M ⎤⎦

⎡ −I I 0 ⎢ 0 0 I ⎢ ×⎢ (A1 + αI )X1 B1CW X2 e dα B1DW C1X1 ⎢ ⎢⎣ (AW + αI )X2 e dα BW C1X1 0

Remark 3. It's noted that the condition in Theorem 2 involves tuning parameters θ1, … , θ6 and ε1, … , ε6 . When these scalars are given, the condition in Theorem 2 becomes strict LMI and can be easily solved by using the LMI toolbox. The issue one then faces is how to find these scalars such that the LMI (16) has feasible solutions, but there is no good basis for theory and it is also very difficult to apply optimization-search algorithms and genetic algorithms to choose them. Thus, the trial-and-error method is adopted to solve this difficulty in this paper.

0 ⎤ ⎥ 0 ⎥ 0 −X1 0 ⎥ ⎥ 0 0 −X2 ⎥⎦ (19)

0 −I

0 0

Let M be

⎡ θ X −1 θ X −1 θ X −1 θ X −1 θ X −1 θ X −1⎤T 2 2 3 1 4 2 5 1 6 2 ⎥ ⎢ 1 1 , ⎢⎣ ε X −1 ε X −1 ε X −1 ε X −1 ε X −1 ε X −1⎥⎦ 1 1 2 2 3 1 4 2 5 1 6 2 where θ1, … , θ6 and ε1, … , ε6 are given constants. ^ Substituting M, (18), (19) into (17) and defining A = (AW + αI )X2 , ^ ^ ^ , , and other new variables in TheB = BW C1X1 C = CW X2 D = DW C1X1 orem 2, we can easily know that (17) is equivalent to (16). The proof is completed. □ Remark 2. If the order number of the reduced-order power system is m, then the order number of the extended closed-loop system is 2m. Thus, in LMI (16), the dimension of positive definite ^ ^ ^ symmetric matrices P , Q , R is both 2m × 2m and the dimension of invertible matrices X1, X2, any appropriately dimensioned matrices ^ ^ ^ ^ ^ N , A , B , C , D is m  m, m  m, 6m × 2m, m  m, m  m, 1 × m, 1 × m, respectively. So the total number of variables is 22m2 + 5m and the

Case studies are carried out on the four-machine two-are benchmark power system, as shown in Fig. 2. The detailed parameters of the test system are given in [6]. This is a benchmark test system implemented in Matlab, which is widely used for studying the low frequency oscillation. Despite its small size, it can mimic very closely the behaviour of typical systems in actual operation. The test system consists of two areas connected by two parallel 220-km-long tie-lines between buses 7 and 9. Each area is composed of two generators, each equipped with a standard governor, automatic voltage regulator and IEEE ST1A-type static exciter. The load is represented as constant impedances and split between the areas in such a way that area 1 is exporting 413 MW to area 2. The reference load-flow with M2 considered as the slack machine is such that all generators are producing about 700 MW each. Electromechanical oscillation modes of open-loop system and dominant machine obtained by calculation of residue matrix are shown in Table 1. It can be found that the first is the inter-area mode, poorly damped, with Generator 1 and 2 of Area 1 swing against Generator 3 and 4 of Area 2. The other two are local modes. Local mode 1 is the oscillation between Generator 1 and 2 in Area 1 and local mode 2 is the oscillation between Generator 3 and 4 in Area 2. Thus, G1 and G3 are equipped with a PSS to damp the local mode oscillation. The transfer function of PSS is the same as the one in [1,6], which is expressed as follows

HL(s ) = 30

10s 1 + 0.05s 1 + 3s 1 + 10s 1 + 0.02s 1 + 5.4s

(20)

and its output is limited by ±0.15 pu . It's obvious that the configuration of local PSS makes the real part of all the eigenvalues of the system is negative, but the system still has a inter-area low-frequency oscillation mode with frequency 0.6567 and damping ratio 0.0823. Therefore, it is necessary to configure wide-area damping controller to restrain the low frequency oscillation. The wide-area damping controller designed as DOFC is

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

M. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 1 Eigenvalue analysis of the two-area system without controller. Mode no.

1

Mode type Eigenvalue Frequency Damping ratio Dominant machine

Inter-area Local Local 0.10797 j4.0265
0.67607 j7.0458
0.6684 7 j7.2672 0.6408 1.1214 1.1566
0.0268 0.0955 0.0916

1#

2

3#

3

1# , 3#

5

Table 2

¯ Delay margin d/ms influenced by different prescribed degree of stability α. α d¯

0 502.7

0.05 434.8

0.1 381.9

0.15 345.4

0.2 314.6

0.25 286.5

0.3 260.4

0.35 241.0

⎡ −0.04568⎤ ⎥ ⎢ ⎢ 0.2717 ⎥ ⎢ 0.04462 ⎥ ⎢ 0.5423 ⎥ B1 = ⎢ ⎥, ⎢ 1.441 ⎥ ⎢ 0.09726 ⎥ ⎥ ⎢ ⎢ 0.6739 ⎥ ⎣ −0.03412⎦ C1 = ⎣⎡ −0.01526 − 0.016 − 0.1932 − 0.1051 0.04531 − 0.009117 0.01094 − 3.173e − 08⎤⎦. For given θ1 = θ3 = θ5 = ε2 = ε4 = ε6 = 1, θ2 = ε1 = − 1 and θ4= θ6 = ε3 = ε5 = 0 , delay margin d¯ with respect to prescribed degree of stability α by Theorem 2 is shown in Table 2. It is clear that the delay margin d¯ reduces when prescribed degree of stability α increases, which implies the delay-dependent stability of the power system decreases. When α = 0, the allowable bound of d¯ obtained by the theorem in [6] and theorem 2 in our paper is 461.2 ms and 502.7 ms, respectively. Assuming α = 0.1 and d ¼0.08 s and solving LMI (16), the parameters of dynamic output feedback controller are obtained as follows:

Fig. 3. Frequency responses of the full-order and reduced-order four-machine twoarea system.

installed at the first generator G1, whose input is rotor speed difference between G1 and G3. The open-loop system excluding the WADC is a 76th-order system. The Schur balanced model reduction method is adopted to obtain the reduced-order system model. The frequency response of the reduced-order system and the full-order system are shown in Fig. 3. It can be found that, with the order more than 8, the frequency response of reduced-order system is very close to that of the full-order system over the frequency range from 0.1 to 50 Hz, which covers the concerned interarea low-frequency oscillation range between 0.2 to 2.5 Hz. Hence, the full-order power system can be reduced to 8th-order system for the studying of low-frequency oscillation. The details of the 8th-order system are as follows:

⎡ −4.57 −3.654 ⎢ − 2.523 6.091 ⎢ ⎢ 0.6902 −2.033 ⎢ 0.5076 1.555 A1 = ⎢ ⎢ −0.1943 −1.645 ⎢ −0.2169 0.5344 ⎢ ⎢ −0.04341 −0.01904 ⎣ 0 0 −3.676 1.076 3.093 8.456 17.45 −19.81 −10.05 0

−50.53 −30.42 12.01 27.4 −17.18 0.005099 −5.708 1.536 1.081 10.37 0.1735 −9.968 −5.469 −4.728 −21.12 −1.075 3.027 13.99 −2.46 1.722 8.914 0 0 0 ⎤ 3.38 0 ⎥ −10.24 0 ⎥ ⎥ 7.903 0 ⎥ −10.8 0 ⎥, −27.6 0 ⎥ ⎥ 23.96 0 ⎥ −0.07181 0 ⎥ 0 1.42e − 09⎦

AW

BW

CW

⎡ −1.4703 −0.0528 0.1052 0.1437 ⎢ ⎢ 0.4281 −1.5953 −0.3967 1.2664 ⎢ −0.0857 −0.1004 −2.4707 1.1206 ⎢ −0.6193 0.4232 0.4167 −5.3520 =⎢ ⎢ 0.0526 0.0724 0.0862 1.3744 ⎢ −0.3422 0.3274 −0.1190 −0.4306 ⎢ ⎢ −0.1287 0.1184 −0.3439 −0.2158 ⎣ −0.0073 0.0016 0.0035 0.0043 −0.1764 −0.0260 0.0048 ⎤ ⎥ 0.1520 0.1195 0.0015 ⎥ −0.2378 −0.6694 −0.0063⎥ −0.2841 −0.3829 −0.0027⎥ ⎥, 0.0790 0.0442 −0.0052⎥ −1.1206 −0.0779 −0.0035⎥ ⎥ −0.2324 −1.2832 −0.0079⎥ −0.0037 0.0042 −1.1546⎦

0.0444 −0.3281 −0.6567 1.5438 −1.4363 −0.0281 −0.0996 −0.0007

⎡ 19.0425 ⎤ ⎥ ⎢ ⎢ 4.3875 ⎥ ⎢ −107.1825⎥ ⎢ −82.9415 ⎥ =⎢ ⎥, ⎢ −2.9316 ⎥ ⎢ −8.2562 ⎥ ⎥ ⎢ ⎢ −72.3384 ⎥ ⎣ 0.1595 ⎦ = ⎣⎡ −0.0689 0.0751 0.6435 0.2029 0.1303 0.0130 0.4674 − 0.0019⎤⎦,

DW ¼5.7239. The wide-area closed-loop system responses, which is equipped with the proposed DOFC when d¼ 500 ms and the wide-area PSS in [6] when KW ¼ 10 and d ¼400 ms, respectively, are shown in Fig. 4.

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

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Fig. 4. System response equipped with two different wide-area controllers when three-phase short-circuit fault.

Fig. 6. System responses with three-phase short-circuit fault.

DOFC over the wide-area PSS in [6] are verified by the following three scenarios. Scenario I is that the tripped faulty transmission line is not re-closed after the fault is cleared. Scenario II is the reference terminal voltage of G1 increases þ5% at t¼ 1 s. Scenario III is the total power flow of tie-lines increases from 413 to 460 MW. Figs. 6–8 illustrate the system responses to Scenarios I, II and III, respectively, and Tables 3–5 present their corresponding dynamic performance datum. Fig. 6 and Table 3 show that our proposed DOFC, compared with the wide-area PSS in [6], can rapidly stabilize the abovementioned system with smaller overshoot, less oscillation and shorter stabilization time when it is subjected to three-phase short-circuit. Fig. 7 and Table 4 illustrate our system responses are with faster stabilization and less fluctuation when the increase of

Fig. 5. System response with three-phase short-circuit fault under different delays.

It can be seen from the Fig. 4 that the delay bound obtained by our proposed method is greatly increased indeed. Though the state response in [6] is almost critical stability when d ¼400 ms, the simulation power system in this paper can be still stabilized even if the signal transmission delay reaches d ¼500 ms (Fig. 5). When the three-phase short-circuit fault occurs at t¼1 s and the fault is removed at t ¼1.2 s, a lot of simulation results under different delays d have been done and the part of system responses are shown in Fig. 4. It can be seen that, for the closed-loop power system equipped with the proposed DOFC, the inter-area low-frequency oscillation can be effectively damped within the delay margin d ¼0.3819 s. The closed-loop system is stabilized within 9 s and the number of oscillations is very small. When d increases, the oscillation amplitude of the system response decreases, regulating time and the oscillation frequency increase. In addition, when d is greater than 0.3819 s, the closed-loop power system becomes unstable with violent oscillations. Under time delay d ¼0.15 s, the advantages of our proposed

Fig. 7. System responses with increasing þ 5% voltage reference of G1.

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i

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is not sensitive to time delay within the upper bound of delay. The main results in this paper may be further extended to general nonlinear systems based on fuzzy dynamic models. What's more, it will also be our main activities to deal with the problem of optimal design and the application in practical power systems in the next step.

Acknowledgements The authors express their sincere gratitude to the editor and reviewer for their constructive suggestions which help improve the presentation of this paper. This work is supported by the National Natural Science Foundation of China under grant 61403425 and 61473314. Fig. 8. System responses when the active power of tie-line is 460 MW.

References Table 3 The key datum δ13/ω13 of state responses in Fig. 6.

Ref. [6] Our paper

Overshoot( deg/pu )

Settling time(s)

34.9698/0.0013 34.5467/9.7282e
4

7.4833/13.7333 4.6833/6.5167

Table 4 The dynamic performance index δ13/ω13 of state responses in Fig. 7.

Ref. [6] Our paper

Overshoot( deg/pu )

Settling time(s)

25.5967/2.4914e
4 25.2385/1.0596e
4

7.7333/6.5667 7.0167/6.1333

Table 5 The key datum δ13/ω13/Power of state responses in Fig. 8.

Ref. [6] Our paper

Overshoot( deg/pu )

Settling time(s)

29.0601/3.4955e
4/482.9722 29.0577/2.9632e
4/482.9722

8.2883/8.6500/9.6000 4.8014/4.9521/8.8500

reference voltage of G1 occurs. Fig. 8 and Table 5 indicate our DOFC has better dynamic performances, such as overshoot, stabilization time and number of oscillations when the active power of tie-line is 460 MW. Thus, it can be concluded that, compared with the wide-area PSS in [6], our DOFC can provide more sufficient damping to the system and present more satisfactory interarea oscillation damping performance in different operating conditions.

5. Conclusion In this paper, a wide-area dynamic feedback control scheme with prescribed degree of stability is proposed to damp inter-area low-frequency oscillation for interconnected power system. The sufficient condition of augmented closed-loop power system asymptomatic stability is presented. The LMI method is used to obtain the control gain matrices of DOFC and calculate the upper bound of delay d¯ . It is shown from the simulation results that the presented control scheme is efficient and permits the rapid convergence of the closed-loop system. The comparisons in different operation conditions with the Ref. [6] show that our method can better damp inter-area low frequency oscillation, allow greater communication delay and have better stable effect. In addition, it

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Miaoping Sun received the B.S. and M.S degree in electrical engineering and automation from South-west Jiaotong University, Chengdu, China, in 2000 and 2003, respectively and the Ph.D. degree in control science and engineering from Central South University, Changsha, China, in 2013. Currently, she is a lecturer in the College of information science and engineering, Central South University, Changsha, China. Her research interests include time-delay system control, stability analysis and advanced control in power system.

Xiaohong Nian received the B.S. degree from Northwest Normal University, Lanzhou, China, in 1985, the M.S. degree from Shandong University, Jinan, China, in 1992, and the Ph.D. degree from Peking University, Beijing, China, in 2004. Currently, he is a Professor in the College of information science and engineering, Central South University, Changsha, China. His research interests cover theory of decentralized control for networked control of multi-agent systems, induction motor control, and converter technology and motor drive control.

Liqiong Dai M.S. candidate at the College of information science and engineering, Central South University, Changsha, China. She received her B.S. degree in automation from Wuhan University of Science and Technology, Wuhan, China, in 2013. Her research interests include stability analysis and advanced control in power system.

Hua Guo M.S. candidate at the College of information science and engineering, Central South University Changsha, China. She received her B.S. degree in mathematics and applied mathematics from Northwest Normal University, Lanzhou, China, in 2013. Her research interests include time-delay systems and fuzzy control systems.

Please cite this article as: Sun M, et al. The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.003i