# A Robust Online Identification of Sustained Low Frequency Oscillation in Steady-State Power Systems

Authors: Zhaobi Chu; Yan Wang; Min Zhu; Xueping Dong; Hua Li
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

For sustained low frequency oscillations in steady-state power systems, an algorithm is proposed for precise online identification of oscillation frequency, oscillation amplitude, and fundamental amplitude. The algorithm consists of a robust low frequency estimator and a notch filter in parallel. The asymptotical convergence property is analyzed by slow integral manifold, averaging method, and Lyapunov stability theorem sequentially. The steady-state antinoise property is investigated by perturbed system theorem. The robust advantages of the proposed algorithm are embodied in the following aspects: the fundamental amplitude identification is little influenced by oscillation frequency and oscillation amplitude, both oscillation frequency identification and oscillation amplitude identification have small steady-state errors under high order harmonics or bounded noises, the ramp variations of both fundamental amplitude and oscillation amplitude are also significantly tracked, and three design parameters have different effects on identification performance of oscillation frequency, oscillation amplitude, and fundamental amplitude, respectively. Simulation results verify the validity.


---

## Body

## 1. Introduction

Low-frequency oscillations (LFOs) in power system are usually in range of 0.1-2.5Hz and may result in unstable and unsecure operation, so its real-time identification is essential [1–3]. Different methods have been proposed for this goal.LFO is usually modeled as exponentially damped sinusoid whose primary parameters are frequency, amplitude, and damping. Accordingly, the well-known Prony method described in [4], sometimes in company with other methods such as sparse eigenvalue analysis [5], was widely used to analyze LFOs. The class of Prony method is an integral-based method using all data in a time interval to estimate each parameter, which needs much intensive computations.One of the techniques with reduced calculation is a kind of differential algorithm instead of integral algorithm, such as the recursive least square-based algorithm in [6].The key technique of LFO identification is to precisely estimate oscillation frequency. Different types of frequency estimators have been developed and used for identification of LFOs, for instance, the adaptive notch filter [7, 8], the enhanced phase lock loop [9, 10], and the second-order general integrator [11, 12].From the view of convergence speed, frequency estimators can be divided into two categories: the normalized estimators and the nonnormalized estimators. Compared with the non-normalized methods, the normalized methods provide the advantage that the amplitude actual value has much less effect on frequency identification convergence speed. The methods in [9, 11] are nonnormalized algorithms and those in [7, 12–14] are normalized ones. From robustness perspective, an inherent disadvantage of the estimators proposed in [7, 9, 11–13] is that the low frequency convergence property conflicts to the high frequency convergence speed. This was investigated and modified in [14] by a robust technique to precisely identify very low frequency.Many algorithms, such as those in [1–3, 5, 6, 8, 10, 12, 15], aimed to identify the damped oscillations in power systems. From an engineering point of view, the damped oscillation will ultimately vanish after enough long time. However, there always exist some low frequency oscillations with very pool even zero damping because of asynchrony of generators between long distance, interarea interactions of grid, load adding or removing, and renewable energy integration. So, from the point of analysis of power system in steady state, it is significant to identify these type of sustained LFOs.The sustained LFO can be regarded as a sinusoid of small values of amplitude and frequency, which is added to the fundamental voltage signals or current signals. To identify the LFO, the fundamental component should be identified simultaneously. This task can be completed by use of parallel algorithm, such as multiple adaptive notch filters in parallel proposed in [16, 17]. Accordingly, in this paper, for the sustained LFOs, we employ the robust low frequency estimator proposed in [14] in parallel to a notch filter for certain frequency sinusoid to online identify the oscillation frequency, the oscillation amplitude, and the fundamental amplitude.In addition to much less calculation than Prony methods, the advantages include the following aspects: the fundamental amplitude identification is little influenced by oscillation parameters, the steady-state oscillation identification errors are small under bounded noises, the amplitude ramp variations are also significantly tracked, and the three design parameters, respectively, have different effects on identification performance.The rest of this paper is organized as following. The proposed algorithm is introduced in Section2. The convergence property and the anti-noise property are investigated in Section 3. Some simulation results are given to verify the validity in Section 4. Conclusion is presented in Section 5.


## 2. Framework of the Proposed Algorithm

Consider some measured voltage signals or current signals with sustained low frequency oscillation in steady-state power systems expressed in timet as (1)yt=y1t+yot+nt=K1sin⁡Ω1t+δ1+Kosin⁡ωot+δo+ntwhere y1(t) and yo(t) denote the fundamental component and the sustained oscillation component, respectively, and n(t) denotes the measure noise including the high frequency harmonic components. Except for the known fundamental frequency Ω1 with value of 50Hz or 60Hz, the fundamental amplitude K1, the oscillation amplitude Ko, the oscillation frequency ωo, the fundamental phase angle δ1, and the oscillation phase angle δo are all positive and unknown. The two phase angles are both less than value of 2π. Generally, the electromechanical oscillation frequency ωo is normally less than 2.5Hz.For identification of the unknown parameters in power signal (1), the robust low frequency estimator proposed in [14] is used in cooperation with a notch filter equipped with certain frequency modified from [17] to form a new algorithm expressed as the follows:(2)x˙1x˙2x˙3x˙4=00Ω10000θ-Ω10000-θ00x1x2x3x4+μ1Ω1μ2θ00et(3)θ˙=εft,x,θ=εθ2x4etx22+x42(4)et=yt-x1-x2(5)ao=x22+x42(6)a1=x12+x32where x=x1x2x3x4T is four-dimension state variable vector, θ denotes the estimated oscillation frequency, ao represents the estimated oscillation amplitude, and a1 represents the estimated fundamental amplitude.ε andμ are two positive design parameters that influence transient and steady-state performance of the frequency estimation.Without loss of generality, we restrict the state variable vectorx∈X and the estimated frequency θ∈Θ. The two compact sets are defined as(7)X=x∣0<Kmin≤x12+x32≤Kmax,x22+x42≤KoM,∀t≥0Θ=θ∣0<ωmin≤θ≤ωmax,∀t≥0where Kmin, Kmax, KoM, ωmin, and ωmax can be roughly preset according to experience and Kmin should normally be far greater than KoM.We assume the state vector has initial valuesx0=x10x20x30x40T and the estimated frequency has initial value θ0.


## 3. Characteristics of the Proposed Algorithm

The convergence property and the antinoise property of the algorithm consisting of (1)--(6) can be characterized as the following theorem.


### 3.1. Convergence Property Analysis

Theorem 1.
When the measure noisen(t)=0, there exists sufficient small 0<ε∗<2 and μ∗>0 such that if 0<ε≤ε∗ and μ2>μ∗ then the estimate values asymptotically converge to their actual values, i.e.,(8)x1x2x3x4θa1ao→K1sinΩ1t+δ1Kosinωot+δoK1cosΩ1t+δ1Kocosωot+δoωoK1Kox×θ∈X×ΘProof.
The proof is referred to the procedure of [14, 16]. We rewrite the state equation (2) as the following differential equation in form of matrix:(9)x˙=Aθx+bθytwhere(10)Aθ=-b1ΓTC-C0,bθ=b10,b1=μ1Ω1μ2θ,Γ=11,C=Ω1θIf γ=0 then the estimated frequencyθ is frozen and the state equation (9) becomes a Four-dimension linear time invariant (LTI) dynamic system excited by two sinusoidal signal.
For homogeneous differential equationx˙=A(θ)x, let Lyapunov candidate function be V(x)=xTP(θ)x where Pθ=diagμ1-1Ω1-1μ2-1θ-1μ1-1Ω1-1μ2-1θ-1 and then we get(11)V˙x=x˙TPθx+xTPθx˙=-x1+x22≤0If V˙(x)=0 then x1+x2=0. From x˙=A(θ)x, we consequently obtain that (12)x¨1+Ω12x1=0,x¨2+θ2x2=0This shows that x1 and x2 are possibly both sinusoids with frequencies Ω1 and θ, respectively. Because θ≪Ω1, if and only if x1=x2=0 then x1+x2≡0. So the origin point x=0 is the unique point that makes V˙(x)=0 and all the eigenvalues of coefficient matrix A(θ) have negative real parts, which makes the transient response of (9) converges exponentially to zero.
According to inversion of blocked matrix, we get the transfer function of (9) from exciting signaly(t) to state vector x. (13)Hθ,s=sI-Aθ-1bθ=11+μ1Ω1s/s2+Ω12+μ2θs/s2+θ2sμ1Ω1s2+Ω12μ2θss2+θ2-μ1Ω12s2+Ω12-μ2θ2s2+θ2By use of frequency characteristics, let j2=-1, substitute s=jΩ1 and s=jωo into (13), and we get(14)Hθ,jΩ1=10j0,Hθ,jωo=hθ,jωoωoμ1Ω1θ2-ωo2ωoμ2θΩ12-ωo2jμ1Ω12θ2-ωo2jμ2θ2Ω12-ωo2where (15)hθ,jωo=1ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo2-jΩ12-ωo2θ2-ωo2So, whenn(t)=0, the steady-state response of (9) can be represented as(16)x-t,θ=Hθ,jΩ1K1sin⁡Ω1t+δ1+Hθ,jωoKosin⁡ωot+δoi.e.,(17)x-1t,θx-2t,θx-3t,θx-4t,θ=K1sin⁡Ω1t+δ10K1cos⁡Ω1t+δ10+hθωoμ1Ω1θ2-ωo2Kosin⁡ωot+δo+φωoμ2θΩ12-ωo2Kosin⁡ωot+δo+φωoμ1Ω1θ2-ωo2Kocos⁡ωot+δo+φμ2θ2Ω12-ωo2Kocos⁡ωot+δo+φwhere(18)hθ=1Ω12-ωo22θ2-ωo22+ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo22φ=arctan⁡Ω12-ωo2θ2-ωo2ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo2Then, we also obtain(19)e-t,θ=yt-x-1t,θ-x-2t,θ=Kosin⁡ωot+δo-cos⁡φsin⁡ωot+δo+φ=-Kosin⁡φcos⁡ωot+δo+φThe partial derivatives ∂x-/∂t(t,θ), ∂x-/∂θ(t,θ), ∂f/∂x(t,x,θ), and ∂f/∂θ(t,x,θ) are all bounded in the compact domainX×Θ and always satisfy Lipschitz conditions from (1), (2), (3), (4), (5), (6), (8), (9), (13), and (17). This ensures that Assumptions 2.1, 2.2, and 3.1 of [18] are all satisfied. Invoking Theorem 3.1 of [18], we know that for, some sufficiently smallε, there exist aε-family of integral manifolds on which the state vector  x hasTo=2π/ωo period and can always be written as(20)xt,θ,ε=x1t,θ,εx2t,θ,εx3t,θ,εx4t,θ,ε=x-1t,θx-2t,θx-3t,θx-4t,θ+εx~1t,θ,εx~2t,θ,εx~3t,θ,εx~4t,θ,ε,xt,θ,ε=xt+To,θ,εSubstituting (20) into (3), we decouple the normalized frequency updating law from the state equation and obtain aTo-period nonlinear dynamic system with respect to the estimated frequencyθ expressed as(21)θ˙=εFt,θ,ε=εft,xt,θ,ε,θ=εft,x-t,θ+εx~t,θ,ε,θwhere(22)Ft,θ,0=θ2x-4t,θe-t,θx-22t,θ+x-42t,θ=-μ2-1θ2θ2-ωo2cos2ωot+δo+φωo2sin2ωot+δo+φ+θ2cos2ωot+δo+φ=-μ2-1θ21-ωo2ωo2sin2ωot+δo+φ+θ2cos2ωot+δo+φApply averaging method addressed in chapter 10.4 of [19] to (21), we know that (23)
F
-
θ
=
1
T
o
∫
0
T
o
F
t
,
θ
,
0
d
t
=
-
μ
2
-
1
θ
2
1
-
1
T
o
∫
0
T
o
ω
o
2
d
t
ω
o
2
s
i
n
2
ω
o
t
+
δ
o
+
φ
+
θ
2
c
o
s
2
ω
o
t
+
δ
o
+
φ
=
-
μ
2
-
1
θ
2
1
-
ω
o
θ
=
-
μ
2
-
1
θ
θ
-
ω
oThe corresponding averaged dynamical system can be expressed as an asymptotically stable system(24)θ˙a=εF-θa=-εμ2-1θaθa-ωoNow, Theorem 1 has been proved true from Theorem 3.1 in [16].


### 3.2. Antinoise Property Analysis

Theorem 2.
When the measure noise0<nt⩽β, there exist positive constantsκ1,κ2, andκ3 and a finite timeT1 such that the estimate errors have the following ultimate boundaries. (25)θ-ωoao-Koa1-K1≤κ1εωo2Ko-1βκ2μ2ωoβκ3μ1Ω1βt>T1Proof.
Whenn(t)≠0, from the decoupled frequency update law (21), we obtain a perturbed system (26)θ˙=εθ2x2t,θ,εy1t+yotx22t,θ,ε+x42t,θ,ε+εθ2x2t,θ,εntx22t,θ,ε+x42t,θ,ε=εθ2F1t,θ+εθ2g1t,θFrom (5) and (6), considering their inverse function, we get two perturbed dynamical system to update two amplitudes.(27)a˙o=μ2θx2y1t+yot-x1-x2x22+x42+μ2θx2ntx22+x42=μ2θF2t,ao+μ2θg2t,ao(28)a˙1=μ1Ω1x3y1t+yot-x1-x2x12+x32+μ1Ω1x3ntx12+x32=μ1Ω1F3t,a1+μ1Ω1g3t,a1The three perturbation terms can be bounded by the following boundaries:(29)εθ2g1t,θ=εθ2x2t,θ,εntx22t,θ,ε+x42t,θ,ε≤εωo2Ko-1β(30)μ2θg2t,ao=μ2θx2ntx22+x42≤μ2ωoβ(31)μ1Ω1g3t,a1=μ1Ω1x3ntx12+x32≤μ1Ω1βWhen the condition for Theorem 1 is satisfied, the three nominal systems of (26)--(28), θ˙=εθ2F1(t,θ), a˙o=μ2θF2(t,ao) and a˙1=μ1Ω1F3(t,a1) are all asymptotically stable. Invoking lemma 9.2 of [18], we can provide the existence of finite timeT1 and constantsκ1,κ2, andκ3 such that (25) is validated.Remark 3.
From (24), (25), (27), and (28), in the three parameters,μ1,μ2, andε, onlyμ1 has great effect on fundamental amplitude identification performance, onlyμ2 has great effect on oscillation amplitude identification performance, bothμ2 andεhave great influence on tracking performance of oscillation frequency identification, andεhas great influence on anti-noise property of oscillation frequency identification. This type of rather independent effects of parameters on identification of different variables was addressed in [14, 16].Remark 4.
In order to highlight the advantage of the proposed method, we write the frequency update law proposed in [17] as(32)θ˙=γμ2f2t,x,θ=γμ2x4etx22+x42It can be used instead of the update law (3) to form another algorithm in cooperation with (2), (4), (5), and (6). According to Theorem 3.1 and Remark 2 of [14], the frequency update law (32) possesses the contradiction between fast convergence speed for higher frequency and steady-state identification precision for low frequency. This will be revealed by comparison simulation results in the next section.


## 3.1. Convergence Property Analysis

Theorem 1.
When the measure noisen(t)=0, there exists sufficient small 0<ε∗<2 and μ∗>0 such that if 0<ε≤ε∗ and μ2>μ∗ then the estimate values asymptotically converge to their actual values, i.e.,(8)x1x2x3x4θa1ao→K1sinΩ1t+δ1Kosinωot+δoK1cosΩ1t+δ1Kocosωot+δoωoK1Kox×θ∈X×ΘProof.
The proof is referred to the procedure of [14, 16]. We rewrite the state equation (2) as the following differential equation in form of matrix:(9)x˙=Aθx+bθytwhere(10)Aθ=-b1ΓTC-C0,bθ=b10,b1=μ1Ω1μ2θ,Γ=11,C=Ω1θIf γ=0 then the estimated frequencyθ is frozen and the state equation (9) becomes a Four-dimension linear time invariant (LTI) dynamic system excited by two sinusoidal signal.
For homogeneous differential equationx˙=A(θ)x, let Lyapunov candidate function be V(x)=xTP(θ)x where Pθ=diagμ1-1Ω1-1μ2-1θ-1μ1-1Ω1-1μ2-1θ-1 and then we get(11)V˙x=x˙TPθx+xTPθx˙=-x1+x22≤0If V˙(x)=0 then x1+x2=0. From x˙=A(θ)x, we consequently obtain that (12)x¨1+Ω12x1=0,x¨2+θ2x2=0This shows that x1 and x2 are possibly both sinusoids with frequencies Ω1 and θ, respectively. Because θ≪Ω1, if and only if x1=x2=0 then x1+x2≡0. So the origin point x=0 is the unique point that makes V˙(x)=0 and all the eigenvalues of coefficient matrix A(θ) have negative real parts, which makes the transient response of (9) converges exponentially to zero.
According to inversion of blocked matrix, we get the transfer function of (9) from exciting signaly(t) to state vector x. (13)Hθ,s=sI-Aθ-1bθ=11+μ1Ω1s/s2+Ω12+μ2θs/s2+θ2sμ1Ω1s2+Ω12μ2θss2+θ2-μ1Ω12s2+Ω12-μ2θ2s2+θ2By use of frequency characteristics, let j2=-1, substitute s=jΩ1 and s=jωo into (13), and we get(14)Hθ,jΩ1=10j0,Hθ,jωo=hθ,jωoωoμ1Ω1θ2-ωo2ωoμ2θΩ12-ωo2jμ1Ω12θ2-ωo2jμ2θ2Ω12-ωo2where (15)hθ,jωo=1ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo2-jΩ12-ωo2θ2-ωo2So, whenn(t)=0, the steady-state response of (9) can be represented as(16)x-t,θ=Hθ,jΩ1K1sin⁡Ω1t+δ1+Hθ,jωoKosin⁡ωot+δoi.e.,(17)x-1t,θx-2t,θx-3t,θx-4t,θ=K1sin⁡Ω1t+δ10K1cos⁡Ω1t+δ10+hθωoμ1Ω1θ2-ωo2Kosin⁡ωot+δo+φωoμ2θΩ12-ωo2Kosin⁡ωot+δo+φωoμ1Ω1θ2-ωo2Kocos⁡ωot+δo+φμ2θ2Ω12-ωo2Kocos⁡ωot+δo+φwhere(18)hθ=1Ω12-ωo22θ2-ωo22+ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo22φ=arctan⁡Ω12-ωo2θ2-ωo2ωoμ1Ω1θ2-ωo2+ωoμ2θΩ12-ωo2Then, we also obtain(19)e-t,θ=yt-x-1t,θ-x-2t,θ=Kosin⁡ωot+δo-cos⁡φsin⁡ωot+δo+φ=-Kosin⁡φcos⁡ωot+δo+φThe partial derivatives ∂x-/∂t(t,θ), ∂x-/∂θ(t,θ), ∂f/∂x(t,x,θ), and ∂f/∂θ(t,x,θ) are all bounded in the compact domainX×Θ and always satisfy Lipschitz conditions from (1), (2), (3), (4), (5), (6), (8), (9), (13), and (17). This ensures that Assumptions 2.1, 2.2, and 3.1 of [18] are all satisfied. Invoking Theorem 3.1 of [18], we know that for, some sufficiently smallε, there exist aε-family of integral manifolds on which the state vector  x hasTo=2π/ωo period and can always be written as(20)xt,θ,ε=x1t,θ,εx2t,θ,εx3t,θ,εx4t,θ,ε=x-1t,θx-2t,θx-3t,θx-4t,θ+εx~1t,θ,εx~2t,θ,εx~3t,θ,εx~4t,θ,ε,xt,θ,ε=xt+To,θ,εSubstituting (20) into (3), we decouple the normalized frequency updating law from the state equation and obtain aTo-period nonlinear dynamic system with respect to the estimated frequencyθ expressed as(21)θ˙=εFt,θ,ε=εft,xt,θ,ε,θ=εft,x-t,θ+εx~t,θ,ε,θwhere(22)Ft,θ,0=θ2x-4t,θe-t,θx-22t,θ+x-42t,θ=-μ2-1θ2θ2-ωo2cos2ωot+δo+φωo2sin2ωot+δo+φ+θ2cos2ωot+δo+φ=-μ2-1θ21-ωo2ωo2sin2ωot+δo+φ+θ2cos2ωot+δo+φApply averaging method addressed in chapter 10.4 of [19] to (21), we know that (23)
F
-
θ
=
1
T
o
∫
0
T
o
F
t
,
θ
,
0
d
t
=
-
μ
2
-
1
θ
2
1
-
1
T
o
∫
0
T
o
ω
o
2
d
t
ω
o
2
s
i
n
2
ω
o
t
+
δ
o
+
φ
+
θ
2
c
o
s
2
ω
o
t
+
δ
o
+
φ
=
-
μ
2
-
1
θ
2
1
-
ω
o
θ
=
-
μ
2
-
1
θ
θ
-
ω
oThe corresponding averaged dynamical system can be expressed as an asymptotically stable system(24)θ˙a=εF-θa=-εμ2-1θaθa-ωoNow, Theorem 1 has been proved true from Theorem 3.1 in [16].


## 3.2. Antinoise Property Analysis

Theorem 2.
When the measure noise0<nt⩽β, there exist positive constantsκ1,κ2, andκ3 and a finite timeT1 such that the estimate errors have the following ultimate boundaries. (25)θ-ωoao-Koa1-K1≤κ1εωo2Ko-1βκ2μ2ωoβκ3μ1Ω1βt>T1Proof.
Whenn(t)≠0, from the decoupled frequency update law (21), we obtain a perturbed system (26)θ˙=εθ2x2t,θ,εy1t+yotx22t,θ,ε+x42t,θ,ε+εθ2x2t,θ,εntx22t,θ,ε+x42t,θ,ε=εθ2F1t,θ+εθ2g1t,θFrom (5) and (6), considering their inverse function, we get two perturbed dynamical system to update two amplitudes.(27)a˙o=μ2θx2y1t+yot-x1-x2x22+x42+μ2θx2ntx22+x42=μ2θF2t,ao+μ2θg2t,ao(28)a˙1=μ1Ω1x3y1t+yot-x1-x2x12+x32+μ1Ω1x3ntx12+x32=μ1Ω1F3t,a1+μ1Ω1g3t,a1The three perturbation terms can be bounded by the following boundaries:(29)εθ2g1t,θ=εθ2x2t,θ,εntx22t,θ,ε+x42t,θ,ε≤εωo2Ko-1β(30)μ2θg2t,ao=μ2θx2ntx22+x42≤μ2ωoβ(31)μ1Ω1g3t,a1=μ1Ω1x3ntx12+x32≤μ1Ω1βWhen the condition for Theorem 1 is satisfied, the three nominal systems of (26)--(28), θ˙=εθ2F1(t,θ), a˙o=μ2θF2(t,ao) and a˙1=μ1Ω1F3(t,a1) are all asymptotically stable. Invoking lemma 9.2 of [18], we can provide the existence of finite timeT1 and constantsκ1,κ2, andκ3 such that (25) is validated.Remark 3.
From (24), (25), (27), and (28), in the three parameters,μ1,μ2, andε, onlyμ1 has great effect on fundamental amplitude identification performance, onlyμ2 has great effect on oscillation amplitude identification performance, bothμ2 andεhave great influence on tracking performance of oscillation frequency identification, andεhas great influence on anti-noise property of oscillation frequency identification. This type of rather independent effects of parameters on identification of different variables was addressed in [14, 16].Remark 4.
In order to highlight the advantage of the proposed method, we write the frequency update law proposed in [17] as(32)θ˙=γμ2f2t,x,θ=γμ2x4etx22+x42It can be used instead of the update law (3) to form another algorithm in cooperation with (2), (4), (5), and (6). According to Theorem 3.1 and Remark 2 of [14], the frequency update law (32) possesses the contradiction between fast convergence speed for higher frequency and steady-state identification precision for low frequency. This will be revealed by comparison simulation results in the next section.


## 4. Simulation Results

We suppose a measured voltage signal in power system with 50Hz fundamental frequency is(33)yt=K1sin⁡2π×50t+δ1+Kosin⁡ωot+δo+ntwhere the amplitudesK1 andKo are both in per-unit value. For different cases, simulation experiments under Matlab/Simulink using the Forth Runge-Kutta solver are performed to reveal the tracking performance, the antinoise performance, and so on.


### 4.1. Simulation for Tracking Performance without Noise

Whenn(t)=0, for step changes ofωo,K1, andKo, the tracking trajectories equipped with the frequency estimation of [17] are depicted in Figure 1.Figure 1
Identification with frequency update law of [17] usingμ1=0.5 andμ2=2 without noise.In Figure1, whenωo=1.5Hz during the two time intervals both fromt=10s tot=20s and fromt=40s tot=50s, the convergence speeds of the red dashed lines using γ=8 are faster than that of the blue solid lines using γ=4. Contrarily, whenωo=0.5Hz fromt=20s tot=40s, the red dashed lines show much slower convergence speed even unstability. These facts reveal that the algorithm equipped with the frequency estimator of [15] possesses the contradiction between fast transient speed for higher frequency and steady-state convergence property for low frequency.For identical signal simulated in Figure1, the tracking trajectories of the algorithm proposed here are illustrated in Figure 2. Compared with Figure 1, the proposed algorithm can provide fast convergence speed and precise identification for both high frequency and low frequency.Figure 2
Identification of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 without noise.In the upper subplot of Figure2, the convergent times of the estimated oscillation frequencyθ fromt=10s, fromt=20s, and fromt=40s are 1.3s, 3.0s, and 2.5s respectively. In the middle subplot and the lower subplot, among the amplitude transient responses to the step changes of oscillation frequency, the curves fromt=20s overshot much than that fromt=10s while less than that fromt=40s. These facts reveal that both start value and actual value of oscillation frequency have great effect on the convergence speed of the estimated oscillation frequency. This provides the validity of (3) and (24).In the middle subplot of Figure2, the convergent times of the estimated oscillation amplitudeao fromt=15s and fromt=30s are 1.0s and 3.0s, respectively. This shows that the convergence speed of the estimated oscillation amplitude is affected greatly by the actual value of oscillation frequency. This can be concluded from (27).In the lower subplot of Figure2, the convergent times of the estimated fundamental amplitudeao fromt=5s and fromt=35s are 0.05s and 0.04s, respectively. This shows that the actual value of oscillation frequency has little effect on the convergence speed of the estimated fundamental amplitude.


### 4.2. Simulation for Antinoise Performance

Under identical conditions for simulation in Figure2, the identification errors are depicted in Figure 3 whenn(t)=0.1sin(2π×100t+0.1π).Figure 3
Identification errors of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 under harmonic component 0.1sin(2π×100t+0.1π).In the upper subplot of Figure3, the steady-state boundary of the error (θ-ωo) is 0.01Hz beforet=10s, is 0.022Hz fromt=10s tot=15s, is 0.045Hz fromt=15s tot=20s, is 0.005Hz fromt=20s tot=30s, and is 0.0025Hz fromt=30s tot=40s. These facts validate the first conclusion of (25) that the oscillation frequency identification error is nearly in proportion to square of oscillation frequency actual value and is in inverse proportion to oscillation amplitude actual value while is influenced little by fundamental amplitude value.In the middle subplot of Figure3, the steady-state boundary of the error (ao-Ko) is 0.0018pu beforet=10s, is 0.0029pu fromt=10s tot=20s and is 0.001pu fromt=20s tot=40s. These facts validate the second conclusion of (25) that the oscillation amplitude identification error is nearly in proportion to oscillation frequency actual value while is little influenced by values of both oscillation amplitude and fundamental amplitude.In the lower subplot of Figure3, the steady-state boundary of the error (a1-K1) is 0.028pu from start to end. It validates the third conclusion of (25) that the fundamental amplitude identification error is little influenced by values of oscillation frequency, oscillation amplitude and fundamental amplitude.The identification errors under white noise with 20dB signal-noise ratio (SNR) are depicted in Figure4, where similar characteristics of the identification errors can be found to validate the three conclusions of (25) again.Figure 4
Identification errors of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 under white noise with SNR=20dB.


### 4.3. Simulation for Ramp Variations

In Figure5, the results are simulated when the actual values of oscillation frequency, oscillation amplitude and fundamental amplitude vary according to ramp function of timet.Figure 5
Identification of the proposed algorithm usingμ1=0.5,μ2=2.5, andε=1 under ramp variations.For oscillation frequency identification in the upper subplot, the ramp variation with -0.02 slope fromt=10s tot=15s and the ramp variation with 0.01 slope fromt=30s tot=35s are neither tracked because of rather large overshoots. For oscillation amplitude identification in the middle subplot, both the ramp variation with 0.02 slope fromt=20s tot=25s and the ramp variation with -0.01 slope fromt=40s tot=45s are approximately tracked. For fundamental amplitude identification in the lower subplot, both the ramp variation with 0.04 slope fromt=5s tot=10s and the ramp variation with -0.06 slope fromt=45s tot=50s are closely tracked.The fundamental amplitude identification is little affected by the ramp variations of both oscillation frequency and oscillation amplitude.


## 4.1. Simulation for Tracking Performance without Noise

Whenn(t)=0, for step changes ofωo,K1, andKo, the tracking trajectories equipped with the frequency estimation of [17] are depicted in Figure 1.Figure 1
Identification with frequency update law of [17] usingμ1=0.5 andμ2=2 without noise.In Figure1, whenωo=1.5Hz during the two time intervals both fromt=10s tot=20s and fromt=40s tot=50s, the convergence speeds of the red dashed lines using γ=8 are faster than that of the blue solid lines using γ=4. Contrarily, whenωo=0.5Hz fromt=20s tot=40s, the red dashed lines show much slower convergence speed even unstability. These facts reveal that the algorithm equipped with the frequency estimator of [15] possesses the contradiction between fast transient speed for higher frequency and steady-state convergence property for low frequency.For identical signal simulated in Figure1, the tracking trajectories of the algorithm proposed here are illustrated in Figure 2. Compared with Figure 1, the proposed algorithm can provide fast convergence speed and precise identification for both high frequency and low frequency.Figure 2
Identification of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 without noise.In the upper subplot of Figure2, the convergent times of the estimated oscillation frequencyθ fromt=10s, fromt=20s, and fromt=40s are 1.3s, 3.0s, and 2.5s respectively. In the middle subplot and the lower subplot, among the amplitude transient responses to the step changes of oscillation frequency, the curves fromt=20s overshot much than that fromt=10s while less than that fromt=40s. These facts reveal that both start value and actual value of oscillation frequency have great effect on the convergence speed of the estimated oscillation frequency. This provides the validity of (3) and (24).In the middle subplot of Figure2, the convergent times of the estimated oscillation amplitudeao fromt=15s and fromt=30s are 1.0s and 3.0s, respectively. This shows that the convergence speed of the estimated oscillation amplitude is affected greatly by the actual value of oscillation frequency. This can be concluded from (27).In the lower subplot of Figure2, the convergent times of the estimated fundamental amplitudeao fromt=5s and fromt=35s are 0.05s and 0.04s, respectively. This shows that the actual value of oscillation frequency has little effect on the convergence speed of the estimated fundamental amplitude.


## 4.2. Simulation for Antinoise Performance

Under identical conditions for simulation in Figure2, the identification errors are depicted in Figure 3 whenn(t)=0.1sin(2π×100t+0.1π).Figure 3
Identification errors of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 under harmonic component 0.1sin(2π×100t+0.1π).In the upper subplot of Figure3, the steady-state boundary of the error (θ-ωo) is 0.01Hz beforet=10s, is 0.022Hz fromt=10s tot=15s, is 0.045Hz fromt=15s tot=20s, is 0.005Hz fromt=20s tot=30s, and is 0.0025Hz fromt=30s tot=40s. These facts validate the first conclusion of (25) that the oscillation frequency identification error is nearly in proportion to square of oscillation frequency actual value and is in inverse proportion to oscillation amplitude actual value while is influenced little by fundamental amplitude value.In the middle subplot of Figure3, the steady-state boundary of the error (ao-Ko) is 0.0018pu beforet=10s, is 0.0029pu fromt=10s tot=20s and is 0.001pu fromt=20s tot=40s. These facts validate the second conclusion of (25) that the oscillation amplitude identification error is nearly in proportion to oscillation frequency actual value while is little influenced by values of both oscillation amplitude and fundamental amplitude.In the lower subplot of Figure3, the steady-state boundary of the error (a1-K1) is 0.028pu from start to end. It validates the third conclusion of (25) that the fundamental amplitude identification error is little influenced by values of oscillation frequency, oscillation amplitude and fundamental amplitude.The identification errors under white noise with 20dB signal-noise ratio (SNR) are depicted in Figure4, where similar characteristics of the identification errors can be found to validate the three conclusions of (25) again.Figure 4
Identification errors of the proposed algorithm usingμ1=0.5,μ2=2, andε=1 under white noise with SNR=20dB.


## 4.3. Simulation for Ramp Variations

In Figure5, the results are simulated when the actual values of oscillation frequency, oscillation amplitude and fundamental amplitude vary according to ramp function of timet.Figure 5
Identification of the proposed algorithm usingμ1=0.5,μ2=2.5, andε=1 under ramp variations.For oscillation frequency identification in the upper subplot, the ramp variation with -0.02 slope fromt=10s tot=15s and the ramp variation with 0.01 slope fromt=30s tot=35s are neither tracked because of rather large overshoots. For oscillation amplitude identification in the middle subplot, both the ramp variation with 0.02 slope fromt=20s tot=25s and the ramp variation with -0.01 slope fromt=40s tot=45s are approximately tracked. For fundamental amplitude identification in the lower subplot, both the ramp variation with 0.04 slope fromt=5s tot=10s and the ramp variation with -0.06 slope fromt=45s tot=50s are closely tracked.The fundamental amplitude identification is little affected by the ramp variations of both oscillation frequency and oscillation amplitude.


## 5. Summary

For low frequency sustained oscillations in steady-state power systems, the proposed algorithm can precisely identify oscillation frequency, oscillation amplitude, and fundamental amplitude. The convergence speed is little affected by the actual values of oscillation amplitude and fundamental amplitude, while is greatly influenced by the actual value of oscillation frequency whose larger value provides faster speed.Small steady-state identification errors of both oscillation frequency and oscillation amplitude under bounded noise can be obtained when actual value of oscillation frequency is small.The identification performance of fundamental amplitude is little influenced by actual values of both oscillation frequency and oscillation amplitude.The ramp variations of both fundamental amplitude and oscillation amplitude are significantly tracked.The ramp variation of oscillation frequency is not significant tracked and should be further investigated.



---

*Source: 8435838-2019-06-10.xml*