# Closed-Loop Identification of Power System Based on Ambient Data

Authors: Chao Wu; Chao Lu; Yingduo Han
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Small fluctuations caused by random changes of loads exist continuously in power grids, which are called ambient signals. Using time-synchronized phasor measurements, the closed-loop identification of power system based on ambient data is discussed, which can reflect accurate operating conditions currently and provide critical information for system analyzing and controller designing. The closed-loop identification of a power system with multiple disturbances is theoretically studied, including the closed-loop identifiability, the consistency properties, and the convergence properties. The requirements for realizing the closed-loop identification are summarized, and the theoretical research results are validated by simulation examples.


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## Body

## 1. Introduction

Generally, the performance of large interconnected power grids is assessed using simulation methods. In this approach, accurate simulation models and parameters are critical parts to analyzing, controlling, and operating a power system. The identification of models and parameters can be accomplished using two types of data: signals collected from special tests or ringdown signals [1–4]. However, special tests are usually costly, since a lot of preparatory work has to be done in order to avoid negative impact on the system. On the other hand, relatively small quantity of ringdown signals in actual power system and relatively high requirement to calculation speed both limit this approach’s application. Moreover, the identification results may not reflect the accurate operating characteristics of power system currently, which would cause serious adaptability problems in system analysis and regulator design.The development and application of wide area measurement system (WAMS) provides favorable conditions for solving these problems. In actual power system, ambient signals caused by random changes of loads are much easier to be collected. Recently, many publications have offered algorithms for estimating the electromechanical modal properties from ambient data [5–12], which fully demonstrate that this type of signal includes rich information about power grids.In this paper, the closed-loop identification of power system based on wide area measured ambient data is proposed. The identification is carried out during normal operation and provides critical information for the improved operational reliability of interconnected power grids.Nowadays, scholars in automation field have done a lot about the system identification, including the identifiability, the identification algorithm, the identification accuracy evaluation, and so forth [13–20]. However, in power system field, researchers mainly focus on the identification algorithm [1–4, 21–23] whereas ignore some basic problems. In actual power systems, multiple stochastic disturbances always exist, and the wide area damping regulators are gradually put into operation for improving the system’s oscillation characteristic. All these indicate that the closed-loop system is less sensitive to changes, and ambient signals have less information about the system, thus the system identification is much more difficult.In this paper, the closed-loop identification of power system based on ambient signals is theoretically studied. Using “true” system and identification model structure shown in Section2 as the subjects, the closed-loop identifiability is discussed in Section 3, in order to specify the conditions for estimating a unique system model in this situation. Section 4 and Section 5 evaluate the identification accuracy, including the consistency properties and the convergence properties. Then, the necessary conditions for ambient-data-based closed-loop identification of power system are summarized in Section 6, and the simulation examples done in Section 7 are used to validate the theoretical research results.


## 2. True System and Identification Model Structure

“True” system and identification model structure shown in Figure1 are used to discuss the power system’s closed-loop identification problems. To simulate random changes of loads in actual power grids, multiple small amplitude stochastic disturbances are added to the “true” system. Taken two disturbances existed as an example, in the “true” system, u(t) and y(t) denote the input variable and output variable of the controlled system at time t (t=1,2,…), e10(t) and e20(t) are the “true” independent random disturbance variables with zero mean values and variances λ10 and λ20, r is an independent external reference signal, G10(q) and G20(q) denote the “true” controlled system, H10(q) and H20(q) are two inversely stable, monic filters, and C0(q) is the “true” feedback regulator. In the identification model structure, e(t) is a stochastic disturbance variable with zero mean value and variance λ, G(q,θ) represents the transfer function of controlled system model corresponding to the parameter value θ, H(q,θ) and C(q,θ) denote the transfer functions of noise model and feedback regulator model and  q is a forward operator satisfying qu(t)=u(t+1).“True system” (a) and identification model structure (b).
(a)(b)We will assume that the dataZN={y(1),u(1),y(2),u(2)…} of the “true” system Θ are generated as depicted in (2.1):
(2.1)y(t)=G10(q)G20(q)u(t)+H10(q)e10(t)+H20(q)G10(q)e20(t),u(t)=-C0(q)y(t)+r(t).Defining(2.2)G0(q)=G10(q)G20(q),H20′(q)=H20(q)G10(q),H0(q)e0(t)=H10(q)e10(t)+H20′(q)e20(t),then, the “true” system Θ can be described as(2.3)y(t)=G0(q)u(t)+H0(q)e0(t),u(t)=-C0(q)y(t)+r(t).The closed-loop system is stated as(2.4)y(t)=S0(q)G0(q)r(t)+S0(q)H0(q)e0(t),u(t)=S0(q)r(t)-S0(q)C0(q)H0(q)e0(t),
where S0(q)=1/(1+G0(q)C0(q)).The identification model structureΩ is also given:(2.5)y(t)=G(q,θ)u(t)+H(q,θ)e(t),u(t)=-C(q,θ)y(t)+r(t).Similar to (2.4), (2.5) can be rewritten as(2.6)y(t)=S(q,θ)G(q,θ)r(t)+S(q,θ)H(q,θ)e(t),u(t)=S(q,θ)r(t)-S(q,θ)C(q,θ)H(q,θ)e(t),
where S(q,θ)=1/(1+G(q,θ)C(q,θ)).


## 3. Closed-Loop Identifiability Analysis

For the system identification problem, it is nature to check whether the “true” system described by (2.1) belongs to the model set defined by (2.5). We, thus, introduce the concept of identifiability.The datasetZN is the source of information about the “true” system. This be fit to a model structure Ω of our choice. The structure Ω describes a set of models Ω* within which the best one is sought for. Identifiability concerns the question whether different parameter vectors of model structure can describe the same model in the set Ω*, that is to say, whether the dataset ZN allows us to distinguish between different models in the set [13]. The dataset ZN is called to be informative if it is capable of distinguishing between different models [13, 24]. Now, we will discuss whether ambient data is informative enough to closed-loop estimate one unique solution of power system model.An obvious approach is to estimate the model parametersθ by the prediction error method. Assuming two models called W(q,θ1) and W(q,θ2) are identified based on the quasistationary dataset ZN, obviously(3.1)0=E̅[(W(q,θ1)-W(q,θ2))Z(t)]2=E̅|[ΔWuΔWy][u(t)y(t)]|2,
where E̅x=limN→∞(1/N)∑t=1NEx(t),i=1,2,…(3.2)W(q,θi)=[Wu(q,θi)Wy(q,θi)],Wu(q,θi)=H-1(q,θi)G(q,θi),Wy(q,θi)=[1-H-1(q,θi)],ΔWu=Wu(q,θ1)-Wu(q,θ2),ΔWy=Wy(q,θ1)-Wy(q,θ2).
We will discuss the closed-loop identifiability problem in two cases: without reference signal and with reference signal.


### 3.1. Without Reference Signal

When no reference signal is added to the system,W(q,θ1) and W(q,θ2) are identified based on the quasistationary data set ZN, (3.1) establishes.The input and output of the “true” system are stated as(3.3)[u(t)y(t)]=[-S0(q)C0(q)S0(q)]H0(q)e0(t).Then, substituting (3.3) into (3.1),(3.4)E̅|(-ΔWuC0+ΔWy)S0H0e0(t)|2=0.As defined above,e0 is a combination of two filtered uncorrelated white-noise disturbances e10 and e20, and the two corresponding filters are both stable and nonsingular. Clearly, Φei0u(ω)>0  establishes.When the feedback regulator in the “true” system satisfiesC0≠ΔWy/ΔWu, then(3.5)ΔWy=ΔWu=0.It means that(3.6)W(q,θ1)=W(q,θ2).In actual power system, this condition shows up as the feedback regulator should be not too simple. High-order complex or nonlinear or time-varying regulator can ensure the identification data informative enough, and then the model structure is closed-loop identifiability.


### 3.2. With Reference Signal

Once the feedback regulator could not assure the identification data set informative enough, an external reference signal should be added. Likewise,W(q,θ1) and W(q,θ2) are identified based on the quasistationary data set ZN, (3.1) establishes.The input and output of the “true” system are stated as(3.7)[u(t)y(t)]=[1-S0(q)C0(q)G0(q)S0(q)]⋅[S0(q)r(t)H0(q)e0(t)].Substituting (3.7) into (3.1), then(3.8)0=E̅|[W̃uW̃y][S0r(t)H0e0(t)]|2=E̅|W̃yH0e0(t)|2+E̅|W̃uS0r(t)|2,
where
(3.9)[W̃uW̃y]=[ΔWuΔWy][1-S0C0G0S0].
Recalling that the signals r and e0 are uncorrelated by the assumption, apparently the two components in (3.8) both equal zero.(1)
E̅|W̃yH0e0(t)|2=0. The signal e0 is defined as a combination of two filtered uncorrelated white-noise disturbances e10 and e20, and the two corresponding filters are both stable and nonsingular, Φe0(ω)>0 establishes, therefore, W̃y=0.(2)
E̅|W̃uS0r(t)|2=0. Obviously, when E̅|S0r(t)|2>0, W̃u=0 establishes. According to the definition, the analytical function S0 depends on the “true” system G0 and H0, then |S0|2  may be zero at most finitely many points. Only when Φr(ω)>0, E̅|S0r(t)|2>0. Only when the reference signal is persistently exciting, W̃u=0  establishes.On the other hand, the determinant of matrix|1-S0C0G0S0| equals 1, the matrix is always invertible. It indicates that(3.10)[W̃uW̃y]=0.Clearly, (3.5) and (3.6) are established.Besides, it has found that if there is just a simple feedback regulator likeu(t)=ay(t), the data set is not informative enough to realize the closed-loop identification even with a persistently exciting reference signal [13].Therefore, not only a persistently exciting reference signal will assure the informative data set, but also certain complexity feedback regulator should, in general, yield data set informative enough.


### 3.3. Closed-Loop Identifiability Conditions

When multiple random changes exist in the power system, in order to ensure the ambient signals informative enough to realize the closed-loop identification, at least one of the following conditions must be met.(1)
The feedback regulator is high-order complex or time-varying or nonlinear.(2)
A persistently exciting reference signal is added, and the feedback regulator should have a certain complexity.We will now characterize in what sense the identified model approximates to the “true” system, including the consistency properties and the convergence properties. The following discussions are based on the frequency domain expression for the limiting criterion function.


## 3.1. Without Reference Signal

When no reference signal is added to the system,W(q,θ1) and W(q,θ2) are identified based on the quasistationary data set ZN, (3.1) establishes.The input and output of the “true” system are stated as(3.3)[u(t)y(t)]=[-S0(q)C0(q)S0(q)]H0(q)e0(t).Then, substituting (3.3) into (3.1),(3.4)E̅|(-ΔWuC0+ΔWy)S0H0e0(t)|2=0.As defined above,e0 is a combination of two filtered uncorrelated white-noise disturbances e10 and e20, and the two corresponding filters are both stable and nonsingular. Clearly, Φei0u(ω)>0  establishes.When the feedback regulator in the “true” system satisfiesC0≠ΔWy/ΔWu, then(3.5)ΔWy=ΔWu=0.It means that(3.6)W(q,θ1)=W(q,θ2).In actual power system, this condition shows up as the feedback regulator should be not too simple. High-order complex or nonlinear or time-varying regulator can ensure the identification data informative enough, and then the model structure is closed-loop identifiability.


## 3.2. With Reference Signal

Once the feedback regulator could not assure the identification data set informative enough, an external reference signal should be added. Likewise,W(q,θ1) and W(q,θ2) are identified based on the quasistationary data set ZN, (3.1) establishes.The input and output of the “true” system are stated as(3.7)[u(t)y(t)]=[1-S0(q)C0(q)G0(q)S0(q)]⋅[S0(q)r(t)H0(q)e0(t)].Substituting (3.7) into (3.1), then(3.8)0=E̅|[W̃uW̃y][S0r(t)H0e0(t)]|2=E̅|W̃yH0e0(t)|2+E̅|W̃uS0r(t)|2,
where
(3.9)[W̃uW̃y]=[ΔWuΔWy][1-S0C0G0S0].
Recalling that the signals r and e0 are uncorrelated by the assumption, apparently the two components in (3.8) both equal zero.(1)
E̅|W̃yH0e0(t)|2=0. The signal e0 is defined as a combination of two filtered uncorrelated white-noise disturbances e10 and e20, and the two corresponding filters are both stable and nonsingular, Φe0(ω)>0 establishes, therefore, W̃y=0.(2)
E̅|W̃uS0r(t)|2=0. Obviously, when E̅|S0r(t)|2>0, W̃u=0 establishes. According to the definition, the analytical function S0 depends on the “true” system G0 and H0, then |S0|2  may be zero at most finitely many points. Only when Φr(ω)>0, E̅|S0r(t)|2>0. Only when the reference signal is persistently exciting, W̃u=0  establishes.On the other hand, the determinant of matrix|1-S0C0G0S0| equals 1, the matrix is always invertible. It indicates that(3.10)[W̃uW̃y]=0.Clearly, (3.5) and (3.6) are established.Besides, it has found that if there is just a simple feedback regulator likeu(t)=ay(t), the data set is not informative enough to realize the closed-loop identification even with a persistently exciting reference signal [13].Therefore, not only a persistently exciting reference signal will assure the informative data set, but also certain complexity feedback regulator should, in general, yield data set informative enough.


## 3.3. Closed-Loop Identifiability Conditions

When multiple random changes exist in the power system, in order to ensure the ambient signals informative enough to realize the closed-loop identification, at least one of the following conditions must be met.(1)
The feedback regulator is high-order complex or time-varying or nonlinear.(2)
A persistently exciting reference signal is added, and the feedback regulator should have a certain complexity.We will now characterize in what sense the identified model approximates to the “true” system, including the consistency properties and the convergence properties. The following discussions are based on the frequency domain expression for the limiting criterion function.


## 4. Consistency Analysis of Closed-loop Identification

For the “true” system and the identification model shown in Figure1, let us focus on the case with a fixed noise model Hθ=H*, the spectrum of the prediction error is(4.1)Φε(ω,θ)=|G0+Bθ-Gθ|2Φu(ω)|H*|2+|Dθ|2|H*|2(λ20-Φue20(ω)Φe20u(ω)Φu(ω))+|H10-H*|2|H*|2λ10+λ10+λ20-|H10-H*|2Φue10(ω)Φe10u(ω)λ20|H*|2(λ20Φu(ω)-Φue20(ω)Φe20u(ω)),Bθ=Φe20u(ω)(H20′-Hθ)Φu(ω)+Φe10u(ω)(H10-Hθ)Φu(ω),Dθ=(H20′-Hθ)-Φe10u(ω)Φue20(ω)(H10-Hθ)λ20Φu(ω)-Φue20(ω)Φe20u(ω),
where Φu(ω) denotes the spectrum of input u,λ10 and λ20 denote the spectra of disturbances e10 and e20, Φuei0(ω) and Φei0u(ω) denote the cross-spectra between signal u and ei0, i=1,2, and the operator eiω is omitted in the formula.The model parameters are estimated using the prediction error method whose principle is to make the prediction error as small as possible [13]. Therefore, the model Gθ would approximate to the biased transfer function G0+Bθ as well as possible, according to the weighted frequency domain function above. It means that the function Bθ denotes the identification bias.We can split up the input spectrumΦu(ω) into one part that originates from r and two parts that come from ei0:(4.2)Φu(ω)=Φur(ω)+Φue10(ω)+Φue20(ω).The cross-spectrum between the signalu and ei0 satisfies(4.3)|Φei0u(ω)|2=Φuei0(ω)λi0.Let us comment on the bias functionBθ.(4.4)|Bθ|2=|Φe20u(ω)Φu(ω)(H20′-H*)+Φe10u(ω)Φu(ω)(H10-H*)|2≥λ20Φue20(ω)|Φu(ω)|2|H20′-H*|2+λ10Φue10(ω)|Φu(ω)|2|H10-H*|2.From (4.4), we see that the bias inclination will be small in the frequency range where either (or all) of the followings holds.(1)
The noise model is good ((Hθ–H*) is small).(2)
The input signal-to-noise ratio is good.(3)
The feedback contribution to the input spectrum is small.The conditions (2.3) and (2.4) are both associated with the disturbances. In this paper, small amplitude stochastic changes of loads in actual power system are considered to be disturbances, which are hard to be manipulated. Thus, in order to reduce the identification bias, we should focus on improving the noise model’s accuracy.Moreover, in the “true” system, the filters of disturbances are generally different from each other, namely,H10≠H20′, which implies that the noise model H* is impossible equal to the true filters  H10 and H20′ at the same time. Then, it can be deduced from (4.1) that Bθ≠0, the identification bias inevitably exists in this situation.Then, extending to the situation thatL disturbances exist in the “true” system, the bias Bθ is described as(4.5)|Bθ|2=|∑i=1L[Φei0u(ω)Φu(ω)(Hi0′-H*)]|2≥∑i=1L[λi0Φuei0(ω)|Φu(ω)|2|Hi0′-H*|2].Obviously, with the increase in the disturbance number, the identification bias also rises. When we magnify the reference signal’s energy, its contribution to input also increases and the input’s total power amplifies correspondingly, then the identification bias reduces.Therefore, when multiple small amplitude stochastic disturbances exist in a power system, the closed-loop identification bias inevitably exists. It will rise with the increase in disturbance number. In order to improve the consistency prosperities of identification, it is necessary to add a persistently exciting reference signal with certain power.


## 5. Convergence Analysis of Closed-Loop Identification

Let us now consider the convergence prosperities of closed-loop identification in two cases: with unfixed noise model, and with fixed noise model.


### 5.1. Identification with Unfixed Noise Model

It can be deduced that the asymptotic variance of the estimated transfer function is(5.1)cov⁡ĜN≈nNΦv0(ω)Φur(ω),
where N denotes the length of identification data, the signal v0 is defined as(5.2)v0(t)=H10(q)e10(t)+H20′(q)e20(t).Substituting the spectra of associated signals into (5.1), the variance of the estimated transfer function is rewritten as(5.3)cov⁡ĜN≈nN|H20′|2λ20+|H10|2λ10|S0|2Φr(ω).When the noise model is unfixed during the closed-loop identification the following hold.(1)
The variance of estimated transfer function will approximate to infinite if there is no reference signal.(2)
The variance of estimated transfer function is proportional to the model order, and inversely proportional to the length of identification data.Then, extending to the situation thatL disturbances exist in the “true” system, the variance of the estimated transfer function is(5.4)cov⁡ĜN≈nN∑i=1L|Hi0′|2λi0|S0|2Φr(ω).The variance of the estimated transfer function rises with the increase in disturbance number. Magnifying the energy of reference signal leads to a reduction of identification variance. It demonstrates again the necessity of the persistently exciting reference signal.


### 5.2. Identification with Fixed Noise Model

Likewise, in this situation, the variance of the estimated transfer function is(5.5)cov⁡ĜN≈nNΦv0(ω)Φu(ω)=nN|H20′|2λ20+|H10|2λ10Φu(ω).In this case, since the noise model is fixed in advance, the variance of the estimated transfer function would not trend to infinite even no reference signal existed.Then, extending to the situation thatL disturbances exist in the “true” system, the variance of the estimated transfer function is(5.6)cov⁡ĜN≈nN∑i=1L|Hi0′|2λi0Φu(ω).Clearly, the variance of estimated transfer function rises with the increase in disturbance number. Similarly, amplifying the power of reference signal will lead to the reduction of identification variance. The result also demonstrates that, it is necessary to add a persistently exciting reference signal.Then, comparing the identification variances in the two cases, obviously(5.7)nNΦv0(ω)Φu(ω)<nNΦv0(ω)Φur(ω).A fixed noise model is effective to reduce the identification variance.Therefore, when multiple small amplitude stochastic disturbances exist in a power system, the closed-loop identification variance rises with the increase in disturbance number. A persistently exciting reference signal with certain energy and a fixed noise model are both helpful to improve the convergence prosperities of closed-loop identification.


## 5.1. Identification with Unfixed Noise Model

It can be deduced that the asymptotic variance of the estimated transfer function is(5.1)cov⁡ĜN≈nNΦv0(ω)Φur(ω),
where N denotes the length of identification data, the signal v0 is defined as(5.2)v0(t)=H10(q)e10(t)+H20′(q)e20(t).Substituting the spectra of associated signals into (5.1), the variance of the estimated transfer function is rewritten as(5.3)cov⁡ĜN≈nN|H20′|2λ20+|H10|2λ10|S0|2Φr(ω).When the noise model is unfixed during the closed-loop identification the following hold.(1)
The variance of estimated transfer function will approximate to infinite if there is no reference signal.(2)
The variance of estimated transfer function is proportional to the model order, and inversely proportional to the length of identification data.Then, extending to the situation thatL disturbances exist in the “true” system, the variance of the estimated transfer function is(5.4)cov⁡ĜN≈nN∑i=1L|Hi0′|2λi0|S0|2Φr(ω).The variance of the estimated transfer function rises with the increase in disturbance number. Magnifying the energy of reference signal leads to a reduction of identification variance. It demonstrates again the necessity of the persistently exciting reference signal.


## 5.2. Identification with Fixed Noise Model

Likewise, in this situation, the variance of the estimated transfer function is(5.5)cov⁡ĜN≈nNΦv0(ω)Φu(ω)=nN|H20′|2λ20+|H10|2λ10Φu(ω).In this case, since the noise model is fixed in advance, the variance of the estimated transfer function would not trend to infinite even no reference signal existed.Then, extending to the situation thatL disturbances exist in the “true” system, the variance of the estimated transfer function is(5.6)cov⁡ĜN≈nN∑i=1L|Hi0′|2λi0Φu(ω).Clearly, the variance of estimated transfer function rises with the increase in disturbance number. Similarly, amplifying the power of reference signal will lead to the reduction of identification variance. The result also demonstrates that, it is necessary to add a persistently exciting reference signal.Then, comparing the identification variances in the two cases, obviously(5.7)nNΦv0(ω)Φu(ω)<nNΦv0(ω)Φur(ω).A fixed noise model is effective to reduce the identification variance.Therefore, when multiple small amplitude stochastic disturbances exist in a power system, the closed-loop identification variance rises with the increase in disturbance number. A persistently exciting reference signal with certain energy and a fixed noise model are both helpful to improve the convergence prosperities of closed-loop identification.


## 6. Conditions for Realizing Ambient Signals Based Closed-Loop Identification of Power System

We summarize the conditions for ambient signals based closed-loop identification of system as follows.(1)
In the aspect of closed-loop identifiability, the feedback regulator should be high-order complex or nonlinear or time varying, otherwise, a persistently exciting reference signal is added and the feedback regulator has a certain complexity.(2)
In the aspect of identification accuracy, the identification bias always exists, and the identification bias and variance both rise with the increase in disturbance number. For improving the identification accuracy, it is necessary to add a persistently exciting reference signal with certain energy and fix the noise model during the identification process.Taking actual power systems into account, in order to realize the closed-loop identification, a high-order complex feedback regulator, a persistently exciting reference signal with certain power and a fixed noise model are necessary.


## 7. Simulation Examples

A two-area four-machine power system [25] shown in Figure 2 is selected to validate the closed-loop identification theoretical results. Area 1 consists of generators at bus 1 and 2, and load at bus 7; Area 2 consists of generators at bus 3 and 4, and load at bus 8. The two areas are connected through long transmission lines between bus 7 and 8.Figure 2
Two-area four-machine system.Based on the reduced-order model calculated by the MATLAB linearization tool, the electromechanical modal properties of system are estimated, shown in Table1.Table 1
Estimated electromechanical modal properties based on the closed-loop identification model.
Closed-loopidentificationTheoretical calculationerror/%Mode 1frequency/Hz1.1746181.1639300.918268Mode 1damping ratio/%10.961910.03769.208263Mode 2frequency/Hz0.6480410.6534450.827002Mode 2damping ratio/%4.89105.37028.924001For the examples that follows, using MATLAB as the tool, a typical time-domain simulation that consists of driving the system with random load variations is done. Each load in this system is split into a portion consisting of constant power and random power. The random portions of real and reactive loads are obtained by passing independent Gaussian white noises through lowpass filters with very low cutoff frequency. The system’s responses consist of small random variations in the system states.In order to improve the oscillation characteristic of power system, a wide area damping regulator is designed and put into operation. Using the comprehensive dominant mode ratio and the improving residue method [26, 27], the active power of tieline between the two areas and the excitation side of Gen4 are selected to be the feedback signal and the control position. Thus, the input signal on the excitation side of Gen4 and the active power of tieline are collected to closed-loop identify the controlled system.In this paper, the moving data window approach is adopted. The length of each data window is 1 minute, and the time interval between adjacent windows is 30 seconds. With the model order selected automatically by the method proposed in [8], the autoregressive exogenous (ARX) model is used to process ambient signals.Defining the amplitude frequency response fitting degree index (ΔM%) and the phase frequency response variance index (ΔP) to evaluate the closed-loop identification performance, obviously the larger the ΔM% value and the smaller the ΔP value will indicate a more precise closed-loop identification:(7.1)ΔM%=[1-∑i=1Nω|Miden(ωi)-M0(ωi)|2∑i=1Nω|M0(ωi)|2]×100%,ΔP=1Nω∑i=1Nω{ΔPiden(ωi)-∑i=1NωΔPiden(ωi)Nω}2,ΔPiden(ωi)=Piden(ωi)-P0(ωi),whereMiden and M0 denote the amplitude frequency response of the identification model and the “true” system, Piden and P0 represent the phase frequency response of the identification model and the “true” system, ΔPiden is the error of phase frequency response, and Nω is the number of frequency points.First, the effect of disturbances on closed-loop identification is analyzed. One, three and five disturbance signals are added to the controlled system, respectively. Ambient data are collected for identifying the system model. The results are shown in Figure3.Closed-loop identification results with different number of disturbances.
(a)(b)When one disturbance exists in the simulation system, theΔM% value fluctuates between 40% and 60%, and the ΔP value is less than 1; when three disturbances are added to the system, the ΔM% value changes between 20% and 30%, and the ΔP value are about 2; when five disturbances exist, the ΔM% value is less than 20%, and the ΔP value is up to 4. Obviously, when the disturbance number increases, the ΔM% value decreases, whereas the ΔP value rises. The accuracy of closed-loop identification reduces with the increase of disturbance number.As mentioned in Section4, a persistently exciting reference signal is helpful to improve the identification performance. Therefore, when three disturbances exist in the controlled system, a reference signal is added. This signal is obtained by passing Gaussian white noise through a one-order filter with cutoff frequency 2 Hz, and we guarantee that it would not affect the fluctuation amplitude of system’s responses. The identification data are collected and the corresponding results are shown in Figure 4.Closed-loop identification results with∖without reference signal.
(a)(b)When the reference signal is added to the simulation system, theΔM% value fluctuates between 30% and 45%, and the ΔP value is about 1. Comparing with the results of system without reference signal, it is clear that due to the existence of reference signal, the ΔM% value rises and the ΔP value falls. A persistently exciting reference signal is helpful to improve the identification accuracy.Then, the effect of reference signal’s energy is further analyzed. When three disturbances exist in the controlled system, we magnify the power of reference signal gradually, and collect ambient signals to identify the system. The results are shown in Figure5. These reference signals are ensured no obvious affect on the system’s response.Closed-loop identification results with increasing energy reference signals.
(a)(b)When we gradually increase the energy of reference signal, the fluctuation ranges ofΔM% value are 30–45%, 60–75%, 85–95%, respectively. On the other hand, the change ranges of ΔP value are 0.8–1.2, 0.6-0.7, 0.1–0.3. Clearly, with the increase of reference signal’s energy, the ΔM% value rises, and the ΔP value decreases. It demonstrates that appropriately boosting the reference signal’s power is effective to improve the closed-loop identification accuracy.Then, basing on the result corresponding to the reference signal with medium energy, the electromechanical modal properties are estimated, listed in Table1, and the corresponding Bode diagram is shown in Figure 6.Figure 6
Bode diagram of closed-loop identification result.Obviously, the closed-loop identification results are basically consistent with the theoretical calculation results, that is to say, the system is identified accurately based on ambient signals.


## 8. Conclusions

Ambient signals based closed-loop identification of power system is theoretically studied in this paper, including the closed-loop identifiability, the consistency properties, and the convergence properties. It is found that a persistently exciting reference signal and a high-order complex feedback regulator are necessary for realizing the system identification in this situation. Considering that the identification bias inevitably exists, and the identification bias and variance boosts with the increase of disturbance number, it is better to appropriately magnify the reference signal’s energy and fix the noise model. At last, the theoretical results are validated by simulation examples.



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*Source: 632897-2012-05-09.xml*