Closed-Loop Identification of Power System Based on Ambient Data

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.

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 Section 2 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 Figure 1 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).

(a)

(b)

Figure 1 

“True system” (a) and identification model structure (b).

We will assume that the data 𝑍^𝑁={𝑦(1),𝑢(1),𝑦(2),𝑢(2)…} of the “true” system Θ are generated as depicted in (2.1): 𝑦(𝑡)=𝐺₁₀𝑞𝐺₂₀𝑞𝑢(𝑡)+𝐻₁₀𝑞𝑒₁₀(𝑡)+𝐻₂₀𝑞𝐺₁₀𝑞𝑒₂₀(𝑡),𝑢(𝑡)=−𝐶₀𝑞𝑦(𝑡)+𝑟(𝑡). (2.1)

Defining𝐺₀𝑞=𝐺₁₀𝑞𝐺₂₀𝑞,𝐻^′₂₀𝑞=𝐻₂₀𝑞𝐺₁₀𝑞,𝐻₀𝑞𝑒₀(𝑡)=𝐻₁₀𝑞𝑒₁₀(𝑡)+𝐻^′₂₀𝑞𝑒₂₀(𝑡), (2.2)

then, the “true” system Θ can be described as𝑦(𝑡)=𝐺₀𝑞𝑢(𝑡)+𝐻₀𝑞𝑒₀(𝑡),𝑢(𝑡)=−𝐶₀𝑞𝑦(𝑡)+𝑟(𝑡). (2.3)

The closed-loop system is stated as𝑦(𝑡)=𝑆₀𝑞𝐺₀𝑞𝑟(𝑡)+𝑆₀𝑞𝐻₀𝑞𝑒₀(𝑡),𝑢(𝑡)=𝑆₀𝑞𝑟(𝑡)−𝑆₀𝑞𝐶₀𝑞𝐻₀𝑞𝑒₀(𝑡), (2.4) where 𝑆₀(𝑞)=1/(1+𝐺₀(𝑞)𝐶₀(𝑞)).

The identification model structure Ω is also given:𝑦(𝑡)𝑢(𝑡)𝑒(𝑡),𝑢(𝑡)𝑦(𝑡)(𝑡).=𝐺𝑞,𝜃+𝐻𝑞,𝜃=−𝐶𝑞,𝜃+𝑟 (2.5)

Similar to (2.4), (2.5) can be rewritten as𝑦(𝑡)𝐺𝑟(𝑡)𝐻𝑒(𝑡),𝑢(𝑡)𝑟(𝑡)𝐶𝐻𝑒(𝑡),=𝑆𝑞,𝜃𝑞,𝜃+𝑆𝑞,𝜃𝑞,𝜃=𝑆𝑞,𝜃−𝑆𝑞,𝜃𝑞,𝜃𝑞,𝜃 (2.6) 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 dataset 𝑍^𝑁 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 𝑍^𝑁 allows us to distinguish between different models in the set [13]. The dataset 𝑍^𝑁 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 parameters 𝜃 by the prediction error method. Assuming two models called 𝑊(𝑞,𝜃₁) and 𝑊(𝑞,𝜃₂) are identified based on the quasistationary dataset 𝑍^𝑁, obviously0=𝐸𝑊𝑞,𝜃₁−𝑊𝑞,𝜃₂𝑍(𝑡)²=𝐸Δ𝑊_𝑢Δ𝑊_𝑦𝑢(𝑡)𝑦(𝑡)², (3.1) where 𝐸𝑥=lim_𝑁→∞(1/𝑁)∑^𝑁_𝑡=1𝐸𝑥(𝑡),𝑖=1,2,…𝑊𝑞,𝜃_𝑖=𝑊_𝑢𝑞,𝜃_𝑖𝑊_𝑦𝑞,𝜃_𝑖,𝑊_𝑢𝑞,𝜃_𝑖=𝐻⁻¹𝑞,𝜃_𝑖𝐺𝑞,𝜃_𝑖,𝑊_𝑦𝑞,𝜃_𝑖=1−𝐻⁻¹𝑞,𝜃_𝑖,Δ𝑊_𝑢=𝑊_𝑢𝑞,𝜃₁−𝑊_𝑢𝑞,𝜃₂,Δ𝑊_𝑦=𝑊_𝑦𝑞,𝜃₁−𝑊_𝑦𝑞,𝜃₂. (3.2) 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 𝑍^𝑁, (3.1) establishes.

The input and output of the “true” system are stated as𝑢(𝑡)𝑦(𝑡)=−𝑆₀𝑞𝐶₀𝑞𝑆₀𝑞𝐻₀𝑞𝑒₀(𝑡). (3.3)

Then, substituting (3.3) into (3.1),𝐸−Δ𝑊_𝑢𝐶₀+Δ𝑊_𝑦𝑆₀𝐻₀𝑒₀(𝑡)²=0. (3.4)

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, Φ_{𝑒𝑖0𝑢}(𝜔)>0 establishes.

When the feedback regulator in the “true” system satisfies 𝐶₀≠Δ𝑊_𝑦/Δ𝑊_𝑢, thenΔ𝑊_𝑦=Δ𝑊_𝑢=0. (3.5)

It means that𝑊𝑞,𝜃₁=𝑊𝑞,𝜃₂. (3.6)

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 𝑍^𝑁, (3.1) establishes.

The input and output of the “true” system are stated as𝑢(𝑡)𝑦(𝑡)=1−𝑆₀𝑞𝐶₀𝑞𝐺₀𝑞𝑆₀𝑞⋅𝑆₀𝑞𝑟(𝑡)𝐻₀𝑞𝑒₀(𝑡). (3.7)

Substituting (3.7) into (3.1), then0=𝐸𝑊_𝑢𝑊_𝑦𝑆₀𝑟(𝑡)𝐻₀𝑒₀(𝑡)²=𝐸𝑊_𝑦𝐻₀𝑒₀(𝑡)²+𝐸𝑊_𝑢𝑆₀𝑟(𝑡)², (3.8) where 𝑊_𝑢𝑊_𝑦=Δ𝑊_𝑢Δ𝑊_𝑦1−𝑆₀𝐶₀𝐺₀𝑆₀. (3.9) Recalling that the signals 𝑟 and 𝑒₀ are uncorrelated by the assumption, apparently the two components in (3.8) both equal zero.(1)𝐸|𝑊_𝑦𝐻_0𝑒0(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, Φ_𝑒0(𝜔)>0 establishes, therefore, 𝑊_𝑦=0.(2)𝐸|𝑊_𝑢𝑆₀𝑟(𝑡)|²=0. Obviously, when 𝐸|𝑆₀𝑟(𝑡)|²>0, 𝑊_𝑢=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 Φ_𝑟(𝜔)>0, 𝐸|𝑆₀𝑟(𝑡)|²>0. Only when the reference signal is persistently exciting, 𝑊_𝑢=0 establishes.

On the other hand, the determinant of matrix 1−𝑆₀𝐶₀𝐺₀𝑆₀ equals 1, the matrix is always invertible. It indicates that𝑊_𝑢𝑊_𝑦=0. (3.10)

Clearly, (3.5) and (3.6) are established.

Besides, it has found that if there is just a simple feedback regulator like 𝑢(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 Figure 1, let us focus on the case with a fixed noise model 𝐻_𝜃=𝐻_∗, the spectrum of the prediction error isΦ_𝜀()=𝐺𝜔,𝜃⁰+𝐵_𝜃−𝐺_𝜃²Φ_𝑢(𝜔)𝐻_∗²+𝐷_𝜃²𝐻_∗²𝜆₂₀−Φ_𝑢𝑒20(𝜔)Φ_𝑒20𝑢(𝜔)Φ_𝑢(𝜔)+𝐻₁₀−𝐻_∗²𝐻_∗²𝜆₁₀+𝜆₁₀+𝜆₂₀−𝐻₁₀−𝐻_∗²Φ_𝑢𝑒10(𝜔)Φ_𝑒10𝑢(𝜔)𝜆₂₀𝐻_∗²𝜆₂₀Φ_𝑢(𝜔)−Φ_𝑢𝑒20(𝜔)Φ_𝑒20𝑢(𝜔),𝐵_𝜃=Φ_𝑒20𝑢(𝜔)𝐻^′₂₀−𝐻_𝜃Φ_𝑢(𝜔)+Φ_𝑒10𝑢(𝜔)𝐻₁₀−𝐻_𝜃Φ_𝑢(𝜔),𝐷_𝜃=𝐻^′₂₀−𝐻_𝜃−Φ_𝑒10𝑢(𝜔)Φ_𝑢𝑒20(𝜔)𝐻₁₀−𝐻_𝜃𝜆₂₀Φ_𝑢(𝜔)−Φ_𝑢𝑒20(𝜔)Φ_𝑒20𝑢(𝜔), (4.1) where Φ_𝑢(𝜔) denotes the spectrum of input 𝑢,𝜆₁₀ and 𝜆₂₀ denote the spectra of disturbances 𝑒₁₀ and 𝑒₂₀, Φ_{𝑢𝑒𝑖0}(𝜔) and Φ_{𝑒𝑖0𝑢}(𝜔) denote the cross-spectra between signal 𝑢 and 𝑒_𝑖0, 𝑖=1,2, and the operator 𝑒^𝑖𝜔 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 𝐺_𝜃 would approximate to the biased transfer function 𝐺₀+𝐵_𝜃 as well as possible, according to the weighted frequency domain function above. It means that the function 𝐵_𝜃 denotes the identification bias.

We can split up the input spectrum Φ_𝑢(𝜔) into one part that originates from 𝑟 and two parts that come from 𝑒_𝑖0:Φ_𝑢(𝜔)=Φ^𝑟_𝑢(𝜔)+Φ^𝑒10_𝑢(𝜔)+Φ^𝑒20_𝑢(𝜔). (4.2)

The cross-spectrum between the signal 𝑢 and e_i0 satisfiesΦ_{𝑒𝑖0𝑢}(𝜔)²=Φ^𝑒𝑖0_𝑢(𝜔)𝜆_𝑖0. (4.3)

Let us comment on the bias function 𝐵_𝜃.𝐵_𝜃²=Φ_𝑒20𝑢(𝜔)Φ_𝑢(𝜔)𝐻^′₂₀−𝐻_∗+Φ_𝑒10𝑢(𝜔)Φ_𝑢(𝜔)𝐻₁₀−𝐻_∗²≥𝜆₂₀Φ^𝑒20_𝑢(𝜔)Φ_𝑢(𝜔)²𝐻^′₂₀−𝐻_∗²+𝜆₁₀Φ^𝑒10_𝑢(𝜔)Φ_𝑢(𝜔)²𝐻₁₀−𝐻_∗². (4.4)

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 ((𝐻_𝜃-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≠𝐻^′₂₀, which implies that the noise model 𝐻_∗ is impossible equal to the true filters 𝐻₁₀ and 𝐻^′₂₀ at the same time. Then, it can be deduced from (4.1) that 𝐵_𝜃≠0, the identification bias inevitably exists in this situation.

Then, extending to the situation that 𝐿 disturbances exist in the “true” system, the bias 𝐵_𝜃 is described as𝐵_𝜃²=^𝐿_𝑖=1Φ_{𝑒𝑖0𝑢}(𝜔)Φ_𝑢(𝜔)𝐻^′_𝑖0−𝐻_∗²≥^𝐿_𝑖=1𝜆_𝑖0Φ^𝑒𝑖0_𝑢(𝜔)Φ_𝑢(𝜔)²𝐻^′_𝑖0−𝐻_∗². (4.5)

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 iscov𝐺_𝑁≈𝑛𝑁Φ_𝑣0(𝜔)Φ^𝑟_𝑢(𝜔), (5.1) where 𝑁 denotes the length of identification data, the signal 𝑣₀ is defined as𝑣₀(𝑡)=𝐻₁₀𝑞𝑒₁₀(𝑡)+𝐻^′₂₀𝑞𝑒₂₀(𝑡). (5.2)

Substituting the spectra of associated signals into (5.1), the variance of the estimated transfer function is rewritten ascov𝐺_𝑁≈𝑛𝑁𝐻^′₂₀²𝜆₂₀+𝐻₁₀²𝜆₁₀𝑆₀²Φ_𝑟(𝜔). (5.3)

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 that 𝐿 disturbances exist in the “true” system, the variance of the estimated transfer function iscov𝐺_𝑁≈𝑛𝑁∑^𝐿_𝑖=1𝐻^′_𝑖0²𝜆_𝑖0𝑆₀²Φ_𝑟(𝜔). (5.4)

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 iscov𝐺_𝑁≈𝑛𝑁Φ_𝑣0(𝜔)Φ_𝑢(𝜔)=𝑛𝑁𝐻^′₂₀²𝜆₂₀+𝐻₁₀²𝜆₁₀Φ_𝑢(𝜔). (5.5)

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 that 𝐿 disturbances exist in the “true” system, the variance of the estimated transfer function iscov𝐺_𝑁≈𝑛𝑁∑^𝐿_𝑖=1𝐻^′_𝑖0²𝜆_𝑖0Φ_𝑢(𝜔). (5.6)

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, obviously𝑛𝑁Φ_𝑣0(𝜔)Φ_𝑢(𝜔)<𝑛𝑁Φ_𝑣0(𝜔)Φ^𝑟_𝑢(𝜔). (5.7)

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 Table 1.

Table 1 

Estimated electromechanical modal properties based on the closed-loop identification model.

For 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 (Δ_𝑀%) and the phase frequency response variance index (Δ_𝑃) to evaluate the closed-loop identification performance, obviously the larger the Δ_𝑀% value and the smaller the Δ_𝑃 value will indicate a more precise closed-loop identification:Δ_𝑀∑%=1−^{𝑁𝜔𝑖=1}𝑀_iden𝜔_𝑖−𝑀₀𝜔_𝑖²∑^𝑁𝜔_𝑖=1𝑀₀𝜔_𝑖²Δ×100%,_𝑃=1𝑁_𝜔^𝑁𝜔_𝑖=1Δ𝑃_iden𝜔_𝑖−∑^𝑁𝜔_𝑖=1Δ𝑃_iden𝜔_𝑖𝑁_𝜔²,Δ𝑃_iden𝜔_𝑖=𝑃_iden𝜔_𝑖−𝑃₀𝜔_𝑖, (7.1)

where 𝑀_iden and 𝑀₀ denote the amplitude frequency response of the identification model and the “true” system, 𝑃_iden and 𝑃₀ represent the phase frequency response of the identification model and the “true” system, Δ𝑃_iden is the error of phase frequency response, and 𝑁_𝜔 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 Figure 3.

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(b)

Figure 3 

Closed-loop identification results with different number of disturbances.

When one disturbance exists in the simulation system, the Δ_𝑀% value fluctuates between 40% and 60%, and the Δ_𝑃 value is less than 1; when three disturbances are added to the system, the Δ_𝑀% value changes between 20% and 30%, and the Δ_𝑃 value are about 2; when five disturbances exist, the Δ_𝑀% value is less than 20%, and the Δ_𝑃 value is up to 4. Obviously, when the disturbance number increases, the Δ_𝑀% value decreases, whereas the Δ_𝑃 value rises. The accuracy of closed-loop identification reduces with the increase of disturbance number.

As mentioned in Section 4, 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.

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(b)

Figure 4 

Closed-loop identification results with∖without reference signal.

When the reference signal is added to the simulation system, the Δ_𝑀% value fluctuates between 30% and 45%, and the Δ_𝑃 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 Δ_𝑀% value rises and the Δ_𝑃 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 Figure 5. These reference signals are ensured no obvious affect on the system’s response.

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(b)

Figure 5 

Closed-loop identification results with increasing energy reference signals.

When we gradually increase the energy of reference signal, the fluctuation ranges of Δ_𝑀% value are 30–45%, 60–75%, 85–95%, respectively. On the other hand, the change ranges of Δ_𝑃 value are 0.8–1.2, 0.6-0.7, 0.1–0.3. Clearly, with the increase of reference signal’s energy, the Δ_𝑀% value rises, and the Δ_𝑃 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 Table 1, 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.

Acknowledgment

This work was supported in part by the Major State Basic Research Development Program of China (2012CB215206), National Natural Science Foundation of China (51037002), Natural Science Foundation of Guangdong Province, China (2011040000995), Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (LYM11108), Shenzhen Technology Research and Development Foundation (2011040000995), and Shenzhen University Basic Research Foundation (201117).