1 Introduction

Low frequency oscillation (LFO) is a common phenomenon in power systems. It has been a serious problem limiting the transmission power of tie-lines and threatening the security and stability of power systems [1–7]. It is crucial to discover the origin of LFO accurately and to generate corresponding control strategies effectively.

The LFOs are generally classified into two categories, the weak damping oscillations (WDO) and the forced oscillations (FO), according to different intrinsic inducements [8–11]. WDOs occur in systems with weak or negative damping so that the oscillation can hardly calm down. FOs are caused by oscillatory sources with approximate frequency to a system’s inherent mode. These two kinds of oscillations have similar appearance. However, the corresponding control measures are entirely different. To restrain the WDO, the output power of some strongly correlated generators needs to be decreased temporarily if the oscillation mode is unexpected, and thereafter controllers such as power system stabilizers (PSS) need to be added or retuned; whereas to attenuate the FO, the oscillation source needs to be isolated or tripped. Therefore, it is essential to discriminate oscillation properties when an LFO is detected.

Eigen-analysis is a major approach in LFO study [12–14]. This approach heavily depends on the availability of power system models and parameters. However, with the expansion of system scales, to obtain parameters of all system devices becomes harder than before. Moreover, it is very difficult to ensure computational convergence and accuracy when calculating the modes and mode-shapes of a large system. Recent years, enormous progresses have been made in measurement-based oscillation analysis because of the rapid development of wide-area measurement system (WAMS). In identifying oscillation properties, the characteristics of waveforms in different oscillation period were analyzed in [8]. Reference [9] focused on the initial period of the waveforms, proposing a second order differential method. The oscillation properties were discriminated according to the components of power response in [10]. Reference [11] decomposed the energy supply on port (ESP) and discriminated the oscillation properties according to the features of ESP’s aperiodic component. As for the oscillation source locating, the concept of generator energy was firstly introduced and successfully used to identify the generator where the external disturbance exists in [15]. Further, [16–18] proposed the concept of energy flow in the network, based on which the oscillation sources could be located correctly on a generator or a load bus. Different from the energy definition of the above references, [19] defined the total energy of power system based on Hamiltonian realization, and the external disturbance could be located effectively in large-scale power systems containing complicated control devices.

This paper proposes a method to discriminate oscillation properties. Combining this method with the previous work of the authors’ research group [19–22], this paper forges a measurement-based solution for LFO analysis. There are three main steps in this solution. First, an online discriminating method is developed, which alternately uses the empirical mode decomposition (EMD) / Hilbert transform (HT) and square calculation to process the measurement data. With this method, an oscillation is promptly classified into WDO or FO. In the next step, the oscillation source is located for the FO, or subsystems that contribute more to the underdamped or negative-damping mode are identified for the WDO. The third step is, only for the WDO, to locate generators that are strongly correlated to the oscillation. This solution not only helps to generate prompt control strategies in real time operation, but also helps to identify the generators that should equip a PSS or retune the parameters of the PSS if they have already equipped one. In the final part of this paper, the solution is applied to an oscillation case found in simulation on the isolated Changdu grid in Tibet to demonstrate its effectiveness.

2 Oscillation property discrimination method based on alternate use of EMD/HT and square calculation

A measurement-based discrimination method based on alternate use of EMD/HT and square processing is proposed, which only requires online measurements such as transmission power on tie-lines or voltage angles at buses. It can discriminate the oscillation properties within a very short time and help generating online control strategies.

2.1 Mathematical tools

2.1.1 Empirical mode decomposition

The empirical mode decomposition can decompose sample data into n-intrinsic mode functions (IMFs) and a residue that could be either the mean trend or a constant [23]. The EMD has been widely used in power oscillations or signal analysis combining with the HT [24–28].

The EMD decomposes the sample into

$$x(t) = \sum\limits_{i = 1}^{n} {c_{i} (t)} + r(t)$$
(1)

where \(x(t)\) is the sample data; \(c_{i} (t)\) is i th intrinsic mode function; \(r(t)\) is the residue.

If the sample includes several oscillation modes, IMF results are physically meaningful: the i th IMF is corresponding to the i th oscillation mode, i.e.,

$$c_{i} (t) = A_{i} {\text{e}}^{{\sigma_{i} t}} \sin (\omega_{i} t + \varphi_{i} )$$
(2)

where A i is a constant; σ i and ω i are the real-part and the imaginary part of the i th mode; φ i is the initial phase.

In this section, the EMD is repeatedly used to get the oscillation component of data.

2.1.2 Hilbert transform

The HT of \(x(t)\) is defined in [23], that is

$$\hat{x}(t) = \frac{1}{\pi }x\int_{ - \infty }^{ + \infty } {\frac{x(\tau )}{t - \tau }} {\text{d}}\tau$$
(3)

With the HT, the analytic signal is defined as

$$z(t) = x(t) + {\text{j}}\hat{x}(t) = A(t){\text{e}}^{i\theta (t)}$$
(4)

The instantaneous amplitude of \(x(t)\) is calculated by

$$A(t) = \left( {x^{2}(t) + \hat{x}^{2}(t) } \right)^{{\frac{1}{2}}}$$
(5)

The HT is used to get the instantaneous amplitude of the oscillation components of the data.

2.2 Theoretic foundation of method for oscillation property discrimination

In a practice power system comprising of several interconnected areas, it usually has only one key mode whose damping is the weakest, especially when the connections between areas are weak while the connections within areas are very strong. After the attenuation of modes with large damping ratios, only the weak damping mode, called the dominant mode, or the forced response remains.

For free oscillation, a state variable is as follows.

$$x_{{{\text{d}}\_{\text{WDO}}}} (t) = A{\text{e}}^{\sigma t} \sin (\omega t + \varphi )$$
(6)

For forced oscillation, a state variable is

$$x_{{{\text{d}}\_{\text{FO}}}} (t) = A\sin (\omega t + \varphi )$$
(7)

The amplitude of (6) changes exponentially with the certain directional trend. The amplitude of (7) keeps constant in ideal condition or fluctuating around a constant value due to some noises. Anyway, it varies without any directional trend.

According to (6) and (7), whether the amplitude varies with a certain directional trend could be used as the key discriminative characteristic to identify the oscillation properties. The problem is that the discriminative characteristic is not so prominent in the raw data obtained from the two kinds of oscillations. Additional work has to be done to enlarge the difference.

In this paper, a new method is proposed to process the raw data, which magnifies the difference by alternately using EMD and square calculation. The raw data are decomposed into n-IMFs and a residue. Then the instantaneous amplitude of every IMF is extracted by HT. Choose the IMF whose instantaneous amplitude is the biggest as the dominant mode and then normalize it by its instantaneous amplitude at t = 0.

As for the weak damping oscillation, square \(x_{{{\text{d}}\_{\text{WDO}}}} \left( t \right)\) and multiply the result by 2,

$$2\left( {x_{{{\text{d}}\_{\text{WDO}}}} \left( t \right)} \right)^{2} = A^{2} {\text{e}}^{2\sigma t} \left( {1 - \cos \left( {2\omega t + 2\varphi } \right)} \right)$$
(8)

Abstract oscillatory component from (8) by EMD, then

$$x_{{{\text{d}}\_{\text{WDO}}}}^{\left( 1 \right)} \left( t \right) = - A^{2} {\text{e}}^{2\sigma t} cos\left( {2\omega t + 2\varphi } \right)$$
(9)

We define the processing by (8) and (9) as a round of operation. \(x_{{{\text{d}}\_{\text{WDO}}}}^{\left( k \right)} \left( t \right)\) is the result after k times of operation.

After n times of operation,

$$x_{{{\text{d}}\_{\text{WDO}}}}^{\left( n \right)} \left( t \right) = \left( { - 1} \right)^{n} A^{{2^{n} }} {\text{e}}^{{2^{n} \sigma t}} cos\left( {2^{n} \omega t + 2^{n} \varphi } \right)$$
(10)

The instantaneous amplitude of \(x_{{{\text{d\_WDO}}}}^{\left( k \right)} \left( t \right)\) is

$$Amp\left( t \right) = A^{{2^{n} }} e^{{2^{n} \sigma t}}$$
(11)

It can be seen from (10) and (11) that the damping factor and oscillation frequency have been magnified by 2n. The variation trend of the amplitude has also been amplified and become steeper.

As for the forced oscillation, repeating square calculation and EMD alternately for n times, finally we obtain

$$x_{{{\text{d}}\_{\text{FO}}}}^{\left( n \right)} \left( t \right) = \left( { - 1} \right)^{n} A^{{2^{n} }} cos\left( {2^{n} \omega t + 2^{n} \varphi } \right)$$
(12)

The instantaneous amplitude of oscillation data processed by HT in (12) is

$$Amp\left( t \right) = A^{{2^{n} }}$$
(13)

The oscillation frequency has also been amplified by 2n, however, the instantaneous amplitude remains constant.

The above analysis results are obtained in the ideal conditions. Factually, the practical oscillation data contain disturbances or noises, but thankfully they are generally irregular and nondirective, therefore they will not impact the variation trend of the instantaneous amplitude. So in the practical conditions, (11) and (13) might have the general forms as (14) and (15), respectively.

$$Amp(t) = a{\text{e}}^{bt} + c$$
(14)
$$Amp(t) = at + b$$
(15)

where a, b, c are constants.

After the above process, the discriminative characteristic between the two oscillation properties has become significant enough for discrimination.

2.3 Implementation process of method

Generally, the sample data contain noises, therefore they should be filtered at first. Then if there is a dominate mode, it should be separated from others by EMD. Besides, if a system has multiple modes excited simultaneously, the dominate mode should be separated from others by the EMD. The systematic implementation process of the method proposed above is as following.

  1. 1)

    Choose generators’ active power as input data for oscillation type identification. A low-pass filter is used to filter the input data in order to eliminate the noises and the components whose frequency are beyond the range of electromechanical oscillations.

  2. 2)

    Get the dominant oscillation mode. Repeat square calculation and EMD alternately n times.

  3. 3)

    Extract the instantaneous amplitude Amp(t) by HT and fit it with (14) and (15) respectively. Then, the exponential fitting result Amp e(t), the approximatively linear fitting result Amp l(t) and the goodness-of-fit statistics such as the sum of squares due to error (SSE), the root mean squared error (RMSE) and the coefficient of determination (CD) are obtained. Calculate the correlation coefficient R between the instantaneous amplitude and the fitting data Amp e(t) and Amp l(t) respectively.

  4. 4)

    Discriminate the oscillation properties with the goodness of fit and the correlation coefficients. If SSE and RMSE for exponential fitting are smaller than that for linear fitting and meanwhile CD and R for exponential are larger, then the exponential fitting is more similar to instantaneous amplitude and we can conclude that the oscillation is a weak damping oscillation. Otherwise, if SSE and RMSE for linear fitting are smaller than that for exponential fitting and meanwhile CD and R are larger, the property could be determined as an FO.

3 Measurement-based solution for LFOs analysis in online monitoring and offline research

3.1 Oscillation source locating method in FOs

The foremost affair to restrain LFOs or eliminate their hazards is to discover their properties. After discriminating the properties, as for the WDO, the next step is to find which component contributes most to the negative damping; for the FO, it is important to locate the oscillation source.

The authors’ research group (2012) proposed an oscillation source locating method that could directly locate the oscillation source in the governors or exciters of generators when there is an FO [19–21] (This method can also be used to distinguish the governors or exciters that participate in a local mode with negative damping).

References [19–21] introduced the concept of ESP. It is a kind of transient energy derived from the port-controlled Hamiltonian theory, which is injected into the network at corresponding ports. By calculating ESP, the external disturbances could be distinguished as the oscillation source for a FO, or the subsystems that contribute more to the negative damping could be found out for a WDO.

3.2 Strongly correlated generators identification method in WDOs

When an unexpected weak damping oscillation occurs, which hasn’t been detected in offline study for whatever reasons, the most efficient measure to restrain it is to reduce the output power of strongly correlated generators temporarily. Conventional eigen-analysis could provide information about mode shapes and participation factors (PF). Nevertheless, the oscillation modes in practical grids usually mismatch those in the simulations due to the un-modeled system dynamics or incorrect parameters. Therefore, an online method to identify strongly correlated generators is in urgent need for real-time control strategies generating when a WDO occurs.

The authors’ research group (2015) proposed an online identification method based on the power supply on the port (PSP) [22], which is defined as the rate of change of ESP. A strong correlation index to assess how deep a generator participates in a WDO is defined as

$$R_{\omega Oi} (t) = A_{{R_{\omega Oi} }} (t){\kern 1pt} \sin (2\omega t + 2\varphi_{{\delta_{i} }} + \varphi_{Bi} )$$
(16)

where \(R_{\omega Oi} (t)\) is the i th generator’s periodic component of the PSP in the governor channel; \(A_{{R_{\omega Oi} }} (t)\) is the instantaneous amplitude of \(R_{\omega Oi} (t)\); ω is the frequency; \(\varphi_{{\delta_{i} }}\) is the initial phase of the oscillation; \(\varphi_{Bi}\) is the term occurred in trigonometric calculation. The \(A_{{R_{\omega Oi} }} (t)\) is defined as the strong correlation index, which could be extracted by HT.

Reference [22] also demonstrated that the index has the same distribution with the PF, so that it can be utilized as a standard to determine which generators’ output should be reduced.

3.3 Systemic implementation of solution

With the methods respectively introduced in Sections 2, 3.1 and 3.2, a measurement-based systematic solution for the LFO analysis in online monitoring or offline research has been forged.

When an LFO is detected, the first step should be the oscillation property discrimination with the method based on the alternate EMD/HT and square processing. Next, the oscillation source should be located utilizing the method in Section 3.1 if the oscillation is regarded as an FO. If the oscillation is judged to be a WDO, the strongly correlated generators identification method should be utilized to find out the generators whose output should be reduced. The implementation process is shown in Fig. 1.

Fig. 1
figure 1

Flow chart of measurement-based solution of LFO analysis

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This solution is appropriate to both simulation researches and online monitoring for practical systems. As for the practical systems, the solution provides abilities to rapidly discover and locate the causal factors of an LFO, thus the control measures could be taken as soon as possible. While in offline researches like power system planning, it helps to find the devices with inappropriate models or parameters, so that the system dynamics could be improved.

4 Case study

4.1 General information of Changdu case

In the simulation study on the Sichuan-Tibet interconnection project, an uncommon phenomenon was found in the isolated grid of Changdu in Tibet, where there is a divergent frequency oscillation in the whole grid.

Changdu is located in the east of Tibet near Ganzi autonomous prefecture in Sichuan province. The grid has been operating in a standalone mode for years. The target of the interconnection project is to connect this isolated grid to the Sichuan grid to solve the power shortage in this area radically. In the case where there is a frequency oscillation, the isolated Changdu grid has 8 generators located in 2 plants and each plant has 4 generators respectively, with the total output of 46 MW. Simulation results on a 3-phase-to-ground short circuit fault are shown in Fig. 2 to Fig. 4 (some curves overlap with each other). The fault lasts 5 circles and disappears thereafter.

Fig. 2
figure 2

Speed deviation of generators

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Fig. 3
figure 3

Angle of generators (reference is Jinhe 2)

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Fig. 4
figure 4

Active power of tie-lines

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These figures indicate that the generators do not lose synchronization but the frequency of the system has a swing with peak-to-peak value up to 20 Hz, which means that the system cannot operate. Nonetheless, the actual system in Changdu has been operating in such a situation over years. So, there must be some inaccurate models or parameters in this simulation case and they must be found.

4.2 Locating problem utilizing proposed solution

4.2.1 Oscillation property discrimination

Noticing that the transmission power has less obvious fluctuation, the angles of generators should be chosen as the study object considering the efficiency of the algorithm.

The fit results are shown in Fig. 5, referring to (16) and (17) respectively. The results obviously imply that the fit in exponential function is better than in constant. The goodness of fit is shown in Table 1, which also verifies the conclusion.

Fig. 5
figure 5

Angle of generators (reference is Jinhe #2)

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Table 1 Goodness of fit
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It can be seen in Table 1 that the SSE and RMSE of the exponential fit are smaller than the linear fit, and meanwhile the CD and R of the exponential fit are larger. Therefore, the oscillation is judged to be a WDO.

4.3 Finding out negative damping

Since the oscillation is a WDO, the next step is to find out which device contributes more to the negative damping.

Figure 6 shows that the governors in both plants has increasing and large ESP (generators in the same plant has almost the same response) in this case, where the Jinhe plant’s ESP is larger. On the contrary, the exciters’ ESP has much smaller magnitude. This result indicates that the governors are the devices that contribute more to the negative damping, and Jinhe plant might play a more important role in the oscillation.

Fig. 6
figure 6

ESP of some generators

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4.4 Identifying strongly correlated generators

After the negative damping has been confirmed, the next problem is to determine which plant is the most critical one.

Figure 7 shows that the \(A_{{R_{\omega Oi} }} (t)\) of Changdu is always less than that of Jinhe, which means that the Jinhe plant has stronger correlation to the oscillation.

Fig. 7
figure 7

\(A_{{R_{\omega Oi} }} (t)\) ratio of Changdu #2 to Jinhe #4

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4.5 Comparison to traditional simulation

After all the process, it can be concluded that the governors in Jinhe plant might be the cause for the negative damping.

In fact, the values of soft feedback constant D d (in the numerator of soft feedback) of the governors in Jinhe is finally found to be inappropriate, so that the system forms a positive feedback, thus it becomes unstable. A Simulink model, which contains a synchronous generator and a detailed governor system with the same parameters as the original devices, is built to test the governor. The governor model (with the corresponding hydro-turbine) is shown in Fig. 8. This simulation is driven by a step change in rotating speed are shown in Fig. 9. It can be seen that with the original D d the system is unstable, whereas the system is stable with a ten times larger D d.

Fig. 8
figure 8

Model of governor with hydro-turbine

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Fig. 9
figure 9

Test results of different D d utilized in governor

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The results of eigen-analysis of the system with original D d are shown in Table 2. There is a pair of eigen-values with positive real-part, which means the system is unstable. Meanwhile, the mode shapes information shown in Table 3 indicates that the modules of right eigenvector of all the generators are adjacent to 1 and the angles are adjacent to 0, which means all the generators swing synchronously, complying with the phenomena observed and the conclusions obtained from the proposed measurement-based methods.

Table 2 Modes of isolated Changdu grid
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Table 3 Mode shape of unstable mode
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5 Conclusion

In this article, an oscillation property discrimination method based on alternate EMD and HT is introduced, and a measurement-based solution for the LFO online monitoring and offline research is proposed. The solution includes a series of concrete methods from the oscillation property discrimination to the oscillation source location and the strongly correlated generator identification. With this solution, the causal factor of low frequency oscillation could be discovered and located rapidly. It is a compatible assistant for control strategies generating to restrain the oscillation or improve inappropriate models and parameters in simulation researches. The case study on Changdu grid proves the effectiveness of the solution.

Considering that multi oscillation modes might be excited after some certain faults and sometimes they could be too similar to be distinguished, this solution could be further refined if there is a methodology to effectively separate them, so that targeting control strategies could be obtained respectively.