The analysis of the long-term frequency stability using full power system models and conventional simulation tools may be rather involved due to the slow nature of the frequency 2 dynamics, as they require large simulation times. Moreover, full scale models often give little insight as to what parameters Here is a vector of steady-state values of fast dynamics have the most effect on frequency dynamics.
On the other hand, state variables; is a set of nonlinear equilibrium equations the very nature of the long-term frequency stability allows for for fast state variables that can be combined with network equa- temporal decomposition of system dynamics and utilization tions and solved using the Newton-Raphson method. Such ap- variables can be found using numerical integration techniques, proximations have already been successfully used for adequate such as the trapezoidal rule.
However, QSS is not limited to the analysis of voltage dynamics. Vournas and Mantzaris described the QSS models 3 of low frequency inter-area oscillations and their application to PSS tuning problem . Grenier et al. The contributions of this paper are twofold. First, it extends the QSS approach to demonstrate the effects of power system 4 static voltage characteristics, system inertia and damping con- trollers on frequency swings.
Furthermore, pro- posed framework provides sound analytical models for coordi- 5 nated design of PSS4B type stabilizers to improve frequency dynamics. General Framework Obtained linear model can provide useful information re- Power systems are described by a system of differential and garding the system dynamic performance and can be utilized algebraic equations for controller design purposes.
Equation 5 can be re-written in the following way: where is a vector of state variables, is a vector of parameters, 7 and is vector of controlled parameters. The quasi-steady-state approach hinges on the idea of tem- where matrices and represent sensitivities of the net- poral decomposition of system dynamics.
Suppose that vector work parameters with respect to state and control variables re- is decomposed into two subsets: state variables that corre- spectively. On each integration step the changes in can be spond to fast dynamics and those that correspond to slow estimated using 7. This approach will yield only an approxi- dynamics. Either different initial conditions for a new operating point to ensure network convergence are required, or sensitivity matrices for the new system conditions must be re-calculated.
Next, a set of power system dynamics assumed critical for the analysis of frequency oscillations is developed and discussed in detail. However, some small time constants can be removed its response to frequency swings. The details conventional PSSs PSS1A are tuned so they provide necessary with respect to simplifying common hydraulic governors can be phase compensation in the range of high frequencies local and found in . The behavior near very low frequencies is Coherent i. Various inter-machine steady-state PSS gain.
However, depending on the spread frequency assumption . Experience with tuning MB-PSSs suggests that per- formance in the low range can be very well approximated by retaining only low and intermediate bands. In this paper in- the time constants . However, aggregated loads in bulk transmission systems are usually represented in the form of static voltage and frequency characteristics. In this paper static load models of the form 9 are assumed . It comprises the vector for generator buses and assumed static load voltage characteristics.
Damping controllers current injections of the load. Vector is derived in a sim- either on synchronous generators or FACTS devices tend to am- ilar manner. Slow speed oscillations can Following a frequency deviation, generators in the system be translated into terminal voltage modulation through the ac- pick up the active power imbalance according to their values tion of the damping controllers.
Common frequency of oscil- of inertia constant, irrespective of the generators' location . Therefore, in the described formulation there is no single generator acting as a slack bus. However, the rotor angle of one of the generators is taken as a reference. This formulation is adopted from  and re-written in the current balance form, as it is more robust to varying initial con- ditions, less nonlinear with respect to voltage magnitudes and generally converges faster than the power balance form.
For the purpose of demonstrating how the effects of PSS can be coupled with nonlinear equilibrium equations, expressions for active and reactive current injections from the generator at bus are shown: Fig. Schematic diagram of the proposed QSS model. The selected settings result in an unstable frequency os- obtained in a similar manner. Details of deriving those can be cillation for the case of constant power load model Fig.
In many follows: cases the main control objective—especially for renewable gen- eration—is keeping the active power output at an optimal level. Stiff load along with assumed governor settings modeling the voltage control similar to the synchronous gener- result in unstable frequency deviations with a period of about ation. It should be noted however that the assumption of con- 10 s. Discrepancy in the equately represent response to frequency deviations. This introduces an additional damping III.
Also note A. Detailed description of the model 9 are assumed zero. Frequency deviation following a load increase. Under the conditions of constant power load the load modula- tion effects are not present, and the action of PSSs destabilizes the frequency mode. Coherent voltage decrease at the gener- ator buses caused by PSSs following a frequency drop results in Fig.
QSS simulations of the generator outage for different reactive power transmission losses increase, as they are inversely proportional load models and voltage conditions. This increase is compensated by additional active power drawn from the generators, thus further decreasing the speed, and hence frequency. Additionally, results from the subsystems prior to the islanding is MW. The results EMTP-RV software have been included for the case of the con- aim to demonstrate the effects of system topology and inertia stant impedance model.
They underscore the differences in fre- changes on frequency swings. Clearly, smaller subsystem quency dynamics estimation for different simulation programs. First, a negative larger disturbance is next considered, namely a generator outage impact of reduced inertia on the initial frequency drop can be that provides MW of active power Fig. As can be observed. Furthermore, a somewhat counter-intuitive damping seen, the response is similar to the load increase scenario, al- improvement with reduced inertia constant can be seen in beit system inertia is now reduced.
This does not however sig- Fig. This is gen- oscillation. In general however, these results cannot be gener- erally not true for larger disturbances, such as islanding events, alized, and the effects of system inertia on the mode damping where system topology is very different from the normal oper- depend on topology and system conditions. Results suggest that reactive power verter-interconnected equipment—both on demand-side and load voltage sensitivity has a slightly negative impact on mode supply-side—affects frequency evolution and frequency sta- damping. Reduced reactive power consumption due to the bility during active power imbalance events.
The detailed voltage decrease in fact exhibits a voltage support effect—thus analysis of renewable generation impact on frequency response counteracting the effects of PSSs. Furthermore, reducing the is beyond the scope of the paper. Rather, in the section the capacity of the SVCs to control the voltage improves the proposed QSS approach is extended to include a wind farm response. That demonstrates the notion that the objectives of WF that supplies additional MW.
QSS simulations of the islanding event with WF. Frequency during islanding. QSS simulations for different conditions of system inertia. Effects of system inertia on frequency mode. However, individual switches of the converter are not C. It reveals a pair of complex conjugate eigen- WF. Upon islanding the WF remains connected to the smaller values that correspond to the unstable frequency mode of Fig.
Also, constant active power output of the WF Fig. While it is established that proliferation of renewables D. QSS Approximation for Damping Controller Design without any inertia control loops deteriorates frequency sup- port, their impact on long-term frequency stability depends on In this section an application of QSS models for damping several factors: controllers design to improve frequency stability is explored.
The main criteria for choosing the settings was adequate damping of local and inter-area modes in the benchmark system. Low bands were subjected to the tuning procedure. Note that Fig. A variety of tuning Similarly for the PSS gains: methods can be applied. Nonlinear constrained optimization has been found to be effective in the proposed setup. A fol- 15 lowing optimization approach based on  has been applied in this paper. The objective function is: Here is an entry of the matrix that is an explicit function of the PSS parameters, and are corresponding left and 16 right eigenvector elements.
The results of Table III give an idea of a relative sensitivity of the eigenvalue associated with frequency mode to varying where is the number of poorly damped oscillatory modes; PSS parameters. High amplitudes along with the angles close the modal performance measure can be found as follows: to signify high mode sensitivity and mobility.
Higher sensitivity of the frequency mode to gain variations is apparent 17 from Table III. In the particular case of Table III, higher oscillatory mode; is a response time horizon, s typically a washout time constants slightly increase mode mobility with period of the slowest mode. However, washout time constants are typically The objective function 16 is subject to the constraints that bounded, as high values can result in the overall gain increase in put limits on the low band central frequencies and gains: the low frequency range and large voltage deviations during se- vere transient events .
Moreover, mode sensitivity towards 18 is relatively weak. Re- ysis. Results demonstrate satisfactory accuracy of QSS approximations for various disturbance scenarios. Voltage at bus following a generator outage.
Although the QSS ap- creasing the mode damping without unduly changing its fre- proach has been applied to a problem of damping controller quency. Moreover, described Damping controller action has a side effect of pronounced QSS models can be extended to more realistic scenarios of fre- voltage deviations, especially during large disturbances. Kundur, J. Paserba, V. Ajjarapu, G.
Andersson, A. For each subsample false positives were counted. Performance of phase-amplitude coupling measures were quantified by simulating independent data sets and modifying the parameters 1 modulation strength, and 2 modulation width, 3 multimodality, 4 data length, 5 sampling rate, and 6 noise level within each dataset. Phase-amplitude coupling values did not differ depending on data length, sampling rate, or noise level. Five percent of the simulated data were falsely classified as containing coupling when setting the critical z-value for the PLV at 1. Thus, these values were defined as critical z-values.
There were significantly more false positives during short epochs ms; 2. Medium and long epochs did not differ in their false positive rates. Figure 2. Probability distribution of coupling values under the null hypothesis: phase-amplitude coupling value distribution under the null hypothesis i. These distributions allow defining the significance threshold. There were significantly more false positives during short epochs ms: 2. PLV 1. The PLV was least sensitive to modulation strength 0. Figure 3. Coupling values of all methods increased with increasing modulation strength.
Also, coupling values of all methods increased with increasing modulation width. The red line marks the critical z-value significance level. All values above this line represent significant phase-amplitude coupling. For each effect, all factor levels within a method are significantly different from each other according to post hoc t -tests. As Tort et al. Since researchers do not only want to prove the existence of phase-amplitude coupling, but also differentiate its strength, a measure that can do this is indispensable.
The effect of modulation width was most pronounced for the MVL 0.
The MI was least sensitive to modulation width 0. Of all four methods, MVL differentiates best between the different factor levels of modulation width. Monophasic coupling 4. Biphasic coupling could not be detected by the PLV [2. The MI was larger in monophasic than in biphasic coupling [9. Monophasic coupling 7. Biphasic coupling could not be detected by the PLV [3. The MI was larger in monophasic than in biphasic coupling [ Figure 4.
This factor might turn out to be not as important, as most studies report monophasic coupling. Coupling values of all methods increased with increasing data length and slightly increase with sampling rate. Sampling rate only becomes relevant when analyzing frequencies close to the Nyquist frequency. Of all four methods, MI is least affected from the confounding factor data length.
Coupling values of all methods decreased with increasing noise, while the PLV is least affected from this confounding factor. That is, multimodality influences the four methods very differently. PLV and MVL cannot find biphasic coupling as it was implemented here amplitude of the higher frequency was increased at peak and trough of the lower frequency. Because of the mathematic construct of the MVL Eq. Peak and trough appear on opposite sides in the polar plane: their mean will cancel each other out.
If other forms of biphasic coupling would be present, the MVL could be able to find it, but would probably underestimate its strength and would furthermore return distorted phase information. Similarly, the PLV cannot detect biphasic coupling, as it was implemented here. For biphasic coupling the amplitude envelope oscillates twice as fast as the lower frequency band. Because of this, the phase lag between lower and upper frequency band spans the entire polar plane.
Literature indicates that biphasic coupling plays a minor role in empirical data. To our knowledge only a very small fraction of studies report biphasic coupling e. Most studies report monophasic coupling e. For the shortest epoch of ms, none of the methods could detect significant coupling, even though it was engineered into the data. The data length effect was most pronounced for MVL 0. The MI was least affected by data length 0. This association was found in the data presented here, but must not generally apply.
Here coupling was simulated continuously into the data. If coupling is transient and does not proportionally vary with data length, this relationship does not need to apply. Potentially, the general rule is that the longer the data epochs where coupling occurs, the stronger the phase-amplitude coupling values. This should be tested in a follow-up analysis. This analysis further showed that a minimal data length is required for finding coupling, which should exceed at least ms per trial when including 30 trials also see Cheng et al.
None of the methods were able to detect coupling in the shortest simulated epoch of ms. It might be useful to develop a correction factor e. The factor sampling rate stands out because of its lacking effect for theta-low gamma coupling and comparatively small effect size for alpha-high gamma coupling.
This analysis showed that sampling rate is indeed important, but only if the investigated upper frequency band approaches the Nyquist frequency here Hz. MVL 0. The PLV 0.
This aspect is not desired but plausible. Noise obscures the relation between the phase of the lower frequency and amplitude of the higher frequency. The data as a whole contains phase-amplitude coupling to a lesser extent, as the relative amount of noise compared to the relative amount of signal increases.
Especially multimodality and data length interacted with the remaining factors, as well as interacted with each other and the remaining factors. Sampling rate only showed significant interactions when analyzing frequencies close to the Nyquist frequency. All interactions had a monotone pattern, following the pattern of each main effect. For example, MVL increased the longer the data, but it increased less when also noise increases Figure 5.
This pattern was true for each added factor. Phase-locking value and MVL did not find biphasic coupling at all. Because of this, for these two methods, the described main effect and interaction patterns are only valid for monophasic, but not for biphasic coupling. Figure 5. Interactions had a monotone pattern, following the pattern of each main effect. Depicted here, MVL increased the longer the data, but it increased less when also noise increased. Only values within the ms condition do not differ between the noise levels. We showed empirically that the methods were indifferent to the chosen frequency band combinations.
In order to facilitate the testing of methods, we provide our MATLAB script in Appendix A , in which the chosen frequency band combination and parameters can easily be adjusted. The GLM-CFC behaves best regarding modulation strength and worst regarding noise compared with the three other methods. Regarding the other factors, its performance is in the intermediate range. The most important disadvantage of the GLM-CFC is its extremely high computation time, which exceeds those of the other methods by two without calculating confidence intervals or up to four orders of magnitude with calculating confidence intervals.
Increasing data points increases computation time for all methods in a similar manner e. Assuming, that this time-factor will lead to the exclusion of this method for most researchers, it is not further considered in the conclusion of this manuscript. For a more detailed review of this method, see Kramer and Eden Comparing the remaining three methods it becomes evident that the MI is least affected by the confounding factors multimodality and data length.
For long data epochs, recorded at high sampling rates, with a high signal-to-noise ratio, the use of the MVL is recommended, because it is more sensitive to modulation strength and width than both other methods. For noisier data, shorter data epochs, recorded at a lower sampling rate, the use of the MI is recommended, as it is least influenced by the confounding factors compared with both other methods.
If it is not clear whether cross-frequency coupling will be mono- or bi-phasic, the MI should be used, even though literature suggests that biphasic coupling can be neglected. The PLV does not stand out in comparison to the two other measures. So far, no review evaluated this measure explicitly as positive. Its usage is potentially problematic because phase information is extracted from the amplitude envelope of a signal.
Phase information can only be correctly extracted from truly oscillating signals; this must not be necessarily the case for an amplitude envelope. However this disadvantage can be counteracted by filtering the amplitude envelope first before extracting phase information from it as is described Vanhatalo et al. Because MVL and MI have complementing strengths and weaknesses, it would be advisably to calculate both.
The time-consuming aspect of measuring the two methods is permutation testing. Calculation of both measures on the other hand will not substantially increase the analysis time. However, even despite substantial quantitative differences in values, the qualitative decision for significance of phase-amplitude coupling is the same for all four methods in our simulation.
Nevertheless, comparison of coupling strengths between the methods is problematic and this lack of comparability provides another reason for reporting both, MVL and MI. The entropy value depends on the amount of bins as well as amount of data squeezed into the same amount of bins. This is an undesirable degree of freedom, which is not present when calculating the MVL. Due to the dependency on confounding variables e.
Comparisons within one study, on the other hand, can be done with confidence. Nevertheless, one should make sure that signal-to-noise ratio is comparable within all experimental conditions and over the course of the experiment. Generally, it is advisable to work with standardized phase-amplitude coupling measures via permutation testing. It facilitates the interpretation of the measures, first and foremost, by giving the researcher knowledge about the probability that the observed MI would have been also found under the assumption of the null-hypothesis.
This aspect is often ignored in the literature. Even if it would be ideal, to have a measure that is less susceptible to confounding variables summarizing this analysis, it should be rather concluded that at least two reasonable analysis methods exist. MH conducted the data simulation and performed the statistical analysis. All authors contributed to manuscript revision, read and approved the submitted version. Both funding Institutions had no further role in the study design, the collection, analysis, and interpretation of data, the writing of the manuscript, and the decision to submit the paper for publication.
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