1 Introduction

In recent years, the wide implementation of phasor measurement units (PMU) results in the increasing developments of the power system dynamic security monitoring and control based on PMU [1–4], such as the post-event analysis [5], the low-frequency oscillation monitoring [6], the model validation, the parameter identification [5], and so on. The synchronized, fast and accurate phasor measurements of the fundamental component obtained by PMUs have brought about a revolution in the dynamic security monitoring and control.

However, with the increasing development of the renewable energy and the high voltage direct current (HVDC) transmission, more and more electronic devices are deployed in the power grid. This introduces a large number of inter-harmonics that are not integer multiples of the fundamental component to the power signals, which changes the measurement of the fundamental component and may further threaten the operating security of the power system.

One of threatens is the subsynchronous oscillation, which is of great harm to the power system. Several subsynchronous oscillation events in the wind-integrated system are reported around the world [7], such as the Electric Reliability Council of Texas (ERCOT) in 2009 [8], Oklahoma Gas and Electric Company (OG&E) in 2011 [9] and China Jibei Power Grid in 2013 and 2014 [10]. From the second half of the year 2015, there have been several subsynchronous oscillations caused by the inter-harmonics happened in a certain area of China with a high concentration of wind farms and HVDC transmission lines. Several thermal power generations nearby were tripped by their torsional stress relays (TSR), the power of more than 1 MW was lost. Then, the wind farms were tripped off as well. The affected area was more than 130000 km2. The inter-harmonics included subsynchronous and supersynchronous ones, whose frequency range was wide, from 10 to 100 Hz. Currently implemented harmonic monitoring devices are not suitable.

The discrete Fourier transform (DFT) is widely used for phasor estimation in PMUs [2]. But, DFT cannot extract the phasor precisely under dynamic conditions. With the release of the PMU standards around the world [11–14], many technologies have been developed to enhance the performance of PMUs [15]. A DFT-based algorithm is developed in [16] to obtain phasors when there are direct current (DC) components in the inputs. The Taylor series-based algorithm is carried out in [17] to improve the performance of PMU under dynamic conditions. Nevertheless, none of the references deals with exploring the phasor measurement of signals with inter-harmonics.

In this paper, the phasor expressions of the signals, which contain the fundamental component and the inter-harmonics, are derived. It is found that both subsynchronous and supersynchronous harmonic will result in the subsynchronous oscillation in phasor measurements. The mechanism frequency transmutation between the inter-harmonic and the phasor oscillation is revealed. Then, the recording data and the phasor measurements obtained by PMUs during the subsynchronous oscillation in the area of China with a high concentration of wind farms and HVDC transmission lines are investigated. The analysis results demonstrate the severe consequences of the inter-harmonics and verify the correctness of the theoretical derivation.

2 Phasor expression of signal containing single inter-harmonic

Under dynamic conditions, the pure stationary sinusoidal formula is not suitable even for an envelope of the signal within one calculation data window. The correct dynamic fundamental signal is expressed as:

$$ x(t)\; = \sqrt 2 X_{\rm m} (t)\cos \left( {2\uppi f_{0} t + \left( {2\uppi \int_{0}^{t} {\Delta f(t){\rm d}t} + \varphi_{0} } \right)} \right) $$
(1)

where x(t) is the instantaneous value of the signal; X m(t) the magnitude; φ 0 the initial phase angle; f 0 the nominal frequency; Δf(t) the frequency deviation. The magnitude and the frequency of the signal are the functions with respect to time.

Then (1) is represented by a complex number \( \dot{X} \) known as its dynamic phasor representation:

$$ \begin{aligned} x(t)\overset {} \longleftrightarrow \dot{X} & = X_{\rm m} (t){\rm }{\rm e}^{{{\rm j}\left( {2\uppi \int_{0}^{t} {\Delta f(t){\rm d}t + } \varphi_{0} } \right)}} \\ & = X_{\rm m} (t)\angle \left(2\uppi \int_{0}^{t} {\Delta f(t){\rm d}t + } \varphi_{0} \right) \\ & = X_{\rm m} (t)\left(\cos \left(2\uppi \int_{0}^{t} {\Delta f(t){\rm d}t + } \varphi_{0} \right) +\,\text{j}{\rm \sin} \left(2\uppi \int_{0}^{t} {\Delta f(t){\rm d}t + } \varphi_{0} \right)\right) \\ \end{aligned} $$
(2)

It is customary to suppress the term \( {\text{e}}^{{{\text{j}}(2\uppi f_{0} t)}} \) in the expression above, with the understanding that the reference coordinate rotates with the angular frequency of 2πf 0. Therefore, the measured frequency by using the rotating coordinate is the difference between real frequency of the power system and the rated one. The further frequency measurement transformation is needed to obtain the real frequency of the power system.

Suppose the signal is

$$ x(t)\; = \sqrt 2 X_{{\rm m}} {\rm cos}(2\uppi f_{1} t + \varphi_{0} ) $$
(3)

where f 1 is the signal frequency.

Then, its phasor expression \( \dot{X} \) is

$$ \dot{X} = X_{\rm m} \angle \varphi_{1} = X_{\rm m} \angle \left[ {2\uppi (f_{1} - f_{0} )t + \varphi_{0} } \right] $$
(4)

When the signal is the subsynchronous harmonic or supersynchronous one, f 1 need not be an integer multiple or divisor of f 0. As shown in (4), the phasor of the harmonic rotates with the speed of 2πΔf 1 = 2π(f 1 − f 0).

If the input contains the rated component and a harmonic component, the expression is shown as follows:

$$ x(t)\; = \sqrt 2 X_{\rm m0} { \cos }(2\uppi f_{0} t + \varphi_{0} ) + \sqrt 2 X_{\rm m1} { \cos }(2\uppi f_{1} t + \varphi_{10} ) $$
(5)

where X m0 is the magnitude of the rated component, X m1 the magnitude of the harmonic component.

Then, its phasor expression \( \dot{X}_{\rm c1} \) is the superposition of the two components. In the real power system, all the phasors in this paper is expressed in a synchronously rotating coordinate. The relationship among three phasors is shown in the synchronously rotating coordinate (Fig. 1). It can be seen that \( \dot{X}_{0} \) is static since it rotates with the coordinate at the same speed, while \( \dot{X}_{1} \) rotates with the speed of 2πΔf 1. Therefore, \( \dot{X}_{\rm c1} \) rotates with the same speed. According to the cosine theorem, the formula of \( \dot{X}_{\rm c1} \) is given as:

Fig. 1
figure 1

Relationship of three phasors

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$$ \left\{ \begin{array}{ll} \dot{X}_{{\text{c1}}} \,=\,X_{{\text{mc1}}} - \varphi_{{\text{c1}}} \,=\,\dot{X}_{\text{0}} + \dot{X}_{\text{1}} \,=\, \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;\sqrt {X_{{\text{m0}}}^{\text{2}} + X_{{\text{m1}}}^{\text{2}} + 2X_{{\text{m0}}} X_{{\text{m1}}} {\rm cos(2\uppi }\Delta f_{\text{1}} t\text{)}} \angle \beta \hfill \\ {\kern 1pt} \beta \,=\,\text{arctan}\left( {\frac{{X_{{\text{m0}}} {\rm sin(2\uppi }\Delta f_{\text{1}} t\text{)}}}{{X_{{\text{m1}}} + X_{{\text{m1}}} {\rm cos(2\uppi }\Delta f_{\text{1}} t\text{)}}}} \right) \hfill \\ \end{array} \right. $$
(6)

It can be observed that both the magnitude and the phase angle are the functions with respect to time.

Now suppose the input contains the harmonic whose X m1 is 5.773 V and f 1 is 20 Hz, and the rate signal whose X m0 is 57.73 V and f 0 is 50 Hz, then the magnitude and the phase angle of the superimposed phasor are outlined in Fig. 2. It can be observed that the superimposed phasor oscillates with the subsynchronous frequency of Δf 1 = 30 Hz. However, both of them are not pure but approximate sinusoidal curves, which can also be concluded from (6). That is also the reason why there are other frequency components in the magnitude and the phase angle spectrum, as shown in Fig. 2b, d. Therefore, the subsynchronous or supersynchronous harmonic will result in the subsynchronous oscillation of the superimposed phasor.

Fig. 2
figure 2

Superimposed phasor with single harmonic

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3 Phasor expression containing multiple inter-harmonics

In practice, several harmonics exist simultaneously in the transmission system. As an example, a signal with two harmonics is given as:

$$ \begin{array}{ll} x(t)\; = &\sqrt 2 X_{\rm m0} {\rm cos}(2\uppi f_{0} t) + \sqrt 2 X_{\rm m1} {\rm cos}(2\uppi f_{1} t + \varphi_{10} ) + \hfill \\ &\sqrt 2 X_{\rm m2} {\rm cos}(2\uppi f_{2} t + \varphi_{20} ) \hfill \\ \end{array} $$
(7)

where X m2, f 2 and φ 20 are the magnitude, the frequency and the initial phase angle of the second harmonic. To simplify the derivation process, φ 0 is set as 0°.

The superimposed phasor \( \dot{X}_{\rm c2} \) can be gotten by

$$ \dot{X}_{\rm c2} = X_{\rm mc2} \angle \varphi_{\rm c2} = \dot{X}_{0} + \dot{X}_{1} + \dot{X}_{2} = \dot{X}_{\rm c1} + \dot{X}_{2} $$
(8)

where X mc2, φ c2 are the magnitude and the phase angle of \( \dot{X}_{\rm c2} \).

Since \( \dot{X}_{1} \) and \( \dot{X}_{2} \) keep rotating in the synchronously rotating coordinate, while \( \dot{X}_{0} \) is static, the position of the three phasors keeps changing. Figure 3 shows six possible scenarios. It can be concluded that, since \( \dot{X}_{\rm c1} \) and φ c2 can be calculated by (6), the key point to obtain X mc2 and φ c2 is obtaining \( \varphi_{\rm c2}^{\prime } \) and δ by the triangle formed by \( \dot{X}_{\rm c1} \), \( \dot{X}_{2} \) and \( \dot{X}_{\rm c2} \), as shown in (9) and (10).

Fig. 3
figure 3

Different scenarios for superimposing of \( \dot{X}_{0} \), \( \dot{X}_{1} \) and \( \dot{X}_{2} \)

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$$ X_{{\text{mc2}}} \text{ = }\sqrt {X_{{\text{mc1}}}^{\text{2}} \text{ + }X_{{\text{m2}}}^{\text{2}} - \text{2}X_{{\text{mc1}}} X_{{\text{m2}}} {\rm cos(\uppi } - \delta \text{)}} $$
(9)
$$ \varphi_{\rm c2}^{\prime } {\kern 1pt} = \angle \left( {\text{arctan}\left( {\frac{{X_{\rm m2} \text{sin}\;\delta }}{{X_{\rm mc1} + X_{\rm m2} \text{cos}\;\delta }}} \right)} \right) $$
(10)

However, the formulas of δ and φ c2 are different for different scenarios in Fig. 3, which are given respectively form (11) to (16).

$$ \left\{ \begin{array}{ll} \delta = \varphi_{2} + \varphi_{\rm c1} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c1} - \varphi_{\rm c2}^{\prime } \hfill \\ \end{array} \right. $$
(11)
$$ \left\{ \begin{array}{ll} {\kern 1pt} \delta = \varphi_{2} - \varphi_{\rm c1} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c2}^{\prime } + \varphi_{\rm c1} \hfill \\ \end{array} \right. $$
(12)
$$ \left\{ \begin{array}{ll} \delta = \varphi_{2} + \varphi_{\rm c1} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c2}^{\prime } - \varphi_{\rm c1} \hfill \\ \end{array} \right. $$
(13)
$$ \left\{ \begin{array}{ll} \delta = \varphi_{2} - \varphi_{\rm c1} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c1} - \varphi_{\rm c2}^{\prime } \hfill \\ \end{array} \right. $$
(14)
$$ \left\{ \begin{array}{ll} \delta = \varphi_{\rm c1} - \varphi_{2} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c2}^{\prime } + \varphi_{\rm c1} \hfill \\ \end{array} \right. $$
(15)
$$ \left\{ \begin{array}{ll} \delta = \varphi_{\rm c1} - \varphi_{2} \hfill \\ \varphi_{\rm c2} = \varphi_{\rm c2}^{\prime } - \varphi_{\rm c1} \hfill \\ \end{array} \right. $$
(16)

Now the phase angle of the phasor is considered to be positive, when the phasor advances \( \dot{X}_{0} \). The phase angle of the phasor is considered to be negative when the phasor lags \( \dot{X}_{0} \). Therefore, for (11), \( \dot{X}_{\text{c1}} \) advances \( \dot{X}_{0} \) and \( \dot{X}_{2} \) lags \( \dot{X}_{0} \). Therefore, \( \varphi_{2} \) and δ are negative, φ c1 is positive. δ can be gotten by (17). Then, \( \varphi_{\rm c2}^{\prime } \) computed by (10) is negative, because \( X_{\rm m2} \text{sin }\delta /\left( {X_{\rm mc1} + X_{\rm m2} \text{cos }\delta } \right) \) is negative. φ c2 can be obtained by (18). For (13), \( \dot{X}_{\text{c1}} \) lags \( \dot{X}_{0} \) and \( \dot{X}_{2} \) advances \( \dot{X}_{0} \). Therefore, \( \varphi_{2} \) and δ are positive, φ c1 is negative. δ can be gotten by (17) as well. Then, \( \varphi_{\rm c2}^{\prime } \) computed by (10) is positive, because \( X_{\rm m2} \text{sin}\;\delta /\left( {X_{\rm mc1} + X_{\rm m2} \text{cos}\;\delta } \right) \) is positive. φ c2 can be obtained by (18). It can be verified that formulas of δ and φ c2 are the same as (17) and (18) for other scenarios. The detail process is not outlined here.

$$ \delta = \varphi_{2} - \varphi_{\rm c1} $$
(17)
$$ \varphi_{\rm c2} = \varphi_{\rm c2}^{\prime } + \varphi_{\rm c1} $$
(18)

Consequently, \( \dot{X}_{\rm c2} \) is expressed as:

$$ \begin{array}{ll} \dot{X}_{\rm c2} = \sqrt {X_{\rm mc1}^{2} + X_{\rm m2}^{2} - 2X_{\rm mc1} X_{\rm m2} {\rm cos(\uppi } - \varphi_{2} + \varphi_{\rm c1} \varvec{)}} \angle \left( {\varphi_{\rm c1} \text{ + arctan}\left( {\frac{{X_{\rm m2} \text{sin(}\varphi_{2} - \varphi_{\rm c1} \varvec{)}}}{{X_{\rm mc1} + X_{\rm m2} \text{\rm cos(}\varphi_{2} - \varphi_{{\varvec{c}1}} \varvec{)}}}} \right)} \right) = \hfill \\ \;\;\;\;\;\;\;\;\sqrt {X_{\rm mc1}^{2} + X_{\rm m2}^{2} - 2X_{\rm mc1} X_{\rm m2} {\rm cos(\uppi } - 2\uppi\Delta f_{2} t + \varphi_{\rm c1} \varvec{)}} \angle \left( {2\uppi\Delta f_{1} t\text{ + arctan}\left( {\frac{{X_{\rm m2} \text{sin(}2\uppi\Delta f_{2} t - \varphi_{\rm c1} \varvec{)}}}{{X_{\rm mc1} + X_{\rm m2} \text{cos(}2\uppi\Delta f_{2} t - \varphi_{\rm c1} \varvec{)}}}} \right)} \right) \hfill \\ \end{array} $$
(19)

Figure 4 depicts the magnitude and the phase angle of the superimposition of the rated component, 10% harmonic of 20 Hz and 20% harmonic of 70 Hz. It can be stated that the major frequency components are 20 and 30 Hz. The 20 Hz component is introduced by the harmonic of 70 Hz, while the 30 Hz component is introduced by the harmonic of 20 Hz. This can be distinguished by their respective magnitude. In particular, when the f 1 and f 2 are symmetric with f 0, φ 10 and φ 20 are symmetric with φ 0, and X m1 = X m2, \( \dot{X}_{\rm c2} \) will be magnitude modulated only.

Fig. 4
figure 4

Superimposed phasor with two harmonics

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More generally, if more than two harmonics exit in the input signal, the mechanism of frequency transmutation between the harmonic and the phasor oscillation frequencies is the same as that illustrated by Fig. 2.

The PMU is designed to measure the fundamental component rather than inter-harmonics. But if the inter-harmonics are not eliminated, their existence results in the fundamental component modulating at the subsynchronous frequency. There are two strategies to dispose of the inter-harmonics. The first one is to remove all of them by filters, and calculate the fundamental phasor only; and the second one is to calculate the superimposed phasor that contains the fundamental component and inter-harmonics. By the second way, we can monitor and detect the subsynchronous oscillation caused by the inter-harmonics with the phasor measurements of PMUs. Then corresponding controls can be carried out to prevent further faults, just like the low frequency oscillation detection based on PMUs.

The PMU measurements are used to analyze the subsynchronous oscillation happed in China in 2015.

4 Subsynchronous oscillation monitoring based on PMU measurements

In this section, a subsynchronous oscillation that occurred in western China where there is a large concentration of wind farms and HVDC transmission lines is studied using the recorded PMU data. Results of the analysis show the severe consequences of the inter-harmonics and verify the correctness of the theoretical derivation.

4.1 Distribution of subsynchronous oscillations

From July, 2015, several subsynchronous oscillations caused by the inter-harmonics occurred in western China, which resulted in several thermal power generations and the tripping-off of wind farms. PMUs installed in the area tracked the oscillation processes by both recording the signal instant value and measuring the phasor. The total PMU data is over 40 GB and covers over 140 transmission lines. The subsynchronous oscillations that occurred in September are exemplary, so the PMU data on that day is analyzed.

Figure 5 shows the geographical wiring diagram of the area with the main oscillation components by histograms. In the figure, the different colors of the histograms represent different oscillation frequencies. The height of the histograms bars indicates the oscillation magnitudes. The larger the oscillation magnitude is, the higher the histogram bars are. The exact oscillation frequencies are outlined under the histograms, and the oscillation magnitudes and their respective proportions of the fundamental one are outlined above the histograms. It can be seen that all the oscillation frequencies are around 10.2, 20.6, and 30.7 Hz. The highest oscillation magnitude reached 32.88% of the fundamental component.

Fig. 5
figure 5

Oscillation analysis based on PMU measurements

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The substation A and B are connected with the large scale of wind farms. The substation H is the convertor station. It can be seen that the oscillation components around wind farms are bigger than those at other places. It means that there are more subsynchronous and supersynchronous harmonics around wind farms. The subsynchronous oscillations are propagated along the transmission lines that cover the voltage levels from 35 to 750 kV. The total affected area is over 130000 km2. Farther away and at higher voltage levels, the subsynchronous oscillation magnitudes become smaller. However, the especially high magnitude of the subsynchronous oscillations around the wind farms does not mean that it is the wind farms that cause the inter-harmonics and the subsynchronous oscillation. The exact reason is complicated, and further studies are necessary.

Consequently, the large scale of renewable energy and HVDC introduce a large number of inter-harmonics. They result in the subsynchronous oscillation which can be propagated widely and threaten the security of the power system.

4.2 Inter-harmonics analysis

In addition of phasor the measurements, PMUs also recorded the instant values of the signals for some transmission lines. Figure 6 shows the spectrum analysis of the current instantaneous values of the transmission line C-E, while Fig. 7 shows the spectrum analysis of current phasors measurements from the PMU of the same transmission during the same period. To show the inter-harmonics and their impacts on the phasor measurements clearly, the fundamental component is not plotted in Fig. 6 and Fig. 7.

Fig. 6
figure 6

Spectrum analysis of current instant values of transmission line C-E

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Fig. 7
figure 7

Spectrum analysis of current phasor of transmission line C-E

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Then, all the frequencies that cannot be ignored appeared in Fig. 6 and Fig. 7 are shown in Fig. 8 in two rows. According to the mechanism of frequency transmutation between the harmonic and the phasor oscillation illustrated by Fig. 2, all the oscillation components can be explained by the corresponding harmonics. For example, the phasor oscillation component at 10 Hz is caused by the inter-harmonics of 40 and 60 Hz, because their differences between the rated frequency are both 10 Hz. As revealed in Section 3, one oscillation component can be caused by multiple subsynchronous or supersynchronous harmonics. Therefore, the derivations in Sections 2 and 3 are verified by the real PMU measurements and recorded data. Since it is feasible to identify the oscillation from the active power, Fig. 9 depicts the spectrum analysis of the active power of the same line during the time window measured by PMUs. It can be seen that the oscillation frequencies are the same as that in Fig. 7.

Fig. 8
figure 8

Corresponding relation between harmonic and oscillation frequencies

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Fig. 9
figure 9

Spectrum analysis of active power of transmission line C-E

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It may seem that the 109.5 Hz harmonic appears to be an exception. According to the mechanism, such a harmonic should result in phasor oscillation at 59.5 Hz. But, the highest reporting rate of PMUs in China is 100 Hz, which cannot meet the Shannon sampling theorem. Therefore, the frequency aliasing occurs, the phasor oscillation frequency become 40.5 Hz. Since the frequency aliasing is irreversible, it will be considered that there is more harmonic of 109.5 Hz than there should be.

Consequently, when the frequency aliasing occurs, phasor measurements cannot restore the original harmonics accurately. Measures should be taken to prevent the frequency aliasing from occurring, especially in the case of the subsynchronous oscillation. For example, a digital filter in the PMU algorithm can eliminate the oscillation components which may lead to the frequency aliasing.

As shown in Fig. 8, one subsynchronous oscillation frequency component of the phasors may be caused by two harmonics that are symmetrical about the rated frequency. Based on nowadays PMU, we can only tell if there are subsynchronous oscillation that maybe caused by the subsynchronous or supersynchronous inter-harmonics. But the exact magnitude of the subsynchronous and supersynchronous inter-harmonics cannot be obtained. Further algorithms for the inter-harmonic monitoring based on PMUs are needed to achieve that. In addition, the fast changing phasor makes precise measurement more difficult. Because, DFT assume that the phasor is static in the data window. The faster the phasor changes, the bigger the measurement error is [17]. Therefore, relevant phasor measurement algorithms which can track the phasor modulated in subsynchronous frequency precisely are also needed.

5 Conclusion

In this paper, the impacts of inter-harmonics on synchrophasor measurements are revealed. The phasor expression of a signal that contains the fundamental component and single or multiple inter-harmonics is proposed. It is found that the harmonics can cause the subsynchronous oscillation of the phasor measurements. The frequency of the oscillation of phasor measurements is the difference between the harmonic and the fundamental component.

The instantaneous value of the power signal and the phasor measurements obtained by PMUs during a subsynchronous oscillation from a real power grid that has a large concentration of wind farms and HVDC transmission lines are analyzed. The geographical wiring diagram with the supersynchronous oscillation distribution shows that more inter-harmonics are located near the wind farms, and many voltage levels and a wide area are impacted. Furthermore, the spectrum analysis comparison between the instantaneous value of the power signal and the phasor demonstrate the correctness of the theoretical derivation.

The monitoring of inter-harmonics using PMUs is possible and necessary. However, one subsynchronous oscillation frequency component of the phasors may be caused by two harmonics, and frequency aliasing may occur. Therefore, how to restore the harmonics precisely based on the phasor measurements of PMUs still need further studies.