Tool for detecting waveform distortions in inverter-based Microgrids: a validation study Geir Kulia
Marta Molinas
Lars Lundheim
Department of Electronics and Telecommunications, NTNU, 7491 Trondheim, Email:
[email protected]
Department of Engineering Cybernetics, NTNU, 7491 Trondheim, Email:
[email protected]
Department of Electronics and Telecommunications, NTNU, 7491 Trondheim, Email:
[email protected]
Abstract— Microgrids in isolated communities need accessible tools to detect anomalies that can potentially hinder the stability of their electrical systems. Modern power electronic equipment introduces new types of distortions that can be traced back to the control of the inverters in the microgrids. In this paper, the notion of instantaneous frequency, new to electrical systems, is introduced and its existence proven in the context of a microgrid. Voltage and current waveform distortions observed in a PV Microgrid in the field were analyzed by applying the Hilbert-Huang Transform (HHT) and the Fast Fourier Transform (FFT).The analysis revealed distortions of the fundamental frequency component that typically goes inadvertent in most analysis of such systems. When using HHT on the data, the distortions were manifested as a frequency modulation, while the mathematical derivation hints on amplitude modulation. Using FFT, the distortions appear in the form of a leakage spectrum composed of a multiple numbers of constant frequency components. The different observations from these methods have raised some questions of interpretation of their physical meaning. In an attempt to provide a physical interpretation and to understand the cause of the potential frequency modulation revealed by the HHT, this paper proposes an analytical investigation of the microgrid controller. The paper argues that the amplitude modulation coming from the analytical investigation is the best mathematical interpretation to the frequency modulation observed with the HHT, due to the constraint of a constant frequency in the mathematical modeling of the microgrid controller. The analytically obtained results compared with the simulated and field measurements results hints on the existence of an instantaneous oscillatory frequency in the voltage waveforms of the stand-alone microgrid of this study. The results of this paper will lay the foundation for future studies into more accurate diagnosis tools for electrical systems, and for the design of inverter controllers, enabling more robust and reliable microgrids. Keywords—Instantaneous frequency, Fast Fourier Transform, Hilbert-Huang Transform, Empirical Mode Decomposition, Microgrid, Single-phase Inverter, Amplitude modulation, Frequency modulation.
I.
M OTIVATION
Universal access to affordable modern power services is one of the United Nations sustainable development goals [2]. Electrical power is the foundation of modern business, medicine, education, agriculture, infrastructure, and
Fig. 1: ACME Portables FlexPAC, the core component in the prototype of the real-time analysis device for microgrids based on instantaneous frequency [1].
communications. Lack of electrical energy is, therefore, a severe impediment to economic growth, yet, this is the case for 1.2 billion people worldwide. Of these 1.2 billion without electricity access, 80 % lives in rural areas; 95 % in sub-Saharan Africa or in developing Asia [3]. Applying islanded microgrids with free renewable energy sources is an alternative for providing electricity to isolated communities where extending the main grid is too expensive. Microgrids are electrical systems that locally generate, store and provide power to a small area, such as a village. Traditionally, these grids have often been based on diesel aggregates, but due to the recent fall in prices for photovoltaic (PV) cells, PV based microgrids have become a conspicuous substitute that can provide cleaner and cheaper electricity. Because of the nature of the location for where the islanded microgrids are desirable, the microgrids needs to be near to maintenance free and have a supervisory control system that can handle any faults that might occur. Due to the stochastic nature of PV sources and the nonlinearities of modern power electronic equipment that are essential microgrid components, the need for accurate monitoring and diagnosis devices based on measurement of instantaneous values of fundamental electrical parameters rather than average values has become apparent. The authors are currently developing
a prototype for a real-time monitoring and diagnosis device for detecting the instantaneous frequency and time-dependent distortions due to nonlinearities. The development is carried out by the authors and a team of students and interns at the Norwegian University of Science and Technology (NTNU) (see fig. 1). This initiative is part of a greater partnership with the Royal University of Bhutan (RUB)s College of Science and Technology (CST) that aim at developing reliable and affordable solutions for easy diagnosis and correction of potential problems of microgrids in rural areas. In a previous contribution, the authors presented a method for decomposing and analyzing electrical voltage and current waveforms to obtain the instantaneous frequency [4] by applying the Hilbert-Huang Transform [5]. We then suggested that the electrical waveforms were frequency modulated, of considerable magnitude. This paper presents a validation study for a diagnosis tool for microgrids based on the investigation of the system behind the microgrid and its controllers. The validation procedure is based on providing first an analytical model of the distortion, supported by simulation data from a detailed Matlab model which then are used to discuss and validate the results observed in the field data. The results from the analytical model and the simulation model exhibit seemingly non-stationary instantaneous frequencies when compared with the measured waveforms obtained at RUB CST’s stand-alone microgrid. The expected frequency in such systems should be a stationary 50 Hz, while the observed frequencies in this investigation show a non-stationary oscillatory frequency with a cycle of 100 Hz around the 50 Hz. II.
M ICROGRID MODEL DESCRIPTION
Fig. 2 shows a model of the microgrid that will be used as reference in this paper. In the model, the dc voltage vd (t) is supplied from a dc source. The purpose of the capacitor C is to discard high frequency components from id (t) by filtering i∗d (t). The dc voltage vd (t) is converted to ac, vac (t), by using a dc/ac inverter. The inverter has a controller that uses vd (t) and the iac (t) as inputs.
A. The PV Inverter Controller The origin of the distortions discussed in this paper will be clearer for the reader if the control system is simplified. The assumptions to simplify the inverter control will be verified by comparing it with the measured waveform data in sec. III-C. The simplified inverter controller is based on a typical PV inverter controller [6] and has the purpose of maintaining a stable output voltage and frequency. A detailed model of the inverter controller is shown in fig. 3. The controller’s main blocks are proportionalintegral (PI) controller and a Resonant Controller (RC). In the controller, the dc voltage, vd (t) is compared with a predefined reference dc voltage r and results in a correction signal e(t) = vd (t) − r. The constant r will be equal to Vd , the constant part of vd (t) to simplify the equations in this paper. This will not distort the output signal of the inverter controller. The correction signal e(t) is used as input to the PI controller. The purpose of the PI controller is to maintain a stable output amplitude while there are slow-varying changes in the dc voltage vdc . Equation (2) gives the output of the PI controller. Z t vpi = Kp · e(t) + Ki · e(τ )dτ (2) 0
The resulting signal vpi (t) is used, together with a signal proportional to the ac current to build up the freequency for the alternating current system i(t) as an input to te RC. The input to the resonant controller vin,rc (t) is then given by vin,rc (t) = A · iac + vpi (t) · cos(ωt)
(3)
A is a constant choosen to scale iac (t). The Resonant Controller’s output is vcontrol = Cp · vin,rc (t) + Ci · vin,rc (t) ∗ h(t)
(4)
where h(t) is a bandpass filter’s transfer function with center frequency of ω. The Resonant Controller is a band amplifier that let all frequencies pass, but amplifies the grid frequency ω. Its purpose is to maintain a stable grid-frequency. The signal vcontrol is fed back to the dc/ac inverter and is used as a reference for output voltage vac (t).
Ideally, the ac output voltage should always be (1)
B. Propagation of the oscillatory component of the dc voltage through the dc voltage controller feedback
where ω is the fundamental grid frequency and is considered here to be constant for the purpose of the demonstration. For 50 Hz systems, ω = 2π · f = 2π50.
As shown in app. A, if the load is a passive resistance, the dc voltage will have the form
vac (t) = Vac cos ωt.
vd
id*
inverter control iac vcontrol id
id =
vd
iac
vpwm
C
vac
˜ dc current source
dc/ac Inverter
low-pass filter
load
Fig. 2: High level model of the microgrid investigated in this paper
vd (t) = Vd + v˜d (t) = Vd + V˜ d cos(2ωt)
(5)
where Vd is the constant component of vd (t) and v˜d (t) is the time-varying or oscillatory component. The dc voltage’s oscillatory part, v˜d (t), will be more dominant if the current i∗d (t) is large compared with the dc source’s internal resistor Rs . If the capacitor C in fig. 2 is small, this oscillatory part of vd (t) will not be filtered, and will be dominant in the inverter control. As shown in app. B, the consequence of this will be a control signal to the inverter given by vcontrol (t) = Vg cos(ωt − φg ) + Vam cos(2ωt − φam ) cos(ωt) + vfeedback (t)
(6)
RC Controller
PI Controller
Kp
vr
Cp
+
vd(t)
e(t) -
Σ
Σ
vpi(t)
vin,rc(t)
+
×
Σ
Σ
vcontrol
Bandpass filter
cos(ω t)
A! iac
Fig. 3: Simplified controller for the microgrids inverter consisting of a proportionalintegral and a resonant controller. The bandpass filter in the resonant controller has a center frequency of 50 Hz.
where Vg and Vam2 are constants determined by the grid frequency and the tuning of the controller, and vfeedback (t) is the result of a feedback from the ac current iac through the RC.
and is plotted in fig. 5b. The pwm voltage vpwm (t) is filtered using a low-pass filter to remove harmonics caused by the square pulse. By assuming an ideal filter, the resulting waveform vac (t) will be the avarage value of vpwm (t) so that
We can see from (6) that the reference voltage vcontrol (t) consists of a sinusoidal expression with constant frequency and a part that consists of an amplitude modulation, with a modulation cycle that is twice the grid frequency.
vac (t) ∝ vcontrol (t).
III.
M ETHODOLOGY FOR VALIDATION
Three sections will be presented to verify the characteristics of the output voltage vac (t) as shown in sec. II and discuss its consequences. The first approach is to use the analytical expression from (6), plot it and retrieve the fast-Fourier transform and Hilbert-Huang transform from the data. Then, this result will be compared with the corresponding output voltage obtained with a Matlab Simulink model of the same system. Here, the effects of the pulse-width modulation will be taken into account. The third step will consist on comparing the analytical and simulated models with the with the corresponding output voltage waveform measured at an operating microgrid in the field. A. Analytical Model Since the ac current waveform iac (t) is controlled by vcontrol (t), we will assume that vfeedback (t) has similar characteristics of vcontrol (t). To simplify the equations it will be neglected from the analysis. Fig. 4a is a plot of vcontrol (t) as described in (6) and an ideal ac voltage waveform as in (1) used as reference. By visual inspection of vcontrol (t) we can see that it is not ideal (as the red line). The ac voltage vac (t) is generated from a pulse-width modulation (pwm), vpwm (t) [7, p. 204]. The pwm voltage vpwm (t) is created by comparing vcontrol (t) with a syntetic triangular signal vtri (t) as shown in fig. 5a. The pwm voltage from the inverter vpwm (t) is given by vpwm (t) =
Vd /2, if vd (t) > vtri (t) −Vd /2, if vd (t) ≤ vtri (t)
(7)
(8)
which means that it is meaningful to compare vac (t) and vcontrol (t) directly. However, because the low-pass filters are not ideal, vac (t) will contain distortions caused by the pwm, as we will see in from the simulations and the measured data. 1) Fourier Spectrum: The Fourier transform is a method to decompose any time series down to sinusoidal components with constant amplitude and frequency. By using the identity in (23) we can rewrite vcontrol (t) as vcontrol (t) = Vf cos(ωg t − φg ) + where
r Vf =
and φf = arctan
Vg2 +
Vam cos(3ωg t − φg ) (9) 2 Vam 2 2
(Vam /2) · cos(ωg t − φam ) Vg cos(ωg t − φg )
(10)
(11)
The Fourier Transform of the control signal vcontrol (t) is given by r
π jφg e δ(ω − ωg ) 2 r π −jφg + Vf e δ(ω + ωg ) 2 r π jφam + Vam e δ(ω − 3ωg ) 8 r π −jφam + Vam e δ(ω + 3ωg ) 8
F vcontrol (ω) = Vf
(12)
where F is the Fourier Transform, ω is the frequency variable, and δ is the Dirac delta function. Equation (12) is visualized as a single-sided frequency spectrum in fig.
100
10
vcontrol (t) cos(ωg t)
Fcontrol (ξ)
0
-0.5
80 Frequency [Hz]
0.5
ωcontrol (t) ωg + ǫ · cos(2ωg t)
90
0 Magnitude [dB]
Normalized Amplitude
1
-10 -20 -30
70 60 50 40 30 20
-1
-40
0
20
40
60
0
Time [ms]
(a) vcontrol (t) from (6) against an ideal ac voltage waveform as in (1). Both waveforms are normalized for an easier comparison.
50
100 150 Frequency [Hz]
200
0
20
40
60
Time [ms]
(b) single-sided frequency spectrum of Fcontrol (ω) = F vcontrol .
(c) Instantaneous frequency ωcontrol (t) of vcontrol (t) estimated using Hilbert-Huang Transform. The amplitude of vcontrol (t) was also estimated and was constant.
Fig. 4: Analytical representation of vcontrol (t) together with its Fourier and Hilbert-Huang transform.
Normalized Amplitude
1.5
2) Instantaneous Frequency: As mentioned in sec. I, the motivation for investigating the source of the distortions was a study of electrical waveform data using the Hilbert-Huang Transform (HHT) to obtain the instantaneous frequency and amplitude of measured grid data from microgrids [4]. In general, for a given monocomponent signal x(t) = a(t) cos θ(t) (13)
vtri (t) vcontrol (t)
1 0.5 0 -0.5 -1 -1.5 0
1
2
3
Time [ms]
(a) Comparison between a generated triangular signal Vtri (t) and vcontrol (t). 800 vpwm (t) vac (t)
Voltage [V]
600 400 200 0 -200
the instantanious frequency ω(t) is defined as the changes in the phase θ, as shown in (14). dθ(t) (14) dt ω(t) is the time-varying instantanious frequency and must not be confused with the constant grid-frequency ωg or the frequency variable ω. The variant of HHT used to estimate the phase in this paper is called Quadrature [8]. A median filter with 3 samples width was used to remove artifacts. Fig. 4c shows ωcontrol (t), i.e. the instantanious frequency of vcontrol . By comparing the numerical estimation of ωcontrol (t) with a pure sinusoidal waveform we can approximate ω(t) =
ωcontrol (t) ≈ ωg + cos(2ωg t)
-400 0.5
1
1.5 2 Time [ms]
2.5
(15)
3
(b) Pule-width modulation generated vcontrol (t) and the filtered vac (t).
Fig. 5: Pule-width modulated signal generated by comparing a generated triangular signal vtri (t) with vcontrol (t).
4b. The single-sided frequency spectrum has a fundamental frequency ωg , and one harmonic at 3ωg . This spectrum correspond well with vcontrol (t) given on the form in (9). It is therefore expected that the frequency components ωg and 3ωg will be dominating in the frequency spectra of the simulated and measured ac voltage.
with an error of 11.7 %. The constant corresponds to the magnitude of the frequency modulation. Although the error in (15) is high, it is reasonable to expect a frequency modulation with a periodicity of 2f1g in the simulations and measured voltage waveforms. B. Study of microgrid simulations A simulation of the microgrid system in fig. 2 was implemented using Simulink to verify the finding from sec. III-A. The letter s will be added in subscript to variables to mark that they are estimated using the simulation. The ac voltage vac,s (t) is shown in fig. 6a. vac,s (t) is distorted compared with a pure sinusoidal waveform with frequency ωg , yet in a different way than vcontrol in fig. 4a. Unlike the analytical expression of vcontrol , the ac voltage vac,s (t) curves towards the right compared with a pure sinusoidal signal. It also
vac,s (t) cos(ωg t) 0.5
0
-0.5
cg,s (t) Normalized Amplitude
Normalized Amplitude
1 1 0.5 0 -0.5 -1
-1 20
40 Time [ms]
60
80
(a) Simulated output from an dc/ac inverter, vac,s (t) plotted against an ideal cosine.
20
40 Time [ms]
60
80
(b) The IME of the simulated waveform vac,s (t) that corresponds to the grid frequency.
Fig. 6: Simulated and filtered electrical voltage waveform 90
0 -10 -20
80 Frequency [Hz]
Magnitude [dB]
ωg,s (t)
Fac,s (t)
-30 -40 -50 -60
70 60 50 40
-70
30
-80 0
100
200 300 Frequency [Hz]
400
500
(a) single-sided frequency spectrum of Fac,s (ω) = F vac,m .
20
40 Time [ms]
60
80
(b) Instantaneous grid frequency ωg,s (t) of vac,s (t) estimated using Hilbert-Huang Transform. The amplitude of vcontrol (t) was also estimated and was constant.
Fig. 7: Fourier transform and Hilbert-Huang Transform of a simulated voltage waveform.
contains ripples that are caused by the pwm that was not properly filtered by the low-pass filter. The single-sided frequency spectrum of F vac,s (t) is given in fig. 6b. It was calculated using the Fast Fourier Transform. The spectrum only shows harmonics between 0 and 500 Hz, although harmonics with higher frequencies are present. From the plot, it is possible to identify the two most significant peaks at ωg and 2ωg with a prior knowledge of the distortions resulting from vd (t) and the controller. Without the prior knowledge of the inverter controller, the frequency spectrum becomes more ambiguous as the spectrum is obscured by harmonics caused by the pwm and leakage. In a real system, more harmonics can occure because of nonlinear loads and more complex control systems. The instantaneous frequency is not well defined for multicomponent signals [9]. The Empirical Mode Decomposition (EMD) proposed by Huan at. al. [5] was used to obtain the fewest monocomponents possible to fit vac (t). These nanocomponents are called Intrinsic mode functions (IMFs), and they have in common that they have a zero-crossing between each local extrema. The sum of a given signal x’s
IMFs should ideally be
x(t) = r +
X
ci (t) = r +
X
ai (t) cos(θi (t))
(16)
where ci (t) is IMF number i that x(t) consist of, and r is the residue. The residue r is also the offset or a monotone function; the signal x(t)’s trend. All but one of vac,m (t)’s IMF was discarded as this paper only analyse the monocomponent concerning the gridfrequency: cg,s (t). Fig. 7b shows the ωg,s (t), the instantaneous frequency of cg,s (t). As expected, ωg,s (t) has a semiperiodic behaviour. It has a pattern with repetitions every 10 ms, or written differently: 2f1g . Moreover, ωg,s (t) has a non-stationary behavior, meaning that its frequency spectrum changes for each period of 10 ms. These non-stationarities can be caused by the nonlinear properties of the modern power electronics, and will be the focus for future studies, once the stationary phenomena have been accounted for.
vac,m (t) cos(ωg t) 0.5
0
-0.5
cg,m (t) Normalized Amplitude
Normalized Amplitude
1 1 0.5 0 -0.5 -1
-1 20
40 60 Time [ms]
80
20
(a) Measured output from an dc/ac inverter, vac,m (t) plotted against an ideal cosine.
40 60 Time [ms]
80
(b) The IMF of the measured waveform vac,m (t) that corresponds to the grid frequency.
Fig. 8: Measured and filtered electrical voltage waveform 90 0
80 Frequency [Hz]
-10 Magnitude [dB]
ωg,m (t)
Fac,m (t)
-20 -30 -40 -50
70 60 50
-60
40
-70
30
-80 0
100
200 300 Frequency [Hz]
400
500
(a) single-sided frequency spectrum of Fac,m (ω) = F vac,m .
20
40 60 Time [ms]
80
(b) Instantaneous grid frequency ωg,m (t) of vac,m (t) estimated using HilbertHuang Transform. The amplitude of vcontrol (t) was also estimated and was constant.
Fig. 9: Fourier Transform and Hilbert-Huang Transform of a measured voltage waveform.
C. Field Measurement Analysis
it also has a frequency leakage around the harmonics.
The above analysis is without value if they do not correspond to the real world. Therefore, to validate the results, the analytical solution and the simulation results were compared with the electrical voltage waveform measured on the output of a physical single-bridge inverter, vac,m (t), using a similar setup as in fig. 2. The measured waveform, vac,m (t), is shown in fig. 8a. It is visually similar to vac,s (t), except that its curving is towards the left instead of the right. It also has harmonics caused by non-ideal filtering of vpwm (t). The single-sided frequency spectrum of F vac,m (t) was calculated using FFT, and is given in fig. 9a. The sample was cropped into to be as periodic as possible before computing the FFT, to reduce the frequency leakage. As in the analytical and simulated voltage waveforms, the frequency spectrum of F vac,m (t) has notable peaks around ωg and 3ωg . Its harmonic at 5ωg has alarger magnitude than vac,s (t)’s fifth harmonic. F vac,m (t) also contain many harmonics with higher frequencies that are not displayed in fig. 9a. Besides,
The measured voltage waveform vac,m is a multicomponent signal. The IMF corresponding to the grid, cg,m (t) was found by applying the EMD. Its instantanious frequency ωg,m (t) is shown in fig. 9b. ωg,m (t) has, like ωg,s (t), a semi-periodic behaviour with periods of 10 ms ( 2f1g ) and non-stationary properties. Based on the similarities between the analytical observations, the analysis of the simulation and the data from measured voltage waveforms in an operating microgrid, it is reasonable to say that the hypothesis proposed by the authors in previous works [4] that the semi-periodic fluctuations in the instantaneous frequency observed on microgrid data is the result of an error signal propagating through the microgrid controller. IV.
D ISCUSSION
A validation study for the use of the instantaneous frequency concept on electrical microgrids has been outlined in this
paper. The root cause of the distortions previously detected on microgrids and reported in this paper has been verified by analytically examining how the intrinsic oscillatory component of the dc bus voltage propagates through the controller feedback of the PV inverter system; resulting in an amplitude modulation, according to the analysis of the mathematical expression of the output voltage waveform of the inverter. This analytical expression when compared to the analysis of the simulated system and of the voltage waveforms measured on the grid, give however differing results, depending on the method used to analyze them. Fast Fourier and Hilbert-Huang Transforms were used to analyze the results from analysis, simulation and field measurements. The Fourier spectrum of the analytical results hints on the presence of the fundamental frequency modulated by a third harmonic component (150Hz). These two frequency components, 50 Hz and 150 Hz, are also observed when the simulated and measured field data are analyzed using FFT. However, a clear distinction of these two frequency components was less evident in the above because of the presence of multiple frequency components in the leakage spectrum resulting from the Fourier analysis of the finite data stream. When the Hilbert-Huang Transform was used to analyze the data (from analysis, simulations and field measurement), an oscillatory instantaneous frequency, with a cycle of circa 10 ms (100 Hz) was clearly identified. Although the oscillations were distorted compared with the analytical results, the oscillatory instantaneous frequency was strikingly similar. The root cause of the problem can be attributed to the propagation of the inherent 100 Hz oscillatory power that characterizes single phase electrical systems. It was proven in the analysis that the inverter dc bus is constrained to provide a dc current with an oscillatory component of twice the frequency of the ac current on the inverter side. This double frequency component of the current is the root of the oscillatory component observed in the dc voltage at twice the grid frequency. When this dc voltage is measured and compared with the dc reference voltage and is given as the input to the PI controller, it starts its propagation through the inverter controller and reaches the ac side through the PWM. This component appears as a frequency modulation on the ac voltage of the ac side of the inverter, as observed when analyzed with the HHT. From the observations of these results, the authors argue that, through this case, it comes clear that the Fast Fourier Transform, while being an excellent tool for analyzing data composed of constant frequency components, is not appropriate for analyzing data containing oscillatory instantaneous frequency components. The Hilbert-Huang transform however, by not making any a-priory assumption of constant frequency, is not constraining the interpretation of the data through a set of constant frequency components as Fourier does. Through the concept of instantaneous frequency defined as the derivative of the phase angle with respect to time, the HHT confers the data a degree of freedom and does not remove from the data its intrinsic attribute of instantaneous frequency. This notion of instantaneous frequency is new in
electrical systems and the case presented in this paper is only an example that shows the existence of instantaneous frequency due to the control action of the power electronics converter. A PPENDIX A D C VOLTAGE A. Simulation Model By following the argumentation of Mohan at. al [7, p. 214] and using the notation in fig. 2, we get that i∗d (t) = Id + ˜id (t) = Id + I˜d cos(2ωg t)
(17)
Id is the constant direct current component of id (t) and ˜id (t) is the time-varying part. The simulation of the dc current i∗d (t) is shown in fig. 10b. I˜d is the amplitude of ˜id (t). Equation (17) assumes that Vd v˜d (t)
(18)
The Th´evenin equivalent of the dc power supply in fig. 2 is shown in fig. 10a. The dc voltage supply vd (t) will be described by vd (t) = Vs − i∗d (t) · Rs = Vs − Rs · Id + Rs V˜ d cos(2ωg t) = Vd + V˜ d cos(2ωg t)
(19)
where Vs is the constant dc power supply, Rs is the power supply’s internal resistance, Vd = Vs − Rs · Is , and V˜ d = Rs Iˆd . Fig. 10c shows the simulation of the dc voltage vd (t). A PPENDIX B T HE PV I NVERTER C ONTROLLER In the PI controller (see fig. 3) the correction signal e(t) will be given by (20) if the dc voltage vd (t) = Vd + V˜ d cos(2ωg t) and r = Vd as shown in app. A. e(t) = V˜ d cos(2ωg t)
(20)
Equation (2) gives us that vpi (t) is given by Ki vpi (t) = V˜ d Kp · cos(2ωg t) + sin(2ωg t) 2ωg
(21)
Leading to an input to the RC on the form vin,rc = A · iac + cos(ωg t) B1 cos(2ωg t) + B2 sin(2ωg t) (22) Ki ˜ where B1 = Kp · V˜ d , and B2 = 2ω V d . A is a constant. As g cos(2ωg t) · cos(ωg t) =
1 cos(ωg t) + cos(3ωg t) 2
(23)
1 sin(ωg t) + sin(3ωg t) 2
(24)
and sin(2ωg t) · cos(ωg t) =
Rs Vs
id*
vd(t)
(a) Th´evenin equivalent of dc power supply.
(b) Simulated dc current i∗d (t).
(c) Simulated dc voltage vd (t).
Fig. 10: Th´evenin equivalent of dc power supply and simulated dc current and voltage i∗d (t) and vd (t).
the resulting vcontrol (t) from the Resonant Controller will be vcontrol (t) = Cp · vin,rc (t) B2 B1 (25) cos(ωg t) + sin(ωg t) + Ci 2 2 + vac,rc (t) where vac,rc (t) = A· Ci iac (t)∗h(t)+Cp iac and h(t) is the impulse response to a band-pass filter with center frequency ωg .
Thanks are in order for Tshewang Lhendup and, Cheku Dorji and my travel partner and coworker HaŁkon Duus for their assistance in collecting electrical voltage waveform data. The icons in fig. 2 were designed by Freepik. R EFERENCES [1] [2]
Equation (25) can be rewritten on the form vcontrol (t) = Vg1 cos(ωg t) + Vg2 sin(ωg t) + Vam1 cos(2ωg t) cos(ωg t) + Vam2 sin(2ωg t) cos(ωg t) + vfeedback (t)
[3]
(26)
[4]
where Vg1 = Ci B21 , Vg2 = Ci B22 , Vam1 = B1 · Cp , Vam2 = B2 · Cp , and vfeedback (t) = A · Cp iac (t) + vac,rc (t).
[5]
By using the trigonometric identity that a cos(ωg t) + b sin(ωg t) = R cos(ωg t − φ) (27) √ where R = a2 + b2 and φ = arctan( ab ) we get that vcontrol (t) = Vg cos(ωg t − φg ) + Vam cos(2ωg t − φam ) cos(ωg t) + vfeedback (t) q p 2 +V2, V 2 2 , Vam1 + Vam2 with Vg = Vg1 am = g2
[6]
[7]
(28)
V
g2 am2 φg = arctan( Vg1 ), and φam = arctan( VVam1 ).
ACKNOWLEDGMENT The authors would like to acknowledge the contribution of Jon Are Suul for providing a simulation setup of a single phase inverter. We would also like to thank Norden Huang for personally giving insight into his methods. His open-mindedness and generosity has been highly appreciated.
[8]
[9]
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