1. Introduction

2. Proposed Method

2.1. General Mathematical Mode of Charging Voltage and Current

Due to the wide use of power electronic devices in the charging piles, the charging for the batteries of electric vehicles brings harmonic current and non-stationary signals to the power grid. The general structure of the charging piles is mainly composed of a rectifier circuit, a power converter, and an output filter circuit, as shown in Figure 1. Alternating current (AC) power is rectified to DC power through a three-phase bridge uncontrollable rectifier circuit, and then provided for the DC/DC power converter to output the required DC of the electric vehicles. Finally, DC is filtered to charge the batteries of the electric vehicles.

Figure 1. Topology of the Charging pile of electric vehicles.

Figure 1. Topology of the Charging pile of electric vehicles.

More researches on the load characteristics of electric vehicles show that the voltage and current waveforms on the grid side of the charging piles, collected at charging stations [13,26], contain multiple harmonics and non-stationary signals. Typically, the current waveform mainly includes $6h\pm 1$ order harmonics, and the amplitude of the harmonics gradually decreases with the increase of the harmonic order, which further can be confirmed in the simulation model in Section 3.2.

Generally, the voltage and current on the grid side of the charging piles can be respectively expressed as

$$\begin{array}{cc}\hfill & u\left(t\right)=u_b\left(t\right)+\sum _{h=1}{u}_h\left(t\right)+u_s\left(t\right)\hfill \\ \hfill & =\sqrt{2}{U}_{b}sin(\omega t+{\phi }_b)+\sum _{h=1}\sqrt{2}{U}_{h}sin((6h\pm 1)\omega t+{\phi }_h)+u_s\left(t\right)\hfill \end{array}$$

(1)

$$\begin{array}{c}\mathrm{and}\hfill \\ i\left(t\right)=i_b\left(t\right)+\sum _{h=1}{i}_h\left(t\right)+i_s\left(t\right)\hfill \\ =\sqrt{2}{I}_{b}sin(\omega t+{\varphi }_b)+\sum _{h=1}\sqrt{2}{I}_{h}sin((6h\pm 1)\omega t+{\varphi }_h)+i_s\left(t\right),\hfill \end{array}$$

(2)

where ($u_b$, $u_h$, $u_s$) denote the fundamental, harmonic, and non-stationary components of voltage signal, respectively, and ($i_b$, $i_h$, $i_s$) denote the fundamental, harmonic, and non-stationary ones of current signal, respectively, $\omega$ is the fundamental frequency in radian, (${\phi }_b$, ${\phi }_h$, ${\varphi }_b$, ${\varphi }_h$) are initial phases of the fundamental voltage, the h-th harmonic voltage, the fundamental current, the h-th harmonic current, respectively.

To accurately calculate the charging power of charging piles for the electric vehicles, the ACMD is firstly applied to decompose the fundamental, harmonic, and non-stationary components from $u\left(t\right)$ and $i\left(t\right)$ in our work, and then the power-related quantities are calculated according to IEEE Standard 1459-2010 [12].

3. Experiments and Results

3.1. Non-Stationary Synthetic Signal

To demonstrate the performance of the ACMD method in signal decomposition and power-related estimation, three stationary waveforms, containing the fundamental and two harmonics, and one non-stationary component, are considered in synthetic signals of the voltage and current, expressed as

$$\begin{array}{cc}\hfill u\left(t\right)& =u_1\left(t\right)+u_2\left(t\right)+u_3\left(t\right)+u_4\left(t\right),\hfill \\ \hfill & =\sqrt{2}\left(220sin(2\pi \times 50t)+20sin(2\pi \times 250t+2\pi )+5sin(2\pi \times 350t+\pi )\right)\hfill \\ \hfill & +e^{-0.3t}cos\left(2\pi (450t+0.5\pi cos(2\pi \times 30t))\right)),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill i\left(t\right)& =i_1\left(t\right)+i_2\left(t\right)+i_3\left(t\right)+i_4\left(t\right),\hfill \\ \hfill & =\sqrt{2}\left(25sin(2\pi \times 50t)+10sin(2\pi \times 250t)+5sin(2\pi \times 350t)\right)\hfill \\ \hfill & +2e^{-0.3t}cos\left(2\pi (450t+0.5\pi cos(2\pi \times 30t))\right)),\hfill \end{array}$$

(23)

where the fundamental frequency is 50 Hz, the sampling frequency is 1500 Hz for discretizing $i\left(t\right)$ and $u\left(t\right)$, and the time interval of the synthetic signal is 0.5 s in this experiment.

To visually evaluate the decomposition performance of ACMD and DWT methods, we show the resultant synthetic current waveform and each component in Figure 2a as the ground-truth ones, based on (22). After the DWT with mother wavelet ’db4’ and ACMD methods are used to decompose the current $i\left(t\right)$, the decomposed components and reconstructed signal are shown in Figure 2b,c. According to the principle of the DWT method, the fundamental component can be extracted from three-level DWT decomposition, and its frequency is within that of the highest-level approximation coefficients, i.e., ca3 in Figure 2b. From the decomposed results of the DWT method shown in Figure 2b, we observe that there is a component that does not already exist in (22) within [93.75,187.5]Hz. In addition, the obtained components within [0, 93.75] Hz is not pure fundamental component as expected but mixed with the leakage components from other frequency ranges. Similar patterns exist in other frequency ranges. Whereas, the ACMD method obtains each component, more closely to the ground-truth ones, compared with the results from the DWT method.

Figure 2. Current synthetic waveform and the corresponding decomposed results by using DWT and ACMD methods. (a) synthetic current and its components; (b) decomposed and reconstructed results of the DWT method, (c) decomposed and reconstructed results of the ACMD method.

Figure 2. Current synthetic waveform and the corresponding decomposed results by using DWT and ACMD methods. (a) synthetic current and its components; (b) decomposed and reconstructed results of the DWT method, (c) decomposed and reconstructed results of the ACMD method.

The calculated quantities and relative errors (‰) of the fundamental, harmonics, and non-stationary components are shown in Table 1, where the reference values are directly calculated from the analytic expressions in (22), agreeing with IEEE Standard 1459-2010. The estimated quantities for three-type components are calculated from the decomposed results from ACMD and DWT methods. The results in Table 1 show that the DWT-based method causes small estimations for the fundamental and large ones for harmonics, especially for non-stationary component, resulting in bring large estimated errors for each component, the main reason is its leakage issue as above mentioned. Whereas, compared with the DWT-based method, the ACMD-based method brings quantities closer to the reference values with fewer estimation errors.

Table 1. Comparisons of estimated quantities of ACMD-based and DWT-based methods for synthetic signal.

Table 1. Comparisons of estimated quantities of ACMD-based and DWT-based methods for synthetic signal.

Quantity		Reference values	ACMD		DWT
Symbols	Parameters	Analytical	Value	Error(‰)	Value	Error(‰)
U	Voltage RMS (V)	220.97	220.82	0.68	220.87	0.45
I	Current RMS (A)	27.39	27.38	0.36	27.38	0.36
P	Active power (W)	5625.86	5618.00	1.40	5602.85	4.09
$U_b$	Fundamental voltage RMS (V)	220.00	219.85	0.68	217.96	9.27
$I_b$	Fundamental current RMS (A)	25.00	24.98	0.80	24.80	8.00
$P_{bb}$	Fundamental active power (W)	5500.00	5492.78	1.31	5399.46	18.28
$U_h$	Harmonic voltage RMS (V)	20.62	20.59	1.45	35.13	703.68
$I_h$	Harmonic current RMS (A)	11.18	11.17	0.89	10.85	29.52
$P_{hh}$	Harmonic active power (W)	125.00	124.66	2.72	186.88	495.04
$U_s$	Non-stationary voltage RMS (V)	1.31	1.07	183.21	6.46	3931.29
$I_s$	Non-stationary current RMS (A)	0.66	0.53	196.96	4.14	5272.72
$P_{ss}$	Non-stationary active power (W)	0.86	0.57	337.21	16.52	18209.30

3.2. Simulation Signal

The performance of the proposed ACMD-based calculation method is further validated by using a simulation model in MATLAB/SIMULINK, as shown in Figure 5. The current and voltage waveforms at the grid side of the charging pile are shown in Figure 3a,b, respectively. From Figure 3, we observe that the charging current is similar to a square wave, deviating from the standard sine signal, while the voltage contains fewer harmonics, closer to the sine signal. Further, considering there are non-linear elements in the charging piles of electric vehicles, the non-stationary signals are introduced to the current and voltage for realistically simulating the charging characteristics of electric vehicles. In this experiment, sawtooth impulse and high-frequency attenuation signals are used for the non-stationary signals $u_s$ and $i_s$, expressed as

$$u_s\left(t\right)=\left\{\begin{array}{cccc}\hfill & 8\times sawtooth(2\pi \times 1800t+0.5\pi ,0.5),\hfill & \hfill & t\in (0.4,0.55)s,\hfill \\ \hfill & 8\times e^{-40t}sin(2\pi \times 1500t),\hfill & \hfill & t\in (0.75,0.9)s.\hfill \end{array}\right.$$

(24)

and

$$i_s\left(t\right)=\left\{\begin{array}{cccc}\hfill & 10\times sawtooth(2\pi \times 1800t+0.5\pi ,0.5),\hfill & \hfill & t\in (0.4,0.55)s,\hfill \\ \hfill & 10\times e^{-40t}sin(2\pi \times 1500t),\hfill & \hfill & t\in (0.75,0.9)s.\hfill \end{array}\right.$$

(25)

respectively. Figure 3c shows the waveform of $i_s\left(t\right)$ . To observe the components of current, the spectrum of the current $i\left(t\right)$ is shown in Figure 3d, which demonstrates that $6h\pm 1$ order harmonics with successively attenuated amplitude appear on the corresponding spectral bins beside the fundamental, and the non-stationary signals appear on multiple high-frequency ranges, as shown in the enlarged drawing.

Figure 3. Waveform generated from simulation model and the spectrum of current. (a) current; (b) voltage; (c) non-stationary $i_s$; (d) the spectrum of the current.

Figure 3. Waveform generated from simulation model and the spectrum of current. (a) current; (b) voltage; (c) non-stationary $i_s$; (d) the spectrum of the current.

The sampling frequency is 4000 Hz in this experiment, and five-level DWT decomposition with the mother wavelet ’db4’ is applied on current and voltage, making that the highest-level frequency range corresponds to that of the fundamental component (i.e., ca5). Figure 4 shows the decomposed results of the current. From the decomposed fundamental and non-stationary components, it is easy to discern that both of them are mixed with other frequency components. Meanwhile, the ACMD method is utilized for current and voltage, and the decomposed results for the current are shown in Figure 6. Since the ACMD method performs decomposition recursively, two non-stationary signals are extracted separately, and close to the raw waveform in the views of waveform trend and occurrence interval. For harmonics, parts of waveform within black windows are enlarged and displayed in small attached windows for comparison. Compared with the results from the DWT method in Figure 4, the decomposed components from the ACMD method in Figure 6 more agree with the real ones.

Figure 4. Decomposed results by using DWT method for current.

Figure 4. Decomposed results by using DWT method for current.

Figure 5. Simulation model in MATLAB/SIMULINK.

Figure 5. Simulation model in MATLAB/SIMULINK.

Figure 6. Decomposed results by using ACMD method for current.

Figure 6. Decomposed results by using ACMD method for current.

Using the decomposed components from DWT and ACMD methods, the RMS values, active power, and relative error (‰) for three components are calculated and shown in Table 2. Since the real value for each component can not be calculated directly as the case of synthetic signals done, the RMS values of the fundamental and harmonics are calculated from the corresponding information of spectral magnitude, while the quantities of the non-stationary component are calculated according to $u_s$ and $i_s\left(t\right)$ in (24) and (25). Similar to the comparison results for synthetic signals, the ACMD-based method achieves better estimation for each component and total active power for simulation data, compared with the DWT-based method.

Table 2. Comparisons of estimated quantities of ACMD-based and DWT-based methods for simulation signals in MATLAB/SIMULINK.

Table 2. Comparisons of estimated quantities of ACMD-based and DWT-based methods for simulation signals in MATLAB/SIMULINK.

Quantity		Reference values	ACMD		DWT
Symbols	Parameters	Analytical	Value	Error(‰)	Value	Error(‰)
U	Voltage RMS (V)	224.76	224.77	0.04	223.88	3.92
I	Current RMS (A)	49.54	49.62	1.61	49.49	1.01
P	Active power (W)	10756.46	10749.33	0.66	10659.48	9.02
$U_b$	Fundamental voltage RMS (V)	224.41	224.46	0.22	200.07	108.46
$I_b$	Fundamental current RMS (A)	47.8200	47.83	0.21	42.74	106.23
$P_{bb}$	Fundamental active power (W)	10686.11	10691.48	0.50	8512.89	203.36
$U_h$	Harmonic voltage RMS (V)	10.49	10.75	24.78	100.39	8570.06
$I_h$	Harmonic current RMS (A)	12.60	12.74	11.11	24.69	959.52
$P_{hh}$	Harmonic active power (W)	48.48	51.07	53.42	2139.69	43135.52
$U_s$	Non-stationary voltage RMS (V)	2.28	2.47	83.33	3.78	657.89
$I_s$	Non-stationary current RMS (A)	2.86	2.82	13.98	3.69	290.21
$P_{ss}$	Non-stationary active power (W)	6.54	5.79	114.48	6.89	53.45

4. Conclusions

Abbreviations

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