Abstract
Direct current (DC) bus voltage stability is essential for the stable and reliable operation of a DC system. If an oscillation source can be quickly and accurately localized, the oscillation can be adequately eliminated. We propose a method based on the power spectral density for identifying the voltage oscillation source. Specifically, a DC distribution network model combined with the component connection method is developed, and the network is separated into multiple power modules. Compared with a conventional method, the proposed method does not require determining the model parameters of the entire power grid, which is typically challenging. Furthermore, combined with a novel judgment index, the oscillation source can be identified more intuitively and clearly to enhance the applicability to real power grids. The performance of the proposed method has been evaluated using the MATLAB/Simulink software and PLECS RT Box experimental platform. The simulation and experimental results verify that the proposed method can accurately identify oscillation sources in a DC distribution network.
IN recent years, the access to distributed generation, energy storage, and diversified loads has transformed direct current (DC) distribution networks from static operation structures to flexible and active intelligent structures. In addition, given the absence of fluctuation in reactive power, the DC bus voltage is an essential index to measure secure and stable operation [
In a DC distribution network, the lack of rotating equipment (e.g., synchronous generators) and its interconnection with an alternating current (AC) power grid through power electronic devices relatively isolate the inertia, leading to the gradual appearance of voltage oscillation problems [
Continuous and unexpected power system oscillations may restrict the transmission capacity in a power grid, possibly leading to serious accidents and large economic losses. To prevent these consequences, an efficient measure is timely localizing the equipment causing the oscillation to take control countermeasures. Research on localization of oscillation source mainly focuses on AC systems. The related methods include the energy function [
Method | Advantage | Disadvantage |
---|---|---|
Energy function method | Clear in concept, convenient for practical application | Dissipated energy can be generated through non-oscillatory sources, which may lead to misjudgment |
Modal estimation method | Suitable for multi-mode oscillation | Identified modal information may be misjudged |
Frequency-domain relation method | Simple and easy to perform | Parameters of the whole network model are required |
Damping torque analysis method | Clear and appropriate physical interpretation | Detailed model and accurate parameters need to be obtained |
In the energy function method, a Lyapunov function is applied to an electric power system [
The main contributions of this paper are summarized as follows.
1) Based on the component connection method, the node impedance of the DC distribution network is separated from each power module (PM), and a two-port model is established. This model reduces the calculation burden. Furthermore, as each PM is a potential oscillation source, the oscillation source can be identified simply and intuitively based on the established model.
2) Based on the PSD, we propose a method for localizing the oscillation source. Compared with a conventional method, the proposed method does not require the parameters of the entire power grid model, and the PSD of each PM can be obtained by using its input signal and transfer function. Furthermore, when combined with a proposed judgment index, the oscillation source can be localized more intuitively and clearly.
3) Equivalent models of diverse DC distribution networks are constructed in the MATLAB/Simulink software using hardware-in-the-loop. Then, the performance of multivariate empirical mode decomposition (MEMD) [
The remainder of this paper is organized as follows. In Section II, a model of the DC distribution network is presented. In Section III, based on the PSD, the method for oscillation source localization is detailed. In Section IV, the theoretical analysis is validated through case studies including simulation and experimental analyses. Finally, conclusions are drawn in Section V.
The research object for this paper is a DC distribution network with multiple sources. The system is powered by an AC system through converters and comprises distributed generation, energy storage (ES), AC loads, electric vehicles (EVs), and other components. A schematic of a typical DC distribution network topology with two sources is shown in

Fig. 1 Schematic of a typical DC distribution network topology with two sources.
The converters that interact between the AC and DC systems adopt the droop control, which has a simple structure and can realize reasonable power allocation [
The schematic of a simplified model of the DC distribution network with multiple sources is shown in

Fig. 2 Schematic of simplified model of DC distribution network with multiple sources.
Because sources and CPLs are connected to a DC distribution network through converters, they can be represented by a PM. Taking the PM as the basic unit, we apply the component connection method to model the DC distribution network [

Fig. 3 Schematic of Norton equivalent model.
Suppose that AC power sources are connected to nodes numbered from 1 to k, and CPLs are connected to nodes numbered from to N. The connection line between the PMs adopts a -type centralized equivalent model. Taking the connection line between nodes 1 and 2 as an example, is the line resistance; is the line inductance; and is the line equivalent capacitance. In each PM, is the DC voltage; is the output impedance of the source; is the input admittance of a CPL; is the node admittance; and are the reference current and voltage, respectively; and and are the closed-loop transfer functions of source and CPL equivalent models, respectively.
The droop control structure adopted in this paper for () is shown in

Fig. 4 Droop control structure adopted in this paper.
For the droop control system illustrated in
(1) |
(2) |
(3) |
where and are the current closed-loop and power open-loop transfer functions of the system, respectively.
The output impedance in the equivalent circuit can be expressed as follows:
(4) |
(5) |
(6) |
where , rj, and Cj are the inductance, resistance, and capacitance of the filter circuit, respectively; and and are the current and voltage controllers, respectively.
For the PMs of the CPLs, the closed-loop transfer function and reactance value can be obtained, as detailed in Appendix A.
The parameters of the equivalent model are obtained using the inverse Laplace transform as follows:
(7) |
(8) |
(9) |
(10) |
The fundamental reason for DC bus oscillation is the imbalance of energy between the power source and loads [
The equivalent circuits of the power source and CPL are shown in

Fig. 5 Equivalent circuits of power source and CPL. (a) Power source. (b) CPL.
According to the equivalent circuit, the input signals of each PM can be expressed as:
(11) |
where is the input signal of .
In (11), the expressions of , , , and are given during the model setup. Therefore, the input signal of each PM is known. Considering as an example, the autocorrelation function of the input signal can be expressed as:
(12) |
where is the autocorrelation function of the input signal; is the independent variable of the autocorrelation function; and is the period of input signal .
During the period of DC system oscillation, the time mean of the input signal autocorrelation function and its PSD are mutual Fourier transforms. Hence, the PSD can be expressed as:
(13) |
where is the expectation; is the PSD of input signal ; and is the independent variable of the PSD.
The DC bus voltage oscillation is a generalized stationary random process that satisfies the Wiener-Khinchin theorem. For a stationary (or generalized stationary) random process, the autocorrelation function and PSD of the process are mutual Fourier transforms [31]. Therefore, the PSD of the input signal can be expressed as:
(14) |
From the analysis above, the PSD of each PM input signal can be expressed as:
(15) |
For a DC distribution network, the input signal of each PM can be decomposed into a superposition of impulse signals with different time delays. When the input signal is known, the output signal of each PM can be obtained using the continuous-domain convolution. Therefore, the output signal can be expressed as:
(16) |
where denotes the convolution operator; is the output signal of ; is the impulse response of when is the impulse signal; and is an integral variable.
Therefore, the autocorrelation function of the output signal can be expressed as:
(17) |
where is the autocorrelation function of the output signal of ; and and are integral variables.
According to (14), the PSD of the output signal of can be expressed as:
(18) |
(19) |
(20) |
Because the system function can be obtained by Laplace transformation of the impulse response, can be converted as follows:
(21) |
where is the transfer function corresponding to the output and input signals of ; and is the conjugate complex of .
Through the relationship between the PSD and autocorrelation function in (14), can be expressed as:
(22) |
Overall, the PSD of an output signal can be expressed as:
(23) |
where is the PSD of the output signal of ; and is the modulus of transfer function .
Therefore, the output signal PSD of each PM can be expressed as:
(24) |
According to (24), the PSD of each PM can be predicted. Ideally, when the DC bus voltage oscillates, if is an oscillation source, the measured PSD includes the PSD of the model response and that caused by the disturbance source. Therefore, the predicted PSD differs from the measured one at the oscillation frequency. This condition can be expressed as:
(25) |
where is the measured PSD of .
If is not an oscillation source, the predicted PSD is assumed to be strictly equal to the measured PSD at the oscillation frequency. This condition can be expressed as:
(26) |
However, because several electronic components are connected to the DC distribution network, there are many nonlinear factors, interference noise, and other factors. Therefore, a deviation remains between the predicted and measured PSD values.
Considering the aforementioned factors, we propose the following index to measure the deviation between the predicted and measured PSD values:
(27) |
The judgment index of each PM can be obtained according to (27). If is an oscillation source, a large deviation between and exists, and the proposed index satisfies . Otherwise, the deviation between and is small, and the proposed index satisfies .
We report simulation and experimental results that demonstrate the effectiveness of the proposed method for oscillation source localization using the MATLAB/Simulink software and PLECS RT Box experimental platform.
To verify the effectiveness of the method to localize an oscillation source, an equivalent model is constructed in MATLAB/Simulink according to the topology of the DC distribution network with two sources, as shown in

Fig. 6 Topology of DC distribution network for simulation analysis.
Parameter | Value | Parameter | Value |
---|---|---|---|
Udc,ref | 0.6 kV | Cdc1, Cdc2 | 4000 μF |
u1, u2 | 0.38 kV | k1,Ip/k1,Ii | 1/12 |
L1, L2 | 2 mH | k2,Ip/k2,Ii | 1/12 |
R1, R2 | 0.04 Ω | k1/k2 | 4.5 |
k1,Vp/k1,Vi | 4/50 | k1,d/k2,d | 5.5 |
k2,Vp/k2,Vi | 4/50 | k1,ga/k2,ga | 7.3 |
Keeping the power consumed by PM3-PM6 constant, a disturbance source is connected to PM1 at 3 s. Because voltage oscillation is induced by the disturbance source connected to PM1, it is the oscillation source. The voltage oscillation curve of the DC bus for this case is shown in

Fig. 7 Voltage oscillation curve of DC bus for simulation case 1.
The measured PSD values of PM1 and PM2 are shown in Figs.

Fig. 8 Measured PSD values of PM1 for simulation case 1.

Fig. 9 Measured PSD values of PM2 for simulation case 1.
The corresponding judgment indices of PMs are shown in

Fig. 10 Judgment indices of PMs for simulation case 1.
MEMD is another method to localize an oscillation source by calculating the variation in DEF, and it combines the energy function method with modal analysis [

Fig. 11 DEF curves of PMs for simulation case 1.
Control parameters of PM3-PM6 are set to be consistent. The power consumed by PM4-PM6 is set to be 30 kW. The initial power consumed by PM3 is set to be 10 kW, and then the power consumed suddenly changes to 30 kW at 0.6 s, leading to DC bus voltage oscillation.
The voltage oscillation curve of the DC bus is shown in

Fig. 12 Voltage oscillation curve of DC bus for simulation case 2.
The measured PSD values of PM3-PM6 for simulation case 2 are shown in Figs.

Fig. 13 Measured PSD values of PM3 for simulation case 2.

Fig. 14 Measured PSD values of PM4 for simulation case 2.

Fig. 15 Measured PSD values of PM5 for simulation case 2.

Fig. 16 Measured PSD values of PM6 for simulation case 2.
The judgment indices and DEF curves of PMs for simulation case 2 are shown in Figs.

Fig. 17 Judgment indices of PMs for simulation case 2.

Fig. 18 DEF curves of PMs for simulation case 2.
Therefore, when the parameters of each PM are consistent, MEMD could not identify the oscillation source. In contrast,
The power consumed by PM4-PM6 is set to be 30 kW, 40 kW, and 50 kW, respectively. The initial power consumed by PM3 is set to be 10 kW and suddenly changed to 60 kW at 0.6 s, leading to voltage oscillation.
The voltage oscillation curve of the DC bus is shown in

Fig. 19 Voltage oscillation curve of DC bus for simulation case 3.
The measured PSD values of PM3-PM6 for simulation case 3 are shown in Figs.

Fig. 20 Measured PSD values of PM3 for simulation case 3.

Fig. 21 Measured PSD values of PM4 for simulation case 3.

Fig. 22 Measured PSD values of PM5 for simulation case 3.

Fig. 23 Measured PSD values of PM6 for simulation case 3.
The judgment indices and DEF curves of PMs for simulation case 3 are shown in Figs.

Fig. 24 Judgment indices of PMs for simulation case 3.

Fig. 25 DEF curves of PMs for simulation case 3.
By comparing the performances of MEMD and the proposed method, the following results are obtained.
1) For the DC bus voltage oscillation induced by the connected disturbance source, both MEMD and the proposed method can identify the oscillation source. However, MEMD requires relevant parameters of the oscillation dissipation capacity, transient energy variation, and modal decomposition. The proposed method can calculate the PSD for each PM solely by its input signal and transfer function, being more practical for engineering applications.
2) For the DC bus voltage oscillation induced by a sudden change in the load consumption power, cases 1 and 2 show that it is difficult or even impossible to identify the oscillation source using MEMD. In contrast, the proposed method can accurately, clearly, and intuitively identify the oscillation source.
The PLECS RT Box experimental platform is equipped with rich digital and analog interfaces, and field-programmable gate array (FPGA) integrated operation modules. It can quickly process real-time models with a hardware-in-the-loop architecture and perform rapid control prototyping. Its results are close to those obtained from an actual power grid. The implementation of RT Box experimental platform used in this paper is shown in Appendix A Fig. A1.
The topology of the DC distribution network for experiment is shown in

Fig. 26 Topology of DC distribution network for experiment.
The power consumed by PM4-PM7 is constant, and the disturbance source is connected to PM1 at 1 s. Because voltage oscillation is induced by the disturbance source connected to PM1, it is the oscillation source. The voltage oscillation curve of the DC bus in this case is shown in

Fig. 27 Voltage oscillation curve of DC bus for experimental case 1.
The measured PSD values of PM1-PM3 are shown in Figs.

Fig. 28 Measured PSD values of PM1 for experimental case 1.

Fig. 29 Measured PSD values of PM2 for experimental case 1.

Fig. 30 Measured PSD values of PM3 for experimental case 1.
The judgment indices are shown in

Fig. 31 Judgment indices of PMs for experimental case 1.
The power consumed by PM5-PM7 is set to be 24 kW, 18 kW, and 24 kW, respectively. The initial power consumed by PM4 is set to be 18 kW and suddenly changed to 36 kW at 5 s, leading to voltage oscillation.
The voltage oscillation of the DC bus is shown in

Fig. 32 Voltage oscillation of DC bus for experimental case 2.
The measured PSD values of the PMs are shown in Fig.

Fig. 33 Measured PSD values of PM4 for experimental case 2.

Fig. 34 Measured PSD values of PM5 for experimental case 2.

Fig. 35 Measured PSD values of PM6 for experimental case 2.

Fig. 36 Measured PSD values of PM7 for experimental case 2.
The judgment indices of PMs are shown in

Fig. 37 Judgment indices of PMs for experimental case 2.
To address voltage oscillation in a DC distribution network, an immediate and efficient method is localizing the component causing the oscillation timely and accurately to take control countermeasures. Accordingly, we propose an oscillation source localization method for a DC distribution network with multiple sources. The following conclusions can be drawn from the simulation and experimental results.
1) The proposed method does not require a detailed model of the system and can estimate the PSD solely using the input signal of each PM to accurately localize the oscillation source.
2) Compared with a conventional method, the proposed method is suitable for generic DC distribution networks and can accurately identify different types of oscillation sources. Furthermore, when combined with the proposed judgment index, the oscillation source can be identified intuitively and clearly.
3) The proposed method considerably improves the accuracy and efficiency of oscillation source localization, likely providing a reference for power grid operators to formulate relevant control strategies with practical engineering applicability.
Appendix
The Norton equivalent admittance of a CPL is given by:
(A1) |
(A2) |
(A3) |
(A4) |
(A5) |
(A6) |
(A7) |
(A8) |
(A9) |
(A10) |
(A11) |
(A12) |
where is the input admittance of ; is the voltage controller; is the duty cycle of pulse-width modulation; is the equivalent gain of the converter; is the equivalent output resistance; , , and are the inductance, capacitance, and resistance of the filter circuit, respectively; and is the current of the filter inductance.
(A13) |
(A14) |
(A15) |
(A16) |
(A17) |
where is the closed-loop transfer function from the reference value in of the output variables.
The implementation of RT Box experimental platform used in this paper is shown in Fig. A1.

Fig. A1 Implementation of RT Box experimental platform.
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