A gray-box harmonic resonance frequency identification method of multiple-inverter-fed power system oriented to system designers

Abstract

This paper presents a gray-box harmonic resonance frequency identification method of multiple-inverter-fed power system, which enables modal analysis oriented to system designers based on only frequency response data provided by diverse vendors or measured by frequency scanning. First, admittance transfer functions of all grid-connected inverters (GCIs) are fitted using Matrix Pencil Method-Vector Fitting (MPM-VF) combined method. Then, node admittance matrix (NAM) is formed according to the topology of whole system. Finally, harmonic resonance frequency along with changes in number of GCIs are identified by NAM-based modal analysis (MA). The proposed gray-box identification method is implemented in a typical multiple-inverter-fed power system. The correctness of harmonic resonance frequency identification results and the effectiveness of the presented method are verified by simulation results obtained in Matlab/Simulink platform and OPAL-RT digital real-time simulation platform. Based on the identification results, a more stable and better power quality multiple-inverter-fed power system can be built by system designers though avoiding the appearance of harmonic sources with corresponding resonance frequency.

Keywords

Harmonic resonance multiple-inverter-fed power system matrix pencil method-vector fitting combined method node admittance matrix modal analysis

Introduction

In recent years, the penetration level and installed capacity of renewable energies (Singh and Sundaram, 2020, 2021), such as wind power (Yue et al., 2021; Zhang et al., 2011), photovoltaic power, and so on, are continuously increasing with the rapid and high quality development of distributed generation technologies (Singh and Sundaram, 2020; Zhou et al., 2018). As an efficient interface between the dispersed generation units and the weak grid, grid-connected inverters (GCIs) are widely adopted (Lei et al., 2014; Singh et al., 2019a) for their superior sustainability, efficiency, and control flexibility. However, complex interactions between GCIs and weak grid may produce harmonic instability in wide frequency ranges, which include low-frequency oscillation (Singh et al., 2019b; Wang and Blaabjerg, 2019) usually occurred in transmission network and high-frequency harmonic resonance (Yan et al., 2018) generally happened in distribution network. These interactions, especially the latter, may interrupt normal system operation and reduce power quality (Liu et al., 2016; Singh et al., 2019c). Therefore, it is important to identify harmonic resonance frequency at the system design stage in order to avoid the appearance of the corresponding harmonic resonance excitation during actual operation.

Two main approaches of identifying harmonic resonance frequency including state-space matrix (SSM)-based eigenvalues analysis (Liserre et al., 2006) and node admittance matrix (NAM)-based modal analysis (MA) have been developed (Wang et al., 2014; Yang et al., 2018). On the one hand, SSM is established in the light of state variables to be analysis, on which eigenvalues of the whole system are calculated based (Yang et al., 2013). On the other hand, NAM is developed according to the topology of the system to be designed, of which eigenvalues of the whole system can also be obtained in view (Xu et al., 2013). Nevertheless, NAM is quiet distinct from and more popular than the other one because SSM with more state variables is often more complex than NAM, which lead to the order of SSM will be higher bringing heavy computational burdens when a multiple-inverter-fed power system include a huge number of GCIs (He et al., 2013; Kuang et al., 2016).

Although the principle of NAM-based MA have been well understood and applied in many researches, a kind of practical engineering scene that system designers identify potential harmonic resonance through this approach is often overlooked. Due to intellectual property rights and the need for conduct confidentiality, internal detailed information is often unknown for a system designer, which indeed obstacle the formation of NAM (Hu et al., 2014; Xu et al., 2005) and the application of MA (Huang et al., 2007; Tang and Yang, 2017) since the admittance of GCIs cannot be modeled by common methods in favor of a set of discrete frequency response data provided by those vendors or measured by frequency scanning.

To fill in this gap, this paper presents a gray-box harmonic resonance frequency identification method for system designers. At first, the problem that this paper would like to investigate is elaborated. Then, the principle of gray-box method presented in this paper is explained. What’s more, harmonic resonance frequency are explicitly identified as the number of GCIs vary from 1 to 4. Simulation results obtained in Matlab/Simulink platform and OPAL-RT digital real-time simulation platform are provided in order to verify the correctness of identification results. Conclusions are drown at last.

Problem statement

In this section, the problem that this paper would like to investigate is elaborated. Figure 1 shows the schematic diagram of a multiple-inverter-fed power system (), which is a simple but typical topology representing photovoltaic power station, wind farm, or other types of distributed generation systems. n GCIs exists in this system, which are labels as #1, #2, …, #n, respectively. Point of Coupled Connected (PCC) is labeled as node # (n + 1).

Where LCL filter is applied to attenuate the high-frequency switching harmonics. L_I, L_G, and C represent the inverter-side inductance, Grid-side inductance and capacitance in LCL filter, respectively. In addition, Z_Line = s × L_line + R_line is line impedance in the s-domain. Z_g = s × L_g + R_g represents grid impedance in the s-domain.

In practice, the multiple-inverter-fed power system similar to Figure 1 built by system designers such as the STATE GRID in China commonly consists of various GCIs provided by diverse vendors. Although it is clear for the harmonic resonance frequency in every individual GCI, the interconnected system may emerge so many undesired interactions among these components. Hence, the designers hope to identify the potential harmonic resonance frequency at the initial system design stage in order to prevent harmonic resonance happening.

However, due to the intellectual property right and the need for conduct confidentiality, detailed controller parameters, types of controller and filter, and so on of these GCIs are unknown. The model of the output admittance cannot normally be obtained via common methods by system designers in the real engineering scenes. Instead, terminal admittance frequency response data can be provided by these vendors or measured by frequency scanning, that is, only gray-box models of GCIs can be used. Therefore, a practical issue arises in real engineering scenes, that is, how to perform NAM-based MA and identify the harmonic resonance frequency of multiple-inverter-fed power systems when it is oriented to system designers?

To extract the harmonic resonance frequency in gray-box approach, and better improve power quality of designed system, this paper presents a gray-box harmonic resonance frequency identification method for system designers, as shown in next section. One possible engineering scene where the method proposed in this paper can be applied is replicated here. Frequency response data of GCIs are delivered to designers by multiple vendors or measured by frequency scanning at the initial design stage. System designers can identify harmonic resonance frequency with the help of the method proposed in this paper. Therefore, the potential harmonic resonance frequency in different number of GCIs are identified. With the identification results, the system designer can add targeted filtering devices, adjust the framework of the system, and even feedback the results to the manufacturers so as to build a more stable and better quality multiple-inverter-fed power system.

Proposed gray-box harmonic resonance investigation method

In this section, equivalent circuit of the multiple-inverter-fed power systems are first introduced. Then, the principle of the MPM-VF combined method is explained. Next, the details of MA are also elaborated. Finally, flowchart of the proposed gray-box harmonic resonance identification method is given, shown as Figure 3, followed by statement of each step in detail.

Equivalent circuit of the multiple-inverter-fed power system

The whole system in Figure 1 can be transformed into an equivalent circuit as shown in Figure 2 which includes n parallel Norton equivalent circuits standing for n GCIs with independently controlled current source I_s and output admittance Y_s and one series Thevenin equivalent circuit representing grid with independently voltage source and impedance. The ith i $\in (1, n)$ line impedance Z_Linei is connected between terminal node of ith GCI and PCC as individual series component. The admittance of each GCI can be fitted with frequency response data by MPM-VF combined method, which will be explained next.

Where I_gi represent current from ith GCI output port.

According to the equivalent circuit above, NAM (Wu et al., 2018) of multiple-inverter-fed power system can be formulated as Y in (1).

Y = [\begin{matrix} Y_{s 1} + Y_{line 1} & 0 & 0 & 0 & - Y_{line 1} \\ 0 & Y_{s 2} + Y_{line 1} & 0 & 0 & - Y_{f 2} \\ 0 & 0 & ⋱ & 0 & ⋮ \\ 0 & 0 & 0 & Y_{s n} + Y_{line n} & - Y_{line n} \\ - Y_{line 1} & - Y_{line 2} & \dots & - Y_{line n} & Y_{g} + \sum_{k = 1}^{n} Y_{line k} \end{matrix}]

(1)

Where Y_line = 1/Z_line and Y_g = 1/Z_g.

MPM-VF combined method

A set of discrete admittance frequency response data can be fitted as a continuous transfer function in form of Poles-Residue representation using MPM-VF combined method, shown as (2).

Y_{s i} = \sum_{k = 1}^{N} \frac{R_{k}}{s - λ_{k}} + D + sE .

(2)

Where λ_k and R_k are the kth pole and residue, respectively. N is the order of the fitted transfer function. D and E are real constant.

Simplified calculation process of MPM-VF combined is explained as follow. First, frequency responses data are converted to time-domain by inverse Fourier transform and applied by MPM in order to obtain the initial poles. These poles are then refined by applying the pole relocation process in the VF algorithm. Then residues and constant terms are computed by solving some least-squares functions which is same as the traditional VF algorithm. Finally, the root mean square error is compared with a given threshold. The fitting will be repeated until the error is smaller than the threshold.

The following issues should be addressed in the fitting process.

(1) Up to which frequency should the fitting process be done?

Due to digital nature of the controller in GCI, it is suggested to approximate the transfer function up to the Nyquist frequency, which is half of the sampling frequency.

(2) How to deal with the noise pollution in the frequency response data?

Consider that there are different statistical algorithms to minimize the noise effect in the fitting. And it is the responsibility for vendors to provide pure and correct data. As a result, in this paper, the noise is not considered for the sake of simplicity and clearer presentation.

(3) How the passivity and stability enforcement are set?

Passivity enforcement meaning that results of fitting hold positive real part at all frequency and stability enforcement representing that all poles are supposed to be refined in the Left Half Plane are ensured in normal use. Howbeit, these enforcements are canceled because the real part of the admittance of GCI can be negative and the system might be unstable in some cases.

Modal analysis method

The node voltage function as shown in (3) will be satisfied if NAM at frequency f formulated as Y_f can be obtained.

U_{f} = Y_{f}^{- 1} I_{f} .

(3)

Where superscript −1 represents matrix inversion calculation. U_f and I_f are the node voltage vector and node current vector, respectively.

Y _f is supposed to be decomposed based on (4) for further harmonic resonance analysis (Liu et al., 2017).

Y_{f} = L Λ T .

(4)

Where $Λ$ = diag ( $Λ$ ₁… $Λ m$ …) represent the diagonal eigenvalue matrix, among which $Λ m$ is mth eigenvalue. L = [L1…Lm…] and T = [T1…Tm…] are the left and right eigenvector matrix that satisfy L = T − 1. Similarly, Lm and Tm are mth left and right eigenvector corresponding to the mth eigenvalue, respectively.

Equation (3) can be reformulated in form of (5) by substituting (4), shown as

T U_{f} = Λ^{- 1} T I_{f} .

(5)

Thus, the modal voltage vector V_f = TU_f and modal current vector J_f = TI_f are defined (Shu et al., 2018), respectively. Furthermore, the node voltage function in (3) can be described as (6).

\begin{matrix} [\begin{matrix} V_{f 1} \\ V_{f 2} \\ ⋮ \\ V_{fn} \end{matrix}] = [\begin{matrix} Λ_{1}^{- 1} & 0 & 0 & 0 \\ 0 & Λ_{2}^{- 1} & 0 & 0 \\ 0 & 0 & ⋱ & 0 \\ 0 & 0 & 0 & Λ_{n}^{- 1} \end{matrix}] [\begin{matrix} J_{f 1} \\ J_{f 2} \\ ⋮ \\ J_{fn} \end{matrix}] \\ ⇓ \\ V_{f} = Λ^{- 1} J_{f} \end{matrix} .

(6)

Where $Λ_{i}^{- 1}$ is defined as modal impedance Z_i i $\in (1, n)$ .

It can be seen that a multiple-inverter-fed power system may exhibit n modes at frequency f from (5) and they are independent from each other. As a result, the modal voltage V_fi related only to the corresponding modal impedance Z_i and is free from other modes, so is the modal current I_fi. Moreover, a normal I_fi will lead to an amplifying V_fi led by a high Z_i. Therefore, the corresponding frequency of the peak of the modal impedance curve is the harmonic resonance frequency.

Proposed gray-box method

Based on the above principle, the flow chart of gray-box method proposed in this paper is given in Figure 3, which consists two main steps, that is, formation of NAM (Step 1), and identification of harmonic resonance frequency (Step 2).

Step 1: Formation of NAM

Terminal admittance frequency response data of all GCIs are first obtained from vendors or measured using the frequency scanning. GCIs admittance transfer function in form of (2) is then generated from the frequency response data using MPM-VF combined method, which includes three steps, that is, compute initial order and poles using MPM, apply VF pole relocation and compute residues and constant terms. Finally, NAM is formed according to the structure of whole system on which steps 2 will be performed based.

Step 2: Identification of harmonic resonance frequency

Harmonic resonance frequency related to inverter number is analyzed. The harmonic resonance modal impedance curves is first plotted. Then, the corresponding frequency of the peak of the modal impedance curves are identified. Finally, all picked harmonic resonance frequency are simulated and verified by Matlab/Simulink platform and OPAL-RT digital real-time simulation platform.

Implementation of the proposed method

In this section, the proposed gray-box harmonic resonance frequency identification method is implemented in a simplified but representative multiple-parallel GCI system referring to Figure 1, where GCIs are regarded as gray-boxes. The admittance transfer function is fitted first by MPM-VF combined method based on frequency response data in order to form NAM. Then, harmonic resonance frequency related to variations of inverter number is identified.

Formation of NAM

One type of GCI with capacitor-current-feedback active damping is taken as example to verify the correctness and effectiveness of the proposed method in this paper. The critical parameters are shown in Table 1. Bode diagrams of the output admittance frequency response data of GCI obtained by frequency scanning method are shown as the pink dashed line labeled as “Original Data” in Figure 4. The threshold of fitting is set as 1 × 10⁻⁶%. What’s more, the Bode diagrams of the fitted output admittance transfer function using MPM-VF combined method with the order chosen as 3, 4, 5, 6 are also plotted which are shown as the yellow, green, blue, purple solid line, respectively. Among them, the order of 3 is chosen as initial order in VF pole relocation identified by MPM.

Table 1.

Parameters of GCI.

Parameters	A
Grid voltage U_g/V	220
DC input voltage U_dc/V	720
Grid fundamental angular frequency ω₁	50 Hz
Inverter-side filter inductance L_I/mH	3
Grid-side filter inductance L_g/mH	0.2
Filter capacitance C/µF	30
Proportional gain of QPR controller K_p	5
Resonance gain of QPR controller K_r	250
Cut-off frequency of QPR controller ω_i/(rad s⁻¹)	5
Capacitor-current proportional feedback active damping H_iC	5

It can be seen that the fitting accuracy increases as the fitting order increases before the best order occur. The fitting accuracy decreases when the fitting order is over the best for over fitting. The Bode diagrams near the peak frequency (1900–2200 Hz in Figure 4) are further plotted in the zoomed view. It can be seen that the Bode diagrams of the original data (pink dashed line) and the fitted 5-order transfer function (blue solid line) are highly overlapped, which indicates that the best order is 5. Actually, the fitting RMS of GCI is 2.374 × 10⁻¹⁴%, which are minimal and indicate that the MPM-VF combined method is suitable for continues transfer function of GCI by frequency response data. Based on the result of fitting, the NAM of the whole system can be established as defined in (1).

Variations of GCI number

One case is discussed based on variations of GCI number by selecting GCI in Table 2 as example. It is supposed to illustrate that the grid equivalent impedance is set as Z_g = 0.5 + 1 × 10⁻³ × s and the line equivalent impedance is set as Z_line = 0.5 + 0.5 × 10⁻³ × s. They are assumed to remain invariant even if the GCI number is increasing. The result of the MA are depicted in Figure 5 as the number of inverters parallel to the PCC increases from n = 1 to n = 4.

Table 2.

Harmonic resonance frequency variations as the GCI number vary from 1 to 4.

GCI number	Low frequency (Hz)	High frequency (Hz)
1	703	—
2	558	1095
3	477	1095
4	423	1095

It can be seen from Figure 5 that variations of GCI number have distinct influences on harmonic resonance frequency, which can be analyzed and summarized as follows.

Only a relatively low frequency harmonic resonance (LFHR) is observed when single GCI connect to the grid.

Two typical harmonic resonance, that is, LFHR and a relatively high frequency harmonic resonance (HFHR) can be observed when n ≥ 2.

As the number of GCIs increases, the central frequency of LFHR decreases and the modal impedance increases.

The central frequency of HFHR remains unchanged and the modal impedance also increases as the number of GCIs increases.

It can be inferred from the above analysis that there may exist two kinds of harmonic resonance interactions among GCIs and the grid, that is, HFHR existing between diverse GCIs and LFHRs only existing between GCIs and grid. What’s more, the system is tougher to be excited by harmonic sources as the number of GCIs increases. Assume the amplitude of the harmonic source is the same, as a result, the more the number of parallel connections, the lighter the degree of harmonic resonance.

Real-time simulation verification

In this section, the correctness of harmonic resonance frequency identified by proposed gray-box method is validated by the time-domain simulation results obtained on an OPAL-RT digital real-time simulator platform and results of Fast Fourier Transform (FFT) in Matlab/Simulink platform.

Introduction of experimental platform

Figure 6 shows configuration of the OPAL-RT digital real-time simulator platform. The OP5600 researched by OPAL-RT Technologies in Canada realize a wide range of rapid control prototyping applications with its RT-LAB software platforms. Code is generated and downloaded into OP5600 based on the Matlab/Simulink-based model established in the RT-LAB software. High-speed and real-time simulation are allowed in both software and hardware platforms which satisfy the requirements in the whole experiment. By the way, the real-time simulation results obtained by OPAL-RT platform are viewed in scope provided by Tektronix.

It is necessary to explain that the main aim of the time domain real-time simulation in this paper is examining the correctness of harmonic resonance frequency identification. As a result, the harmonic currents, which coincides with the frequency identified, with the same magnitude are injected into the PCC and the voltages are measured in PCC or terminal of GCIs. The same magnitude of each time injected harmonic current is 1.5 A (Peak amplitude), which satisfy IEEE Standard 1547. And spectrums are plotted by FFT analysis based on the sampled voltage signals.

Variations of GCI number

First, the harmonic resonance frequency when the number of GCIs is increased from 1 to 4 is verified. In the whole experimental, the combination of harmonic currents in 8th, 10th, 11th, 14th (LFHR), and 22th (HFHR) are injected at the PCC. The time-domain waveform of U_GCI1 is plotted in Figure 7(a) and spectrums of U_GCI1 and U_PCC are shown in Figure 7(b) and (c), respectively.

It can be seen from Figure 7 that variations of GCI number indeed have different influences on harmonic resonance frequency, which can be analyzed and summarized as follows.

Only 14th harmonic are excited, which can be observed whether in UGCI1 or UPCC. So, when single GCI connect to the grid, the whole system will occur harmonic resonance at a specific and relatively low frequency.

When n ≥ 2, two typical harmonics, that is, 8th, 10th, 11th and 14th (LFHR), respectively and 22th (HFHR) are excited. However, they can be viewed in UGCI1 instead of UPCC. Though, it is confirmed that both LFHR and HFHR exist among GCIs and LFHRs only occur between GCIs and grid.

As the number of GCIs increases, the central frequency of LFHRs and peak value decreases, however, the central frequency of HFHR remains unchanged and the peak value also decreases. As a result, the more the number of parallel connections, the reliably lighter the degree of harmonic resonance.

The phenomenon observed in Figure 7 agree well with the harmonic resonance frequency analyzed and summarized from Figure 5.

Conclusion

This paper presents a gray-box harmonic resonance frequency identification method for multi-inverter-fed power system, which is able to perform NAM-based MA based on only frequency response data when it is oriented to system designers. The main findings of this work are as follows.

When only frequency response data can be provided or measured, NAM of the whole system to be designed can be built by gray-box models. As a result, harmonic resonance frequency can be identified for system designers by NAM-based MA.

NAM-based MA approach can be applied to explicitly identify harmonic resonance frequency and explore variation characteristics of harmonic resonance modes related to number of GCIs.

MPM-VF combined method can be used in the form of admittance transfer function of GCIs in order to construct NAM. The model order can be identified and the initial poles can be selected automatically. The capability of the MPM-VF combined method is further explored.

Damping representing the transient process and participant factors (PFs) dividing responsibilities of GCIs and grid can also be gotten and analyzed by method presented in this paper, which will be investigated in future works.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Yue Wang

Binyuan Wu

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