Modeling method of sequence admittance for three-phase voltage source converter under unbalanced grid condition
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Abstract
The admittance is a strong tool for stability analysis and assessment of the three-phase voltage source converters (VSCs) especially in grid-connected mode. However, the sequence admittance is hard to calculate when the VSC is operating under unbalanced grid voltage conditions. In this paper, a simple and direct modeling method is proposed for a three-phase VSC taking the unbalanced grid voltage as a new variable for the system. Then coupling in the three-phase system can be calculated by applying the harmonic linearization method. The calculated admittance of three-phase VSCs is verified by detailed circuit simulations.
Keywords
Unbalanced grid voltage Three-phase voltage source converter Harmonic linearization Positive and negative sequence admittance Stability1 Introduction
In recent years, renewable energy is widely used in the power system to solve the increasing energy crisis and environmental problems. In a modern power system, the penetration of renewable energy is much higher than ever before. The three-phase voltage source converter (VSC) is one of the key pieces of equipment to make full use of renewable energy because it can provide a sinusoidal current to the grid and flexible power control [1, 2]. However, the stability of the grid-connected VSC will affect the safety and stability of the power system in such a situation [3].
There are two main ways to perform the stability analysis of grid-connected converter systems [4]. One is the time domain method based on the state space model [5] and the other one is the frequency domain method which is based on the impedance model [6]. The time domain method is mainly suitable when the root locus analysis determines that the frequency is not changing, given a unified model of the grid-connected system [7, 8]. Thus, the stability analysis will be very complex when considering the influence of the phase-locked loop (PLL) and the grid impedance [9].
The frequency domain method is based on the impedance model which includes the impedance of the grid and the power converter and uses them to represent their external characteristics. Then, the stability of the system can be judged by stability criteria like the Nyquist criterion [10]. The impedance of the grid can be easily obtained by measuring through experiments or from a given X/R ratio [11]. However, the impedance of the converter is hard to obtain because of the various and complex control strategies and circuit parameters.
Much research effort has been devoted to impedance modeling of three-phase voltage source converters. In [12], several small-signal analysis methods used for modeling electric ship power systems are discussed. The conventional small-signal linearization techniques cannot be directly used because of the lack of a constant operating point during the periodic steady-state operation trajectory. Belkhayat linearized the system in the dq-coordinate reference frame by transforming the fundamental component into DC quantities in [13]. Then the nonlinearity can be eliminated to develop a small-signal model which can be used for control design and system stability analysis at a given operation point. But it is not suited for unbalanced conditions and provides no clear physical meaning for the d-axis and q-axis impedances. In [14], an impedance model of a three-phase VSC is proposed in the dq-frame considering the effect of PLL’s and the control strategy’s parameters. But the impedance in the dq-frame is hard to measure directly, and the stability criterion in the dq-frame is too complicated to be used in practice [4]. In [15], a small-signal model of a PV inverter containing the PLL and DC-side voltage and inner loops was proposed. It illustrates the PLL impacts on the output q-channel impedance of the inverter which can lead to instability. However, the coupling between the d-q channels has not been considered. In [16], a d-q impedance matrix of the inverter was proposed to analyze the stability of the system. In [17], the same model was used for analyzing stability of three-phase paralleled converters. The q-q channel impedance interaction leads to instability of the system. However, it is hard to obtain such impedances without some special equipment and instruments. The research above mainly focusses on situations of balanced grid voltage. However, there is still a gap because there is no mature method for impedance modeling of the VSC under practical three-phase unbalanced condition.
Harmonic linearization [18] is a method to transform a nonlinear periodically time varying system into a small-signal linear model, and has been successfully realized to get the impedance of uncontrolled diode rectifiers [6, 19] and single-phase PFC converters [20] as well as grid-connected inverters [21, 22]. Particularly, in [21, 22], the positive and negative sequence impedance model is proposed in a static frame and then can be applied in stability analysis. In [22], an impedance-based stability analysis method is studied initially for a grid-connected inverter working under unbalanced grid voltage. However, the method in this paper neglects the coupling effects between the positive and negative sequence impedances. Knowing the sequence admittance of a VSC under sustained grid faults is helpful for judging the stability of the grid-connected converter system [23, 24]. It is also helpful for the design of the control system and for admittance shaping to manage power quality of the converter [25, 26, 27]. Therefore, the harmonic linearization method should be improved for unbalanced situations.
In this paper, a simple and direct modeling method of a three-phase VSC under unbalanced grid voltage is proposed. Based on harmonic linearization, the coupling of the positive and negative sequence admittances is considered and the admittance matrix is given. This method can be used with some kinds of sustained grid faults such as unbalanced grid voltage. This would be caused by unbalanced sources or loads in the grid. These faults can be sustained, so the operation of the VSC will be highly affected by them. In our opinion, the impedance of the VSC would be changed by an unbalanced grid fault. However, some fault conditions may occur very quickly, in milliseconds, and the fault can be eliminated. These are transient faults. Actually, in responding to these transient faults, the VSC would be quite slow. For instance, the PLL is usually used with a bandwidth of 10–100 Hz, like a general used synchronous reference frame PLL (SRF-PLL). Therefore, during these kinds of transient fault, the output of PLL would be only slightly affected and in many cases the converter will keep operating. The rest of this paper is organized as follows: Section 2 proposes the improved sequence admittance of the three-phase VSC under unbalanced grid conditions by harmonic linearization. The unbalanced grid voltage can be introduced as a new variable of the original model. Section 3 shows the calculation method of the sequence admittance of the VSC based on this model. Section 4 verifies the admittance of the VSC by detailed circuit simulations and analyzes the influence of the admittance on the system stability. Section 5 concludes this paper.
2 Sequence admittance modeling
2.1 Topology and control strategy of VSC
The inner current controller is shown as Fig. 2b. The I_{ d } and I_{ q } are the current component on the d-axis and the q-axis obtained by transforming the grid side current using the Park Transformation. H_{ i } is the compensation factor for the inner current controller, which is usually a PI or a PR controller, and K_{ d } is its decoupling factor. c_{ d } and c_{ q } are the d-axis and q-axis current control signals after decoupling. The current control signals are obtained by the inverse Park Transformation.
2.2 Sequence admittance modeling for balanced grid
However, a disturbance will be introduced at the output of the PLL due to the grid voltage, and this leads to nonlinearity in the Park Transform. So, the effect of PLL should be taken into consideration when modeling the admittance of the VSCs.
Thus, the equation of Δθ in the frequency domain is
Thus, the realistic d-axis and q-axis current components I_{ d } and I_{ q } at the frequency ±(f_{ p }−f_{1}) and ±(f_{ n }+f_{1}) may be obtained, and are given by (A1) in Appendix A.
Relationship of frequency components
C_{ d }[f], C_{ q }[f] | T^{−1}(θ_{PLL}) | C_{a}[f], C_{b}[f], C_{c}[f] |
---|---|---|
dc | ± f_{ p } | ± f_{ p } |
± (f_{ p }− f_{1}) | ± f_{1} | ± f_{ p } |
2.3 Proposed sequence admittance modeling for unbalanced grid conditions
Reference [21] proposes a modeling method for the sequence impedance of VSCs under balanced grid conditions using harmonic linearization. All variables of the system are transformed into the frequency domain firstly, such that V_{1} represents V_{PCC,a} at ± f_{1} and V_{ p } represents V_{PCC,a} at ± f_{ p }, and so on. Then, the output disturbance of the PLL due to the harmonic voltage can be modeled as shown in (27–35). This establishes the use of the Park Transformation considering the effect of Δθ, which is applied below under unbalanced grid conditions. Then, the realistic d-axis and q-axis current I_{ d } and I_{ q } are obtained in (A1). These allow the output signal of the inner current controller in dq-frame to be easily calculated using (3). Finally, the output signal of the inner current controller c_{a} at the relevant harmonic frequency (± f_{ p } or ± f_{ n }) can be calculated as shown in (37), which is derived using the convolution relationships in Table 1 and the inverse Park Transform. The sequence admittance can be obtained by putting the calculated C_{a} into the circuit equation of the system as shown in (2).
Then, the output signals of the inner current controller in the dq-frame can be easily obtained. In order to compare with the balanced condition, some new frequency components such as C_{ dq }[±(f_{ p }+f_{1})], C_{ dq }[±(f_{ n }−f_{1})] and C_{ dq }[±2f_{1}] will be introduced. C_{a} can be calculated at the relevant frequencies such as ± f_{ p } and ± f_{ n } by considering all the possible convolution relationships of the frequency components.
Therefore, there are some new elements in C_{a} due to the unbalanced grid voltage V_{2}. By putting the positive and negative components of C_{a} into each sequence circuit equations of the system, and solving them, the sequence admittance of VSCs can be obtained. The detailed step-by-step calculation is shown in Section 3.
3 Step-by-step admittance calculation of VSCs under unbalanced grid conditions
Thus, we can find C_{a}[f] which is the phase A output control signal of the inner current controller in the frequency domain.
Convolution relationship of frequency components
C_{ d }[f], C_{ q }[f] | T^{−1}(θ_{PLL}) | C_{a}[f], C_{b}[f], C_{c}[f] | Sequence |
---|---|---|---|
dc | ± f_{ p } | ± f_{ p } | Positive |
dc | ± f_{ n } | ± f_{ n } | Negative |
±(f_{ p }− f_{1}) | ± f_{1} | ± f_{ p } | Positive |
±(f_{ p }− f_{1}) | ∓ f_{1} | ± f_{ p } | Negative |
±(f_{ n }+ f_{1}) | ± f_{1} | ± f_{ n } | Negative |
±(f_{ n }+ f_{1}) | ∓ f_{1} | ± f_{ n } | Positive |
±(f_{ p }+ f_{1}) | ± f_{1} | ± f_{ p } | Negative |
±(f_{ p }+ f_{1}) | ∓ f_{1} | ± f_{ p } | Positive |
±(f_{ n }− f_{1}) | ± f_{1} | ± f_{ n } | Positive |
±(f_{ n }− f_{1}) | ∓ f_{1} | ± f_{ n } | Negative |
±2f_{1} | ±(f_{ p }−2f_{1}) | ± f_{ p } | Negative |
±2f_{1} | ±(f_{ n }+2f_{1}) | ± f_{ n } | Positive |
4 Simulations and verifications
4.1 Detailed circuit simulations
Parameters of system
Symbol | Description | Value | |
---|---|---|---|
V _{dc} | Magnitude of DC side voltage | 800 V | |
V _{1} | Magnitude of fundamental positive sequence voltage at frequency f_{1} | 311 V | |
V _{2} | Magnitude of fundamental negative sequence voltage at frequency f_{1} | 31.1 V | |
I _{1} | Magnitude of fundamental positive sequence current at frequency f_{1} | 20 A | |
φ _{i1} | Initial phase of I_{1} | 0° | |
L | Inductance of output filter | 3 mH | |
T _{ m } | Period of voltage and current sampling | 50 μs | |
ω _{ m } | Cut-off frequency of analog-digital converter | 10 kHz | |
K _{ m } | Modulator gain | 0.5 | |
H _{ i } | Proportionality coefficient of compensation factor of inner current controller | 5 | |
Integral coefficient of compensation factor of inner current controller | 15 | ||
H _{PLL} | Proportionality coefficient of forward loop gain of PLL | 20 | |
Integral coefficient of forward loop gain of PLL | 5 | ||
K _{ d } | Decoupling factor of inner current controller | 0.9425 |
4.2 Sensitivity analysis
This indicates that the magnitude of V_{2} and the transfer function of the PLL will have big impacts on the coupled admittances. The transfer function of the PLL determines its bandwidth.
5 Conclusion
The sequence admittances of a three-phase VSC are independent of each other when it is operating under normal conditions. However, the positive and negative sequence harmonic components will be coupled with each other if three-phase grid-connected converter is running with unbalanced grid voltages. This is mainly caused by the negative sequence voltage at the fundamental frequency. This paper proposes a step-by-step modeling method to calculate the matrix of positive and negative sequence admittances of three-phase grid-connected converters under unbalanced grid voltage conditions, based on the method of harmonic linearization. The calculated admittances agree with simulated results within 1.07 dB of magnitude and 2.56° of phase, and can be easily applied in stability analysis.
Notes
Acknowledgement
This work was supported by National Natural Science Foundation of China (Nos. 51637007, 51507118).
Supplementary material
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