Unbalanced voltage control of virtual synchronous generator in isolated micro-grid

Virtual synchronous generator (VSG) control is recommended to stabilize the voltage and frequency in isolated micro-grid. However, common VSG control is challenged by widely used unbalance loads, and the linked unbalance voltage problem worsens the power quality of the micro-grid. In this paper, the mathematical model of VSG was presented. Based on the analysis of positive- and negative-sequence equivalent circuit of VSG, an approach was proposed to eliminate the negative-sequence voltage of VSG with unbalance loads. Delay cancellation method and PI controller were utilized to identify and suppress the negative-sequence voltages. Simulation results verify the feasibility of proposed control strategy.


Introduction
Micro-grid that using wind, solar and other renewable energy as main source is an effective local power supply, which has broad application prospects in the mountain area and island [1]. However, isolated micro-grid often has no synchronous power supply to support its voltage and frequency. Therefore, the virtual synchronous generator(VSG) control is usually adopted in the isolated microgrid inverter [2][3]. The existing VSG works as voltage source, and the output voltage is usually openloop controlled. When unbalanced loads connect to the isolated micro-grid, the unbalance degree of output voltage is usually higher than national standard-2%.
The circuit topology of the neutral-splitting circuit or the three-phase four-leg circuit has solved the problem brought by unbalance loads. But the applications are not common in existing micro-grid inverter products, as this type of topology increases the cost of the system and introduces the problem of neutral point voltage balance control. For the control of unbalance voltage in isolated micro-grid, the existing research mainly focused on the compensation of negative-sequence voltage [4][5]. Reference [6] derived universal transformation from synchronous to stationary reference frame, then proposed the concept of virtual composite impedance to control the positive-and negative-sequence voltage respectively. References [7][8] pointed that the negative-sequence voltage caused by unbalance loads can be controlled indirectly by controlling negative-sequence admittance. Reference [9] proposed a virtual negative impedance control method to compensate the voltage drop on the inverter output impedance, which effectively reduced the voltage unbalance degree. However, the existing relevant papers had no analysis of the generation mechanism of VSG output voltage imbalance, and the unbalanced voltage control method remains to be further studied.
This paper aims at solving the electrical power quality problem of isolated micro-grid inverter operating with VSG control strategy under unbalanced loads situation. On the basis of the positive- and negative-sequence equivalent circuit of VSG and negative-sequence voltage depressing methods, an improved VSG voltage control algorithm has been proposed. Simulation results have verified that using this control strategy can promote the ability of carrying unbalanced load and improve the power supply quality.

Basic principle of VSG
The typical main circuit topology of three-phase three-wire inverter is shown in Figure 1.

VSG ontology model
The motion equation of the rotor and the electrical equation of the stator are respectively shown in equation (1), which using the mathematical language to describe the rotary synchronous generator electromagnetic changes and mechanical motion.
In equation (1), ωg is the synchronous angular velocity of the AC bus; E0 is the open-circuit voltage amplitude of the synchronous generator; LSG is the synchronous reactance, and rSG is the stator resistance.
The excitation equation of the synchronous generator is shown in equation (2), where if0 is the excitation current required to synchronize the generator to establish the rated no-load electromotive force when the rotor is running at rated speed.
But in VSG, the excitation current is not an intuitive controllable variable, which is not easy to set the parameters. In the case of neglecting the saturation of the magnetic circuit, the open-circuit voltage E0 and the excitation current if is approximately proportional relationship [4]. Therefore, we can see that the no-load EMF Es0 is: Es0=ωMf if0, Mf is the inter-rotor mutual inductance. By multiplying both sides of equation (2) by the constant term ω0Mf , we can obtain equation (3): That is the same expression as equation (4): Thus, the adjustment of the excitation current if can be converted into a direct adjustment to the open-circuit voltage E0. The excitation process of real synchronous generator has a certain hysteresis, so join the first-order inertia link to make sure that the excitation system of VSG is closer to the excitation system of real synchronous generator. Since the realization of the synchronous reactance is behind the combined of EMF, it is placed outside the VSG ontology model. Together with virtual speed regulator and virtual excitation regulator, the VSG ontology model can be shown in Figure 2. Figure 2. Structure diagram of VSG ontology model.

Realization of synchronous reactance
The synchronous reactance is an important parameter in synchronous generator. In order to simulate the synchronous generator accurately, the synchronous reactance must be introduced as a control parameters of VSG. If the equation in (1) is rewritten as a current integral form, as shown in equation (5), the output can the stator current of synchronous generator.
By using the stator current as the reference current of VSG, and introducing current closed-loop, the output current of VSG can be consistent with the output current of synchronous generator ,which ensuring the VSG can accurately simulate the running characteristics of synchronous generator.

Negative-sequence voltage control of VSG
There is a large amount of single-phase loads in micro-grid. The load current presents asymmetry when common VSG connects to these loads in island mode, which leads to output voltage unbalanced.

Analysis of VSG under unbalance loads
Based on symmetric component method, we can know that three-phase unbalanced phasors can be decomposed into three sets of symmetric phasors: positive-sequence, negative-sequence and zerosequence component. For the three-phase inverter without neutral line, the zero-sequence is usually neglected.
According to the deduction and analysis in reference [10], we can obtain positive-and negativesequence equivalent circuit of VSG in island mode, as shown in Figure 4.
positive-sequence equivalent circuit (b) negative-sequence equivalent circuit If the negative-sequence voltage in VSG can be controlled to zero, only leaving the positivesequence component, it can be ensure that the output terminal voltage is three-phase balanced. which will improve the voltage quality of power supply in isolated micro-grid.

The extraction of negative-sequence component
In order to suppress the negative-sequence component, we need to extract it from unbalanced voltages first. Three-phase asymmetric voltage can be decomposed according to equation (6).  (6) where Um+ ,Um-respectively means the positive-and negative-sequence peak value of fundamental voltage. And φ+,φrespectively means the initial phase angle of the positive-and negative-sequence of fundamental wave voltage. Transform the positive-and negative-sequence components into α-β axis, then equation (7) and (8) (8) We can find that uα+ and uβ+ have the cosine-sine function relationship with the same amplitude and frequency. Thus, the delay cancellation method [11] can be used to acquire the uα-,uα+ and uβ-,uβ+ component, where they are still AC values in the α-β axis. In this paper, the positive-and negativesequence components are transformed into the d-q axis rotating coordinate to control. The control diagram of negative-sequence component extraction method is shown in Figure 5.  Figure 5. Control diagram of negative-sequence component extraction method.
In the above figure, θ is the angle between d-axis and a-axis, R(θ) represents the transformation matrix of positive-sequence from α-β axis to d-q axis ,while R(-θ) represents the negative one. In order to achieve 90° phase shift, the SOGI phase shift system is adopted, and its transfer function can be reorganized as equation (9): When k =1, ω =100π, D(s) has a gain of 1dB at 50Hz and the phase offset is 0°, while Q(s) has a gain of 1dB and the phase offset is 90° at the same frequency.

Unbalance voltage control strategy
In order to suppress the negative-sequence voltage and increase the unbalance load capacity of VSG, the unbalance voltage control strategy for VSG is designed (shown in Figure 6 ). The measured output terminal voltage uCabc are calculated through delay cancellation method and thus the positive-and negative-sequence components in the α-β axis are obtained. The negativesequence component uCα-and uCβ-are transformed into d-q axis to achieve the direct-current negativesequence components uCd-and uCq-. According to the classical control theory, in the synchronous rotating coordinate system, the negative sequence output voltage can track the reference quantity without static error using PI controller. The reference DC negative-sequence components uCd-and uCqare set to 0 and the PI controller output values are added on the current loop reference instructions id * and iq * as compensations. This improved VSG control can suppress the negative-sequence voltages to zero.

Simulation results
The simulation was performed in Matlab/simulink environment to verify the feasibility of proposed unbalance voltage control strategy. Simulation parameters are shown in Table1. Unbalance condition (1)Inserting 5Ω load between A and C phase (2)B phase load out of running Unbalance degree is usually adopted to indicate how bad the unbalance condition is. It is defined as the amplitude ratio of negative-sequence and positive-sequence component. Unbalance degree is expected to keep as small as possible.
The output voltage and load current waveform under unbalance condition(1) can be seen in Figure  7. When inserting 5Ω load between A and C phase, the output voltage of common VSG become unbalanced distinctly. The unbalance degree of output voltage comes to 7.5%, which is higher than relevant standard. In contrast, the improved VSG outputs three-phase balanced voltage, whose unbalance degree is 0.17%.   Figure 8 shows the output voltage and load current waveform under unbalance condition (2) . The unbalance degree of output voltage comes to 12.4% when B phase load is out of running, which is too much higher than 2%. In contrast, the improved VSG outputs three-phase balanced voltage, whose unbalance degree is 0.21%.  By comparing the simulation results, we can draw a conclusion that the proposed unbalance voltage control strategy can make VSG adapt to various kind of unbalance loads, even the worst condition: one phase load out of running. The unbalance degree of output voltage can be kept under 0.5%.

Conclusion
In isolated micro-grid, the output voltage unbalance degree of common VSG is difficult to fit the relevant technical standards because of the existence of a large number of asymmetric loads. In this paper, an improved VSG voltage control algorithm has been proposed based on the positive-and negative-sequence equivalent circuit of VSG and negative-sequence voltage depressing methods. The simulation results verify the feasibility and effectiveness of the proposed VSG unbalance voltage control strategy. However, the problem of voltage unbalance in multiple VSG co-governed island mode is still to be further studied.