The output active power P _e and reactive power Q _e of single VSG can be calculated in the virtual two-phase system by sampling the output voltage u _C and load current i _d (Suul et al., 2016). The virtual voltage and current will be 90° phase shifted in stationary conditions, and the vector amplitudes of the voltage and current are equal to the amplitude of the measured signals of the voltage and current. Then, the amplitude E ₀ and frequency ω of the output voltage reference can be obtained based on the VSG algorithm. The VSG algorithm includes active-frequency (P − f) and reactive-voltage (Q − U) control method by mimicing the traditional SG (Zhong and Weiss, 2011). In convenience of practical engineering applications, the second-order model of traditional SG is expressed as T m − T e = P m ω − P e ω = J d ω d t + D p ( ω − ω r e f ) d θ d t = ω (1) where T _m and T _e are the mechanical and electrical torques respectively, P _m and P _e are the mechanical and electrical power. J is the moment of inertia, D _p is the damping coefficient, ω is the rotor angular velocity, and ω _ref is the grid synchronous angular velocity. When the single-phase VSG operates in grid-connected mode, the value of ω _ref is the power grid frequency ω _g which can be obtained through a phase-locked loop (PLL). When the VSG works in island mode, its value is equal to the microgrid reference frequency.

In order to mimic the primary frequency function of the SG, the primary frequency controller is substituted into the power frequency Eq. 1. The primary frequency controller can be expressed as P m − P r e f = k p ( ω r e f − ω ) (2) where k _p is the P − f droop coefficient, P _ref is the active power reference value. By combing Eqs 1–3 can be deduced as P r e f − P e = ω J d ω d t + ω D p ( ω − ω r e f ) − k p ( ω r e f − ω ) (3)

From the above Eq. 3, it can be seen that the P − f controller of the VSG includes the characteristics of the droop control, moment of inertia and damp, which can enhance the stability of the power grid.

The Q − U controller of the VSG mimics the excitation regulation function of the SG to realize the drop characteristics of reactive power and voltage amplitude (Zhong and Weiss, 2011), and the equation can be expressed as: E 0 = E r e f + k q ( Q r e f − Q e ) (4) where E ₀ and E _ref are the actual output and rated voltage amplitude of the VSG respectively, k _q is the Q − U droop coefficient, Q _ref and Q _e are the reference and actual output reactive power respectively. However, according to Eq. 4, output reactive power Q _e cannot achieve the zero-steady-error tracking of the reference reactive power Q _ref based on the proportional gain k _q when the VSG operates in grid-connected mode. Therefore, a modified Q-U controller with integral term is obtained as E 0 = E r e f + k q ( Q r e f − Q e ) + k i ∫ ( Q r e f − Q e ) d t (5) where k _i is the integral coefficient of the Q-U controller. It should be noted that when the single VSG operates in grid-connected mode, Eq. 5 is used for achieving the zero-steady-error tracking of the reactive power, and the voltage amplitude E _ref is equal to the grid voltage amplitude U _gm . When the VSG operates in island mode, Eq. 4 is used for Q-U controller, the reference reactive power Q _ref is equal to zero and the voltage amplitude E _ref is equal to rated value.

Based on the analysis of the above mentioned section, the amplitude and frequency of the reference voltage E _ref can be derived from the outer VSG power loop. In this section, the inner voltage regulation loop based on capacitor voltage and inductor current feedback achieves the tracking of the reference voltage, and the control block diagram is shown in Figure 2.

The capacitor voltage transfer function u _c (s) of the closed-loop system can be derived as u C ( s ) = G ( s ) u r e f ( s ) − Z o ( s ) i d ( s ) (6)

For simplicity, the resistance R _s is ignored because the value is small, and the voltage gain G(s) and output impedance Z _o (s) can be respectively expressed as G ( s ) = k u p k L k P W M s + k u i k L k P W M L C s 3 + k L k P W M C s 2 + ( k u p k L k P W M + 1 ) s + k u i k L k P W M (7) and Z o ( s ) = L s 2 L C s 3 + k L k P W M C s 2 + ( k u p k L k P W M + 1 ) s + k u i k L k P W M (8) where k _up and k _ui are the proportional coefficient and integral coefficient of the capacitor voltage feedback loop controller respectively, k _L is the proportional coefficient of the VSG inductor current feedback loop controller. According to Eq. 6, the output impedance of the system can be adjusted by changing the proportional and integral (PI) coefficient. Therefore, different output impedance can be obtained by selecting the appropriate PI parameters.

The detailed parameters of the whole system are given in Table 1 and Figure 3 shows bode diagram of output impedance Z _o (s) with different PI parameters. As the value of the proportionality coefficient k _up increase, it can be seen that the inductive proportion of the output impedance of the system gradually increases at the low frequency as shown in Figure 3A. However, large value of parameter k _up will cause the instability of the system. Therefore, the parameter k _up should be chosen based on the tradeoff the dynamic performance and the stability of the system. Figure 3B shows that the inductive proportion of the output impedance of the system gradually increases at the low frequency with the decreasing value of parameter k _ui , so a small value of parameter k _ui is preferred.

Although the inertia and damping are provided for the single-phase inverter by mimicking the traditional SG, the effects of virtual moment of inertia J, damping coefficient D _p , P − f and Q − U droop coefficients k _p and k _q on the stability of the VSG should be further analyzed. Next, a small signal model of the modified VSG strategy in grid-connected mode is established, and the influence of key parameters on the stability of the system is introduced.

Figure 4 shows the equivalent circuit when VSG operates in grid-connected mode. Where E and U represents the amplitude of the VSG output terminal voltage and the grid voltage respectively. Assuming that the phase angel of the power grid voltage is zero, δ is the phase angle of the VSG output voltage, the total impedance of the line between the VSG and the grid can be equivalent to Z∠α = R + jX, where α is the impedance angle. Therefore, the apparent power output of the VSG can be obtained, S = U ̇ I ̇ * = P e + j Q e = E U c o s ( α − δ ) − U 2 c o s α Z + j E U s i n ( α − δ ) − U 2 s i n α Z (9)

Assuming that the line impedance is mainly inductive and active power P _e and reactive power Q _e can be derived as (Liu et al., 2020b) P e = E 2 R − E U R c o s δ + E U X s i n δ R 2 + X 2 = E U s i n ( δ ) X Q e = E 2 X − E U X c o s δ − E U R s i n δ R 2 + X 2 = E 2 − E U c o s ( δ ) X (10)

The small signal model of the modified VSG can be obtained based on Eq. 8. Δ P e = ∂ P e ∂ E Δ E ( s ) + ∂ P e ∂ δ Δ δ ( s ) = K P E Δ E ( s ) + K P δ Δ δ ( s ) Δ Q e = ∂ Q e ∂ E Δ Q ( s ) + ∂ P e ∂ δ Δ δ ( s ) = K Q E Δ E ( s ) + K Q δ Δ δ ( s ) (11) where K _PE and K _Pδ are the partial derivative of active power P _e to voltage amplitude E and power angle δ respectively. K _QE and K _Qδ are the partial derivative of reactive power Q _e to voltage amplitude E and power angle δ respectively, and they can be expression as K P E = U s i n δ X K P δ = E U c o s δ X K Q E = 2 E − U c o s δ X K Q δ = E U s i n δ X (12)

As the output power of a single-phase VSG contains secondary power oscillations, a first-order low-pass filter is used to filter the secondary power oscillations contained in the output power. Small signal model of the output voltage amplitude and frequency of the VSG can also be obtained based on Eqs 1, 2, 5, Δ ω ( s ) = − ω c s + ω c k p k p J ω e r f s + k p D p ω r e f + 1  ⋅ ( K P E Δ E ( s ) + K P δ Δ δ ( s ) ) Δ E ( s ) = − ω c s + ω c k q + k i s ( K Q E Δ E ( s ) + K Q δ Δ δ ( s ) ) (13) where, ω _c is the cut-off frequency of the low-pass filter. Since Δ ω = s Δ δ (14)

According to Eqs 8, 9, 12, it implies that s 4 Δ δ ( s ) + a s 3 Δ δ ( s ) + b s 2 Δ δ ( s ) + c s Δ δ ( s ) + d Δ δ ( s ) = 0 (15) where a = ( K q K Q E + 1 ) J ω c ω r e f b = D p ω r e f + k q D p ω r e f K Q E + 1 k p + k i K Q E J ω r e f + k q k p k Q E ω c c = ( k q K Q E K P δ + K P δ − k q K P E K Q δ ) ω c 2 + D p ω r e f + 1 k p k i K Q E ω c d = K i ( K Q E K P δ − K P E K Q δ ) ω c 2 (16)

Eq. 15 describes the characteristics of the system with the small disturbances around the equilibrium point. Figure 5 shows the root locus diagram with different inertial J, damp coefficient D _p , droop parameters k _p and k _q .

It can be seen that with the increase of inertia J, droop parameters k _p and k _q , oscillation will appear and the system will become unstable. On the contrary, the VSG system will become more stable with the damp coefficient D _p increases.

In order to reduce the sensor counts, a state observer is introduced here to evaluate the actual inductor current i _L value. By the kirchhoff′s law of voltage and current, the single VSG system is modeled by C d u c d t = i L − i d L s d i L d t = U A − u C − R s i L (17) And the state-space mode can be derived by Eq. 17 x ̇ = A x + B u + P ω y = C x (18) where A = 0 1 / C − 1 / L s − R s / L s , B = 0 1 / L s T , P = − 1 / C 0 T , C = 1 0 , and state variable x, input variable u, disturbance signal ω, and output variable y can be expressed as x = u c i L T , u = U A , ω = i d , y = u C ,

As the load current i _d and capacitor voltage u _C must be obtained to calculate the output active power P _e and reactive power Q _e , an observed inductor current i L ̂ is used to replace the real inductor current to reduce the sensor counts for the inner voltage regulation. Eq. 18 can be rewritten as x ̇ = A ′ x + B ′ u ̄ y = C ′ x (19) and matrix A′ = A, C′ = C, B′ and the input variable u ̄ can be expressed as B ′ = 0 − 1 / C 1 / L s 0 , u ̄ = u c i d T ,

It is easy to prove that Eq. 19 satisfies the observability condition, that is r a n k = C ′ C ′ A ′ = 2 .

Therefore, the observed inductor current i L ̂ can be obtain based on Eq. 20. x ′ ̂ ̇ = A ′ x ′ ̂ + B ′ u + L ( y ′ − C ′ x ′ ) y ′ ̂ = C ′ x ′ ̂ (20) where L is gain matrix. The block diagram of the observer is shown in Figure 6. By subtracting Eq. 20 from Eq. 19, the state error matrix is introduced: e ̇ x = ( A ′ − L C ′ ) e x = e ( A ′ − L C ′ ) e x (21) where e _x represents the state error matrix. When the eigenvalue of the matrix (A′ − LC′) is in the left side of the complex plate, the error of the observer will approach zero. Therefore, it is necessary to adjust the parameters of the state feedback matrix L.

In this section, the optimal pole assignment of the state observer is designed based on the optimal pole assignment principle (Lin, 2007). The distribution of the position of the characteristic root of the system in the complex plane determines the dynamic response speed and the degree of oscillation of the system, the farther the pole position of the transfer function is from the imaginary axis, the faster the dynamic response speed of the system.

Based on Eq. 19, the state variables of the capacitor voltage u _C and the inductor current i _L have the greatest impact on the system, the poles related to these two state variables are selected as the dominant poles. The original dominant pole and the expectation dominant pole after the optimal pole placement are shown in Figure 7. It is found that the original system is relatively close to the imaginary axis, the convergence speed of the system is slow, and it may cause instability of the system. However, the expectation pole is farther from the imaginary axis, which improves the dynamic response speed of the system. As the inductor current must be used for feedback control, the dynamic speed for the state observer should be faster than the controller, that is, its pole is generally 4 to 10 times that of the controller.

When VSG operates in island mode, the switch S ₂ is disconnected and the VSG works in P − f. The switch S ₅ is connected to the rated angular frequency ω _ref . The switches S ₁, S ₄ are disconnected and VSG works in Q − U, the switch S ₃ is connected to the rated voltage amplitude E _ref . At this time, the output power of the VSG is the load power.

When the VSG operates in grid-connected mode, the switch S ₂ is disconnected and the switch S ₅ is connected to the grid and the reference frequency is angular frequency ω _g . The switch S ₄ is disconnected, the switch S ₁ is connected to make the integration term works, the switch S ₃ is connected to the grid voltage amplitude U _gm .

When the VSG receives a signal that it needs to switch from grid-connected mode to island mode, since the VSG was previously in grid-connected operation and there is an inertia link in the VSG, the output voltage of the VSG theoretically will not have a transient sudden change at the moment of switching. At this time, the voltage, angular frequency, power reference value in the VSG control will automatically change to the given amount when the island is isolated, so the switch from grid-connected to island operation of the VSG can be completed without adding a complicated control strategy.

When the VSG switches from grid-connected mode to island mode, it starts the pre-synchronization process. At this time, the switches S ₂ and S ₄ are closed, the VSG gradually adjusts the phase, frequency and amplitude information of the output voltage in the island operation mode. When the grid-connected standard is reached, the pre-synchronization process finished and the grid-connected switch is closed. The reference power P _ref and Q _ref in the VSG control loop will be replaced by the grid’s active power P _g and reactive power Q _g . Due to the inertia of the VSG, the power output by the VSG will not undergo a step change, the VSG will complete the transfer from island to grid-connected with seamless switching feature.

In order to verify the effectiveness of the above-mentioned luenberger observed-based VSG strategy, a 3kVA simulation model based on the single-phase inverter was built. The key parameters are given in Table 1. The whole control diagram of the proposed modifier single-phase VSG control strategy is shown in Figure 8.

The single VSG operates in island mode initially, and the grid-connected signal is triggered at 0.35s, the island mode signal is triggered at 1.4s. In grid-connected mode, the VSG reference active power and reactive power is 3kW and 500var respectively.

Figure 9 shows the output active power P _e and reactive power Q _e of single-phase VSG when operates in grid-connected mode and island mode. The simulation results demonstrate that when the single-phase VSG switches between the two operating modes, there is no obvious power mutation and the output power of the VSG can reach the reference power in grid-connected mode.

The seamless transfer capability of the single-phase VSG based on the proposed method is also verified by simulation results as shown in Figure 10. Figure 11A shows the waveform of capacitor voltage u _C , power grid voltage U _g , and the load current i _d when the VSG operates in island mode. As the initial phase difference between the power grid voltage and capacitor voltage u _C is 30°, the voltage deviation Δu between the grid voltage and the inverter output voltage exists.

Figure 10B shows the pre-synchronization process when single-phase VSG switches from island mode to grid-connected mode. At time 0.3s, the trigger signal enables, and the pre-synchronization process starts, the phase angle difference Δθ between the VSG output voltage phase and the grid voltage phase gradually decreases. When the phase difference Δθ is less than the threshold value of 3° at time 0.45s, the pre-synchronization process ends, and the VSG switches to the grid-connected mode.

Figure 10C,D shows the voltage and current waveforms of seamless transfer between the island mode and grid-connected mode at time 0.45 and 1.4 s respectively. Simulation results demonstrate that smooth switching between island mode and grid-connected mode can be achieved, and the inrush current is small. Figure 10E shows the partial enlarged view of VSG voltage u _C and load current i _d , it demonstrates that the voltage sags is small when VSG switches from grid-connected mode to island mode.

Figure 11 shows the simulation waveform of the observed inductor current i _L and the actual inductor current i _L , and Δi _L is the deviation between them. The gain matrix L = [−500,−500,−500,−500,−500,−500,−500,−500,−500,−500] ^T . It can be seen that the estimation error of the observer is small and the dynamic response speed is fast when the VSG operates in grid-connected mode and island mode.

A 3kW single-phase full-bridge inverter prototype is built to verify the feasibility of the proposed control method. The main parameters of the system use are shown in Table 1. The performance of the proposed control strategy is validated by dSPACE SCALEXIO system for real-time implementation, as shown in Figure 12. The prototype is mainly composed of single-phase full-bridge inverter main circuit, A/D sampling circuit, DSPACE transfer board, auxiliary switching power supply and host computer. The output voltage and current waveforms are displayed through the oscilloscope YOKOGAWA-DLM2024. The DC side voltage source adopts programmable DC power supply model Chroma62150H-1000s, and the AC power supply adopts the programmable AC power supply model Chroma61845.

Figure 13 shows the experimental results of seamless transfer process between the island and grid-connected based on the proposed control strategy. The output voltage u _C and load current i _d experimental waveform is demonstrated in Figure 13A when the VSG is switched between grid-connected and island modes. The output voltage almost unchanged and the grid-connected current fluctuates very little when the operating mode of the single-phase VSG changes. High performance of dynamic waveform can be achieved. Figure 13B,C show the partial enlarged experimental waveform of the dynamic process between the island mode and the grid-connected mode, and the voltage waveform changes little throughout the process. Figure 13D shows the output active power P _e and reactive power Q _e of the VSG reached 3kW and 500var respectively when connected to the power grid, which is consistent with the reference active and reactive power.

Figure 14 shows the experimental waveforms of the actual inductor current i _L and the observed current i L ̂ when the single-phase VSG works in island and grid-connected modes. The blue waveform is the actual value of the inductor current, the brown waveform is the observed value of the inductor current, and the green waveform is the observation error Δi. The experimental results demonstrate that the observed inductor current can track the actual value well based on the proposed luenberger state observer.