The topology structure of the grid-connected inverter is shown in Figure 2 where C is the DC filter capacitor, U _dc is the DC voltage, L _f is the filter inductance of inverter, L _g is the grid inductance, U _t and U _g are rms values of the grid voltages.

From Figure 2, the mathematical model is shown in Eq. 1 L d i d d t = U d − U t d + ω L i q L d i q d t = U q − U t q − ω L i d U d c ⋅ C d U d c d t = i P V ⋅ U d c − U t d ⋅ i d (1)

The inverter generally adopts the vector control strategy based on grid voltage orientation, as shown in Figure 3. The power outer loop control generates d axis and q axis current command values respectively according to the demand of active and reactive power, and regulates the active and reactive power injected into the grid by adjusting the current values of d axis and q axis. In Figure 3, P _ref and P are the reference and actual values of the active power respectively; Q _ref and Q are the reference value and actual value of reactive power respectively; i _dref and i _qref are the reference values of d axis current and q axis current, respectively; i _d and i _q are the d axis and q axis components of the inverter output current, respectively; U _d and U _q are the d axis and q axis components of the modulated voltage output by the current controller.

In the DG grid connection inverter, the phase-locked loop (PLL) is needed to generate the reference phase of the power grid. The PLL usually uses the q-axis component generated by coordinate transformation and obtains the reference phase by controlling the q-axis voltage to zero (Nabil et al., 2022). The PLL adopted in the grid-connected inverter is given in Figure 4 where U _tq is the q-axis component of the voltage vector at the grid-connection point. The working principle is that the instantaneous value of grid voltage is transformed to generate U _tq . Through the PI controller and coefficient K _V , the angular frequency difference Δω can be obtained. Then the angular frequency ω _tra is obtained by adding Δω to the rated angular frequency ω₀, and the measured value angle θ _tra of the phase is obtained by the integration process.

According to Figure 4, the power grid frequency f obtained from the PLL is given as f = K V ⋅ U tq 2 π ⋅ K P + ∫ K I d t + ω 0 2 π (2) where K _P and K _I are the proportional and integral coefficients; K _V is the gain coefficient. By taking the derivative of Eq. 2, the frequency change rate df/dt of the power grid is obtained as d f d t = U tq ⋅ K V K I 2 π (3) Then the active power increment ΔP _inertia is further obtained as Δ P inertia = 2 H inertia d f d t = 2 H inertia ⋅ U tq ⋅ K V K I 2 π (4) where H _inertia is the virtual inertia time constant. Therefore, the virtual inertia control of DGs based on the PLL control of the grid-side inverter is realized.

The above virtual inertia based secondary control method for DGs actively participating in power grid frequency regulation is shown in Figure 5.

The system small-signal model of hierarchical control scheme is firstly established in this part. The mathematical model of the grid-connected inverter in the synchronous rotating d-q reference frame is given as L d i d d t = U d − U t d + ω L i q L d i q d t = U q − U t q − ω L i d (5) From the primary PQ control in Section 3.1, the inverter current i _d and i _q are controlled to follow the current references i _dref and i _qref through the PI controller, and it yields U d = U t d − ω L i q + K PId + K IId s i dref − i d U q = U t q + ω L i d + K PIq + K IIq s i qref − i q (6) where K _PId and K _IId, K _PIq and K _IIq are the proportional and integral parameters of the PI controllers. The outer power loop controls are also based on the PI controllers which has i dref = P ref − P K PP + K IP s i qref = Q ref − Q K PQ + K IQ s (7) where K _PP and K _IP, K _PQ and K _IQ are the proportional and integral parameters of the power loop PI controllers. Then the following differential and algebraic equations can be obtained as d x P dt = P ref − P , d x Q dt = Q ref − Q d x id dt = i dref − i d , d x iq dt = i qref − i d (8) and i dref = K IP x P + P ref − P K PP i qref = K IQ x Q + Q ref − Q K PQ U d = U t d − ω L i q + K IId x id + K PId i dref − i d U q = U t q + ω L i d + K IIq x iq + K PIq i qref − i q (9) where x _P , x _Q, x _id and x _iq are the intermediate state variables. Based on the above derivation, the small signal model of the grid-connected inverter with primary PQ control is formulated as Δ x ˙ = A Δ x (10) where x = x P , x Q , x i d , x iq , P , Q , i d , i q , i d r e f , i q r e f T ; here Δ stands for the small-signal components and A is the system matrix which can be obtained from Eqs 8, 9.

Besides the primary PQ control, the virtual inertia based secondary control method of DGs is carried out for the fast frequency enhancement. Once receiving the measurements sent from the inverters, the secondary controller generates the supplementary power signal for frequency regulation. According to (4), the power reference command with the virtual inertia based secondary control is given as P ref = P n + Δ P inertia = P n + H inertia K V K I π U tq (11) where P _n is the power reference command from primary PQ control. Defining the control input as u sec = H inertia K V K I π U tq (12)

Then the Small signal modeling of DGs with the hierarchical control strategy is Δ x ˙ = A Δ x + B u (13) where the control input u = u _sec, and the matrix B is given as B = H inertia K V K I π 0 1 × 9 T (14)

However, the unavoidable time delay in signal sampling, measurement and execution may reduce the fast control performance of the virtual inertia based secondary control. The control structure diagram and tested time delay of some practical DGs integration system can be seen in Figure 6, which shows that the total time delay in the signal communication can be up to tens to hundreds of milliseconds. That is to say, the problems caused by the time delay may make the virtual inertia based secondary controller not only unable to achieve the expected control objectives, but also may lead to the deterioration or even instability of the dynamic performance of the inverter systems.

Considering the time delay, the system model can be written as Δ x ˙ = A Δ x t − τ 1 + B u t − τ 2 (15)

The equivalent delays τ ₁ and τ ₂ contain both transmission delays of state variables and control inputs and satisfy 0 ≤ τ 1 t ≤ τ max 0 ≤ τ 2 t ≤ τ max (16) where τ _max is a positive constant. It can be seen from Eq. 13 that the system stability under the control input u can be guaranteed with the small disturbance analysis method. However, the existing of delays τ ₁ and τ ₂ in Eq. 14 may destroy the system stability and make the stability condition no longer satisfied. Therefore, the effects of time delays in system state transmission and control inputs are simultaneously considered in the model and corresponding improved control is proposed in the next section.

In this section, an improved hierarchical control strategy considering communication delay is designed based on the robust passivity method to compensate the equivalent delay disturbance. The main problem to design the improved control is how to deal with the time delay term both in state variable and control input properly. To handle this, the system dynamic equation is transformed in the following form. Δ x ˙ = A Δ x t + B u t + D t (17) where the equivalent delay disturbance is defined as D t = A Δ x t − τ 1 − A Δ x t + B u t − τ 2 − B u t . In this part, the robust passivity method is proposed to compensate the equivalent delay disturbance D(t). For the equivalent delay disturbance D(t), the expansion expression can be given according to Taylor’s formula D t = − τ 1 A Δ x ′ t + τ 1 2 2 ! A Δ x ″ t − ⋯ ⏟ T a y l o r   e x p a n s i o n   o f   τ 1   i n   Δ x t − τ 2 B u ′ t + τ 2 2 2 ! B u ″ t − ⋯ ⏟ T a y l o r   e x p a n s i o n   o f   τ 2   i n   u t (18)

Since the equivalent delay time τ ₁ and τ ₂ are time-varying, the high-order polynomial with respect to time t can be used to express the equivalent delay disturbance D(t), which is given as D t = ∑ i = 0 ∞ a i t i (19) where a _i (i = 0,…) is the constant coefficient of the polynomial. Considering that the noise becomes serious and the structure becomes complicated in the controller design, the high order dynamic behavior of the delay disturbance is ignored and it has d dt D t = 0 (20)

Then the robust passivity method to compensate the equivalent delay disturbance is proposed in this paper. For the system dynamic model (15), take the following variable transformation Δ x t = Δ x * t + Δ x ∼ t (21) where Δ x * t and Δ x ∼ t are the ideal value at the equilibrium point and the corresponding error. By taking (19) into (15), it has d dt Δ x * t − A Δ x * t − B u t − D ^ t = − d dt Δ x ∼ t + A Δ x ∼ t + D ∼ t (22) where D ^ t and D ∼ t are the dynamic estimate and estimate error of the unknown equivalent delay D t , which means D ∼ t = D t − D ^ t . To stabilize the system with the existence of communication delay, the virtual damping matrix R t is introduced in Eq. 20 to compensate the equivalent delay disturbance, which has d dt Δ x * t − A Δ x * t − B u t − D ^ t + R t Δ x ∼ t = − d dt Δ x ∼ t + A Δ x ∼ t + D ∼ t + R t Δ x ∼ t (23) If the actual control law u t is taken as u t = B + d dt Δ x * t − A Δ x * t − D ^ t + R t Δ x ∼ t (24) where B + is the inverse matrix of B, it has − d dt Δ x ∼ t + A Δ x ∼ t + D ∼ t + R t Δ x ∼ t = 0 (25) Then the system Lyapunov function with equivalent time delay is chosen as H Δ x ∼ = 1 2 Δ x ∼ t T Q x Δ x ∼ t + 1 2 D ∼ t T Q D D ∼ t (26) where Q _x and Q _D are the positive coefficient matrix. Together with Eq. 23, the derivative of the Lyapunov function H Δ x ∼ is d dt H Δ x ∼ = Δ x ∼ t T Q x A Δ x ∼ t + D ∼ t + R t Δ x ∼ t + D ∼ t T Q D D ∼ ˙ t (27) From Eq. 18 it is seen that the equivalent delay disturbance is assumed to be first-order. Thus it yields d dt H Δ x ∼ = Δ x ∼ t T Q x A + Q x R t Δ x ∼ t + Δ x ∼ t T Q x D ∼ t − D ∼ t T Q D D ^ ˙ t (28) If the adaptive law is designed as D ^ ˙ t = Q D − 1 D ∼ t T − 1 Δ x ∼ t T Q x D ∼ t (29) then the derivative of the Lyapunov function H Δ x ∼ can be written as d dt H Δ x ∼ = Δ x ∼ t T Q x A + Q x R t Δ x ∼ t (30)

To ensure the system Lyapunov function H Δ x ∼ with the virtual damping matrix R t can be convergent, the virtual damping matrix R t should be determined to ensure that the term Q x A + Q x R t < 0 is satisfied. Then the Lyapunov function H Δ x ∼ satisfy the system asymptotic stability condition and the asymptotic stability of the system can be guaranteed, which means the improved hierarchical control strategy based on virtual inertial considering communication delay is realized with the robust passivity control law (22) and the equivalent delay disturbance compensation law (27).

In this case, the initial active load in the system is 40 kW. At time 3 s, the load power increases to 60 kW. Besides, there exists time delay in the communication of the virtual inertia based secondary controller. The system simulation results with and without the proposed improved control are shown in Figures 8–10.

The system simulation results in Figures 8, 9 show the ac frequency and voltage response of the DG integration system during the load fluctuation in different situations. In Figure 8, the conventional virtual inertia control without any time delay compensation is adopted with no time delay in the signal communication. It can be seen that the system frequency and voltage can transition to a new steady state after the load increase at 3 s. However, when there exists time delay in the signal communication of the virtual inertia based secondary controller, the system response gets worse and even becomes unstable. From the system response under conventional control in Figure 9, it is seen that when the time delay is 10 ms, the system frequency is oscillating and the oscillation continues to 3.4 s. When the time delay increases to 12 ms, the dynamic response of system frequency and voltage gets larger oscillation with the conventional virtual inertia control. More serious is that the system loses stability when the time delay is 15 ms from Figures 9E, F with the conventional virtual inertial control. The simulation results under proposed robust passivity control method in Figure 9 show the ac frequency and voltage with different time delays to compensate the equivalent delay disturbance. It is seen that the ac frequency and voltage all respond satisfactorily no matter how much the communication time delay is. Figure 10 gives the simulation results of the equivalent delay disturbance of frequency, which indicates that the proposed control can track the equivalent disturbance caused by time delay and then compensate for the delay.

In this case, the active load in the system is still 40 kW. At time 3 s, a single-phase ground fault occurs and lasts for 10 ms. It is also assumed that there exists different time delay in the communication of the virtual inertia based secondary controller. The system simulation results with and without the proposed control are shown in Figure 11. Besides, the system performance is also compared with the time delay compensation control in the existing literature (Natoriand and Ohnishi, 2008).

The simulation results show that during the single-phase ground fault, the transient process is deteriorated when there exist time delays in the signal communication with the conventional virtual inertia control. However, when the proposed robust passivity method works to compensate the equivalent delay disturbance, the oscillations of ac frequency and voltage can still be decreased and the impact of time delay can be well suppressed. The system response is also compared with the time delay compensation control in (Natoriand and Ohnishi, 2008), which shows that although the oscillations can be suppressed, large overshoots still exist. It can be concluded from the above cases that even the system suffers large disturbances, the improved hierarchical control strategy based on the robust passivity method can significantly enhance the robustness of the DG integration system for superior dynamic response when there exists time delay in the communication channel.