3.1. Mathematical Modeling of HGS.
The bandwidth of the current loop in the grid-following converter (GFL) is significantly larger than the bandwidth of the phase-locked loop (PLL). By ignoring the dynamic process of the current loop, the GFL can be approximated as a current source. Similarly, in the grid-forming converter (GFM), the adjustment speed of the inner voltage and current loops is much faster than that of the active and reactive power loops. By neglecting the dynamic process of the inner voltage and current loops, the GFM can be approximated as a voltage source. Circuit impedance is considered negligible, and the phase angle of the grid voltage is taken as the reference angle. The equivalent circuit diagram of
Figure 1 is shown in
Figure 4.
In this figure,I1 is the RMS value of the output current of the current-controlled converter; U1 is the RMS value of the output voltage of the GFL;θpll is the phase angle difference between the PLL and the grid, corresponding to the power angle of a synchronous generator, and for convenience, it will be referred to as the power angle hereafter; φ1 is the phase angle difference between voltage U1 and current I1; U2 is the RMS value of the output voltage of the GFM; δvsg is the power angle of the GFM; Vg is the RMS value of the grid voltage.
According to the superposition theorem, the PCC voltage
Upcc∠
δpcc in
Figure 4 can be expressed in the stationary reference frame as:
In the equation:.
The point where the phase-locked loop (PLL) measures the voltage is
U1∠
δpll, which can be expressed as:
In the equation:.
Transforming equation(5) to the dq rotating reference frame based on the phase-locked loop (PLL), the q-axis voltage of
U1 can be obtained as:
In equation (6): typically,δvsg,δpll∈[0,π/2].
Comparing equation (6) with the case of a single grid-following converter (GFL) connected to the grid, it can be observed that the q-axis voltage uq1 of the GFL has an additional impedance drop coupling term. The magnitude of this coupling term is related to k2, U2, δvsg, and δpll, indicating that the power angle of the phase-locked loop (PLL) is also influenced by the GFM power angle δvsg.
From
Figure 4, the single-phase output complex power of the GFM can be expressed as:
In the equation, X2 = ωn L2, where ωn is the nominal angular frequency.
Substituting equation (4) into equation (7), we obtain:
From equations (8) and (9), it can be seen that the GFL affects both the active power loop and the reactive power loop of the GFM. Compared to the single machine case, an additional active power coupling term and a reactive power coupling term are introduced. The magnitude of this impact is related to
k1、
U2、
I1、
X2、
δvsg,
δpll, and
φ1. The power angle of the GFM is also influenced by the output current
I1 of the GFL. By calculating the three-phase active power using equation (8) and substituting it into equation (2), it can be obtained that the GFM in steady state satisfies equation (10).
In summary, the transient analysis model of the HGS can be obtained, as shown in
Figure 5. It can be seen that the integration of the GFL adds a power coupling term to the GFM; similarly, the GFM adds an impedance drop coupling term to the GFL.
3.2. Transient Stability Analysis Considering Coupling Effects
Setting equation (6) to zero, the condition for the PLL to have an equilibrium point can be obtained as:
When the GFM operates stably,δvsg ∈[0, π/2].If ,then equation (11) cannot be satisfied, and there is no equilibrium point for the PLL. If , then equation (11) can be satisfied, and an equilibrium point exists for the PLL.
From equation (10), it can be seen that the presence of I1 increases δvsg, reducing the stability margin of the GFM. The magnitude of δvsg is directly proportional to k1, U2, and I1, and inversely proportional to X2. If I1 is too large, it may cause the initially stable GFM to lose power angle stability. If the GFM loses power angle stability due to a fault or other reasons, its output angle δvsg increases indefinitely. As indicated by equation (11), when δvsg=2kπ+π, k∈Z, the impact on the PLL is maximized, leading to two possible situations: ① If equation (11) is not satisfied, the GFL becomes unstable.② If equation (11) is satisfied, since δvsg increases indefinitely, equation (6) remains in a state of adjustment and cannot stabilize or converge to 0, indicating that the GFL becomes unstable.
Therefore, transient instability in the GFM will trigger a chain reaction, causing transient instability in the GFL.
From equation (10), the GFM has a steady-state point when the following condition is satisfied:
When the GFL operates stably,δpll ∈ [0, π/2] and φ1 ∈ [-π/2, 0].If , equation (12) cannot be satisfied, and there is no equilibrium point for the GFM.If , then equation (12) can be satisfied, and an equilibrium point exists for the GFM.
From equation (6), it can be seen that when δvsg > δ pll, δ pll increases, and the stability margin of the GFL decreases. When δvsg < δpll, δpll decreases, and the stability margin of the GFL increases. When δvsg = δpll, the coupling term is zero, and δpll is not affected by the GFM. If the GFL loses synchronization stability due to a fault or other reasons, the PLL will have no equilibrium point, causing the power angle δpll to increase indefinitely, thereby affecting the GFM. There are two possible scenarios:① If equation (12) is not satisfied, the GFM becomes unstable. ② If equation (12) is satisfied, due to δpll increasing indefinitely, equation (10) remains in a state of adjustment and cannot converge to zero, indicating that the GFM becomes unstable.
Therefore, when the GFL experiences transient instability, the increase in its output phase also causes the GFM to experience transient instability.
In summary, it is essential to ensure the simultaneous stability of both converters; instability in either one will affect the stability of the other.
Additionally, during grid voltage sag, the GFM, being a voltage source type, can experience transient overcurrent and steady-state overcurrent. Under fault conditions, the circuit satisfies:
In the equation:i2F is the instantaneous value of the GFM output current during a fault;u2F is the instantaneous value of the GFM output voltage during a fault;vgF is the instantaneous value of the grid voltage during a fault;i1 is the instantaneous value of the output current of the current-controlled converter.
From equation (13), it can be seen that the presence of the GFL can reduce the fault current of the GFM.
During a fault, reducing the GFL output current
id1 not only ensures the existence of the PLL equilibrium point, but also, as seen from equation (6), the reduction of
id1 can slow down the acceleration process of the GFL output
ωpll during the fault, thereby reducing the deviation of
ωpll. Since the inertia of the GFM is much larger than the equivalent inertia of the GFL, the increase rate of
δvsg is slower than that of
δpll during a fault. Additionally, because the fault duration is short, during the fault, (
δvsg −
δpll) ∈ [-π/2, 0] [
18]. For purely active power output,
φ1 = 0, and the reduction in
id1 is much greater than the change in the cosine function, so the effect of the cosine function can be ignored. Therefore, when
id1 is reduced, the power coupling term in equation (8) is also reduced. From equation (2), it can be seen that this also helps to slow down the acceleration of the GFM output
ωvsg during the fault, reducing the deviation of
ωvsg. The phase plane diagram of the HGS under different
id1 conditions during a fault is shown in
Figure 6.
From
Figure 6, it can be observed that reducing the active current of the GFL can slow down the acceleration process of both the GFL and GFM during a fault, thereby improving the synchronization stability of the two converters. Moreover, the smaller the
id1, the better the synchronization stability of the two converters; conversely, the larger the
id1, the worse the synchronization stability of the two converters.
During a fault, reducing the input mechanical power
Pset of the GFM not only ensures the existence of the GFM’s equilibrium point but, as shown in equation (2), also slows down the acceleration process of the GFM’s output
ωvsg during the fault, reducing the deviation of
ωvsg. A decrease in
Pset during a fault causes the
δvsg of the GFM, which has inertia, to increase more slowly than
δpll. The smaller the
Pset, the slower the increase of
δvsg, the larger the |
δvsg −
δpll |, and the smaller the impedance drop coupling term in equation (6), resulting in a smaller deviation of the GFL’s output
ωpll. The phase plane diagram of the HGS under different
Pset conditions during a fault is shown in
Figure 7.
From
Figure 7, it can also be seen that reducing the input mechanical power
Pset of the GFM can similarly slow down the acceleration process of both the GFM and GFL during a fault, thereby enhancing the synchronization stability of the two converters. Moreover, the smaller the
Pset, the better the synchronization stability of the two converters; conversely, the larger the
Pset, the worse the synchronization stability of the two converters.