Long-Term Voltage Stability Bifurcation Analysis and Control Considering OLTC Adjustment and Photovoltaic Power Station

: The inﬂuence of photovoltaic (PV) output with stochasticity and uncertainty on the grid-connected system’s voltage stability is worth further exploration. The long-term voltage stability of a 3-bus system with a large-scale PV power station considering the adjustment of an on-load tap changer (OLTC) was studied. In this typical system, two supercritical Hopf bifurcation (SHB) points are found using the bifurcation calculation. At the SHB point that appears ﬁrst, a small sudden increase in reactive load power or a sudden increase in PV active power P pv can eventually cause a voltage collapse after a long increasing oscillation. The long-term collapse phenomenon shows that SHB cannot be ignored in the PV grid-connected system. Meanwhile, the time constant of OLTC can affect the progress of long-term voltage collapse, but it has different effects under different disturbances. When P pv drops suddenly at the SHB point, due to the adjustment of OLTC, the load bus voltage can recover to near the target value of OLTC after a long period of time. Similarly, the time constant of OLTC can affect the progress of long-term voltage recovery. To prevent the long-term voltage collapse when P pv increases suddenly at the SHB point, a new locking-OLTC index I lock , depending on the value of P pv corresponding to the SHB point, and a locking OLTC method are proposed, and the voltage can be recovered to an acceptable stable value quickly. Compared with the system without OLTC, OLTC adjustment can effectively prevent long-term voltage oscillation instability and collapse, so that PV power can play a bigger role in power systems.


Introduction
From the perspective of protecting the environment and reducing smog pollution, the development of renewable energy and clean energy generation has become a mainstream trend, and vigorously developing the new energy is an important strategy for achieving the goal of "carbon peaking and carbon neutrality" [1,2].Photovoltaic (PV) power generation is an important method of solving the problem of carbon emissions [3].At present, PV power generation has been vigorously promoted and widely used around the world.
PV power generation has the characteristics of volatility and intermittency.When large-scale PV stations are connected to the power grid, it will inevitably have an adverse impact on the stability of the power system, increasing the complexity and uncertainty of grid operation [4,5].Therefore, the influence of PV output on the grid-connected system's voltage stability must be studied [6][7][8][9].
Long-term voltage stability belongs to large-disturbance voltage stability.The large disturbances in the power system refer to a short circuit, a disconnection fault, or a largescale new energy off-grid, etc.The large-disturbance voltage stability includes two aspects: the transient process and the long-term process [10].The time frame for the transient voltage stability investigations is usually 10-15 s, while the time frame for the long-term voltage stability investigations is usually from a few minutes to tens of minutes with Energies 2023, 16, 6383 2 of 17 the role of the slow-regulating devices, such as the on-load tap changer (OLTC) and the over-excitation limiter (OEL), etc.
Currently, several studies on long-term voltage stability have been reported.The literature [11] proposed a remedial measure system to prevent medium-term and longterm voltage instabilities in power systems.When the operating point deviates from the normal operating state, the system will select a required procedure, power generation rescheduling or load shedding, based on the concept of electrical distance.According to the selected procedure, the system implements the corresponding remedial measure to bring the operating point back to a normal operating state.The literature [12] proposed a coordinated decentralized emergency control strategy for long-term voltage instability by dividing the power system into multiple local areas based on the concept of electrical distance; and designing a performance indicator based on the load bus voltage and the generator reactive power to assess the severity of disturbances and emergency risks in each local area.
The literature [13] proposed the concepts of long-term voltage stability PDR (Reactive Power Reserve) and short-term voltage stability PDR and established a dual-objective reactive power reserve optimization model to coordinate long-term and short-term voltage stability simultaneously.The literature [14] proposed a long-term voltage stability index that can identify the critical load power of voltage collapse, and a transmission-distribution distinguishing index (TDDI) is also proposed to identify whether the voltage stability limit is due to the transmission or distribution network.Both VSI-index and TDDI-index indicators are calculated based on the measured value of the phasor measurement unit, which is convenient for online applications.The literature [15] designed a long-term voltage stability monitoring index based on the measured value of voltage amplitude and the calculation value of the Thevenin equivalent angle using a voltage trajectory method.The literature [16] studied the coordinated scheduling problem of gas-electricity integrated energy systems including wind power, taking the long-term voltage stability as a constraint condition.
In recent years, machine learning and data mining technologies have developed rapidly, and machine learning algorithms have also been applied to long-term voltage stability research.The literature [17] used the random forest algorithm to predict the load margin to monitor long-term voltage stability in real-time, and a variety of different voltage stability indexes were used as the input variables of the machine learning integration model.The literature [18] proposed a data-based learning and control method based on offline knowledge accumulation and feature acquisition to solve the long-term voltage stability problem caused by emergency online events.The literature [19] proposed an auto-encoder constructed by long short-term memory networks combined with a fully connected layer, which only needs to train the data from a safe operating state to evaluate the long-term voltage stability of power systems.
OLTC is a kind of typical slow-adjustment device and it completes one-tap adjustments with a required time of 10 s to 100 s.When studying long-term voltage stability, the dynamic adjustment process of OLTC cannot be neglected.On the other hand, the dynamic load also has a greater impact on voltage stability, and the adjustment effect of OLTC on voltage stability is related to the dynamic load characteristics [20].Generally, when the system has sufficient reactive power, OLTC adjustment can contribute to the system's voltage stability; if the system reactive power is insufficient, OLTC will have a negative voltage adjustment effect [21].
The literature [22] studied the impact of a variable-speed wind turbine on long-term voltage stability, considering the actions of OEL and OLTC.When the doubly fed induction generator adopts the grid-side converter to control the reactive power, the long-term voltage stability can be effectively improved.The literature [23] took the Nordic 32-bus test system including AVR (Automatic Voltage Regulator), OEL, and OLTC as an example to investigate the impact of PV power generation on long-term voltage stability.With the large-scale access to new energy power stations, the impact of wind power and PV power generation on long-term voltage stability has also begun to receive attention.In a large-scale PV grid-connected system, it is worth discussing the mechanisms of sudden changes in PV power (including sudden increases and sudden drops) on long-term voltage instability, especially under extreme weather conditions.The impacts of OLTC adjustment and dynamic load characteristics on long-term voltage stability under sudden PV power changes also deserve further study.
Because bifurcation parameters correspond to uncontrollable parameters such as load power and PV power variations, bifurcation theory and its analysis method are suitable for studying voltage stability in the PV grid-connected system.The literature [24] analyzed in detail the bifurcation behaviors of a typical 3-bus system with a PV station.When the bifurcation parameter changes gradually, once the behavior of the grid-connected system changes, the system will bifurcate, which may be saddle-node bifurcation (SNB) or other forms of bifurcation, such as Hopf bifurcation (HB), including supercritical Hopf bifurcation (SHB) and subcritical Hopf bifurcation (alternatively termed unstable Hopf bifurcation (UHB)), etc.These bifurcations may have adverse impacts on the system's voltage stability.For example, at the UHB point, a small increase in load power or a sudden change in PV active power can cause a long-term voltage collapse or a long-term voltage oscillation [24,25].In the literature [25], an index-based predictive control approach for UHB was proposed.However, the impact of OLTC adjustment is not considered in the literature [24,25].This may cause a one-sided bifurcation analysis result, as well as a poor understanding of the mechanism of long-term voltage instability.
The biggest factor affecting PV active power is the intensity of solar light.When encountering extreme weather conditions, such as solar eclipses, snowstorms, and sandstorms, etc., it is highly likely that a sudden and significant drop or even complete loss of PV power will occur, but whether it leads to long-term voltage instability should be further investigated.
This paper mainly investigates, in detail, the long-term voltage instability mechanism of the PV grid-connected system, considering OLTC adjustments when the PV output has a sudden change by using bifurcation theory, explores OLTC's key influencing factors on long-term voltage instability, and considers the influence of load dynamic characteristics at the same time.In addition, this paper attempts to design a novel long-term voltage stability index that can predict the long-term voltage instability phenomena, so as to provide an adaptive control scheme.

Brief Introduction of the Hopf Bifurcation Theory
The PV grid-connected system can be described by the single-parameter family form of ordinary differential equations (ODEs). .
where x represents the state variables; µ represents the bifurcation parameter such as the load power or the PV power, etc., and f is a continuously differentiable function.
The eigenvalues of the state matrix A of Equation (1) at the system's equilibrium point (x 0 , µ) can describe the grid-connected system's dynamic stability.With the gradual change of µ, if a pair of conjugate eigenvalues α(µ) ± jβ(µ) of A pass through the imaginary axis in the complex plane, and satisfy the next equation at µ = µ 0 [26]: Then a HB happens at the equilibrium point (x 0 , µ 0 ) in the system, and the point corresponding to the pure imaginary eigenvalues ±jβ(µ 0 ) is namely the HB point [26].According to the stability of the limit cycle arising from the HB point, HB can be divided into SHB (the corresponding first Lyapunov coefficient (FLC) is less than 0) and UHB (the corresponding FLC is greater than 0) [27].

Model of the 3-Bus System with OLTC and PV
As shown in Figure 1a, the traditional 3-bus system (not including the OLTC and PV stations), has been widely used in traditional voltage stability bifurcation analysis [28,29].Now a PV station and an OLTC station are introduced into the system.n is the transformation ratio of OLTC, in general, 0.8 pu ≤ n ≤ 1.2 pu.The PV station is connected to the load bus to investigate the impact of PV power on the load bus's voltage stability.For facilitating the bifurcation calculation, the PV transmission line is omitted.P pv + jQ pv is the PV power.E m ∠δ m is the equivalent generator potential, E 0 ∠0 is the infinite bus voltage, and V∠δ is the load bus voltage.Y m and Y 0 are the grid equivalent admittance modulus, θ m and θ 0 are the corresponding admittance angles, and Y m includes the equivalent admittance of OLTC.The load P D + jQ D adopts the Walve dynamic load model [28,29].The load P 1 + jQ 1 is a constant power load.C is the parallel capacitor bank.In the traditional 3-bus system without OLTC and PV, C is used to raise the load bus voltage.Due to the intermittency of P pv , this setting is retained in this case.d μ  Then a HB happens at the equilibrium point (x0, μ0) in the system, and the poin responding to the pure imaginary eigenvalues ±jβ(μ 0 ) is namely the HB point [26].Ac ing to the stability of the limit cycle arising from the HB point, HB can be divided SHB (the corresponding first Lyapunov coefficient (FLC) is less than 0) and UHB (the responding FLC is greater than 0) [27].

Model of the 3-Bus System with OLTC and PV
As shown in Figure 1a, the traditional 3-bus system (not including the OLTC an stations), has been widely used in traditional voltage stability bifurcation analysis [2 Now a PV station and an OLTC station are introduced into the system.n is the tran mation ratio of OLTC, in general, 0.8 pu ≤ n ≤ 1.2 pu.The PV station is connected t load bus to investigate the impact of PV power on the load bus's voltage stability facilitating the bifurcation calculation, the PV transmission line is omitted.
is the equivalent generator potential, 0 0 E ∠ is the infinite bus age, and V δ ∠ is the load bus voltage.Ym and Y0 are the grid equivalent admittance ulus, θm and θ0 are the corresponding admittance angles, and Ym includes the equiv admittance of OLTC.The load PD + jQD adopts the Walve dynamic load model [28,29] load P1 + jQ1 is a constant power load.C is the parallel capacitor bank.In the tradition bus system without OLTC and PV, C is used to raise the load bus voltage.Due t intermittency of Ppv, this setting is retained in this case.
To simplify Figure 1a, the generator branch is converted to the load bus side, an capacitor banks and the infinite system branch are subjected to a Thevenin equiv transformation, as shown in Figure 1b.To simplify Figure 1a, the generator branch is converted to the load bus side, and the capacitor banks and the infinite system branch are subjected to a Thevenin equivalent transformation, as shown in Figure 1b.
The inverter of the PV station adopts a 1-order dynamic model [30], and its dynamic characteristic equation is as follows [31]: Energies 2023, 16, 6383 5 of 17 where i d , i q are the d-axis and q-axis components of the PV inverter's output current, and T p , T q are the responsive time constants of the PV inverter.The PV station is set as a PQ-type station operating in a constant power factor mode [9], so we can assume: where k is the tangent of the PV station's power factor angle.For example, if k is 0.2, the PV power factor is 0.98.The discrete model of OLTC is as follows [32]: where d is the tap step length, and V ref is the load bus reference voltage (namely the adjustment target value of OLTC).
The function f (V ref − V) can be expressed as follows: where ∆V is the adjustment dead zone.
In order to facilitate the bifurcation calculation with the numerical bifurcation analysis software, in this paper, OLTC adopts a continuous adjustment dynamic model, ignoring the adjustment dead zone and the tap time delay [33]: where T n is the time constant of OLTC (s).To approach the discrete model, T n can be expressed as T n = τ/d, τ is the tap time delay.
To simplify the analysis process, the equivalent generator uses the 2-order model: .
where δ m and ω are the generator power angle and the rotor angular velocity, respectively, M is the inertia, D is the damping coefficient, and P m and P G are the mechanical power and output electric power of the generator, respectively.Derived from Figure 1b, the output active power of the generator can be given below: The power supplied by the grid to all loads is: (10)   The dynamic load model adopts the Walve load model, then the ODEs of the system shown in Figure 1 can be given below: where T is the dynamic load time constant (s), k pω , k qω , k pV , k qV, and k qV2 are the load coefficients of the Walve load, and P 0 and Q 0 are the constant powers of the Walve load.The values of the power supply and network parameters are shown in Table 1, and the load parameters are shown in Table 2 (part of the data from the literature [24,29]).In this paper, all electrical quantities are in per-unit value and the phase angle's unit is rad, the time's unit is s.We set τ = 10 s and d = 0.625%, then T n = τ/d = 1600 s, as listed in Table 1.

The Equilibrium Point Curves
Keep P pv = 1 pu, and take Q 1 as the bifurcation parameter, the numerical bifurcation calculation software Matcont (Version 6.2, A. Dhooge, W. Govaerts, etc., Universiteit Gent, Belgium; Utrecht University, The Netherlands) was used to calculate the equilibrium point curves and bifurcation points of the ODEs shown in Equation (11).We can obtain the equilibrium point curves of the system, as shown in Figure 2. In the Q 1 -n curve shown in Figure 2a, the lower half of the curve is a stable equilibrium point curve.Before Q 1 reaches the SNB point named "LP" (corresponding to Q 1 = 12.6347 pu), two Hopf bifurcation points appear: H 1 (corresponding to Q 1 = 10.3418pu) and H 2 (corresponding to Q 1 = 11.9544pu), FLC's values are both less than 0 (FLC H1 = −0.00876and FLC H2 = −2.21),so the two Hopf bifurcation points are both SHB points.Now keep Q 1 = 10.3418pu and take P pv as the bifurcation parameter.The P pv -n curve is shown in Figure 2b, and the lower half of the curve is the stable equilibrium point curve.H 1 (corresponding to P pv = 1 pu) and H' 2 (corresponding to P pv = 16.7645pu) are both SHB points.The installed capacity of a large-scale PV station can reach several pu (namely several hundred MW) currently, so it is possible that the system operates at the SHB point H 1 , and the impact of this SHB point on the system's long-term voltage stability is noteworthy.Now keep Q1 = 10.3418pu and take Ppv as the bifurcation parameter.The Ppv-n c is shown in Figure 2b, and the lower half of the curve is the stable equilibrium point c H1 (corresponding to Ppv = 1 pu) and H'2 (corresponding to Ppv = 16.7645pu) are both points.The installed capacity of a large-scale PV station can reach several pu (namely eral hundred MW) currently, so it is possible that the system operates at the SHB poin and the impact of this SHB point on the system's long-term voltage stability is notewo   Take the SHB point H 1 as the initial operating point, and the parameters of this bifurcation point are: (δ m ω δ V n i d i q Q 1 ) = (0.3057 0 0.1227 1 0.8478 0.0761 1.0170 10.3418).Assume that Q 1 has a forward small disturbance (a small increase) at time t = 50 s suddenly, and it increases from 10.3418 pu to 10.3918 pu.That is, the increment ∆Q 1 is 0.05 pu.The time-domain change curves are shown in Figure 3.The load bus voltage V eventually collapses at about t = 975 s after a long increasing oscillation, as shown in Figure 3a.

Impact of Load Power Disturbance on Long-Term Voltage Stability
Assume that Q1 has a forward small disturbance (a small increase) at time t = 50 s denly, and it increases from 10.3418 pu to 10.3918 pu.That is, the increment ΔQ1 is pu.The time-domain change curves are shown in Figure 3.The load bus voltage V e tually collapses at about t = 975 s after a long increasing oscillation, as shown in Figu Figure 3b is the time-domain change curve of the OLTC transformation ratio n. the disturbance occurs, the transformation ratio n keeps increasing to increase V, bu voltage collapse is unavoidable, and n no longer increases.Now assume that starting from the SHB point H1, the load power increment Δ 0.1 pu ~ 0.4 pu, respectively.It can be seen from Figure 3a that the larger ΔQ1 is, the f V collapses, and the time frame becomes smaller as the long-term voltage collapse gr ally evolves into the short-term voltage collapse.Therefore, it is recommended tha maximum value of Q1 should not exceed the value of Q1 corresponding to the SHB H1. Figure 3b is the time-domain change curve of the OLTC transformation ratio n.After the disturbance occurs, the transformation ratio n keeps increasing to increase V, but the voltage collapse is unavoidable, and n no longer increases.Now assume that starting from the SHB point H 1 , the load power increment ∆Q 1 is 0.1 pu~0.4 pu, respectively.It can be seen from Figure 3a that the larger ∆Q 1 is, the faster V collapses, and the time frame becomes smaller as the long-term voltage collapse gradually evolves into the short-term voltage collapse.Therefore, it is recommended that the maximum value of Q 1 should not exceed the value of Q 1 corresponding to the SHB point H 1 .
In general, it is believed that if there is a UHB point in power systems, it will pose a threat to the voltage stability and lead to oscillation-type voltage instability and collapse [24,25].For this example, it can be seen from the above analyses that at the SHB point, a forward small or large disturbance of the load power can also cause long-term voltage oscillation instability and collapse.This phenomenon regarding SHB deserves more attention.

Impact of OLTC's Time Constant-Key Influencing Factor Investigation
The impact of OLTC's time constant T n is investigated on long-term voltage collapse when the system operates at the SHB point H 1 .Changing the tap step length d or the tap time delay τ can change the time constant of OLTC.Now set τ = 10 s and d = 1.25%, then T n = τ/d = 800 s.Set ∆Q 1 = 0.05 pu, the time to reach the voltage collapse increases, as shown in Figure 4.Moreover, set τ = 20 s and d = 0.625%, T n = τ/d = 3200 s, it can be seen from Figure 4 that the time taken to reach voltage collapse gets shorter.Therefore, increasing the time constant T n can shorten the process of the long-term voltage collapse, that is, a larger T n is not conducive to voltage stability.
In general, it is believed that if there is a UHB point in power systems, it will pose a threat to the voltage stability and lead to oscillation-type voltage instability and collapse [24,25].For this example, it can be seen from the above analyses that at the SHB point, a forward small or large disturbance of the load power can also cause long-term voltage oscillation instability and collapse.This phenomenon regarding SHB deserves more attention.

Impact of OLTC's Time Constant-Key Influencing Factor Investigation
The impact of OLTC's time constant Tn is investigated on long-term voltage collapse when the system operates at the SHB point H1.Changing the tap step length d or the tap time delay τ can change the time constant of OLTC.Now set τ = 10 s and d = 1.25%, then Tn = τ/d = 800 s.Set ΔQ1 = 0.05 pu, the time to reach the voltage collapse increases, as shown in Figure 4.Moreover, set τ = 20 s and d = 0.625%, Tn = τ/d = 3200 s, it can be seen from Figure 4 that the time taken to reach voltage collapse gets shorter.Therefore, increasing the time constant Tn can shorten the process of the long-term voltage collapse, that is, a larger Tn is not conducive to voltage stability.

Sudden Drop of PV Power-Long-Term Voltage Recovery
Assume that the system shown in Figure 1 is operating at the SHB point H1, the initial Ppv is 1 pu, and Q1 is maintained at 10.3418 pu.Considering extremely bad weather conditions, such as a total solar eclipse, at time t = 50 s, Ppv suddenly drops from 1 pu to 0 pu, 0.5 pu, and 0.8 pu, respectively.The drop depth dPpv is 1 pu, 0.5 pu, and 0.2 pu, respectively.It can be seen from Figure 5a that the load bus voltage V drops after the disturbance occurs, but an OLTC adjustment can finally recover V to near the target value Vref (1 pu) after a long period of time.As shown in Figure 5b, the drop depth dPpv is 1 pu and it is big enough.When the disturbance occurs, the transformation ratio n continues to increase to raise the load bus voltage and eventually stabilizes at a value of 0.884 pu after a very long time.
It can be determined by calculation that when Ppv drops suddenly, OLTC can all adjust V to near the target value, and V can be recovered faster under the smaller drop depth dPpv.Assume that the system shown in Figure 1 is operating at the SHB point H 1 , the initial P pv is 1 pu, and Q 1 is maintained at 10.3418 pu.Considering extremely bad weather conditions, such as a total solar eclipse, at time t = 50 s, P pv suddenly drops from 1 pu to 0 pu, 0.5 pu, and 0.8 pu, respectively.The drop depth dP pv is 1 pu, 0.5 pu, and 0.2 pu, respectively.It can be seen from Figure 5a that the load bus voltage V drops after the disturbance occurs, but an OLTC adjustment can finally recover V to near the target value V ref (1 pu) after a long period of time.As shown in Figure 5b, the drop depth dP pv is 1 pu and it is big enough.When the disturbance occurs, the transformation ratio n continues to increase to raise the load bus voltage and eventually stabilizes at a value of 0.884 pu after a very long time.
It can be determined by calculation that when P pv drops suddenly, OLTC can all adjust V to near the target value, and V can be recovered faster under the smaller drop depth dP pv .

Sudden Increase in PV Power-Long-Term Voltage Collapse
Now assume that the system shown in Figure 1 is operating at the SHB point H 1 , P pv increases suddenly from 1 pu to 1.1 pu, 1.3 pu, and 1.5 pu, respectively.The increment ∆P pv is 0.1 pu, 0.3 pu, and 0.5 pu, respectively.It can be seen from Figure 6 that the voltage V collapses after a long period of time.The larger ∆P pv is, the faster V collapses, but the time frame is still in the category of long-term voltage stability.

Sudden Increase in PV Power-Long-Term Voltage Collapse
Now assume that the system shown in Figure 1 is operating at the SHB point H increases suddenly from 1 pu to 1.1 pu, 1.3 pu, and 1.5 pu, respectively.The incr ΔPpv is 0.1 pu, 0.3 pu, and 0.5 pu, respectively.It can be seen from Figure 6 that the v V collapses after a long period of time.The larger ΔPpv is, the faster V collapses, b time frame is still in the category of long-term voltage stability.
According to the above analyses, in the absence of any control measures taken preferably not greater than the PV active power corresponding to the SHB point H1 ( In practice, Ppv is generally less than the PV-installed capacity PpvN, hence a solutio ensure that PpvN ≤ Ppv.H1.However, this solution limits the use of PV power generati According to the above analyses, in the absence of any control measures taken, P pv is preferably not greater than the PV active power corresponding to the SHB point H 1 (P pv.H1 ).In practice, P pv is generally less than the PV-installed capacity P pvN , hence a solution is to ensure that P pvN ≤ P pv.H1 .However, this solution limits the use of PV power generation.

Impact of OLTC's Time Constant-Key Influencing Factor Investigation
Different time constants T n can make different impacts on long-term voltage stability when P pv has a sudden change.In Figure 7, when P pv drops from 1 pu to 0 pu, a smaller T n is conducive to the recovery of the voltage V.However, in Figure 8, when P pv increases from 1 pu to 1.3 pu, a smaller T n is not conducive to voltage stability.

Impact of OLTC's Time Constant-Key Influencing Factor Investigation
Different time constants Tn can make different impacts on long-term voltage stability when Ppv has a sudden change.In Figure 7, when Ppv drops from 1 pu to 0 pu, a smaller Tn is conducive to the recovery of the voltage V.However, in Figure 8, when Ppv increases from 1 pu to 1.3 pu, a smaller Tn is not conducive to voltage stability.

Impact of OLTC's Time Constant-Key Influencing Factor Investigation
Different time constants Tn can make different impacts on long-term voltage stability when Ppv has a sudden change.In Figure 7, when Ppv drops from 1 pu to 0 pu, a smaller Tn is conducive to the recovery of the voltage V.However, in Figure 8, when Ppv increases from 1 pu to 1.3 pu, a smaller Tn is not conducive to voltage stability.

Comparison with the PV Grid-Connected 3-Bus System without OLTC
As shown in Figure 1, the PV grid-connected 3-bus system without OLTC is equivalent to the system with OLTC being locked in n = 1 pu, and the system's ODEs take the first four equations of Equation ( 11) [24].Keeping Ppv = 1 pu, the bifurcation calculation

Comparison with the PV Grid-Connected 3-Bus System without OLTC
As shown in Figure 1, the PV grid-connected 3-bus system without OLTC is equivalent to the system with OLTC being locked in n = 1 pu, and the system's ODEs take the first four equations of Equation ( 11) [24].Keeping P pv = 1 pu, the bifurcation calculation results show that on the stable half of the Q 1 -V equilibrium point curve, there is a UHB point H 1 (corresponding to Q 1 = 11.1309pu), a SHB point H 2 (corresponding to Q 1 = 11.4999pu), and a SNB point LP (corresponding to Q 1 = 11.5069pu) [24].
Taking the UHB point H 1 as the initial operating point, since it is a subcritical Hopf bifurcation point, a small sudden increase in the reactive load Q 1 can cause long-term voltage collapse.As shown in Figure 9, assuming that P pv is maintained at 1 pu, Q 1 increases suddenly from 11.1309 pu to 11.1809 pu at time t = 50 s (namely ∆Q 1 = 0.05 pu), and the load bus voltage V collapses at about t = 708 s.

Comparison with the PV Grid-Connected 3-Bus System without OLTC
As shown in Figure 1, the PV grid-connected 3-bus system without OLTC is equivalent to the system with OLTC being locked in n = 1 pu, and the system's ODEs take the first four equations of Equation (11) [24].Keeping Ppv = 1 pu, the bifurcation calculation results show that on the stable half of the Q1-V equilibrium point curve, there is a UHB point H1 (corresponding to Q1 = 11.1309pu), a SHB point H2 (corresponding to Q1 = 11.4999pu), and a SNB point LP (corresponding to Q1 = 11.5069pu) [24].
Taking the UHB point H1 as the initial operating point, since it is a subcritical Hopf bifurcation point, a small sudden increase in the reactive load Q1 can cause long-term voltage collapse.As shown in Figure 9, assuming that Ppv is maintained at 1 pu, Q1 increases suddenly from 11.1309 pu to 11.1809 pu at time t = 50 s (namely ΔQ1 = 0.05 pu), and the load bus voltage V collapses at about t = 708 s.Compared with Figure 3a, due to the lack of OLTC adjustment, the time to reach voltage collapse has been reduced, but it still belongs in the category of long-term voltage stability.
Assume that Q 1 remains unchanged (keeping 11.1309 pu), and the impact of the sudden change of P pv on long-term voltage stability is investigated.As shown in Figure 10, P pv drops suddenly from 1 pu to 0 pu (namely dP pv = 1 pu) at time t = 50 s.After V drops, the increasing oscillation is carried out firstly and the continuous oscillation (namely the constant-amplitude oscillation) begins at about t = 350 s after the disturbance occurs, and the oscillation amplitude of V is (0.9942 − 0.9344) × 100% = 5.98% > 5%.For the continuous oscillation, if the voltage's oscillation amplitude is greater than or equal to 5%, it can be considered that the load bus is experiencing long-term voltage oscillation instability.
The smaller the drop depth dP pv is, the longer the time for the voltage V to begin the continuous oscillation is, but the larger the oscillation amplitude of V will be [24].For example, when P pv suddenly drops from 1 pu to 0.8 pu (namely dP pv = 0.2 pu), the time for the load voltage V to begin the continuous oscillation is about t = 4800 s after the disturbance occurs, and the oscillation amplitude of V reaches (1.072 − 0.8562) × 100% = 21.58%>> 5%.
On the other hand, as shown in Figure 10, when P pv suddenly increases from 1 pu to 1.5 pu (namely ∆P pv = 0.5 pu), after the voltage V increases, it quickly enters a continuous oscillation, and the oscillation amplitude is (1.105 − 1.099) × 100% = 0.6%, far less than 5%, which does not pose a threat to voltage stability.
On the other hand, as shown in Figure 10, when Ppv suddenly increases from 1 1.5 pu (namely ΔPpv = 0.5 pu), after the voltage V increases, it quickly enters a conti oscillation, and the oscillation amplitude is (1.105 − 1.099) × 100% = 0.6%, far less tha which does not pose a threat to voltage stability.The operation comparison results of the system with OLTC and without OLT listed in Table 3.According to the comparative analyses, in order to avoid the long voltage oscillation instability when Ppv has a sudden drop, it is advisable to ado scheme of installing OLTC.For the sudden increase in Ppv, measures such as limiti installed capacity of the PV station or locking OLTC can be used to solve the po threat.The operation comparison results of the system with OLTC and without OLTC are listed in Table 3.According to the comparative analyses, in order to avoid the long-term voltage oscillation instability when P pv has a sudden drop, it is advisable to adopt the scheme of installing OLTC.For the sudden increase in P pv , measures such as limiting the installed capacity of the PV station or locking OLTC can be used to solve the potential threat.

Prevention of Voltage Collapse Caused by a Sudden Increase in PV Power
For the 3-bus system with OLTC and PV shown in Figure 1, the installed capacity P pvN can be limited according to the value of P pv at the SHB point H 1 (P pv.H1 ) so that P pv does not exceed P pv.H1 .However, in practice, P pvN may be much larger than P pv.H1 , so limiting the installed capacity is not an economic method.
In Figure 10 and Table 3, at the UHB point, when P pv increases suddenly from 1 pu to 1.5 pu (∆P pv = 0.5 pu), long-term voltage oscillation with minimal amplitude will occur in the system without OLTC.Therefore, the locking OLTC method (namely maintaining a constant transformation ratio) can be used when P pv increases suddenly in the system with OLTC, in order to prevent long-term voltage collapse.
For a better implementation of this locking OLTC method, a locking-OLTC index is designed: Under the premise of a constant Q 1 , where the value of the I lock index is less than or equal to 0, OLTC should be locked in time.To keep a certain margin, the threshold can even be raised to 5~10%.
For example, when the system with OLTC operates at the SHB point H 1 , assuming that Q 1 is maintained at Q 1.H1 and P pv fluctuates, once the I lock index value is less than or equal to 5~10%, the locking OLTC method can be used to avoid long-term voltage collapse.
Assume that the system with OLTC operates at the SHB point H 1 , and Q 1 is maintained at 10.3418 pu.When P pv increases suddenly from 1 pu to 1.1 pu, 1.3 pu, and 1.5 pu, respectively, I lock is −0.1, −0.3, and −0.5, respectively.Now OLTC is locked in n = 0.9 pu, it can be seen from Figure 11 that V can quickly recover to an acceptable stable value.The flow chart of the locking OLTC method is shown in Figure 12.Since the voltage collapse caused by the sudden increase in P pv is a long process (up to hundreds of seconds or more), the method of locking OLTC is completely feasible.
Considering the fluctuation of P pv , the I lock index should be calculated and judged twice or even more consecutively, with a delay of 0.5~1 min between the two consecutive calculations and judgments.Once the two (or more) consecutive judgment results are the same, the command of locking OLTC should be issued, and n will be locked at a constant value.

Prevention of Voltage Collapse Caused by a Sudden Increase in Load Power
The traditional load-margin index [7] can be modified as follows: where Q 1.H1 is the reactive power of the SHB point H 1 .
Once the I LMHB index value is less than 0, the load-shedding method can be taken to improve the load margin.A detailed load-shedding scheme is referred to in the literature [7].Since the impact of the sudden increase in P pv at the SHB point can be solved by using the locking OLTC method, the value of I LMHB can be 0.
Load shedding is certainly a conservative method, and it may also cause a smallamplitude oscillation in the load bus voltage.

Conclusions
The typical 3-bus system containing an OLTC and a PV station is investigated for long-term voltage stability using bifurcation theory, and a new index and a locking OLTC method are used to prevent long-term voltage collapse.Using the bifurcation calculation and the time-domain calculation, and by comparison with the system without OLTC, the innovation conclusions are stated as follows: (1) There are two SHB points and an SNB point on the stable half of the equilibrium point curve in the 3-bus system with OLTC and PV, and at the SHB point, a small sudden increase in reactive load power Q 1 , or a sudden increase in PV active power P pv can all cause long-term voltage collapse.This phenomenon suggests that SHB is also potentially threatening to the voltage stability.
(2) Aiming to change of P pv at the SHB point suddenly, when P pv drops suddenly, an OLTC adjustment can recover the load bus voltage to near the target value after a long period of time; when P pv increases suddenly, locking OLTC and taking a suitable ration n can recover the voltage to a stable value.The impact of PV power disturbance at the SHB point on long-term voltage stability can be eliminated through the OLTC adjustment, and the system's long-term voltage stability will focus on the impact of load power disturbance.
(3) The time constant T n of OLTC is an important influencing factor in long-term voltage stability, as it can affect the process of long-term voltage collapse and long-term voltage recovery.However, under different disturbances, the magnitude of T n can cause different effects.
(4) A new locking OLTC index I lock based on PV active power was proposed.It can be taken as the benchmark of locking OLTC when P pv increases suddenly at the SHB point.
Moreover, a modified load-margin index I LMHB is proposed to prevent long-term voltage collapse caused by the small sudden increase in load power at the SHB point.
Obviously, the discrete model of OLTC and the continuous changes of P pv are not considered in the bifurcation analysis of the present case, and the calculation workload of bifurcation is very heavy for the multi-bus grid-connected system.The above topics require further study.

Figure 1 .
Figure 1.The 3-bus system with OLTC and PV.(a) The initial model.(b) The simplified mode

Figure 1 .
Figure 1.The 3-bus system with OLTC and PV.(a) The initial model.(b) The simplified model.

2. 4 .
Impact of Load Power Disturbance on Long-Term Voltage Stability 2.4.1.Sudden Increase in the Reactive Load Q 1 -Long-Term Voltage Collapse

Figure 3 .
Figure 3.The time-domain change curves when Q1 increases suddenly from the SHB point H 1600 s).(a) t-V curves at different increments of Q1.(b) t-n curve (ΔQ1 = 0.05 pu).

Figure 3 .
Figure 3.The time-domain change curves when Q 1 increases suddenly from the SHB point H 1 (T n = 1600 s).(a) t-V curves at different increments of Q 1 .(b) t-n curve (∆Q 1 = 0.05 pu).

Figure 4 .
Figure 4. t-V curves at different time constants of OLTC when Q1 increases suddenly from the SHB point H1 (ΔQ1 = 0.05 pu).

Figure 4 .
Figure 4. t-V curves at different time constants of OLTC when Q 1 increases suddenly from the SHB point H 1 (∆Q 1 = 0.05 pu).2.5.Impact of PV Power Disturbance on Long-Term Voltage Stability 2.5.1.Sudden Drop of PV Power-Long-Term Voltage Recovery

Figure 5 .
Figure 5.The time-domain change curves when Ppv drops suddenly from the SHB point H 1600 s).(a) t-V curves at different drop depths of Ppv.(b) t-n curve (dPpv = 1 pu).

Figure 5 .
Figure 5.The time-domain change curves when P pv drops suddenly from the SHB point H 1 (T n = 1600 s).(a) t-V curves at different drop depths of P pv .(b) t-n curve (dP pv = 1 pu).

Figure 7 .
Figure 7. t-V curves at different time constants of OLTC when Ppv drops suddenly from the SHB point H1 (dPpv = 1 pu).

Figure 6 .
Figure 6.t-V curves when P pv increases suddenly from the SHB point H 1 (T n = 1600 s).

Figure 7 .
Figure 7. t-V curves at different time constants of OLTC when Ppv drops suddenly from the SHB point H1 (dPpv = 1 pu).

Figure 7 . 18 Figure 8 .
Figure 7. t-V curves at different time constants of OLTC when P pv drops suddenly from the SHB point H 1 (dP pv = 1 pu).Energies 2023, 16, x FOR PEER REVIEW 12 of 18

Figure 8 .
Figure 8. t-V curves at different time constants of OLTC when P pv increases suddenly from the SHB point H 1 (∆P pv = 0.3 pu).

Figure 8 .
Figure 8. t-V curves at different time constants of OLTC when Ppv increases suddenly from the SHB point H1 (ΔPpv = 0.3 pu).

Figure 10 .
Figure 10.t-V curves when P pv changes suddenly from the UHB point H 1 (dP pv = 1 pu and ∆P pv = 0.5 pu).

Figure 11 .
Figure 11.t-V curves when P pv changes suddenly from the SHB point H 1 (OLTC is locked in n = 0.9 pu).

Figure 12 .
Figure 12.The flow chart of the locking OLTC method (t and i are the sampling time and times).

Table 1 .
The power supply and network parameters.

Table 2 .
The load and PV parameters.

Table 3 .
Comparison of the PV grid-connected 3-bus system with OLTC and without OLTC, taking the HB point H 1 as the initial operating point.