Stability analysis and optimal control strategy of DC microgrids

The basic structure of the DC microgrid in islanding mode [23] includes components such as photovoltaic (PV) units, energy storage units, constant power loads, and power electronic converters. Among them, the PV unit is connected to the bus through a Boost converter as the microsource of the DC microgrid in this paper: the energy storage unit is connected to the bus through a Bi-DC/DC converter to realize the bidirectional flow of power in two states of charging and discharging, and the energy storage converter operates in the Buck mode when the DC microgrid is charging the energy storage unit, and in the Boost mode when the energy storage unit is discharging to the DC microgrid, which is used to balance the The difference between the power of the PV unit and the load unit. The load unit is connected to the bus through the Buck converter, and the output voltage is precisely and quickly controlled through the voltage closed-loop control, which can be regarded as a constant power load, and the load characteristics are expressed as a power load that obeys the law of electricity consumption and changes slowly. Figure 1 shows the DC microgrid in islanding mode, which is taken as the research object in this paper, and the rated bus voltage is set to 400V, and the allowable operating range is 380V~420V.

Figure 1.

Architecture of the DC microgrid

According to analyze the working principle of each component converter can get the average state equation of each unit converter, for the large signal stability analysis to establish the large signal average model of each converter to establish the basis for the subsequent establishment of the large signal model based on the analysis process in this section.

According to the working principle of Boost converter, the average state equation of PV interface converter can be obtained as: (1){ LpvdipvLdt=Upv−(1−dpv)udc Cpvdudcdt=(1−dpv)ipvL−ipv$$\left\{ {\begin{array}{*{20}{l}} {{L_{pv}}\frac{{d{i_{pvL}}}}{{dt}} = {U_{pv}} - (1 - {d_{pv}}){u_{dc}}} \\ {{C_{pv}}\frac{{d{u_{dc}}}}{{dt}} = (1 - {d_{pv}}){i_{pvL}} - {i_{pv}}} \end{array}} \right.$$

According to the Bi-DC/DC converter operating principle, the average state equation of the energy storage interface converter can be obtained as: (2){ LbdibLdt=Ub−(1−db)uo−rbibL Cbdudcdt=(1−db)ibL−ib$$\left\{ {\begin{array}{*{20}{l}} {{L_b}\:\frac{{d{i_{bL}}}}{{dt}} = {U_b} - (1 - {d_b}){u_o} - {r_b}{i_{bL}}} \\ {{C_b}\:\frac{{d{u_{dc}}}}{{dt}} = (1 - {d_b}){i_{bL}} - {i_b}} \end{array}} \right.$$

According to the Buck converter operating principle, the average state equation of the load interface converter can be obtained as: (3){ LcpldicplLdt=dcpludc−ucpl Ccpldudcdt=icplL−icpl$$\left\{ {\begin{array}{*{20}{l}} {{L_{cpl}}\:\frac{{d{i_{cplL}}}}{{dt}} = {d_{cpl}}{u_{dc}} - {u_{cpl}}} \\ {{C_{cpl}}\:\frac{{d{u_{dc}}}}{{dt}} = {i_{cplL}} - {i_{cpl}}} \end{array}} \right.$$

In this paper, the DC bus voltage is used as the control signal to divide the DC microgrid into two working modes, in order to maximize the use of renewable energy, set the photovoltaic unit to give priority to the load power supply, when the load power required power and the photovoltaic unit power imbalance, the energy storage unit for charging and discharging power compensation, to achieve the balance of the system power, set the rated voltage of the DC bus to 400V, the bus voltage is lower than 400V when the load power is bigger, and the bus voltage is higher than 400V when the load power is smaller. The rated DC bus voltage is set to 400V, the bus voltage is lower than 400V when the load power is large, the bus voltage is higher than 400V when the load power is small, and the fluctuation range of the bus voltage is ±5%. According to the change in bus voltage, the isolated DC microgrid has two working modes.

Mode 1: The bus voltage is 380V~400V, at this time, the power required by the constant power load is larger, the bus voltage is lower than 400V, the Boost converter of the PV unit adopts the MPPT control strategy to send out the maximum power, and the power shortfall of the constant power load is compensated by the discharge of the storage unit, and the Bi-DC/DC converter of the storage unit adopts the P-U droop control strategy to stabilize the DC bus voltage. Mode 2: The DC bus voltage is at the same level as that of the storage unit.

Mode 2: The bus voltage is 400V~420V, at this time, the power required by the constant power load is small, the bus voltage is higher than 400V, the Boost converter of the PV unit adopts the P-U sag control strategy to stabilize the DC bus voltage, and the excess power of the PV unit is absorbed by the charging of the storage unit, and the Bi-DC/DC converter of the energy storage unit adopts the constant current control strategy.

The mixed potential function theory allows to establish the energy function of Liapunov type, to obtain the criterion that ensures the stable operation of the system under large perturbations, and to give the stability condition in analytical form. The steps of applying the mixed potential function theory to determine the stability of the system are as follows: 1)

Establish the mixed potential function model of the system.

The hybrid potential function is denoted by P, which is modeled with respect to the circuit structure and is the sum of the current potential function and the voltage potential function.

With u_u, i_u representing the voltage and current of the non-storage element in the system, i₁, …, i_r representing the current through the rth inductor, and u_r+1, …, u_r+s representing the voltage across the sth capacitor, the energy potential function P is shown below: (4)P=P(i,u)=∫∑μ>r+suμdiμ+∑σ=r+1r+suσiσ$$P = P(i,u) = \int {\sum\limits_{\mu > r + s} {{u_\mu }} } {\text{d}}{i_\mu } + \sum\limits_{\sigma = r + 1}^{r + s} {{u_\sigma }} {i_\sigma }$$

where Σ_σ = r + 1^{r + s}u_σi_σ represents the sum of the capacitive energies of the system; and ∫∑μ>r+suμdiμ$$\int {\sum\limits_\mu > } r + s{u_\mu }d{i_\mu }$$ represents the current potential function of the non-energy storage element.

Some non-energy storage elements cannot be constructed with their current potential functions, which can be represented in the form of a voltage potential function by the relation ∫uμdiμ=uμiμ+iμduμ$$\int {{u_\mu }} d{i_\mu } = {u_\mu }{i_\mu } + {i_\mu }d{u_\mu }$$.

The unified model of the hybrid potential function P is shown below: (5)P=P(i,u)=−A(i)+B(u)+(i,γ−αu)$$P = P(i,u) = - A(i) + B(u) + (i,\gamma - \alpha u)$$

The correctness of P can be checked by whether Eq. (6) is satisfied between the steady-state model of the circuit and the model P of the established hybrid potential function. (6){ Lρdiρdt=∂P∂iρ,ρ=1,2,⋯,r −Cσduσdt=∂P∂uσ,σ=r+1,r+2,⋯,r+s$$\left\{ {\begin{array}{*{20}{l}} {{L_\rho }\frac{{d{i_\rho }}}{{dt}} = \frac{{\partial P}}{{\partial {i_\rho }}},\rho = 1,2, \cdots ,r} \\ { - {C_\sigma }\frac{{d{u_\sigma }}}{{dt}} = \frac{{\partial P}}{{\partial {u_\sigma }}},\sigma = r + 1,r + 2, \cdots ,r + s} \end{array}} \right.$$

According to the hybrid potential function in model P of A(i), B(u) in the characterization, there exist three different stability criteria to describe the stability of the system: 1)

If A(i) is related to all parameters in i₁, ⋯, i_r, the range of values of the parameters for the system to maintain stable operation under large perturbations can be obtained by the first stability theorem of the mixed potential function theory;

In order to analyze in detail the influence of the line parameters of constant power loads on the stability of large signals in DC microgrids, the equivalent constant power load model shown in Fig. 2 is now taken as the object of study.

Figure 2.

Equivalent model of constant power load

3.2.2

Mixed potential function model for constant power loads

A damping filter circuit needs to be added to the filtering link of the constant power load to suppress resonance spikes, with the objective of improving the power quality of the DC bus voltage. Three types of damping filter structures are usually used: shunt RC structure, shunt RL structure, and series RL structure. However, only the shunt RC structure can effectively stabilize the system. The study of the effect of damping filter parameters on the steady state performance of the system can provide a reference basis for parameter selection for the DC microgrid.

The equivalent circuit model of the constant power load after adding the series damping circuit is shown in Fig. 3.

Figure 3.

Constant power load equivalent model with damping filter circuit

The damping filter circuit does not affect the steady state characteristics of the constant power load, so the steady state equilibrium operating point criterion of the system remains unchanged.

For the circuit shown in Fig. 3, model its hybrid potential function. The current potential function of the equivalent supply voltage u_dc, line resistance R₁ and damping resistance R₂ is: (18)udcicpl−12R1icpl2+(uCl−uC2)2R2$${u_{dc}}{i_{cpl}} - \frac{1}{2}{R_1}i_{cpl}^2 + \frac{{{{({u_{Cl}} - {u_{C2}})}^2}}}{{{R_2}}}$$

The current potential function of constant power load P_cpl is constant and the energy of capacitor C₁, C₂ is: (19)−uC1(icpl−(uC1−uC2)R2−PcpluC1)−uC2(uC1−uC2)/R2$$ - {u_{C1}}({i_{cpl}} - \frac{{({u_{C1}} - {u_{C2}})}}{{{R_2}}} - \frac{{{P_{cpl}}}}{{{u_{C1}}}}) - {u_{C2}}({u_{C1}} - {u_{C2}})/{R_2}$$

The unified model of the hybrid energy potential function is shown below: (20)P(i,u)=−12R1icpl2+udcicpl+(uC1−uC2)22R2+∫0uC1(Pcplu)du−uC1icpl$$P(i,u) = - \frac{1}{2}{R_1}i_{\:cpl}^2 + {u_{dc}}{i_{cpl}} + \frac{{{{({u_{C1}} - {u_{C2}})}^2}}}{{2{R_2}}} + \int_0^{{u_{C1}}} {(\frac{{{P_{cpl}}}}{u})du - {u_{C1}}{i_{cpl}}}$$

Examination of the developed hybrid energy potential function model, based on the model parameters with the circuit shown in Fig. 3, can be obtained: (21){ ∂P∂icpl=−R1icpl+udc−uCl=L1dicpldt ∂P∂uCl=uCl−uC2R2−icpl+Pcpl/uCl=−CduCldt ∂P∂uC2=−uC1−uC2R2=−C2duC2dt$$\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial P}}{{\partial {i_{cpl}}}} = - \:{R_1}{i_{cpl}} + {u_{dc}} - {u_{Cl}} = {L_1}\:\frac{{d{i_{cpl}}}}{{dt}}} \\ {\frac{{\partial P}}{{\partial {u_{Cl}}}} = \frac{{{u_{Cl}} - {u_{C2}}}}{{{R_2}}} - {i_{cpl}} + {P_{cpl}}/{u_{Cl}} = - \:C\:\frac{{d{u_{Cl}}}}{{dt}}} \\ {\frac{{\partial P}}{{\partial {u_{C2}}}} = - \:\frac{{{u_{C1}} - {u_{C2}}}}{{{R_2}}} = - \:{C_2}\:\frac{{d{u_{C2}}}}{{dt}}} \end{array}} \right.$$

The hybrid potential function of this system can be rewritten as: (22){ A(i)=12R1icpl2 B(u)=(uCl−uC2)22R2+∫0uCl(Pcpl/u)du (i,γ−αu)=udcicpl−uClicpl$$\left\{ {\begin{array}{*{20}{l}} {A(i) = \frac{1}{2}{R_1}i_{cpl}^2} \\ {B(u) = \frac{{{{({u_{Cl}} - {u_{C2}})}^2}}}{{2{R_2}}} + \int_0^{{u_{Cl}}} {({P_{cpl}}/u)} du} \\ {(i,\gamma - \alpha u) = {u_{dc}}{i_{cpl}} - {u_{Cl}}{i_{cpl}}} \end{array}} \right.$$

Solve for the value of A_ii(i), B_uu(u) as shown in Eq: (23){ Aii(i)=R1 Buu(u)=−PcpluCl2+1R2=−1Rcpl+1R2$$\left\{ {\begin{array}{*{20}{l}} {{A_{ii}}(i) = {R_1}} \\ {{B_{uu}}(u) = - \frac{{{P_{cpl}}}}{{{u_{Cl}}^2}} + \frac{1}{{{R_2}}} = - \frac{1}{{{R_{cpl}}}} + \frac{1}{{{R_2}}}} \end{array}} \right.$$

The values of the minimum eigenvalues μ₁ and μ₂ can be obtained as shown below: (24){ μ1=R1/L1 μ2=min(1C1(1R1+1−Rcpl),1C2(1R2+1−Rcpl))$$\left\{ {\begin{array}{*{20}{c}} {{\mu _1} = {R_1}/{L_1}} \\ {{\mu _2} = \min (\frac{1}{{{C_1}}}(\frac{1}{{{R_1}}} + \frac{1}{{ - {R_{cpl}}}}),\frac{1}{{{C_2}}}(\frac{1}{{{R_2}}} + \frac{1}{{ - {R_{cpl}}}}))} \end{array}} \right.$$

For the system working in the steady state equilibrium point, in order to make it can be restored to the stable operation state after being subjected to large perturbations, parameters μ₁ and μ₂ need to meet the requirements, and the solution is obtained: (25){ C1L1>1R1(−1R2−1−Rcpl) C2L1>1R1(−1R2−1−Rcpl)$$\left\{ {\begin{array}{*{20}{l}} {\frac{{{C_1}}}{{{L_1}}} > \frac{1}{{{R_1}}}( - \frac{1}{{{R_2}}} - \frac{1}{{ - {R_{cpl}}}})} \\ {\frac{{{C_2}}}{{{L_1}}} > \frac{1}{{{R_1}}}( - \frac{1}{{{R_2}}} - \frac{1}{{ - {R_{cpl}}}})} \end{array}} \right.$$

The large-signal stability criterion of the system after adding the damping filter link before the constant power load is shown in Eq. (26), which intuitively gives the design requirements of the damping filter circuit parameters. Combining the large-signal stability criterion formula with Eq. (3), it can be obtained: (26){ C1L1>1R1(−1R2−1−Rcpl) C2L1>1R1(−1R2−1−Rcpl) Pcpl<udc24R1$$\left\{ {\begin{array}{*{20}{l}} {\frac{{{C_1}}}{{{L_1}}} > \frac{1}{{{R_1}}}( - \frac{1}{{{R_2}}} - \frac{1}{{ - {R_{cpl}}}})} \\ {\frac{{{C_2}}}{{{L_1}}} > \frac{1}{{{R_1}}}( - \frac{1}{{{R_2}}} - \frac{1}{{ - {R_{cpl}}}})} \\ {{P_{cpl}} < \frac{{u_{dc}^2}}{{4{R_1}}}} \end{array}} \right.$$

If the values of the filtering parameters and the constant power load are satisfied, the system can maintain good steady state performance under large disturbances.

The equivalent power weight method considers the equivalent discharge amount of the battery during the charging and discharging cycles, and in the optimized operation of low-voltage DC microgrids, the battery has different charging and discharging powers and discharge depths at different moments, and the use of the equivalent power weight method can be used to effectively evaluate the battery life, and the battery life loss is calculated by the equivalent cumulative discharge amount: (27)Lloss=AcAtotal$${L_{loss}} = \frac{{{A_c}}}{{{A_{total}}}}$$

The equivalent cumulative discharge is expressed as: (28)Ac=λsoc⋅As$${A_c} = {\lambda _{soc}} \cdot {A_s}$$

Where, λ_soc is the weighting factor; A_s is the actual discharge (kW-h).

Weighting factor λ_soc is obtained from the following equation: (29)λsoc={ d1 0<SOC(t)≤0.5 d2⋅SOC(t)+d3 0.5<SOC(t)≤1$${\lambda _{soc}} = \left\{ {\begin{array}{*{20}{l}} {{d_1}}&{0 < SOC(t) \le 0.5} \\ {{d_2} \cdot SOC(t) + {d_3}}&{0.5 < SOC(t) \le 1} \end{array}} \right.$$

When 0 < SOC(t) ≤ 0.5, the weighting factor is related to the value of d₁ only. When 0.5 < SOC(t) ≤ 1, the weighting factor is related to the values of d₂ and d₃. In operation, the life loss of the battery is accumulated to get the battery life after t moments as: (30)Us=1∫01λsoc⋅As(t)Atotaldt$${U_s} = \frac{1}{{\int\limits_0^1 {\frac{{{\lambda _{soc}} \cdot {A_s}(t)}}{{{A_{{\text{total}}}}}}dt} }}$$

Objective function: the objective function is to maximize the daily net gain, minimize the total equivalent battery charge/discharge, and minimize CO₂ emissions. The overall objective function expression is as follows: (31){ f1=max∑t=1T(F(t)−Ccon(t)−Cq(t)) f2=min∑t=1T|PBd(t)|⋅Δt f3=min∑t=1N[(Pcon(t))+(Pcon(t))22Δt⋅eω2]$$\left\{ {\begin{array}{*{20}{c}} {{f_1} = \max \sum\limits_{t = 1}^T {(F(} t) - {C_{con}}(t) - {C_q}(t))} \\ {{f_2} = \min \sum\limits_{t = 1}^T {\left| {{P_{Bd}}(t)} \right|} \cdot \Delta t} \\ {{f_3} = \min \sum\limits_{t = 1}^N {\left[ {\frac{{\left( {{P_{con}}(t)} \right) + \sqrt {{{\left( {{P_{con}}(t)} \right)}^2}} }}{2}\Delta t \cdot {e^{{\omega _2}}}} \right]} } \end{array}} \right.$$ 1)

Maximum daily net gain f₁

After determining the capacity configuration of each unit, the cost of purchasing PV panels and batteries in the investment has been fixed, and in actual operation, the load, PV output and real-time electricity price are different every day, resulting in the purchase and sale of electricity and the amount of discarded light are different every day. Therefore, the daily net revenue is mainly related to the daily revenue from selling electricity to the customer load and the cost of purchasing and selling electricity from the grid through the grid converter and the cost of discarding light, and the objective function is: (32)f1=max∑t=1r(F(t)−Ccon(t)−Cq(t))$${f_1} = \max \sum\limits_{t = 1}^r {(F(} t) - {C_{con}}(t) - {C_q}(t))$$ (33)F(t)=Pload(t)⋅Δt⋅Rpricc(t)$$F(t) = {P_{load}}(t) \cdot \Delta t \cdot {R_{pricc}}(t)$$ (34)Ccon(t)={ Cbuy(t)⋅Pcon(t)⋅Δt,Pcon(t)≥0 Csale(t)⋅Pcon(t)⋅Δt,Pcon(t)<0$${C_{con}}(t) = \left\{ {\begin{array}{*{20}{l}} {{C_{buy}}(t) \cdot {P_{con}}(t) \cdot \Delta t,\quad {P_{con}}(t) \ge 0} \\ {{C_{sale}}(t) \cdot {P_{con}}(t) \cdot \Delta t,\quad {P_{con}}(t) < 0} \end{array}} \right.$$ (35)Cq(t)=c⋅Pq(t)⋅Δt$${C_q}(t) = c \cdot {P_q}(t) \cdot \Delta t$$

1)

Power balance constraints: (40)Pload(t)=Ppv(t)+Pcon(t)+PB(t)$${P_{load}}(t) = {P_{pv}}(t) + {P_{con}}(t) + {P_B}(t)$$

At t moment the load power is equal to the sum of the energy storage charging and discharging power, the PV output power and the power purchased and sold from the grid through the grid-connected converter.

Where I_c is the charging current of the battery (A); I_d is the discharge current of the battery (A); U_EV is the voltage of the battery during discharge (A); E_b is the rated capacity of the battery (kW-h). 3)

Battery charging and discharging power limit constraints

The charging and discharging power of the battery should be less than the rated power of the energy storage in the configuration results in Chapter 2, and charging and discharging power higher than the rated power will lead to high temperature of the battery and even explosion, which will have an impact on the safety and stability of operation.

where P_ratc is the power rating of the storage battery (kW). 4)

SOC constraints of the battery

where SOC_min, SOC_max is the upper and lower limits of the storage battery charge state, respectively; η_de is the energy conversion efficiency when the storage battery is discharged; η_ch is the energy conversion efficiency when the storage battery is charged. 5)

Grid-connected converter constraints

The grid-connected converter regulates the size of its transmission power based on the power command, and operates at the rated power when its power limit is reached.

Where P_con(t) is the power of the grid-connected converter at the moment of t (kW); P_reon is the rated power of the grid-connected converter (kW).

Multi-objective’s Gray Wolf Algorithm (MOGWO) [25] Compared with most of the traditional multi-objective algorithms, MOGWO has the advantages of fast convergence, high accuracy, easy to implement, etc., and it can solve the high-dimensional multi-objective optimization problems better. The flow of the multi-objective gray wolf algorithm is shown in Fig. 5.

Figure 5.

Multi-target gray Wolf algorithm flow

The specific process of MOGWO algorithm is as follows 1)

Set the control parameters, initialize the position of the wolves, and calculate the fitness.

(4) Update the iterative selection of wolf leaders (optimal, second and third), select the wolf with greater crowding distance by calculating the crowding distance, and perform the fitness calculation again to obtain the Pareto optimal solution set until the number of cycles is reached and the output is terminated.

The Pareto optimal solution set for multi-objective planning can be obtained by the multi-objective gray wolf algorithm, but the optimal solution cannot be obtained directly, and the optimal solution needs to be obtained by using the TOPSIS method, which is an objective weight assignment method that determines the size of the weights in accordance with the correlation between the objective values or the degree of change in the objective values when processing the data. The optimal solution is selected by evaluating the positive and negative ideal solutions of the problem and calculating the relative proximity of each solution to the positive or negative ideal solution. The specific steps are as follows: 1)

Form a normalized decision matrix based on the objective values of the Pareto optimal solution set generated by multi-objective optimization.

Substituting the relevant parameters into the large-signal stability criterion, the CPL values Re₁ and Re₂ of the two sub-microgrids are calculated to be limited to a minimum value of 7 Ω. Four DC microgrid large-signal perturbation scenarios are simulated in the experimental simulation, and the four perturbation scenarios are shown in Table 1.

Next, simulation experiments are carried out to validate the four cases according to Table 1, in which constant power loads CPL₁ and CPL₂ are connected at 1.5s and the two sub-microgrid bus voltages are observed to see if they can return to the equilibrium point after the perturbation. Figures 6 to 9 show the bus voltages for cases 1 to 4.

Figure 6.

Bus voltage in Case 1

Figure 7.

Bus voltage in Case 2

Figure 8.

Bus voltage in Case 3

Figure 9.

Bus voltage in Case 4

In Case 1, after the system is connected to the constant power loads CPL₁ and CPL₂ for 1.5 s, both sub-microgrid 1 and sub-microgrid 2 satisfy the large-signal stability criterion, and the bus voltage of sub-microgrid 1 decreases to 490 V and then rises to recover to 500 V. The bus voltage of sub-microgrid 2 decreases to 638 V and then rises to recover to 650 V.

After the system in Case 2 is connected to the constant power loads CPL₁ and CPL₂ at 1.5 s, sub-microgrid 1 satisfies the large-signal stability criterion, sub-microgrid 2 does not, and the bus voltage of sub-microgrid 1 hardly fluctuates and remains at 500 V. The bus voltage of sub-microgrid 2 plummeted to 150 V and did not recover to 650 V even after fluctuations. In case 3, after the system is connected to the constant power loads CPL₁ and CPL₂ for 1.5 s, sub-microgrid 1 does not satisfy the large-signal stability criterion, and sub-microgrid 2 does, and the bus voltage of sub-microgrid 1 drops to 170 V, and it does not recover to 500 V after fluctuations. The bus voltage of sub-microgrid 2 fluctuated slightly, but still remained at 650V overall. In case 4, after the system is connected to the constant power loads CPL₁ and CPL₂ at 1.5 s, both sub-microgrid 1 and sub-microgrid 2 do not satisfy the large-signal stability criterion, and the bus voltage of sub-microgrid 1 drops to about 160 V and does not recover to 500 V. The bus voltage of sub-microgrid 2 dropped to about 160V and did not recover to 650V.

Based on the above analysis, it can be seen that when the equivalent load in the sub-microgrid satisfies the large-signal stability criterion of the sub-microgrid, the bus voltage of the sub-microgrid returns to the equilibrium point after fluctuation and the system is stable. When the equivalent load in the sub-microgrid is less than the minimum allowable value, the bus voltage of the sub-microgrid will not return to the equilibrium point after fluctuation, and the system is in a collapsed state. Therefore, the calculated minimum allowable value of equivalent load R is correct, thus verifying the correctness of this stability criterion.

According to the obtained large-signal stability criterion formula, it can be seen that the improvement of the system large-signal stability can be realized by increasing the capacitance value or reducing the inductance value, and this conclusion is verified next. Firstly, to verify the effect of capacitance value on the system large-signal stability, set the system constant power load Re₁=5Ω, Re₂=21Ω, and on the basis of the experiments designed in the previous chapter, set the capacitance Cb₂ to 0.1F, 0.3F, 0.4F, 0.5F in turn, and other parameters remain unchanged, and the corresponding sub-microgrid 1 bus voltages in the four cases and the sub-microgrid 2 bus voltages in the four cases are shown in Figures 10 and Fig. 11.

Figure 10.

Busbar voltage of submicrogrid 1

Figure 11.

Busbar voltage of submicrogrid 2

When Cb₂=0.1F, the bus voltage of sub-microgrid 1 keeps dropping and is collapsed. When Cb₂=0.3F, the bus voltage drops up to 480 V and then returns to the equilibrium point of 500 V. When Cb₂=0.5F, the bus voltage drops up to 480 V and then returns to the equilibrium point of 500 V. It can be seen that, as the value of system capacitance increases, the system’s large signal stability increases. The bus voltage of sub-microgrid 2 in all four cases is stable because the CPL value of sub-microgrid 2 is 21Ω, which satisfies the stability criterion of sub-microgrid 2.

Then, to verify the effect of inductance value on the stability of large signal of the system, set the system constant power load Re₁=16Ω, Re₂=9Ω, and set the value of inductance L_t4 to 3mH, 2.9mH, 2.8mH, and 2.7mH in turn on the basis of the original experiments, other parameters remain unchanged, and the corresponding sub-microweb 1 bus voltage and sub-microweb 2 bus voltage of the four cases are shown in Fig. 12 and Fig. 13 shown.

Figure 12.

Busbar voltage of submicrogrid 1

Figure 13.

Busbar voltage of submicrogrid 2

The bus voltage of sub-microgrid 1 in all four cases is stable because the CPL value of sub-microgrid 1 is 16Ω, which satisfies the stability criterion of sub-microgrid 1. When L_t4=3mH, the bus voltage drops up to 632.5 V and then returns to the equilibrium point of 650 V. When L_t4 is reduced to 2.9 mH, the bus voltage drops to 634 V and still manages to return to the equilibrium point. When L_t4 is reduced to 2.8 mH and 2.7 mH, respectively, the bus voltage returns to the equilibrium point of 650 V after a slight drop. It can be seen that decreasing the value of the inductance of the system, can lead to an increase in the large signal stability of the system. This shows that it is effective to improve the system’s large-signal stability according to the obtained criteria, and indirectly proves the correctness of the criterion.

Based on the above analysis, it can be seen that the system can be improved based on the derived criterion of large signal stability, i.e., the large signal stability of the system can be improved by increasing the capacitance value of the system or decreasing the inductance value of the system.