In this paper, the doubly-fed induction generator (DFIG) with certain reactive power regulation ability is taken as the research object. Its output active power is described according to the change of wind speed, which can be expressed as (Kuang et al., 2022): P W T = 0 ,   v < v i n , v > v o u t P W T , r a t e v − v i n v r a t e − v i n ,   v i n ≤ v < v r a t e P W T , r a t e ,   v r a t e ≤ v < v o u t (1) where P _WT is the output active power of the WT; P _WT,rate is the rated active power of the WT; v _in and v _out are cut-in and cut-out wind speed, respectively; v _rate is rated wind speed.

As shown in Figure 1, when DFIG participates in reactive power regulation of the power system, its regulation ability is mainly determined by the capacity of the stator side and grid side rectifier. The output power constraints on the stator side are (Zhou et al., 2020): P W T , s 2 + Q W T , s 2 = 3 U W T , s I W T , s 2 (2) P W T , s 2 + U W T , s + 3 U W T , s 2 X W T , s 2 = 3 X W T , m X W T , s U W T , s I W T , R 2 (3) where P _WT,s and Q _WT,s are the active and reactive power output of the stator side of the WT, respectively; U _WT,s and I _WT,s are the voltage and current of the stator side of the fan, respectively; X _WT,s and X _WT,m are the leakage reactance and excitation reactance of the stator side of the fan, respectively; I _WT,R is the rectifier current on the rotor side of the fan. Ignoring the reactive power consumed by the rectifier, the actual active and reactive power outputs of the DFIG are: P W T , T = 1 − w P W T , s , Q W T , T ≈ Q W T , s (4) where w is the slip rate of DFIGURE After the actual output power of DFIG is determined according to Eq. 1, its reactive power regulation capability (Q _WT,T,min, Q _WT,T,max) can be determined as: Q k 1 − Q k 2 ≤ Q W T , T ≤ Q k 1 + Q k 2 (5) Q k 1 = − 3 U W T , s 2 X W T , s Q k 2 = 3 X W T , m X W T , s U W T , s I W T , R , max 2 − P W T , T 1 − s (6) where I _WT,R,max represents the maximum rectifier current on the rotor side.

As shown in Figure 2, the photovoltaic unit is mainly composed of a photovoltaic panel, AC-DC rectifier, DC-AC inverter, filter circuit and control system. The AC-DC rectifier is used for boosting to realize the maximum power point tracking (MPPT) control, and then connected to the grid through a DC-AC inverter. Its output power at the grid-connected point can be expressed as (Ai and Ding, 2020): P P V = k P V U P V U P C C G P V ⁡ cos θ P V − θ P C C + B P V ⁡ sin θ P V − θ P C C (7) Q P V = k P V U P V U P C C G P V ⁡ sin θ P V − θ P C C − B P V ⁡ cos θ P V − θ P C C (8)

In the above equations, k _PV is the modulation ratio of the photovoltaic inverter, which is used to restrict the output voltage of the photovoltaic unit and ranges from (0, 1); U _PV and U _PCC are the voltage amplitudes of the photovoltaic unit and its grid-connected point, respectively; G _PV and B _PV are the equivalent conductance and susceptance of the photovoltaic unit, respectively. In addition, the output power of the photovoltaic unit is also constrained by the grid-connected point voltage and capacity.

Defining the equivalent impedance of the photovoltaic unit as: Z P V ∠ θ Z = 1 G P V + j B P V (9)

We can formulate the power constraint of the grid-connected point as follows. P P C C + j Q P C C = k P V U P C C U P V Z P V ∠ θ P C C + θ Z + θ P V − U P C C 2 / Z P V ∠ θ Z (10)

The renewable-based power system model mainly describes the power balance at the buses and branches. Considering the photovoltaic and wind power integration, they can be expressed as: P i = U i ∑ j U j G i j ⁡ cos ⁡ θ i j + B i j ⁡ sin ⁡ θ i j P i = P G , i + P P V , i + P W T , i − P L , i (11) P i = U i ∑ j U j G i j ⁡ cos ⁡ θ i j + B i j ⁡ sin ⁡ θ i j P i = P G , i + P P V , i + P W T , i − P L , i (12) P i j = U i U j G i j ⁡ cos ⁡ θ i j + B i j ⁡ sin ⁡ θ i j − G i j U i 2 (13) Q i j = U i U j G i j ⁡ sin ⁡ θ i j − B i j ⁡ cos ⁡ θ i j + B i j U i 2 (14) where U _i , θ _i are the voltage amplitude and phase angle of bus i; P _i and Q _i are the net injected active and reactive power of bus i; P _G,i , Q _G,i are the active and reactive power outputs of the generator at bus i; P _L,i , Q _L,i are the active and reactive loads at bus i; P _PV,i , Q _PV,i are the active and reactive power outputs of the photovoltaic unit at bus i; P _WT,i , Q _WT,i are the active and reactive power output of the WT at bus i; G _ij and B _ij are the conductance and susceptance between buses i, j; P _ij , Q _ij are the active and reactive power flow between buses i, j.

Considering the stability, cleanliness and economic requirements of power system operation, this paper establishes an objective function with the objectives of minimizing operational costs, renewable curtailment, and voltage offset.

The objective function f ₁ represents the objective of minimizing the operational cost, which is expressed as: min f 1 = ∑ i α 1 , i P G , i 2 + α 2 , i P G , i + α 3 , i (15) where α _1-3 are the constant coefficients of the generator set.

The objective function f ₂ represents the objective of minimizing the renewable curtailment, which is expressed as: min f 2 = ∑ t 1 − ∑ η P V , t P P V , t + ∑ η W T , t P W T , t ∑ t P P V , t + ∑ t P W T , t (16) where η _PV,t and η _WT,t are the curtailment rates of photovoltaic units and wind turbines at time t, respectively.

The objective function f ₃ represents the objective of minimizing voltage fluctuation, which is expressed as: min f 3 = ∑ i U i − U i , r a t e 2 (17) where U _i,rate represents the reference voltage amplitude of bus i. In this paper, the penalty factors are introduced to integrate the three types of objective functions, i.e., renewable curtailment and voltage offset will undertake certain economic losses. In this condition, the multiple objective functions can be reconstructed as: min   f = ∑ t c P V , t 1 − ∑ η P V , t P P V , t ∑ t P P V , t + ∑ t c W T , t 1 − ∑ η W T , t P W T , t ∑ t P W T , t + ∑ i c i U i − U i , r a t e 2 + f 1 (18) where c _PV, t and c _WT,t are the unit penalty costs of abandoning solar power and wind power at time t, respectively; c _i is the unit penalty cost of the voltage fluctuation at bus i.

In large-scale power systems, the computational burden of the optimization problem considering the detailed AC power flow model is very considerable, which is not conducive to a direct solution. Therefore, this paper adopts an improved DC power flow model for transmission networks to simplify the non-linear equations in Eqs 11–14. In the transmission network, the differences between phase angles of different buses are usually very small, and the voltage amplitude is often close to 1.0 p.u. Based on these simplifications, we can use Taylor expansion to simplify Eqs 11–14, as shown below. sin ⁡ θ i j ≈ θ i j , cos ⁡ θ i j ≈ 1 − θ i j 2 2 (19) U i U j θ i j ≈ θ i j , U i U j θ i j 2 ≈ θ i j 2 (20)

Substituting Eqs 19, 20 into the first equation in Eqs 11, 12, we can get (Zhang et al., 2022): P i j = G i j U i 2 − U j 2 2 − B i j θ i j + G i j 2 θ i j 2 + U i j 2 Q i j = B i j U j 2 − U i 2 2 − G i j θ i j − B i j 2 θ i j 2 + U i j 2 (21) P i = ∑ j P i j + ∑ j G i j U i 2 , Q i = ∑ j Q i j − ∑ j B i j U i 2 (22)

In addition, the reactive power regulation of the power system needs to meet the following security constraints: U i , min ≤ U i ≤ U i , max (23) P G , i , min ≤ P G , i ≤ P G , i , max , Q G , i , min ≤ Q G , i ≤ Q G , i , max (24) P i j 2 + Q i j 2 ≤ S i j , max 2 (25) where U _i,max and U _i,min represent the upper and lower limits of the voltage amplitude of bus i, respectively; P _G,i,max and P _G,i,min represent the upper and lower limits of the active power output of the generator at bus i, respectively; Q _G,i,max and Q _G,i,min represent the upper and lower limits of the reactive power output of the generator at bus i, respectively; S _ij,max represents the maximum branch apparent power flow between buses i, j.

Besides the reactive power constraints, the power system operator can determine the actual consumed wind power at each time step, which is expressed as: 0 ≤ P W T , t ≤ P W T , t , max (26) where P _WT,t,max represents the maximum output active power of WT at time t. For photovoltaic units, the model described in Section 2.2 presents the strong non-linearity, which is not conducive to solving the optimization model. Therefore, this section further analyzes the operating characteristics of the grid-connected photovoltaic unit, to appropriately simplify the model and improve the solvability of the corresponding optimization problem.

In general, the equivalent impedance Z _PV of the photovoltaic unit is a constant, and the voltage amplitude U _PCC of the grid-connected point fluctuates very little. Therefore, the term (U _PCC )²/Z _PV can be regarded as a constant (Mao et al., 2019). By characterizing Eq. 10 to the polar coordinate domain, the left side of the equal sign can be regarded as the operating domain composed of the capacity of the photo-voltaic unit, as shown in the blue solid line section of Figure 3. The right side of Eq. 10 can be regarded as the operating domain formed by the voltage amplitude constraint of the grid-connected point.

Mathematically, the operating domain described on the right side of Eq. 10 can be understood as follows. When (U _PCC )²/Z _PV is fixed, the operating domain of the photo-voltaic unit is a circle formed by taking (U _PCC )²/Z _PV ∠θ _Z as the center and k _PV U _PCC U _PV /Z _PV as the radius. The radius of the operating interval is determined by the modulation ratio of the photovoltaic inverter and the voltage amplitude of the photovoltaic unit, as shown in the blue dotted part of Figure 3. Therefore, the basic operating range of the photovoltaic unit can be regarded as the intersection of the blue solid line and the blue dotted line. Further, considering the harmonic and frequency constraints (Zhao et al., 2018; Zhao and Chen, 2021), the operating constraints of the PV unit can be further characterized as Eqs 27, 28. Q P V = S P V , max 2 − P P V   P P V > α S P V , max 0 , P P V ≤ α S P V , max (27) 0 ≤ P P V , t ≤ P P V , t , max (28) where S _PV,max represents the maximum rated capacity of rectifier and inverter considering grid-connected harmonic and frequency constraints, P _PV,t,max represents the maximum output active power of photovoltaic unit at time t, α represents the critical ratio of output active power considering grid-connected harmonic and frequency constraints, as shown in the red part of Figure 3.

Combined with Section 3.1 to Section 3.3, the multi-objective reactive power optimization model of power system suitable considering large-scale renewable integration in this paper can be summarized as follows: min   f P G , i , t , Q G , i , t , P P V , i , t , Q P V , i , t , P W T , i , t , Q W T , i , t s u b j e c t   t o   21 , 22 , 23 , 24 , 25 5 , 6 , 26 27 , 28 (29) where, the first row of the constraint corresponds to those from the power system side, the second row corresponds to those from the wind turbine side, and the third row corresponds to those from the photovoltaic unit side. The variables to be optimized include the active and reactive power output of the generator at each time step, the actually-consumed active power of the WT and PV unit, and the output active power of the renewable energy participating in the voltage regulation.

In this part, the impacts of renewable integration on reactive power optimization and voltage regulation are analyzed. The simulation results are shown in Figure 6 and Table 1. According to Table 1, almost all the renewable generation realizes 100% consumption of solar and wind power in different scenarios. This is because the use of photovoltaic units and WTs in the cases does not need to undertake its operational costs, and the unit penalty of renewable curtailment is distinct. As a result, the operators are more inclined to consume renewable energy for power supply, thereby reducing the operational cost of the traditional generators. However, due to the integration of renewable energy, the system presents a multi-source state, and its voltage level fluctuates more significantly. Therefore, the voltage offset in scenario 2 is larger and the maximum value is up to 0.054 p.u. Correspondingly, the penalty of voltage offset in scenario 2 is higher than that in scenario 1. On the contrast, scenario 1 makes full use of the reactive power regulation capability of the WTs and PV units. In this condition, the active and reactive output of the WTs and PV units are optimized according to the operational constraints in the power system. Moreover, due to the renewables’ participation in reactive power compensation and voltage regulation, the maximum voltage deviation in the power system is reduced by 67%, as shown in Figure 6A. As the voltage offset penalty in scenario 1 is reduced by 597.81$, the total operational cost is also significantly reduced by about 6.4%. In particular, the impact of renewable integration on voltage regulation also varies distinctly according to the different constraints of voltage amplitude and their reference values. For example, the photovoltaic units connected to each bus all act as the reactive load and participate in the voltage regulation. On the contrary, the WTs connected to the bus 5 and bus 30 act as the reactive power sources and then participate in the voltage regulation, as shown in the comparison between Figures 6B, C.

According to the simulation results, although the large-scale integration of renewable energy will affect the active power distribution in the power system, the rational use of its reactive power regulation ability can effectively improve the overall voltage level, thereby improving operational stability.

As for the solution time, the solver IPOPT takes 44.3 and 38.7 s to solve the problems in scenarios 1 and 2, respectively. Although the non-linearity in the power flow equations will bring some computational burden, the results indicate that the optimization problem can still be solved effectively with IPOPT. Besides, there are no distinct differences between scenarios 1 and 2 in solution time. This is understandable because the reactive power regulation of the renewables only adds several linear constraints into the optimization problem, while the problem scale is slightly influenced.

This section further analyzes the impact of renewable energy participation in reactive power regulation on power system operation by setting different unit penalties of renewable curtailment and voltage offset. Firstly, setting the unit penalty of renewable curtailment as constant, we test how total operational cost varies with the unit penalty of voltage offset. The optimized results are shown in Figure 7.

According to Figure 7, the various operational costs in scenario 1 all gradually increase with the increment of unit penalty of voltage offset. When the unit penalty of voltage offset increases to 550$/p.u., the increment of f ₃ is greater than the sum of those corresponding to f ₁ and f ₂. Therefore, in the subsequent operational optimization, the power system aims to minimize the voltage offset cost, and different state variables mainly remain stable because the traditional generators have reached their maximum ability for regulation. In scenario 2, since the renewable energy is not involved in reactive power optimization, the power system’s ability to regulate the voltage level is limited. As a result, the overall reactive power capacity of conventional units in the power system cannot meet the demand of minimizing voltage offset. Therefore, as the unit penalty of voltage offset increases, all kinds of operational costs continue to increase, and the corresponding growth rates are larger than those in scenario 1. To summarize, large-scale renewable integration provides a certain reactive power capacity for operational optimization and voltage regulation of the power system. This helps to improve the operation economy of the system, and the improvement is gradually obvious with the increasing demand for voltage stability.

Then, we firstly set the unit penalty cost of voltage offset to 400 $/p.u. and keep it constant. By gradually increasing the unit penalty cost of renewable curtailment, the impact of renewable integration on reactive power optimization is studied. The optimization results are shown in Figure 8. It can be seen from the figure that the variation in unit penalty cost of renewable curtailment has little effect on the operational cost in this case. Consequently, its impact on the total operational cost is almost negligible. This is because, in this case, the consumption of solar and wind power does not need to undertake additional costs. In this context, the power system is always in the operation state of 100% renewable energy consumption. Therefore, no matter how the unit penalty cost of renewable curtailment changes, the required output of conventional generating units is almost constant and the total operational cost remains constant. However, as the unit cost of renewable curtailment gradually increases, the power system gradually optimizes its output active and reactive power to reduce the penalty of solar and wind power curtailment in scenario 1. This adjustment affects the voltage level correspondingly, causing the voltage offset penalty in scenario 1 also gradually increases, as shown in Figure 8B. In summary, without considering the cost of renewable energy generation, the voltage offset penalty is the main factor affecting the system operational cost. From this point of view, the participation of renewable energy in reactive power optimization has a significant improvement in the economy and stability of operation as well.