In recent years, with the application and popularization of automatic voltage control technology, system operators can achieve voltage control of distribution network feeder nodes by adjusting the reactive power output of generator units in the transmission network. Due to the continuous regulation of reactive power, the shortcoming of OLTC, i.e., discontinuous voltage regulation, is overcome, and the mechanical equipment damage caused by multiple OLTC switching is also avoided. At this point, the VSLs in the main network can be seen as a ZIP load and connected to the transmission network through OLTC.

In previous studies on voltage reduction, energy conservation, and load side frequency response (Wang and Wang, 2014), the ZIP model has been widely used to characterize the relationship between voltage changes and power adjustment in VSLs. This paper mainly considers the real-time steady-state scheduling problem, and the load structure can be considered to remain unchanged in the short term. Therefore, the ZIP model is used to characterize the electricity consumption characteristics of VSLs, which can meet the accuracy requirements of actual scheduling. Its mathematical expression is shown in Eqs 1, 2: P i , t L = P i , t B a i , t P k 2 + b i , t P k + c i , t P Q i , t L = Q i , t B a i , t Q k 2 + b i , t Q k + c i , t Q (1) k = U i , t / U N , i = 1 , 2 , . . . , N B (2)

Here, a i , t P , b i , t P and c i , t P represent the coefficients of the quadratic, linear, and constant terms of the active power injected by nodes in the ZIP model, with values of 0.2, 0.3, and 0.5 respectively in this paper. a i , t Q , b i , t Q and c i , t Q represent the coefficients of the quadratic, linear, and constant terms of the reactive power injected by nodes in the ZIP model, with values of 0.2, 0.3, and 0.5 respectively according to reference (Tang et al., 2018) and this is a typical set of ZIP modeling data for analyzing the scheduling potential of VSLs. P i , t L and Q i , t L represent the active/reactive injection of load node i at time t, respectively; U N represents the rated voltage value; U i , t represents the voltage amplitude of VSL access node i at time t; N B represents the number of VSL nodes in the power system.

Although continuous control of node voltage can be effectively achieved by controlling the generator units and reactive power regulation equipment in the transmission network, unreasonable voltage control may affect the safe operation of the system. Meanwhile, the integration of renewable energy such as wind power further reduces the margin for safe operation. Therefore, voltage safety related constraints must be considered in the real-time scheduling process of VSLs. This section designs the voltage safety margin (VSM) index and derives the sensitivity relationship between node power changes and VSM index. The specific mathematical derivation process is as follows:

Firstly, based on the node admittance matrix relationship, establish the system voltage current network equation, as shown in Eq. 3: I G I L = Y GG Y GL Y LG Y LL U G U L (3)

Here, I G and U G respectively represent the current and voltage vectors of the generator node; I L and U L respectively represent the voltage and current vectors of load nodes. The matrix elements Y GG , Y GL , Y LG and Y LL can be calculated according to reference (Kessel and Glavitsch, 1986). After matrix transformation, Eq. 3 can be equivalently written in the form shown in Eq. 4: I G U L = Y GG − Y GL Y LL − 1 Y LG Y GL Y LL − 1 − Y LL − 1 Y LG Y LL − 1 U G I L (4)

Let F = − Y LL − 1 Y LG , and the calculation formula for VSM is shown in Eq. 5: VSM j = 1 − ∑ k ∈ a G F j k U k / U j (5)

It can be observed in Eq. 5 that the VSM index needs to calculate the modulus of the complex number 1 − ∑ k ∈ a G F j k U k / U j . In order to facilitate analysis, the complex term VSM C , j is first defined as Eq. 6: VSM C , j = 1 − ∑ k ∈ a G F j k U k / U j (6)

According to the complex modulus calculation rule, Eq. 5 can be transformed into the form shown in Eq. 7: VSM j = Re VSM C , j 2 + Im VSM C , j 2 (7)

When the system control variables change, the small change △VSM at the current operating point can be calculated according to Eq. 8: Δ VSM j = Re VSM C , j Δ Re VSM C , j + Im VSM C , j Δ Im VSM C , j Re VSM C , j 2 + Im VSM C , j 2 = Re VSM C , j VSM j Δ Re VSM C , j + Im VSM C , j VSM j Δ Im VSM C , j (8)

The real and imaginary parts of VSM C , j are shown in Eq. 9: Re VSM C , j = 1 − ∑ k ∈ a G F j k U k / U j ⁡ cos δ j k + δ k − δ j Im VSM C , j = − ∑ k ∈ a G F j k U k / U j ⁡ sin δ j k + δ k − δ j (9)

The partial derivative accumulation of state variables, namely, voltage and phase angle, can be obtained as Eq. 10: Δ Re VSM C , j = ∑ k ∈ a G F j k U k / U j 2 ⁡ cos δ j k + δ k − δ j Δ U j + ∑ k ∈ a G F j k U k / U j ⁡ sin δ j k + δ k − δ j Δ δ k − Δ δ j Δ Im VSM C , j = ∑ k ∈ a G F j k U k / U j 2 ⁡ sin δ j k + δ k − δ j Δ U j − ∑ k ∈ a G F j k U k / U j ⁡ cos φ j k + δ k − δ j Δ δ k − Δ δ j (10)

Substituting Eq. 10 into Eqs 8, 11 can be obtained: Δ VSM j = 1 VSM j U j Re VSM C , j ∑ k ∈ a G Re F j k U k / U j + Im VSM C , j ∑ k ∈ a G Im F j k U k / U j Δ U j − 1 VSM j Re VSM C , j ∑ k ∈ a G Im F j k U k / U j − Im VSM C , j ∑ k ∈ a G Re F j k U k / U j Δ δ j + ∑ k ∈ a G 1 VSM j Re VSM C , j Im F j k U k / U j − Im VSM C , j ∑ k ∈ a G Re F j k U k / U j Δ δ k (11)

Eq. 11 describes the sensitivity relationship between node voltage and phase angle changes and VSM index changes, where K represents the determinant and matrix elements can be obtained according to Eq. 11. Here, Eq. 11 is written in the compact form shown in Eq. 12: Δ VSM j = K Δ δ , Δ U T , K = K 1 , K 2 (12)

In order to intuitively characterize the impact of node power changes on VSM index, the power flow equation is first introduced here, as shown in Eq. 13: Δ δ = M Δ P + N Δ Q Δ U = L Δ P + S Δ Q (13)

The sensitivity relationship between node power variation and VSM index can be obtained by combining Eqs 12, 13, as shown in Eq. 14: Δ VSM = K 1 M + K 2 L Δ P + K 1 N + K 2 S Δ Q (14)

As mentioned earlier, the regulation of VSLs is mainly achieved through voltage control. During steady-state operation, the adjustable range of voltage at each node is generally allowed to be between 0.95p.u. and 1.05p.u., with the limitation of VSM index, it will further lead to a reduction in the adjustable range of voltage. Therefore, it is necessary to first evaluate the adjustable range of voltage at each node under the current operating point. Corresponding mathematical model for VSLs’ adjustable margin evaluation is shown in Eqs 16–23: o b j : min ∑ i = 1 N B P i , t L and max ∑ i = 1 N B P i , t L (16) s . t . Δ P Δ Q = J Δ δ Δ U / U (17) Δ P i , t Δ Q i , t = P i , t G Q i , t G + Δ P i , t G Δ Q i , t G − P i , t S Q i , t S , i = 1 , 2 , . . . , N G (18) R i G , d ≤ Δ P i , t G ≤ R i G , u , i = 1 , 2 , . . . , N G (19) U i min δ i min ≤ U i , t δ i , t + Δ U i , t Δ δ i , t ≤ U i max δ i max , i = 1 , 2 , . . . , N A (20) P i G , ⁡ min Q i G , ⁡ min ≤ P i , t G Q i , t G + Δ P i , t G Δ Q i , t G ≤ P i G , ⁡ max Q i G , ⁡ max , i = 1 , 2 , . . . , N G (21) VSM + Δ VSM ≤ VSM max (22) Δ P i , t L Δ Q i , t L = 2 P i , t B a i , t P Δ U i , t U N 2 Q i , t B a i , t Q Δ U i , t U N + P i , t B b i , t P Δ U i , t U N Q i , t B b i , t Q Δ U i , t U N , i = 1 , 2 , . . . , N B (23)

The objective function (16) and related constraints (17)–(23) construct a VSLs’ adjustable range evaluation model. The meaning of objective function (16) is to obtain the lower and upper boundary of the adjustable power of VSLs at the current operating point, in order to ensure the maximum power adjustable range of VSLs; (17) represents the power flow constraint. In (17), J represents the Jacobian matrix at the convergence of the power flow calculation iteration; (18) represents the balance constraint of active and reactive power at the generator node, where Δ P i , t G and Δ Q i , t G represent the reactive power and active output change of the generator, P i , t S and Q i , t S represent the active and reactive power of the load connected to the generator node; (19) represents the climbing constraint of generator units, where R i G , d and R i G , u represents the lower and upper climbing limits of the generator units, respectively; (20) represents the adjustment constraints for node voltage and phase angle and N A represents the number of all nodes; (21) represents the active and reactive power output constraints of generator units, respectively; (22) represents the related constraints of VSM index. The system dispatcher ensures voltage safety during the VSLs regulation process by setting a VSM index in advance; (23) represents the linear relationship constraint between active power, reactive power, and voltage in the equivalent ZIP model.

By combining the real-time adjustable range evaluation model of VSLs in Section 3.1, the upper and lower power limits of for VSLs can be obtained. For VSLs’ regulation, its essence is to use adjustable VSL resources as flexible backup and supplementary resources for units within the system. Here, the discussion is divided into two scenarios:

Scenario 1: when the actual output of wind power is lower than the short-term forecast value, the power demand on the load side is larger than the sum of the planned output of the units and wind power. If VSLs do not participate in economic dispatch, in order to maintain real-time power balance, it is necessary to increase the output of units in the system or carry out load shedding operations, which increases the operating cost of the system and reduces user satisfaction with electricity consumption. When VSLs participate in scheduling, by reducing the voltage value of VSLs access nodes, the power consumption of VSLs is reduced and real-time power balance is achieved, which is both economical and environmentally friendly.

Scenario 2: When the actual output of wind power is higher than the short-term forecast value, the sum of the planned output of units and wind power is larger than the power demand on the load side. At this time, the conventional operation is to reduce the output of units in the system to maintain real-time power balance. In this case, if VSLs participate in scheduling, power control can be achieved by increasing the voltage of the VSLs’ access node. However, there are two drawbacks: Firstly, for the user side, VSLs’ regulation is a passive scheduling method. When VSLs’ power is increased, it will increase the user’s electricity consumption and cost, which is unreasonable; secondly, for the power supply side, increasing VSLs’ power to achieve renewable energy consumption has poor economic and environmental benefits. However, in some extreme scenarios, such as when the system communication fails and the output of the generators cannot be adjusted, it can also be coordinated by increasing the VSLs’ power for scheduling. However, the probability of these extreme scenarios occurring is very low. In order to maintain generality, in this paper, the focus is on examining and analyzing the VSLs’ scheduling effect under scenario 1.

Based on the adjustable range of VSLs obtained in the previous section, a real-time economic dispatch model can be constructed, as shown in Eqs 24–31: min   ∑ i = 1 N G F i G P i , t G + ∑ i = 1 N B l C P i , t C , L + ∑ i = 1 N S l C P i , t S , L (24) s . t . 17 , 19 ‐ 23 (25) ∑ i = 1 N G P i , t G + ∑ i = 1 N W P i , t W = ∑ i = 1 N B P i , 0 B + Δ P i , t L − P i , t C , L + ∑ i = 1 N S P i , t S − P i , t C , S + P t loss (26) 0 ≤ P i , t C , L ≤ P i , 0 B + Δ P i , t L , i = 1 , . . . , N B 0 ≤ P i , t C , S ≤ P i , t S , i = 1 , . . . , N S (27) 0 ≤ P i , t W ≤ P i , t W , ⁡ max , i = 1 , . . . , N W (28) Δ P i , t Δ Q i , t = P i , t G Q i , t G + Δ P i , t G Δ Q i , t G − P i , t S − P i , t C , S tan φ i , t P i , t S , C , i = 1 , 2 , . . . , N S (29) Δ P i , t Δ Q i , t = − Δ P i , t L Δ Q i , t L + P i , t C , L tan φ i , t P i , t C , L , i = 1 , 2 , . . . , N B (30) P i , t L , ⁡ min ≤ P i , t L + Δ P i , t L ≤ P i , t L , ⁡ max , i = 1 , 2 , . . . , N B (31)

The objective function (24) represents the minimization of operating costs, consisting of three parts. The first term represents the operation cost of generator units, which can be calculated based on the quadratic function of the generation cost. The second and third terms represent the cost of load shedding penalty, where l C represents the load shedding penalty coefficient and P i , t C , L represents the load shedding amount of the ith VSL node at time t; P i , t C , S represents the load shedding amount of the connected load on the ith generator node; (25) include some constraints that have already appeared in the VSLs’ adjustable range evaluation process, including the ZIP model, constraint of VSM index, power flow constraint, generator ramping constraint, generator output constraint, node voltage and angle adjustment range constraints; (26) represents power balance constraint. (27) represents the load shedding range constraint; (28) represents wind power output constraint; (29) represents the power balance constraint of the generator node; (30) represents the power balance constraint of VSLs’ nodes; (31) represents the adjustable range constraint of VSLs, where P i , t L , ⁡ min and P i , t L , ⁡ max respectively represent the lower and upper limits of power regulation for the ith VSL’s node at time t. The specific values can be calculated according to Eqs 16–23.

By combining the current power imbalance deficit and wind power consumption demand of the system, solving models Eqs 24–31 can obtain the optimal VSL control value under the current operating point. To gain a better understanding, a detailed flow chart of optimal economic dispatching strategy considering VSLs is shown in Figure 1.

This paper uses an improved IEEE-30 node system for case verification. The testing system topology is shown in Figure 2, and some load nodes are regarded as controllable VSLs (i.e., red triangles in the figure). The power consumption of each VSL is controlled by adjusting the voltage amplitude. The base-state active reactive power of each VSL access node is shown in Table 1. The proposed real-time evaluation scheduling model is written using MATLAB 2016b nested YALMIP and solved using the CPLEX 12.10.0. The desktop computer installed with the software is configured with an Intel TM i5-10400 CPU and a processor of 2.90 GHz.

This section focuses on analyzing the scheduling potential of VSLs and the impact of VSM index on the regulation range of VSLs. Firstly, Figures 3, 4 respectively show the VSLs’ power and corresponding access point voltage in the base-state operation scenario 1, maximizing VSLs’ adjustable power scenario 2, and minimizing VSLs’ adjustable power scenario 3.

Observing Figures 3, 4, combined with Eq. 1, it can be concluded that the adjustable range of VSLs mainly depends on two factors: firstly, the power of each VSL fluctuates upper and below the base-state operating point, and for VSLs with larger base-state operating power, their adjustable range is often larger; secondly, some nodes have a higher risk of voltage instability, so their voltage adjustable range is smaller, and the controllable power range of VSL is smaller. In order to provide a more intuitive analysis of the scheduling potential of VSL, Table 2 shows the VSLs’ power regulation ranges for three different scenarios when the value of VSMmax is 0.15. During maximum upper power operation, the total power of VSL increases by 5.31 MW compared to base-state operation, with a power variation of 4.55%. When operating at maximum lower power operation, the total power of VSLs decreases by 5.20 MW compared to ground state operation, with a power variation of 4.45%. Overall, through voltage control, while ensuring system safety, the newly added regulation range of VSLs accounts for 9% of the total power of base-state operating load, which has great scheduling potential and value. In addition, with the help of reactive power control equipment, node voltage can be adjusted on a second level time scale. The current real-time scheduling time scale in China is 5–15 min, therefore VSLs have a fast response speed and can meet the needs of real-time applications without involving privacy protection issues.

Previously, scheduling strategies for VSLs often ignored the impact of the regulation process on voltage safety (Lauria, 1987; Wang and Wang, 2014). In order to quantitatively analyze the impact of VSM index on the VSLs’ power regulation range, Table 3 provides the maximum upper power regulation amount and maximum lower power regulation amount of VSLs under different VSM index.

Under different values of VSM index, the maximum power of VSLs remains basically unchanged, but the maximum power decreases as the value of VSM index decreases. In order to facilitate the explanation of the reasons for this phenomenon, Figure 5 provides a schematic diagram of the voltage feasible region under different constraints. The voltage feasible range of each load node is determined jointly by constraints (20) and (22). The voltage feasible range in the blue area of Figure 5 is determined by constraint (20). When the VSM index is not set in the system or the setting value is large, the voltage feasible range determined by voltage safety constraint (22) is in the right area (i.e., green line in the figure). At this time, constraint (22) is ineffective constraint, and the maximum reduction power of VSLs remains unchanged; as the VSM index gradually decreases, both constraints (22) and (20) are effective constraints. The voltage feasible range determined by constraint (22) is V 2 , + ∞ , which is the yellow line in the figure. The intersection with the voltage feasible range determined by constraint (20) is V 2 , V max , indicating that the voltage feasible range is compressed, the lower boundary of the voltage feasible range is raised, and the maximum power reduction of VSLs is reduced. Therefore, for power system dispatchers, the value of the VSM index should not be too small. When its value is too small, the voltage feasible range determined by constraint (22) is V 3 , + ∞ , as shown in the red line. At this point, the intersection with the voltage feasible range determined by constraint (20) is an empty set, and the optimization problem has no solution. Especially in scenarios with multiple renewable energy inputs and heavy loads, the critical safety value of voltage is already at a high level. In this case, setting a smaller VSM index can easily lead to unsolvable economic scheduling problems. Combined with the data given in Table 3, it can be found that when the VSMmax value is within the range of [0.15, 0.20], the voltage feasible region is the blue region in Figure 5. In this case, the maximum upper and lower power adjustment amounts remain basically unchanged. When the VSMmax value is within the range of [0.15, 0.043), the maximum low power adjustment amount gradually decreases. When the VSMmax value is less than 0.043, there is no feasible solution, indicating that the scheduling decision is too conservative.

In order to provide a more intuitive explanation of the real-time scheduling effect of VSLs, two common operating scenarios with different power shortages have been set up here. The specific information is as follows:

Case 1: Due to the daily prediction error of wind power output and load power, the demand for load side power consumption is 10 MW larger than the sum of the planned output of units and the actual wind power; Case 2: Due to the daily prediction error of wind power output and load power, the demand for load side power consumption is 4 MW greater than the sum of the planned output of units and the actual wind power; In the above two operating scenarios, the power adjustment of VSLs and the output of generator units within the system are shown in Figures 6, 7, respectively. In Case-1, the power shortfall on the power and load sides is 10 MW. Based on Table 2, it can be seen that the maximum reduction power of VSLs under the current operating point is 5.20 MW. This indicates that even when all VSLs’ power is adjusted to the minimum value, it is still unable to fill the current power shortfall, and there is a power shortfall of 4.80 MW. Therefore, it is necessary to adjust the units’ output in the system to maintain real-time power balance. As shown in Figure 6B, the output of units when VSLs participate in regulation is significantly lower than that when VSLs do not participate in dispatch. This is because after VSLs participate in dispatch, the real-time power shortfall of the system decreases, and the generator units only need to slightly increase its output to maintain real-time power balance. From the perspective of economic investment, when VSLs do not participate in scheduling, the operating cost under the current point is 579.75 $. However, when VSLs participate in scheduling, the operating cost of the system decreases to 555.83 $, which means that the power supply pressure is reduced through VSL regulation, while reducing the cost by 23.92 $, and avoiding frequent adjustments to the planned output of units. Due to the low base-state operating point of the VSLs, the corresponding adjustable power of VSLs is also small. In heavy load scenarios, the economic benefits brought by VSL regulation will be more considerable. In Case 2, the current power shortfall is only 4 MW, and VSLs’ own power adjustment capacity is sufficient to cover the current power shortfall. Therefore, as shown in Figure 7B, there is no need to increase the output of the generator units, which not only ensures the safe operation of the power system but also reduces operation costs and greenhouse gas emissions. In fact, VSLs’ flexibility resources can be seen as virtual generators, serving as backup flexible resources for traditional generator units: increasing VSLs’ own electricity consumption is equivalent to reducing the output of units in the system. The evaluation and real-time control process of VSLs is essentially a voltage reduction and energy-saving project that takes into account system voltage safety.