Abstract
Photovoltaic (PV) inverter, as a promising voltage/var control (VVC) resource, can supply flexible reactive power to reduce microgrid power loss and regulate bus voltage. Meanwhile, active power plays a significant role in microgrid voltage profile. Price-based demand response (PBDR) can shift load demand via determining time-varying prices, which can be regarded as an effective means for active power shifting. However, due to the different characteristics, PBDR and inverter-based VVC lack systematic coordination. Thus, this paper proposes a PBDR-supported three-stage hierarchically coordinated voltage control method, including day-ahead PBDR price scheduling, hour-ahead reactive power dispatch of PV inverters, and real-time local droop control of PV inverters. Considering their mutual influence, a stochastic optimization method is utilized to centrally or hierarchically coordinate adjacent two stages. To solve the bilinear constraints of droop control function, the problem is reformulated into a second-order cone programming relaxation model. Then, the concave constraints are convexified, forming a penalty convex-concave model for feasible solution recovery. Lastly, a convex-concave procedure-based solution algorithm is proposed to iteratively solve the penalty model. The proposed method is tested on 33-bus and IEEE 123-bus distribution networks and compared with other methods. The results verify the high efficiency of the proposed method to achieve power loss reduction and voltage regulation.
SOLAR photovoltaic (PV) units have been growingly installed in microgrids. However, the high penetration of PV may cause voltage issues such as voltage rise [
To conquer this challenge, voltage/var control (VVC) is an effective measure that can reduce network power loss or regulate bus voltage. Conventional mechanical VVC resources, including capacitor banks (CBs) and transformer on-load tap changers (OLTCs), operate in a discontinuous manner and thus cannot respond to voltage fluctuation timely [
Generally, the VVC method can be categorized into central control and local control. As implied by the name, the former optimizes the operating decisions of VVC resources based on several global information including the prediction of renewable generation and load demand as well as network parameters. In [
Alternatively, the local VVC device responds to the local measurement such as bus voltage based on the built-in control strategy. In [
Recently, the integration of the central and local VVC methods has drawn more attention. The parameters of local droop control functions such as droop slope are optimized simultaneously with the central VVC decisions based on the system-wide information, forming a hierarchical VVC framework. In [
In addition, except for reactive power, active power also plays an important role in microgrid voltage profiles and power losses during operation. Price-based demand response (PBDR) is a typical demand response scheme that aims to encourage users to shift their partial electricity consumption via scheduling time-varying prices [
However, most existing PBDR-related research works focus on cost minimization, and the linkage between PBDR and grid voltage control in the microgrid is often ignored. Moreover, the PBDR prices are generally scheduled in advance, giving users sufficient time for response, while the inverter reactive power is managed at a short time interval. Therefore, based on their different characteristics, a multi-stage hierarchical framework is required to systematically coordinate PBDR with VVC.
Through the above literature review, it can be found that droop control functions are either modelled with the pre-determined parameters, or modelled in a nonlinear manner that requires an efficient solving method. Also, a multi-stage hierarchical framework is needed to coordinate PBDR and inverter-based VVC for microgrid operation. To bridge the research gap, this paper proposes a PBDR-supported three-stage hierarchically coordinated voltage control (HCVC) method. To efficiently optimize the droop control function with the bilinear constraints, a convex-concave procedure (CCP) [
1) A PBDR-supported three-stage HCVC method is proposed to efficiently coordinate the PBDR and inverter-based VVC resources at different hierarchies and timescales, aiming to minimize the microgrid power loss and voltage deviation. The droop slope and a pair of base coordinates consisting of the expected bus voltage and reactive power setpoint are fully modelled, such that the key parameters of the local Q-V droop control function can be integrated into the central hierarchy for intraday optimization.
2) To address the bilinear constraints of droop control function model in the second-stage hierarchical VVC, a second-order conic relaxation (SOCR) model is introduced. Then, considering the relaxation errors, a penalty convex-concave model is proposed to reformulate and convexify the relaxed constraints. Accordingly, a CCP-based solution algorithm is developed to efficiently solve the second-stage optimization.
A PBDR-supported three-stage HCVC method is proposed to minimize microgrid power loss and voltage deviation by co-optimizing the operation of both active and reactive power resources. Its framework is illustrated in

Fig. 1 Framework of proposed PBDR-supported three-stage HCVC method.
In the first stage, with the day-ahead prediction of PV outputs and loads, the day-ahead PBDR prices are determined and scheduled for load shifting, aiming to achieve peak shaving and valley filling. These prices are scheduled one day ahead in order to give consumers enough response time. Moreover, the intraday hour-ahead reactive power dispatch of PV inverters is taken into account under uncertainties in this stage, forming central coordination between the first and second stages.
Then, with more accurate hourly prediction of PV outputs and loads, the second stage optimizes the parameters of the Q-V droop control functions as well as reactive power setpoints of PV inverters on an hourly basis. The results are delivered to the corresponding PV inverters for the local VVC. Since the system-wide information and possible scenarios of uncertainty realizations are used to determine the droop control functions, it hierarchically coordinates the central and local layers.
Lastly, in the third stage, with the local RT voltage measurement, PV inverters generate RT reactive power based on the optimized setpoint and droop control functions. In other words, inverters can generate either constant or dynamic reactive power based on the optimized droop control function.
It is worth noting that as the size and scale of the microgrid increase, the three-phase unbalance issue becomes gradually evident. This could be tackled by scheduling PBDR price levels and optimizing PV reactive power generation on the basis of each phase.
The purpose of PBDR is to motivate users to shift their partial electricity loads by scheduling different hourly prices one day ahead. A high price will encourage consumers to reduce their load demand and vice versa. Since there is a strong connection between load demand and network bus voltage or power loss, it is expected to increase the load during the peak PV generation period and decrease the load during the peak load period.
According to [
(1) |
To efficiently implement PBDR, five price levels are designed based on (1), forming a stepwise price-elastic demand model [
Price level | Price rate (%) | Demand response rate (%) |
---|---|---|
1 | 60 | 121 |
2 | 80 | 108 |
3 | 100 | 100 |
4 | 130 | 93 |
5 | 170 | 88 |
By setting different price levels, the microgrid load demand is expected to change accordingly. Thus, the active and reactive power of the loads with PBDR can be modelled as:
(2) |
(3) |
It is worth noting that the load uncertainties such as prediction errors and non-responsive loads can be considered in and . Other constraints for PBDR implementation are given as:
(4) |
(5) |
(6) |
(7) |
where is set to be 1 hour in the day-ahead stage.
Constraint (4) ensures that only one price level of PBDR can be activated every hour. Constraint (5) indicates that the consumers’ electricity bills with the scheduled PBDR cannot be greater than their original bills. Constraint (6) states that the consumers’ electricity consumption should not be compromised by PBDR. Lastly, the total hourly load after PBDR is constrained by (7), indicating that a new peak load cannot be created.
The above PBDR implementation is used to achieve load shifting from the perspective of the microgrid system by scheduling different PBDR price levels. The uncertainties of loads will be further addressed by a stochastic optimization (SO) in Section III-C.
The day-ahead PBDR-supported central voltage control coordinately optimizes the PBDR scheduling and inverter reactive power dispatch to achieve the reduction of microgrid power loss and voltage deviation. This day-ahead model is formulated as:
(8) |
s.t.
(2)-(7)
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
The objective function (8) minimizes the total microgrid power loss and average bus voltage deviation with the assigned weighting factor . The PBDR-related constraints can be referred to (2)-(7). Furthermore, (9) constrains the inverter reactive power supply. can be calculated first based on the predicted PV generation and inverter settings. Note that even though the rated active power is output, the PV inverters can still generate extra reactive power [
To efficiently solve the day-ahead model, some constraints can be further linearized. Firstly, the absolute expression in (13) can be linearized by introducing a slack variable . Thus, the absolute expression can be reformulated as and , and the objective function will minimize instead of .
Secondly, the quadratic inequality in (16) is linearized by a polygonal inner approximation approach [

Fig. 2 Illustration of polygonal inner approximation approach using a regular dodecagon.
The feasible region formed by active and reactive power is approximated by a regular polygon. A regular dodecagon is used in this paper. Based on the coordinates of vertices, (16) can be converted into the following forms:
(17) |
With the above linearization approaches, the day-ahead PBDR-supported central voltage control model including (2)-(15), (17) forms a mixed-integer quadratic programming problem. The primary decision variables are and , and the uncertainties include and .
To address the uncertainties of the PV outputs and loads, a scenario-based SO method [
(18a) |
s.t.
(18b) |
(18c) |
where denotes the day-ahead decision variable ; and denotes the intraday decision variables including and other dependent variables in scenario .
It is noteworthy that after solving (18), only is scheduled and will be re-optimized in the intraday stage according to more accurate hourly predictions.
The inverter reactive power output can be fixed at the setpoint or adjusted based on the droop control function. Thus, this dual output mode is considered in the inverter reactive power generation.
The conventional Q-V droop control function embedded in the PV inverter is illustrated in

Fig. 3 Conventional Q-V droop control function and linear droop control function. (a) Conventional Q-V droop control function. (b) Linear droop control function.
However, when the bus voltage stays inside the deadband, the inverter does not react to the system variation, resulting in inadequate utilization of the inverter reactive power capacity. To make full use of inverter reactive power capacity, a linear droop control function is employed [
(19) |
(20) |
(21) |
(22) |
(23) |
where the subscript denotes the index of RT time interval in the hour-ahead stage.
The droop control function of the inverter is described by (19)-(23).
Based on the inverter reactive power generation, the hour-ahead hierarchically coordinated VVC is modelled as:
(24a) |
s.t.
(24b) |
Considering more accurate hour-ahead prediction of PV generation and loads, i.e., these uncertainties will be realized in a shorter predicted time interval, the SO method is applied to simulate the RT uncertainty realization through scenario generation. The slope of droop control function , reactive power setpoint , and the corresponding expected bus voltage are optimized in the generated scenarios. It is noted that this hour-ahead model (24) is nonlinear due to the bilinear term in (21), which will be addressed in the following subsections.
The bilinear term in (21) significantly decreases the solving efficiency. To solve this problem, firstly, (21) is converted into a SOCR model, offering an initial approximate solution. Then, a penalty CCP model is introduced to recover a feasible solution.
For illustration, the subscripts are temporally neglected at this stage. The bilinear term can be converted into an equivalent form, shown as:
(25a) |
(25b) |
Then, the quadratic equality constraint (25b) can be reformulated as:
(26a) |
(26b) |
Constraint (26a) is convex while constraint (26b) is concave. Multiplying both sides of (26a) by 4, it can be rewritten into a second-order cone form, given as:
(27a) |
(27b) |
Lastly, the concave (26b) is removed and a McCormick envelope [
(28a) |
(28b) |
(28c) |
(28d) |
Thus, by replacing (21) with (25a), (27), and (28), the original hour-ahead problem forms a SOCR model. It is noted that, since the objective function is not strictly increasing in and , this SOCR model is not exact and it only provides a lower bound to the original problem.
The above SOCR model is loose for the bilinear term and may cause relaxation errors, because constraint (27) is only restricted on one side. To address this issue, a CCP-based reformulation is proposed to transfer the SOCR model into a penalty convex-concave model by convexifying the concave constraints. The original CCP is introduced to seek the solution to the difference of convex programming problems [
(29a) |
s.t.
(29b) |
It is noted that is concave. This concave term is replaced by a convex upper bound, which obtains the following problem.
(30a) |
s.t.
(30b) |
(30c) |
where is the linearized function of , which can be regarded as a tangent plane of , at the current solution point . As a result, this linearized model (30) can be solved iteratively to obtain the solution.
In line with the form of (30), the concave (26b) can be reformulated into the difference between and , as shown in (31).
(31a) |
(31b) |
(31c) |
(31d) |
Since the relaxation errors need to be reduced, slack variables are introduced to represent the difference, as given in (32).
(32a) |
(32b) |
Then, the slack variables are added to the objective function, regarded as penalty terms, to tighten the relaxation.
Finally, the penalty convex-concave model for the hour-ahead VVC is given as:
(33a) |
s.t.
(33b) |
where should increase as the iteration continues for convergence improvement [
A CCP-based solution algorithm is developed in this subsection to iteratively solve this penalty convex-concave model, as presented in

Fig. 4 CCP-based solution algorithm for solving penalty convex-concave model.
The SOCR model provides a lower bound to the original hour-ahead problem, which can serve as an initial solution. Compared with the SOCR model, the feasible region of the penalty convex-concave model is the same. Due to the convexification of (31b), (31d), and the penalty terms, the objective function of (33) is non-increasing [
Accordingly, the following termination criteria are considered.
(34a) |
(34b) |
determines the difference of the original objective function (24) between the current and last iterations, while checks the sum of the slack variables to ensure equality constraints.
Note that the penalty convex-concave model is a second-order cone programming problem, which can be efficiently solved by off-the-shelf solvers. The logic and procedure of the solution algorithm for dealing with the bilinear constraints can also be applied to other practical optimization problems such as microgrid planning and operation.
A 33-bus distribution network [

Fig. 5 33-bus distribution network with 10 PV units.
PV No. | Rated active power (kW) | PV No. | Rated active power (kW) |
---|---|---|---|
1 | 600 | 6 | 600 |
2 | 550 | 7 | 650 |
3 | 600 | 8 | 550 |
4 | 600 | 9 | 600 |
5 | 500 | 10 | 650 |

Fig. 6 Predicted profiles of PV output and load response before PBDR.
This numerical simulation is conducted on a 64-bit PC with a 2.5 GHz CPU and 16 GB RAM using YALMIP [
By solving the day-ahead PBDR-supported central voltage control model, the results of scheduled day-ahead PBDR price levels are obtained, as shown in

Fig. 7 Scheduled day-ahead PBDR price levels and expected load response.
Besides, it is worth noting that without the PBDR and inverter-based VVC, the voltage range under the expected condition is [0.92, 1.07]p.u., which is out of the allowed voltage range. However, with the coordination of PBDR and inverter reactive power supply, the voltage range under the expected condition is [0.99, 1.02]p.u., indicating a significant improvement in voltage deviation reduction.
The hour-ahead hierarchically coordinated VVC is solved by the proposed CCP-based solution algorithm. Since the voltage profile has been improved in the day-ahead stage, only power loss reduction is considered as the objective in the hour-ahead stage.
Taking the period of 12:00-13:00 as an example, the total expected PV generation and loads with PBDR implementation are 5.9 MW and 2.63 MW, respectively.

Fig. 8 Solving process of proposed CCP-based solution algorithm during period 12:00-13:00 in 33-bus distribution network. (a) C1 and C2 values. (b) Objective value.
By iteratively solving the hour-ahead penalty convex-concave model, the average power loss during this period is 125 kW. Note that since only power loss is considered in the hour-ahead stage, the results can achieve a better power loss reduction compared with that in the day-ahead stage.
Moreover, the results of reactive power setpoints , expected bus voltage , and the slope of droop control functions are obtained. Thus, the associated Q-V droop curves of the inverters with the reactive power setpoints can be plotted, as shown in

Fig. 9 Q-V droop curves of PV inverters with reactive power setpoints during period 12:00-13:00. (a) PV1-PV5. (b) PV6-PV10.
Besides, the Q-V droop curves of the PV inverters with the reactive power setpoints during the peak load period 20:00-21:00 are given in

Fig. 10 Q-V droop curves of inverters with reactive power setpoints during period 20:00-21:00. (a) PV1-PV5. (b) PV6-PV10.
In the RT stage, the inverters that operate under droop control mode can change the reactive power supply in response to the measurement of local bus voltage based on the optimized droop control function.
To verify the effectiveness of the droop control function and the proposed PBDR-supported three-stage HCVC method, 5000 scenarios of PV outputs and loads are randomly generated by Monte Carlo sampling during the periods 12:00-13:00 and 20:00-21:00. These two periods correspond to the peak PV generation period and the peak load period, respectively, which can be regarded as two typical cases of the whole day. Then, 20% PV output scenarios are randomly selected and set to be 30% of the original value to simulate PV ramping events such as cloud passing. These scenarios are regarded as RT uncertainty realizations.
In addition, two other voltage control methods are applied for comparison, which are given as follows.
1) Method 1: HCVC without PBDR implementation. The inverter reactive power setpoint and droop control function are optimized every hour. Then, with local RT bus voltage measurement, each inverter changes its reactive power output based on the optimized droop control function.
2) Method 2: two-stage central coordination of PBDR and inverter reactive power generation without local control. The inverter reactive power dispatch is re-optimized every hour and fixed.
The comparison results during the periods 12:00-13:00 and 20:00-21:00 are given in Tables
Method | Average power loss (kW) | Voltage range (p.u.) | Voltage violation rate (%) |
---|---|---|---|
1 | 110.4 | [0.997, 1.048] | 0 |
2 | 95.8 | [0.966, 1.044] | 0 |
Proposed | 80.0 | [0.996, 1.045] | 0 |
Method | Average power loss (kW) | Voltage range (p.u.) | Voltage violation rate (%) |
---|---|---|---|
1 | 105.3 | [0.947, 0.998] | 2.08 |
2 | 81.2 | [0.949, 0.998] | 0.12 |
Proposed | 81.0 | [0.953, 0.998] | 0 |
During the peak PV generation period 12:00-13:00, Method 1 has the largest average power loss, followed by Method 2, indicating the significance of PBDR for power loss reduction. However, the voltage range of Method 2 is larger than that of Method 1, but both are within the allowed voltage range. For the proposed method, it obtains the lowest average power loss, and the voltage range is similar to that of Method 1. Also, all the methods have no voltage violations.
During the peak load period 20:00-21:00, the average power loss, voltage range, and voltage violation rate of Method 1 are the largest, indicating that only using inverter reactive power compensation cannot fully address the voltage issue. By contrast, the average power loss and voltage violation rate in Method 2 are greatly reduced due to the consideration of PBDR, but voltage violation still occurs. Regarding the proposed method, although its average power loss and voltage range are slightly less than those of Method 2, no voltage violations are observed.
In addition, based on the simulation results of 5000 scenarios, the probability distribution of the microgrid power loss during the period 12:00-13:00 for Method 2 and the proposed method is given in

Fig. 11 Probability distribution of microgrid power loss during period 12:00-13:00 for Method 2 and proposed method.
Moreover, the probability distribution of the voltage profile at Bus 18 (end bus along feeder) during the period 20:00-21:00 for all methods are plotted in

Fig. 12 Probability distribution of voltage profile at Bus 18 during period 20:00-21:00 for all methods.
As observed from this figure, Method 1 has the widest voltage range and largest voltage violation rate. The voltage profile of Method 2 is significantly improved compared with that of Method 1 due to the coordination of PBDR. For the proposed method, the bus voltage does not exceed the allowed range with a better voltage profile (further from the lower voltage limit) and less deviation range, showing superior performance in voltage regulation against uncertainties.
In summary, the above comparison shows the effectiveness of the PBDR as an active power resource to support voltage control. More importantly, the proposed method takes advantage of the central and local VVC methods with the load shifting function of PBDR, demonstrating the high efficiency for power loss reduction and voltage regulation.
This paper also applies the IEEE 123-bus distribution network [

Fig. 13 IEEE 123-bus distribution network with 14 PV units.
PV No. | PV rated power (kW) | PV No. | PV rated power (kW) |
---|---|---|---|
1 | 230 | 8 | 270 |
2 | 250 | 9 | 250 |
3 | 240 | 10 | 230 |
4 | 260 | 11 | 240 |
5 | 250 | 12 | 240 |
6 | 480 | 13 | 250 |
7 | 250 | 14 | 230 |
The load profiles before and after PBDR implementation are obtained by solving the day-ahead PBDR-supported central voltage control model, which is shown in

Fig. 14 Load profiles before and after PBDR implementation of IEEE 123-bus distribution network.
During the entire day, the proposed CCP-based solution algorithm is used to solve the hour-ahead hierarchically coordinated VVC. Similarly, taking the peak PV generation period 12:00-13:00 as an example, the total expected PV generation and load demand after PBDR are 3.67 MW and 2.58 MW, respectively. The solving process of the CCP-based solution algorithm is shown in

Fig. 15 Solving process of proposed CCP-based solution algorithm during period 12:00-13:00 in IEEE 123-bus distribution network. (a) C1 and C2 values. (b) Objective value.
To further verify the effectiveness of the proposed method as well as the optimized droop control function in the IEEE 123-bus distribution network, 5000 scenarios of PV outputs and loads during the periods 12:00-13:00 and 20:00-21:00 are randomly generated using the same setting of the 33-bus distribution network in Section V-D. Moreover, Methods 1 and 2 in Section V-D are adopted for comparison. The comparison results of average power loss and voltage range during these two periods are given in Tables
Method | Average power loss (kW) | Voltage range (p.u.) |
---|---|---|
1 | 25.73 | [0.984, 1.015] |
2 | 21.28 | [0.980, 1.018] |
Proposed | 20.58 | [0.984, 1.015] |
Method | Average power loss (kW) | Voltage range (p.u.) |
---|---|---|
1 | 58.61 | [0.990, 1.005] |
2 | 52.63 | [0.990, 1.006] |
Proposed | 52.37 | [0.993, 1.005] |
During the peak PV generation period 12:00-13:00, the average power loss of Method 1 is significantly higher than that of Method 2 and the proposed method, indicating that PBDR is effective in reducing network loss. However, the voltage range of Method 2 becomes wider compared with that of Method 1, but both are within the allowed voltage range. Lastly, the proposed method can achieve the lowest average power loss and the smallest voltage range, indicating the effectiveness of the optimized droop control function in the hour-ahead stage.
During the peak load period 20:00-21:00, Method 1 has the largest average power loss, while the power loss of Method 2 significantly decreases after PBDR implementation. The proposed method achieves the smallest average power loss as well as voltage range.
It is worth noting that during these two periods, all the methods do not encounter the violation of voltage constraints.
With the results of these 5000 scenarios, the probability distribution of the voltage of phase C at Bus 31 during the period 12:00-13:00 for all the methods are plotted in

Fig. 16 Probability distribution of voltage of phase C at Bus 31 during period 12:00-13:00.
In addition, the probability distribution of microgrid power loss during the period 20:00-21:00 for Method 2 and the proposed method are given in

Fig. 17 Probability distribution of microgrid power loss during period 20:00-21:00.
In summary, the above comparison shows the superiority of the proposed method in microgrid power loss reduction and voltage control.
In this paper, a PBDR-supported three-stage HCVC method is introduced, aiming to minimize the microgrid power loss and voltage deviation via co-optimizing the operation of both active and reactive power resources. The day-ahead PBDR scheduling is efficiently coordinated with intraday inverter reactive power dispatch in a centralized manner via a SO method. By contrast, the inverter reactive power dispatch and RT local control are hierarchically coordinated considering the possible RT uncertainty scenarios.
Several linearization methods are applied to solve the nonlinear constraints for the day-ahead model. Then, in the second-stage hierarchical VVC, to exploit the PV inverter potentials, the reactive power output is considered to be constant or adjustable based on a linear Q-V droop control function. The key parameters of the droop control function, which include the droop slope and a pair of base coordinates representing the expected voltage and corresponding reactive power setpoint, are fully modelled. More importantly, a CCP-based solution algorithm including model relaxation, reformulation, and solving procedure is developed to efficiently solve the second-stage optimization problem. The case study verifies the effectiveness and high efficiency of the proposed method and solution algorithm.
The proposed method focuses on the coordination of PBDR and PV inverters. However, there are various distributed energy resources and network devices such as battery energy storage and CBs in microgrids. In future research works, the proposed method can be expanded to coordinate with these resources and devices. Moreover, a more general model of the droop control function with deadband and the three-phase unbalance issue of the microgrid can be considered.
A. Sets and Indices | |
Set of branches | |
, | Convex functions |
, , | Indices of microgrid buses |
Set of parent buses of bus | |
Set of microgrid buses | |
Set of child buses of bus | |
Index of iterations | |
, | Set and index of price levels of price-based demand response (PBDR) |
Index of objective function and constraints, starting from 0 | |
, | Set and index of sampled scenarios |
, , | Set, index, and length of time intervals |
Index of regular dodecagon vertices | |
B. Parameters | |
Price elasticity of loads | |
Weighting factor | |
The maximum penalty factor | |
Penalty factor at iteration | |
, | Lower and upper bounds |
, | Uncertainty and expected value of uncertainty |
Obtained result of decision variable | |
, , | Coefficients of dodecagon inner approximation |
Coefficient of electricity price | |
, | Coefficient matrices of day-ahead and intraday decision variables |
Demand response rate at price level (%) | |
Total numbers of buses and scenarios | |
Price level of PBDR ($/MWh) | |
Electricity price before PBDR ($/MWh) | |
The maximum reactive power generation of photovoltaic (PV) inverter (kvar) | |
Resistance and reactance of branch (p.u.) | |
The maximum capacity of branch (kVA) | |
The maximum bus voltage (p.u.) | |
The minimum bus voltage (p.u.) | |
C. Variables | |
Binary decision of price level of PBDR | |
Binary decision of slope of droop control function | |
Slope of droop control function | |
Active and reactive power flows through branch (kW and kvar) | |
Power loss of branch (kW) | |
Active power generation of PV (kW) | |
Active and reactive power of load demand without PBDR (kW and kvar) | |
Active and reactive power of load demand with PBDR implemented (kW and kvar) | |
Reactive power setpoint of inverter (kvar) | |
Reactive power output of inverter (kvar) | |
Reactive power adjustment of inverter responding to voltage variation (kvar) | |
Relaxation errors | |
Voltage variation from expected voltage value (p.u.) | |
Bus voltage (p.u.) | |
Reference voltage (p.u.) | |
Expected bus voltage in expected scenario of uncertainties (p.u.) | |
Average bus voltage deviation (p.u.) | |
Vector of decision variables in convex-concave procedure |
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