Abstract
To optimize the placement of soft open points (SOPs) in active distribution networks (ADNs), many aspects should be considered, including the adjustment of transmission power, integration of distributed generations (DGs), coordination with conventional control methods, and maintenance of economic costs. To address this multi-objective planning problem, this study proposes a multi-stage coordinated robust optimization model for the SOP allocation in ADNs with photovoltaic (PV). First, two robust technical indices based on a robustness index are proposed to evaluate the operation conditions and robust optimality of the solutions. Second, the proposed coordinated allocation model aims to optimize the total cost, robust voltage offset index, robust utilization index, and voltage collapse proximity index. Third, the optimization methods of the multi- and single-objective models are coordinated to solve the proposed multi-stage problem. Finally, the proposed model is implemented on an IEEE 33-node distribution system to verify its effectiveness. Numerical results show that the proposed index can better reveal voltage offset conditions as well as the SOP utilization, and the proposed model outperforms conventional ones in terms of robustness of placement plans and total cost.
RENEWABLE energy can effectively address the conflict between growing load demands and environmental protection, and these topics have received growing attention in recent years [
Numerous studies on the optimal targets of SOPs in ADNs have been conducted. References [
The outputs of renewable energy and load demand in ADNs are characterized by a certain degree of uncertainty. Provided that the optimization objectives are highly sensitive to the fluctuation of uncertainty factors, the effectiveness of the SOP planning is weakened. Previously, the generation of stochastic distribution functions, typical uncertainty scenarios, or robust optimization was adopted to address uncertain variables. Monte Carlo simulations were conducted in [
For the solution method, the majority of existing multi-objective optimization methods using SOPs typically model multiple objectives as a single target with a weighted sum, such as in [
To address the weaknesses of previous studies, this paper proposes a multi-stage coordinated optimization model for the SOP allocation while considering economic, robust technical, and voltage collapse proximity indices. The main contributions of this paper are summarized as follows.
1) New robust technical indices based on the quantitative robustness assessment method are proposed to evaluate the robust optimality of allocation results. These strengthen the effectiveness of the SOP allocation optimization strategy without requiring assumed probability distributions for uncertain variables.
2) A multi-stage SOP allocation optimization model is established to balance the tradeoff among the investment and operation costs, robust voltage offset index, robust SOP utilization index, and voltage collapse proximity index.
3) A coordinated multi-objective nonlinear model and single-objective linear model solution method are proposed to solve the multi-stage SOP allocation optimization model. Compared with conventional methods that consider single-objective optimization or direct multi-objective optimization, the proposed model not only avoids the subjectivity of the single-objective model but also breaks through the computational burden of direct multi-objective solvers, thereby allowing robustness to be considered.
The remainder of this paper is organized as follows. Section II presents the technical evaluation indices (robustness, robust voltage offset, robust SOP utilization, and voltage collapse proximity indices). Section III establishes the multi-stage robust optimization model. Section IV describes the computational steps, and presents a flow of the proposed solution method. Section V presents case studies, and Section VI concludes this paper.
The uncertainty of the output and load demand of DGs under an ADN operation can deteriorate the performance of SOPs. In this paper, a novel robustness index is proposed to assess the sensitivity of an SOP plan when subjected to uncertain variables. The acceptable sensitivity region (ASR), which represents the maximum acceptable variation of the objective function value subjected to variations in specific uncertain parameters [
Two variations of uncertainties ( and in

Fig. 1 ASR and sensitive directions.

Fig. 2 Range of derived from .
To estimate the ASR for alleviating computational burden, the concept of a worst-case sensitivity region (WCSR) is introduced. WCSR is defined as the largest n-sphere within the ASR [

Fig. 3 WCSR.
Because of the symmetry of the WCSR, its radius RW can be used to represent multi-robust optimality against several uncertainties, which is expressed as:
(1) |
s.t.
(2) |
(3) |
where is an acceptable deviation; is the variation in uncertainty; is the nominal value of objective i, and , , and are the nominal values of the decision, state, and uncertainty variables, respectively; p is a constant that defines RW as the Lp-norm; is the tolerance for an acceptable deviation ; Nu is the number of uncertainties; and Nf is the number of objective functions.
Although RW is effective at estimating the ASR, judging whether a solution is sufficiently robust using RW alone remains difficult. Therefore, a reference robust radius is introduced and determined using an acceptable uncertainty vector. The robustness index FR is proposed to quantitatively evaluate the robust optimality of the SOP allocation decisions under multiple uncertainties.
(4) |
where Rref is the reference robust radius that represents the smallest acceptable radius. Therefore, any RW smaller than Rref should be considered insufficiently robust.
Incorporating the proposed robustness index in the optimization model as an objective is straightforward but not perfect. On one hand, an additional objective increases the computational burden and difficulty in convergence. On the other hand, FR is closely related to other technical indices individually instead of collectively. Alternatively, this paper proposes the following indices to incorporate robustness assessment into different technical indices to maintain the number of objectives and directly correlate the robustness assessment with individual technical objectives.
Based on the robustness index previously introduced, a robust voltage offset index is proposed to minimize the voltage fluctuations and optimize the operating conditions.
(5) |
where is the conventional voltage offset index; is the number of nodes; is the rated voltage; is the voltage of node i at time t; and T is the number of time periods.
Robust SOP utilization index is designed to maximize the utilization of the planned SOP under the same allocation capacity, further reducing the total cost of the distribution network.
(6) |
where is the normal SOP utilization index; , , , and are the active and reactive power injections of voltage source converters (VSCs) 1 and 2 at nodes i and j at time t, respectively; is the number of VSC units in the SOP module; and is the capacity of a single VSC unit.
Line-based voltage collapse proximity index VCPIl is adopted to assess the line voltage stability in failure scenarios based on the concept of maximum power transferable through a line [
(7) |
s.t.
(8) |
(9) |
where is the number of branches; and are the active and reactive power transferred to the receiving end though line l in fault scenarios, respectively; and are the phase angles of the load and line impedance, respectively; and are the maximum active and reactive power transferred at the receiving-end bus, respectively; is the impedance of the sending-end bus; and is the voltage of the sending-end bus.
A multi-stage robust optimization model of an SOP in a PV-penetrated distribution network is established, as shown in

Fig. 4 Multi-stage robust optimization model of SOP.
The first stage is the planning stage, in which the decision variables denote the capacity and locations of the SOP. This is the long-term stage that considers the annual investment cost. The second stage is the normal operation stage, in which the decision variables are the output power of the PV, drop-cut strategy of the CB, active and reactive power of the SOP, active power purchased from the upper substation, and other factors that simulate the actual working conditions of the ADNs. In the third stage, the voltage instability is considered to derive from unexpected contingencies. In this paper, the VCPI is selected to assess the voltage stability, and an SOP coordinated with DR is implemented as a defensive control. The operation and contingency stages are short-term that consider the total cost, robust technical indices, and VCPI.
1) Compact Model
The first and second stages aim to minimize the investment cost CI and operation cost CO and to optimize the robust technical indices, thereby optimizing the operation conditions in normal scenarios. The specific objective functions are as follows:
(10) |
s.t.
(11) |
(12) |
where F1 is the objective, which extends to (13) and (29)-(31); x1 and y1 are the decision and dependent variables, respectively; represents the equality constraints, which extend to (1)-(6), (14)-(18), (24)-(26), and (32)-(36); and represents the inequality constraints, which extend to (19)-(23), (27), (28), and (37).
2) Planning Stage
1) Objective function
The first stage aims to minimize investment costs. The specific objective functions are as follows:
(13) |
① Investment cost of SOP
The investment cost of the SOP can be expressed as:
(14) |
where r is the discount rate; y is the service life of the VSC; cp is the annual investment cost per kVA converter power of the VSC; and xi and xj are the power capacities of the two VSCs corresponding to the SOP.
② Investment cost of CB
The investment cost of the CB can be expressed as:
(15) |
where xi,CB is the power capacity of the CB installed at node i; and cCB is the annual investment cost per kVA of the CB.
2) Constraints
① Constraints of the SOP
The SOP is installed between adjacent feeders in the ADNs, as shown in

Fig. 5 Modeling of SOP.
The SOP power equation at time t can be expressed as:
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
where , , , and are the active and reactive power injections of nodes i and j in the ADNs, respectively; and are the power losses of VSCs 1 and 2 at nodes i and j, respectively; AVSC1 and are the power loss coefficients; and , , , and are the minimum and maximum reactive power injections of VSC1 and VSC2, respectively.
② Upper limits of VSC units
(23) |
where is the upper limit of the number of VSC units of the SOP.
③ Linearized DistFlow equations
The following power flow equations should be satisfied [
(24) |
(25) |
(26) |
where rij is the resistance value of branch ij; xij is the reactance value of branch ij; represents that node i is subordinate to the set u(j) of parent nodes connected to node j; represents that node k is subordinate to the set v(j) of child nodes connected to node j; and are the active and reactive power on branch ij at time t, respectively; and are the active and reactive power losses on branch ij, respectively; and are the active and reactive power injections into node j, respectively (node j is selected in this case as a substitute for other nodes, excluding the balanced one); and are the active and reactive power injections of distributed power supply PV, respectively; and are the active and reactive power injections of the load, respectively; is the reactive power of the CB; is the voltage of node j at time t; and UN is the system-rated voltage.
④ Network operation constraints
The node voltage and branch active power must satisfy the following constraints to ensure safe operation of the system:
(27) |
where and are the minimum and maximum node voltages, respectively; is the active power of branch ij; and is the maximum branch active power.
⑤ CB switching constraints
(28) |
where is the number of CBs in operation; is the capacity of the CB per unit; is the binary variable if the CB is in operation, in which case ; otherwise, ; and is the maximum switching times of the CB.
3) Normal Operation Stage
The operation cost includes the power exchange cost, SOP operation cost, loss cost of the VSC converter, CB drop-cut cost, network loss, and emission cost. The technical indices include the robust offset voltage index and robust SOP utilization index.
1) Objective function
(29) |
(30) |
(31) |
① SOP operation cost
(32) |
where is the annual SOP operation cost coefficient; is the investment cost of the SOP per apparent power; Dy is the number of days in a year adopted to convert the daily cost into the annual cost; and is the capacity of the SOP on branch ij.
② CB drop-cut cost
(33) |
where cCB,O is the CB drop-cut cost.
③ Network loss cost
(34) |
where is the daily network loss cost.
④ Converter loss cost
(35) |
where is the daily converter loss cost.
⑤ Emission cost
Because the PV does not generate carbon emissions, the carbon cost in this study is generated by the upper grid.
(36) |
where cemis is the daily emission cost; Psub,t is the active power obtained from the upper substation; and is the carbon emission intensity per kWh for the upper substation.
2) Constraints
In addition, to satisfy constraints (16)-(28), PV generation constraints must also be satisfied.
(37) |
where is the maximum available active power of the PV units; and is the power factor angle of the PV.
1) Compact Model
With unexpected contingencies in ADNs, the voltage collapse proximity index is a major index for assessing the voltage stability. In this case, the DR and SOP are used to ensure voltage stability. Accordingly, the objectives of the contingency scenario stage can be expressed as:
(38) |
(39) |
s.t.
(40) |
(41) |
where F2 is the objective, which extends to (7)-(9) and (39); x2 and y2 are the decision and dependent variables, respectively; denotes the equality constraints, which extend to (16)-(18), (39), (42)-(44), and (46)-(48); and denotes the inequality constraints, which extend to (19)-(23), (27), (28), and (37).
2) Contingency Stage
1) Objective function
The load reduction cost is expressed as:
(42) |
where is the load-cut cost per kWh; is the load reduction coefficient; and is the node set of the DR.
2) Constraints
In addition to constraints (16)-(23), (27), (28), and (37), the DR constraints must also be satisfied.
① DR constraints
(43) |
(44) |
(45) |
where and are the active and reactive power reductions, respectively; and are the minimum and maximum load reduction coefficients, respectively; and and are the active and reactive power injections of load following load reduction, respectively.
② New linearized DistFlow equations
(46) |
(47) |
(48) |
The single-objective model optimization method (Cplex [

Fig. 6 Computational flow of proposed method.
Step 1: initialization. After the algorithm parameters are set, the first generation of the population is initialized, including decision variables for the SOP, CB installation, and output power of the PV.
Step 2: coordination of planning and contingency stages. The method used in this step involves contingency determination. Power flow calculations are conducted to determine whether contingencies occur in the ADNs. Provided no contingency occurs, we can proceed to Step 3. Otherwise, defensive controls should be implemented and the subroutine should be booted (Step 9).
Step 3: objective calculation. The total cost and robust technical indices are calculated to optimize the capacities and sites of SOPs.
Step 4: individual update. Identify and renew the best individuals of the present generation, including non-dominated sorting, crowding distance and fitness calculation, selection, crossover, and mutation implemented by NSGA-II.
Step 5: coordination of planning and operation stage. The method for this step is as follows: either reaching the maximum number of iterations or finding no other new non-dominated solution in a predefined number of successive iterations to determine the evaluation condition for booting the operation-stage subroutine. Otherwise, proceed to the next iteration.
Step 6: update the optimization capacity and sites of SOP. The fuzzy membership function is adopted to determine the optimal capacity and sites of the SOP. After the conditions for Step 5 are met, the main procedure terminates and the optimization for the operation stage commences (Step 7) with the optimal capacity and sites of the SOP imported.
Step 7: determine the feasible region. The feasible region is formed by constraints (1)-(6), (14)-(24), (26)-(28), and (32)-(37).
Step 8: update the CB drop-cut strategy and SOP timing change. The CB drop-cut strategy and the active and reactive power support from the SOP are optimized by Cplex.
Step 9: generate a new population. The first generation of population in the planning stage is overridden by the decision variables for the SOP, the DR (enabled when the voltage exceeds the threshold), and the output power of the PV following contingency determination.
Step 10: objective calculation. The total cost and VCPI are calculated to optimize the capacity and sites of the SOP as well as the load reduction coefficient.
Step 11: termination criteria. The procedure terminates when the termination criteria of Cplex or NSGA-II are satisfied.
The modified IEEE 33-node distribution system is illustrated in

Fig. 7 Modified IEEE 33-node distribution system.
For a detailed analysis and discussion, the robust SOP utilization index is converted to its reciprocal, . The smaller the value of all objectives, the more optimal the results are.
A Pareto front with 10 robust Pareto optimal solutions of the proposed method is shown in

Fig. 8 Pareto front with 10 robust Pareto optimal solutions of proposed method.
($) | (%) | ||
---|---|---|---|
612101.97 | 0.07 | 1.34 | 74.74 |
The capacities of the SOP of the final optimal solution are shown in
Location | Capacity (kVA) |
---|---|
12-22 | 100 |
8-21 | 100 |
9-15 | 100 |
18-33 | 150 |
25-29 | 100 |
Three cases are selected to investigate the effects of SOPs and CB on voltage stability and network loss.
1) Case 1: PV-penetrated ADNs without SOPs or CB installation.
2) Case 2: PV-penetrated ADNs with CB installation only.
3) Case 3: PV-penetrated ADNs with SOPs and CB installation.

Fig. 9 Voltage profiles for three cases.
Case | Loss cost ($) |
---|---|
1 | 146150.171 |
2 | 117982.442 |
3 | 103317.219 |
Case | Emission cost ($) |
---|---|
4 | 597610 |
5 | 595420 |
6 | 595360 |
This paper presumes that branch 32-33 and branch 3-23 have contingencies in the modified IEEE 33-node system, as shown in

Fig. 10 Modified IEEE 33-node distribution network of contingency stage.
1) Case 4: ADNs with only DR allocation to avoid voltage instability.
2) Case 5: ADNs with only SOP allocation to avoid voltage instability.
3) Case 6: ADNs with coordinated SOP and DR allocation to avoid voltage instability.
The Pareto optimal solutions for the three cases are compared in

Fig. 11 Comparison of Pareto optimal solutions for three cases.
Case | VCPI | Cost ($) |
---|---|---|
4 | 0.3463 | 337500 |
5 | 0.3339 | 475000 |
6 | 0.2812 | 365700 |
Location | Capacity (kVA) | ||
---|---|---|---|
Case 4 | Case 5 | Case 6 | |
12-22 | 100 | 50 | |
8-21 | 150 | 200 | |
9-15 | 100 | 150 | |
18-33 | 100 | 150 | |
25-29 | 300 | 300 |
Location | |||
---|---|---|---|
Case 4 | Case 5 | Case 6 | |
7 | 0.101 | 0.092 | |
10 | 0.123 | 0.110 | |
13 | 0.093 | 0.093 | |
17 | 0.155 | 0.070 | |
23 | 0.155 | 0.113 |
As shown in Tables

Fig. 12 Comparison of voltage profile for Cases 4 and 6.
Two cases are selected to investigate the effects of the robustness index.
1) Case 7: and are used to evaluate the performance of the normal operation stage.
2) Case 8: and are used to evaluate the performance of the normal operation stage.
The Pareto optimal solutions for Cases 7 and 8 are compared in

Fig. 13 Comparison of Pareto optimal solutions for Cases 7 and 8.
In addition, a robustness analysis does not require assumptions about the probability distribution of uncertain variables, which reduces the adverse effects of uncertainties. Under practical operating conditions, the fluctuations of the PV and load deviate from the set value of the planning scenarios. With a typical solution on the Pareto fronts as an example and based on the assumption that has an error of , Foffset worsenes from 0.10 to 0.84, and the SOP utilization becomes increasingly inadequate from 57.9% to 51.2% in Case 7. By contrast, increases from 0.10 to 0.79, whereas the SOP utilization decreases from 72.1% to 71.3% in Case 8, indicating that the proposed quantitative robustness assessment could make the objective function less sensitive and more optimal to variations in uncertainty.
Two cases are selected to investigate the effects of the SOP utilization index.
1) Case 9: and cost are used to determine the optimal allocated SOP capacity without considering .
2) Case 10: , , and cost are used to determine the optimal allocated SOP capacity.
The fuzzy membership function [
Case | ($) | (%) | ||
---|---|---|---|---|
Case 9 | 664194.91 | 0.14 | 1.81 | 55.19 |
Case 10 | 612101.97 | 0.07 | 1.34 | 74.74 |
Location | Capacity (kVA) | |
---|---|---|
Case 9 | Case 10 | |
12-22 | 100 | 100 |
8-21 | 100 | 100 |
9-15 | 50 | 100 |
18-33 | 50 | 150 |
25-29 | 50 | 100 |
This paper proposes a multi-stage coordinated optimization for the SOP allocation in ADNs with PV based on robust technical indices to enhance the effectiveness and robust optimality of the solutions and the SOP utilization. The applicability of the proposed model is verified through case studies. The major conclusions are as follows.
1) The proposed quantitative robustness assessment method could effectively improve the robust optimality and effectiveness of allocation results without requiring assumed probability distributions for uncertain variables.
2) When introducing a robust SOP utilization index, the proposed model could improve the SOP utilization (e.g., by 19.57%) and operation conditions (e.g., voltage offset decreased by 50%) while reducing the investment and operation costs (e.g., by 7.84%).
3) The proposed multi-stage optimization framework and corresponding computational method demonstrated that the introduction of an SOP could adjust the transmitted power and effectively decrease network loss (e.g., by 29.3% compared with no SOP or CB installation), further improving the economy of ADNs. In addition, case studies prove that an SOP can provide reactive power to support the voltage of ADNs in coordination with the CB, thereby enhancing voltage quality. Furthermore, when contingencies occurred, the SOP could reduce both the load demand and voltage collapse risk (e.g., by 18.8% compared with only DR allocation). Thus, the proposed multi-stage allocation model is more comprehensive and sophisticated than the conventional models.
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