Decentralized Voltage Control Strategy of Soft Open Points in Active Distribution Networks Based on Sensitivity Analysis

: With the increasing penetration of distributed generators, various operational problems, especially severe voltage violation, threaten the secure operation of active distribution networks. To effectively cope with the voltage fluctuations, novel controllable power electronic equipment represented by soft open points has been used in active distribution networks. Meanwhile, the communication has dramatically increased due to the rise of the variety and number of devices within the network. This paper proposes a decentralized voltage control method of soft open points based on voltage-to-power sensitivity. The method reduces the burden of communication, storage, and calculation effectively in a decentralized manner and fulfills the rapid requirements of large-scale active distribution networks. First, the network is divided into several sub-areas; each is under the control of one soft open point at most. The initial strategies of soft open points are adjusted by local voltage-to-power sensitivity and the voltage information within the sub-areas. If some nodal voltages still exceed the expected range after the sub-area autonomy, the operation strategies of soft open points are further improved by inter-area coordination with the alternating direction method of multipliers algorithm. The effectiveness of the proposed decentralized control method is verified on the IEEE 33-node system.


Introduction
The growing penetration of distributed generators (DGs), such as wind turbines (WTs) and photovoltaics (PVs), brings a series of operational problems to active distribution networks (ADNs), especially severe voltage violation [1,2]. Novel power electronic equipment represented by soft open points (SOPs) can effectively cope with the problems through accurate active and reactive power flow controls. SOPs are usually installed to replace a normally open point (NOP) for a flexible connection between feeders in ADNs. Compared to conventional regulation devices, such as shunt capacitors, on-load tap changers (OLTCs) and static var compensator (SVCs), SOPs can conduct a fast and precise active and reactive power flow adjustment. Therefore, it is of great significance to integrate SOPs in ADNs to realize a rapid and flexible operation control [3][4][5][6].
Previous studies mainly focused on the central control or local control manners [7][8][9]. However, for centralized control, the central controller needs to collect the information of the whole physical system and the external environments of ADNs. With the remarkable rise in the number and variety of devices in ADNs, the amount of data required for centralized control has dramatically increased, 3) Both the boundary information and the selected important non-boundary information in each sub-area interact with adjacent sub-areas under inter-area coordinated control. Namely, the operation information of SOPs integrated into non-boundary nodes is considered to be reflected at the boundary, based on sensitivity analysis, to approach the global optimization performance as close as possible.
The remainder of this paper is organized as follows. Section 2 presents the framework of decentralized voltage control strategies of SOPs. Section 3 builds an intra-area autonomous voltage control model of SOPs, and the inter-area voltage coordinated control model is established in section 4. Section 5 presents the results and discussion of the IEEE 33-node system to verify the effectiveness of the proposed method. Finally, section 6 concludes the paper.

Framework of Sensitivity-Based Decentralized Voltage Control of SOPs
First, the ADN is decomposed into several sub-areas based on electricity distance, and each area is under the control of one SOP at most. Taking each control period as a unit, the SOP operation strategies of each sub-area in the current period are initially adjusted using intra-area measurements and corresponding voltage-to-power sensitivity of the nodes inside the sub-area. The mutual influences among sub-areas are ignored in the first autonomous control stage. If some nodal voltages are still beyond the expected range after the sub-area autonomy, the operation strategies of SOPs can be improved by further inter-area coordination through boundary information interaction. The interarea coordination control mode is based on the alternating direction method of multipliers (ADMM) algorithm, in which the influence of the SOP in each sub-area is transferred to the boundary nodes by voltage-to-power sensitivities and interacts with the adjacent sub-areas as boundary information. In summary, the proposed decentralized control framework of SOPs is shown in Figure 1.

Intra-area Autonomous Voltage Control of SOPs
This section proposes a sensitivity-based intra-area autonomous voltage control method to reduce communication and calculation burden and improve the system operation performance. The operation strategies of SOPs in each sub-area are obtained only by intra-area measurements and corresponding voltage-to-power sensitivity of the nodes inside the sub-area.

Network Partition
The partition of the ADN can be described as a clustering problem. The central nodes of each sub-area are regarded as the clustering centers, and the electrical distance between the nodes is considered as clustering basis. Since this paper focuses on the optimization control of SOPs for ADNs, partitioning the network centered on SOP can fully utilize its ability of voltage and power flow regulation while reducing the computational complexity. Thus, under this partition principle, there is one SOP at most in each sub-area.
In this paper, the clustering method in [23] is used to divide the ADN with SOPs. Considering that the optimal number of sub-areas of the ADN may be inconsistent with the number of SOPs in the system, to ensure that each sub-area contains one SOP at most, the number of divided sub-areas is defined as: where is given by the distribution system operator (DSO) or refers to the empirical value [24]. ⌊•⌋ indicates that the elements of "•" round to the nearest integer less than or equal to "•".
is the total number of the nodes in the ADN, and represents the total number of SOPs integrated in the system. This paper considers an SOP with two terminals. Therefore, during the clustering calculation, two terminals of one SOP are selected separately as two clustering centers. After the clustering result is obtained, two sub-areas of the terminal nodes are merged into one subarea under the control of the related SOP. Then, the number of cluster centers is determined by For each node , the local density and its distance from nodes of higher density are defined by the electrical distance as [23]: where represents the threshold value of the electricity distance. − = 1 if < , and − = 0 otherwise. is the set of all nodes in the ADN, and is the set of all integration nodes of SOPs.
denotes the set of all nodes whose local density is larger than node . The electricity distance between nodes and is defined as follows [25].
where and are the electrical distance between nodes and defined in terms of the relation between nodal voltage and active/reactive power injection, respectively. The weight coefficients ≔ / + and ≔ / + describe the different influence of active and reactive power to nodal voltage. and represent the overall line resistance and reactance of the overlapped part of the unique path from the source node to node and the unique path from the source node to node [26]. and can also be regarded as the approximate voltage-toactive-power sensitivity and voltage-to-reactive-power sensitivity, respectively, which depend only on the physical topology of the network.
Since the cluster centers are assumed to be a relatively long distance from other points with a higher local density and to be surrounded by neighbor nodes with lower local density, nodes with high and long should be selected as the center nodes. Thus, in order to guarantee that the terminal nodes of SOPs are selected, their local density and distance are set to: where 100 is an empirical value that ensures the value stands out but is not too far from the database. In this case, must be a subset of the set of all cluster centers . A comprehensive index for identifying central nodes is proposed as follows; nodes with a large index value are selected as cluster centers.
(10) In summary, the steps of partitioning include: a) determine and according to (1) and (2); b) select the central nodes based on (3)-(10) and assign each remaining node to the cluster of its nearest neighbor with a higher density; and c) combine the two sub-areas centered at the two terminal nodes of SOPs.

Model of Intra-area Voltage Control
In the stage of initial autonomous voltage control, the minimum voltage deviation in each subarea is taken as the objective function: is the set of nodes in sub-area , , represents the nodal voltage magnitude at node in time period .
, is the desired voltage range. In time period , the assumption is made that there is no variation in active or reactive power at all nodes, except power injections of SOPs. Then, the voltage magnitude at node can be calculated according to its original value , * before the inter-area coordination and corresponding voltage-topower sensitivity.
, is the set of terminal nodes of the SOP integrated in sub-area . ∆ , and ∆ , are active and reactive power injection variations of the SOP at node in time period . , ≔ , , ⁄ denotes voltage-to-active-power sensitivity between , and , . , ≔ , , ⁄ denotes voltage-to-reactive-power sensitivity between , and , . , and , represent active and reactive power injection at node in time period . Specifically, , and , are calculated for every time period using real-time local measurements of the deviation of nodal voltage magnitude to unit changes in nodal active or reactive power output [27]. Alternatively, if there are μPMUs in the system, the sensitivities can also be estimated via network equivalents with μPMU measurements [17]. The nodal voltages should satisfy their operation constraints as below: where , denotes the operation range of nodal voltage. In addition, SOP operation constraints should be included. The control variables of SOPs under intra-area autonomous control are ∆ , and ∆ , , the variation of active and reactive power of the SOP at node in time period .
where , and , represent active and reactive power injection of the SOP at node in time period . , , * and , , * denote active and reactive power injection of the SOP before inter-area coordination at node in time period . The loss constraint of SOPs is as follows: , , , where is the loss coefficient of the SOP at node . represents the capacity of the SOP at node .
, and , denote the absolute value of maximum active and reactive power injection of the SOP at node .
In summary, the autonomous control model of SOPs can be concluded as (26).

Inter-area Coordinated Voltage Control of SOPs
In the stage of autonomous voltage control, operation strategies of SOPs are conducted based simply on the measurements inside the sub-areas to reduce data traffic and the calculation scale. However, it may be hard to fulfill the requirement of global optimization since the operation state of SOPs located in other sub-areas is not accounted for, especially in the neighboring ones. Therefore, this section proposes a coordinated control model based on the ADMM algorithm. The operation condition of the SOP in the current sub-area can be transferred to the boundary nodes by voltage-topower sensitivities and interacts with the adjacent sub-areas as boundary information.

Boundary Information for Interaction
This section conducts the information interaction for inter-area coordination at boundary nodes using nodal voltage-to-power sensitivities obtained from local measurements. By exchanging the information through boundary nodes, the inter-area optimization of each sub-area can be well linked to the global one.
However, based on the partition method in Section 3.1, the boundary between two sub-areas is a branch. Thus, we choose the end node of the boundary branch as the boundary node of the two neighboring sub-areas in the inter-area coordination. The control variables in each sub-area are the variations of operation strategies of the SOP within the sub-area, ∆ , and ∆ , , and the active and reactive power required from SOPs in adjacent sub-areas. For the boundary node between subarea and , the latter variables can be described as ∆ , , and ∆ , , in sub-area in time period . While, in sub-area , the corresponding local information for interacting through boundary node is calculated according to (27) and (28): where , is the set of boundary nodes of sub-area . The nodal voltage magnitude in sub-area can be calculated as: where not only the SOP located in the certain sub-area but also SOPs in its adjacent sub-areas are taken into account to obtain an approximate global optimization in a decentralized way.

Model of Inter-area Voltage Control
Based on the autonomous control model of SOPs, this section establishes a coordinated control model of SOPs based on a fully decentralized form of the ADMM algorithm [15,28] aiming at a globally optimal solution. Each sub-area collects the local nodal voltage information and corresponding voltage-to-power sensitivities and then forms an interaction of operation variables with the adjacent sub-area through boundary nodes. where represents the objective function of each sub-area. and denote the inequality and equality constraints in sub-area proposed in Section 3.2 and Section 4.1.
is the iteration index.
, can be expressed as , , ∆ , , , , ∆ , , , | ∈ , . The iterations consist of the following two steps: where denotes the feasible region of state vector , in area . denotes the set of all subareas include node .
, , is an auxiliary variable that represents , − , , / in the standard form of ADMM in which , is the global variable of node in time period , and , , is the local Lagrangian multiplier corresponding to node in sub-area in the th iteration. The penalty factor is refreshed by: where and are the preset parameters. is defined as the original residual of the feasibility of the original problem, and is defined as the dual residual of the feasibility of the dual problem. The criterion for iterative convergence can be determined by (34) [28]: where is the border residual in the th iteration. is the given tolerance. The flowchart of the proposed decentralized voltage control strategy of SOPs is shown in Figure 2.

Results and Discussion
In this section, the effectiveness of the proposed control strategy of SOPs is verified on the IEEE 33-node system. The model is implemented in the YALMIP optimization toolbox [29] using MATLAB R2014a and is solved by IBM ILOG CPLEX 12.60. The numerical experiments were carried out on a Windows 10 computer equipped with an Intel(R) Xeon(R) CPU E5-1620 processor running at 3.70 GHz, 8 GB of RAM. Figure 3 exhibits the modified IEEE 33-node system [30], of which the rated voltage level is 12.66 kV. The total active and reactive power loads of the system are 3715 kW and 2300 kVar, respectively. One PV and two WTs are integrated into the system. Table 1 presents the detailed installation parameters of DGs.  Given the remarkable development in power electronics technologies, SOPs are gradually being widely used in ADNs. Therefore, this section replaces tie switches TS 1 to TS 4 by SOPs, each with a capacity of 500 kVA. The loss coefficient of each inverter for the SOP is set to 0.02.

Result of the Partition
According to Equation (1), the number of sub-areas is set to 4, and the number of cluster centers is 8 according to Equation (2) 8,12,18,21,22,25,29, and 33 are selected as cluster centers by Equations (3)- (10), colored in blue, as shown in Figure 4. Table 2 presents the distribution result of nodes. Then, nodes in each line in Table  2 are gathered into one cluster since this paper considers two sub-areas centered at two terminal nodes of a certain SOP as one.  Overall, the partitioning result of the system with four SOPs is shown in Figure 5. Nodes 10, 17, 22, 25, 28, and 31 are selected as boundary nodes in inter-area coordination. The total numbers of nodes and branches are relatively small, and the calculation scale is reduced effectively.

Analysis of Optimization Results
Four scenarios are adopted to verify the effectiveness of the proposed decentralized control method of SOPs.
Scenario 1: There is no control strategy conducted on SOPs, and the initial operation state of ADNs is obtained. Scenario 2: The proposed intra-area autonomous control of SOPs is conducted. Scenario 3: The proposed autonomous and coordinated control of SOPs is conducted. Scenario 4: SOPs are centralized controlled to realize the global operation optimization of ADNs based on the method in ref. [5].
Taking an hourly time step over a day, the daily DG and load operation curves are obtained by forecasting, as shown in Figure 6. It is assumed that the range of statutory voltage is [0.90,1.10]. The desired voltage range is set from 0.98 p.u. to 1.02 p.u., which is also the margin of threshold in the coordinated control stage. The penalty factor is initially set to 10 4 in the ADMM algorithm, and

Optimization Results
To consider the severe voltage violation as a result of high penetration of DGs in ADNs, we set the power factors of the DGs to 1.0, and every DG operates at its full capacity. The total active power penetration of DGs reaches 100%. Figure 7 displays the values of nodal voltages, and Figure 8 shows the operation strategies of four SOPs. Table 3 presents the optimization results of the four scenarios. In scenario 1, there is severe voltage violation in the system without the control of SOPs. From Figure 7a, voltage magnitudes at node 18 exceed both the upper and lower desired level initially in scenario 1. Then, after the intraarea autonomous control in scenario 2, voltage violation is significantly mitigated. The operation strategies of SOPs are modified further by inter-area coordination through boundary information interaction in scenario 3 when voltage violations still occur, namely, in time periods 6, 10-14, 17, and 22, as shown in Figure 8. After the coordination in scenario 3, voltage profiles perform much better and more closely approximate the global optimization in scenario 4, as established in Figure 7c. Therefore, given that the proposed method in scenario 3 is based on less measurement information, the proposed decentralized control strategy of SOPs can effectively mitigate voltage violations without a huge communication and calculation burden.  In this section, the DG penetration was varied from 0 to 140% to show the role and performance of inter-area coordination in decentralized control. The power factors of the DGs were all set to 1.0. Scenarios 1, 2, and 3 designed above were adopted in this section. Figure 9 displays the variation in the number of inter-area coordination control times in scenario 3 with different penetration levels. Since DGs were integrated to the end of the branch where nodal voltages decreased sharply in the initial condition, they are helpful to compensate the imbalance between the power supply and load demand to increase the voltage magnitude. Therefore, along with the rise of the penetration from zero, the frequency of coordination decreased initially. However, when the penetration continuously increased, some magnitudes of nodal voltage were no longer lower than the desired range but exceeded the upper limit. SOPs in these sub-areas could not adequately distribute the surplus power in the stage of intra-area autonomous control. Hence, interarea coordination became more frequent to further regulate the power flow with adjacent sub-areas through information interaction. Figure 10 shows the maximum and minimum values of nodal voltages with various penetrations of DGs. The inter-area autonomous control eliminated voltage violations effectively, while the inter-area coordination became more important with increasing DG penetration.

Conclusions
This paper proposes a decentralized voltage control method of SOPs based on voltage-to-power sensitivity to reduce data traffic and fulfill the rapid and reliable control requirements of large-scale ADNs. First, the network is divided into several sub-areas, each of which is under the control of one SOP at most. The initial strategies of SOPs are adjusted by using local voltage-to-power sensitivity and voltage information within the sub-areas. If nodal voltages still exceed the expected range after the intra-area autonomy, the operation strategies of SOPs are further improved by inter-area coordination with the ADMM algorithm. The effectiveness of the proposed decentralized control method is verified on the case studies of the IEEE 33-node system.
With the rapid development of power electronic technology, it is foreseeable that the applications of SOPs will become more and more extensive. The proposed method shows great potential in applications with different integration conditions of DGs, the plug-and-play of SOPs, and the multi-time scale control scheme in ADNs. Additionally, based on limited measurements, sensitivities will be easier to be obtained and will be more precise with the promotion of advanced measurement equipment and technology in the future. The operation strategy based on sensitivities will be more efficient, and the relative error between the solutions of central and decentralized control manners will be further reduced.

Sets
Set of all nodes Set of all nodes whose density is larger than node Set of all integration nodes of SOPs Number of divided sub-areas Number of cluster centers Electrical distance from node to node Electrical distance threshold Electrical distance from node to node defined in terms of the relation between active power injection and nodal voltage Electrical distance from node to node defined in terms of the relation between reactive power injection and nodal voltage Local density of node Distance of nodes with a higher density than node Comprehensive index for identifying center nodes Overall line resistance of the overlapped part of the unique path from the source node to node and the unique path from the source node to node Overall line reactance of the overlapped part of the unique path from the source node to node and the unique path from the source node to node

Conflicts of Interest:
The authors declare no conflict of interest.