The distribution system is the final link in the power system, directly serving the majority of users, and being closely related to population distribution and economic development. With the high-permeability access of DGs and the uneven spatial distribution of industrial, commercial, and residential regions in urban areas, the problem of load unbalanced spatial-temporal distribution has always existed in urban distribution networks. The mismatch between power generation and consumption may cause lower resource utilization, network congestion (Zhao et al., 2022), and increased active power loss, which seriously affects the security and economy of distribution network.

Aiming at the problem of load spatial-temporal unbalanced condition, there are mainly the following solutions. One option is to consider new substations or line expansion (Yao et al., 2014), but this involves high investment costs and is challenging to implement, especially in areas with limited urban land resources. Through flexible regulation of controllable resources such as energy storage devices and flexible load, load curves can be dynamically adjusted by peak shaving and valley filling to alleviate load imbalance and other problems (Cho et al., 2012; Hosseina and Bathaee, 2016; Li et al., 2022). The effectiveness of this method is affected by the capacity and location of controllable resources. Network topology reconfiguration involves altering the open and closed states of switches in the feeders to balance feeder load, adjust power flow distribution, and improve the economic efficiency of distribution network (Baran and Wu, 1989a; Ji, 1997). Generally, distribution network reconfiguration can be divided into two categories: static reconfiguration and dynamic reconfiguration. Static reconfiguration optimizes the topology of the distribution network under a single time section, which is the basis of dynamic reconfiguration. Dynamic reconfiguration takes into consideration the fluctuation of DGs and load demand to dynamically optimize the distribution network structure in a continuous period (Jabr et al., 2012). The optimization objectives for distribution network reconfiguration are usually to minimize the active power loss (Jakus et al., 2017), improve the system reliability (Lotfi et al., 2020) and voltage stability (Zhang and Wang, 2004; Wang, 2012), and balance the feeder load (Gao et al., 2022). The optimization problem in multi-time dynamic reconfiguration involves both continuous and numerous discrete variables, posing a challenging task. Mathematical optimization methods (López et al., 2016; Tian et al., 2016; Aldik and Venkatesh, 2020), intelligent optimization algorithm (Lotfi and Ghazi, 2021; Lotfi, 2022) and heuristic algorithms (Jakus et al., 2020; Silveira et al., 2021; Montoya et al., 2023; Yu et al., 2023) have been proposed to solve the problem. Heuristic algorithms, not necessitating convexity in the mathematical model, have been widely used in solving non-convex problems. However, the results may be local optimal solutions through heuristic algorithms. With the development of optimization theory, mathematical optimization methods have been extensively applied to solve dynamic reconfiguration problems. Specifically, the convex relaxation technique represented by second-order cone programming (SOCP) is used more and more frequently, which can obtain the globally optimal solution and reduce the computation time.

The above studies address the distribution network load balancing problem with traditional single-radial distribution network, most of them start from the spatial dimension to realize load balancing by adjusting the tie switches state. The dispatching means are relatively single, considered only from the feeder level, with less consideration of scenarios where load and DG output change over time, and seldom with the main objective of load balancing of multiple feeders in multiple substations. However, relative to the previous grid structure, snow-shaped distribution network is upgraded based on a 10 kV single-double ring network by adding new station tie switches. The snow-shaped distribution network (SDN) (Wang et al., 2023) constructs a “3-station 6-wire” or “4-station 8-wire” feeder cluster, with the ring main units as the network unit, and each feeder contains multiple tie switches and segment switches. The flexible and controllable ring network structure of SDN establishes a supportive platform for dynamic reconfiguration to achieve multi-level and multi-directional load transfer, and provides an effective means to promote the spatial-temporal balance of load within feeder clusters. It can comprehensively consider a variety of control means to realize the load transfer between two feeders supplied by the same transformer, different transformers, and different substations, to achieve the “feeder-transformer-substation” three levels of load balancing.

To our best knowledge, this is the first study on the load imbalance problem of substation level, transformer level and feeder level in urban SDN. From both time and space dimension, this paper proposes a multi-objective dynamic reconfiguration model for load balancing through collaborative optimal dispatching of dynamic reconfiguration and distributed energy storage system in SDN. Firstly, the grid structure of SDN is analyzed and the power flow model for its dynamic reconfiguration is developed. Secondly, based on the advantages of multiple inter-station and intra-station connections, a dynamic reconfiguration model in SDN for feeder load balancing is established, considering DGs and ESS. In order to take into account the economic efficiency and load balancing, the weighted combination of the minimum active power loss and load balance index is proposed as the optimization objective function. Thirdly, the original non-convex model is converted into a mixed-integer second-order cone programming (MISOCP) model using convex relaxation theory for an accurate and efficient solution. Finally, the proposed dynamic reconfiguration model and method are analyzed and verified in the 10 kV Tianjin snow-shaped distribution network and IEEE 33-node system.

In urban distribution networks, different types of loads have different spatial-temporal asynchrony, and some feeders have heavy load during peak-load period. SDN constructs feeder clusters with four (or three) stations and eight (or six) lines, providing a flexible load transfer supportive platform. Through dynamic reconfiguration, it can realize the flexible spatial-temporal transfer of load in feeder clusters to mitigating the coexistence of light and heavy loads to decrease the security risks in SDN operation and improve economic efficiency.

The dynamic reconfiguration model for SDN is a highly non-convex nonlinear programming problem, which may have multiple locally optimal points and is difficult to solve to obtain the globally optimal solution. In order to solve this problem accurately and efficiently, the original non-convex nonlinear programming (NLP) can be converted into a mixed integer second-order cone programming (MISOCP) by applying the second-order cone relaxation and the big-M method.

Equations 20, 21 are obtained by introducing auxiliary variables i _ij (t), and u _ij (t) for variable transformation, and Eq. 22 is obtained by relaxation through the big-M method. M is an arbitrarily large positive number that is not infinite. i i j t = I i j 2 t (20) u i t = U i 2 t (21) − a i j t M ≤ P i j t ≤ a i j t M − a i j t M ≤ Q i j t ≤ a i j t M (22)

With the above constraints Eqs 20–22, the trend constraint Eq. 1 is transformed into Eq. 23, and Eq. 2 is relaxed by the big-M method to obtain Eq. 24. ∑ i ∈ Ω i n j P i j t − i i j t R i j + P j t = ∑ k ∈ Ω o u t j P j k t ∑ i ∈ Ω i n j Q i j t − i i j t X i j + Q j t = ∑ k ∈ Ω o u t j Q j k t (23) u j t − u i t ≥ − M 1 − a i j t − 2 P i j t R i j + Q i j t X i j + i i j t R i j 2 + X i j 2 u j t − u i t ≤ M 1 − a i j t − 2 P i j t R i j + Q i j t X i j + i i j t R i j 2 + X i j 2 (24)

Equation 7 is transformed to Eq. 25 accordingly. f 2 = ∑ t = 1 T ∑ i j ∈ Ω l i i j t R i j (25)

Equation 3 is transformed into Eq. 26 by the second-order cone relaxation, which is further transformed into the standard second-order cone form as shown in Eq. 27: i i j t ≥ P i j 2 t + Q i j 2 t u i t (26) 2 P i j t 2 Q i j t i i j t − u i t 2 ≤ i i j t + u i t (27)

Similarly, the linearization of Eq. 6 is achieved by introducing the auxiliary variable f _ij (t) for variable substitution. Equation 28 is obtained by using variable substitution of P i j 2 t S i j . ⁡ max 2 , Q i j 2 t S i j . ⁡ max 2 . f i j t = P i j 2 t S i j . ⁡ max 2 + Q i j 2 t S i j . ⁡ max 2 (28)

Equation 6 is transformed into Eq. 29. f 1 = ∑ t = 1 T ∑ i j ∈ Ω l f i j t (29)

Equation 28 is transformed into Eq. 30 with the second-order cone relaxation, which is further transformed into the standard second-order cone form as shown in Eq. 31: f i j t ≥ P i j 2 t S i j . ⁡ max 2 + Q i j 2 t S i j . ⁡ max 2 (30) 2 P i j t S i j . ⁡ max 2 Q i j t S i j . ⁡ max 1 − f i j t 2 ≤ 1 + f i j t (31)

The flowchart of SDN dynamic reconfiguration method for load balancing in this paper is shown in Figure 2. And the specific operation process includes the following five steps:

Step 1: Data inputting. Input the basic SDN data including network parameters, load data, devices data, and so on;

Based on the characteristics of SDN frame structure, this paper establishes a multi-objective dynamic reconfiguration model for load balancing through collaborative optimal dispatching of dynamic reconfiguration and distributed energy storage system in SDN. And the second-order cone relaxation and big-M method are employed to convert the original NLP model into a MISOCP model which can be tractably solved. Case studies in an urban 10 kV distribution network and IEEE 33-node system are performed to show the effectiveness of the proposed model, the following conclusions can be drawn:

(1) The optimization results show that the safety and economy of SDN operation are both included in our formulated model to mitigate the feeder load unbalance, reduce the active power loss and improve voltage profile.

This paper focuses on the dynamic reconfiguration strategy in SDN for feeder load balancing, flexibly transferring load through intra-station and inter-station contacts in all directions to improve system security and economic efficiency. Further research will consider additional elements such as flexible loads and electric vehicles to fully exploit the operational potential of SDN.