An MILP approach for optimal operation of unbalanced distribution networks through coordinated network reconfiguration and SOP utilization

Abstract

Peer-to-peer (P2P) energy trading, a prominent approach to energy exchange, can lead to violations of network security, intensify imbalances, and worsen operational conditions in unbalanced distribution networks (DNs). This paper, focusing on the DN as the physical foundation for P2P trading, recommends the benefits of joint network reconfiguration and soft open point (SOP) utilization to enhance flexibility and improve efficiency. In the proposed method, line mutual impedances are regarded, and the existence of single-, double-, and three-phase buses and branches is considered. Consequently, the single-phase reconfiguration capability is proposed, enabling radiality to be achieved independently for each phase. The AC optimal power flow model, structured to simultaneously determine the optimal network topology and SOP power injections, is formulated as a mixed integer linear programming problem. The model addresses loss reduction, voltage magnitude and angle imbalances mitigation, and P2P energy exchange facilitation. Numerical simulations executed on two IEEE unbalanced test DNs confirm the effectiveness of the proposed method. For the IEEE 13-bus DN, reconfiguration alone caused a 15.38% reduction in the objective, while the combined use of SOP and reconfiguration achieved an 85.51% reduction. For the IEEE 123-bus DN, the joint SOP-reconfiguration utilization led to a 74.46% improvement in the objective.

Introduction

The growing integration of distributed energy resources (DERs) and the rising energy demand are reshaping how modern distribution networks (DNs) operate, paving the way for new energy exchange models. Peer-to-peer (P2P) energy trading is one of the most notable energy exchange models. P2P trading enables local resource utilization and grants consumers the autonomy to select their preferred energy provider. However, the widespread adoption of DERs and P2P energy exchange present challenges beyond just bidirectional power flow. These include instances of voltage limit violations, overloaded lines, increased power losses, and a weakened network stability^1,2.

P2P trading model relies on the existing physical DN to transport energy. This energy trading system essentially comprises a virtual layer, dedicated to data handling and market functions, and a physical layer, where the actual energy exchanges take place. Notably, financial settlements in P2P trading do not ensure the physical delivery of electricity². Several studies have addressed how the P2P energy trading affects the physical system. For instance³, investigates voltage violations and power losses induced by the P2P transactions. Sensitivity analysis was used in Haggi and Sun⁴ to assess the P2P transactions that violate network operational limits and to filter them out. Another study⁵, by incorporating cost for using the DN infrastructures, as a motivation to form traders behavior, tries to adjust P2P transactions based on network limits. To increase flexibility, the Yan et al.⁶ proposes the DN topology modification. As a result, the integration of P2P energy exchange can be expanded without compromising the physical system constraints. Furthermore, the Suthar et al.⁷ proposes network reconfiguration as a tool for distribution system operators (DSOs) to reduce network losses induced by P2P transactions, utilizing the artificial bee colony, a metaheuristic algorithm.

The distribution network reconfiguration (DNR) is a NP-hard optimization problem due to the nonlinear characteristics of DN and presence of switches, which are typically represented by discrete variables. However, some approaches also model switches using continuous variables⁸. The complexity increases when the unbalanced attributes of DN are also considered. While DNs are mostly regarded as balanced systems, the real-world DNs exhibit a significant degree of imbalance. The imbalance is due to several factors, including the non-uniform allocation of loads across the phases, asymmetrical network structures such as single-phase and double-phase laterals, and untransposed line segments⁹. The increasing deployment of single-phase distributed generation (DG) units, particularly solar photovoltaics (PVs) intensifies these imbalances¹⁰, while P2P energy trading further compounds operational security concerns. In the literature various approaches are utilized for solving the DNR problem that can be broadly categorized as heuristic and metaheuristic approaches, artificial intelligence (AI) based approaches and mathematical based methods. Comprehensive review of approaches in solving the DNR can be found in Lotfi et al.¹¹ and Mahdavi et al.¹². The convexified mathematical approaches can acquire global optimal solution but the heuristic and metaheuristic approaches optimality can hardly be guaranteed in solving the DNR optimizations. The AI and machine learning based methods have the disadvantage of needing the vast training process.

While network reconfiguration is a useful method for network operators to improve flexibility, the application of this technique in unbalanced DNs has not been extensively studied. Regarding the minimization of operating costs through reconfiguration in unbalanced DNs, Zhai et al.⁹ and Zhou et al.¹³ proposed a mixed-integer linear programming (MILP) approach. However, they ignored deviations in phase angles from rated values; thus, they could not consider the angle unbalance metrics in the model. For simplification purposes, the impact of voltage magnitude on line losses was not considered in Zhai et al.⁹ and Zhou et al.¹³. The study Zheng et al.¹⁴ employed a deep neural network (DNN) to model the uncertainties associated with DGs and loads, utilizing a two-stage mixed-integer quadratic programming framework for unbalanced DNR. In the proposed method relation of line’s flowing power to the voltage magnitude of connecting busses is over simplified. Besides, it doesn’t consider the existence of single and double-phase lines. The model does not address voltage imbalance reduction and in its objective the impact of voltage magnitude on lines losses is overlooked. The study Paul et al.⁸ takes into account mutual coupling effects and the existence of single- and double-phase lines, while also considering angle imbalances. Instead of using binary variables to represent switch statuses, it introduces a novel method based on continuous variables to enforce system radiality constraints. This approach eliminates the need for a mixed-integer formulation in unbalanced DNR, which is structured as a nonlinear optimization problem with reduced nonconvexity. The study in Cikan and Cikan¹⁵ applied the slime mold algorithm, a metaheuristic technique, to solve the nonlinear optimization problem of unbalanced DNR while incorporating a voltage imbalance index in the objective function. Like other metaheuristic methods, this algorithm has limitations such as the necessity of precise parameter tuning and the risk of convergence to local optima. In Duan et al.¹⁶, a multi-objective framework was introduced for DNR and wind turbine allocation in unbalanced systems, utilizing an improved metaheuristic algorithm. To address the deficiencies of metaheuristics in optimizing solutions with discrete variables, several modifications were proposed, and the algorithm’s performance was compared to four other metaheuristic techniques. However, this study employed the backward-forward power flow method, which is unsuitable for networks with bidirectional power flow. The approach presented in Zheng et al.¹⁷ introduced a two-stage unbalanced reconfiguration framework based on column and constraint generation. In the first stage, the discrete variables related to switch states were determined, followed by the calculation of continuous optimal power flow recourse variables in the second stage once uncertainties were revealed. Although Zheng et al.¹⁷ employs more accurate power flow equations compared to Zheng et al.¹⁴, its linearization method requires prior knowledge of line power flow at the operating point which is often challenging to obtain. Additionally, this method does not account for the phase angle of buses.

Soft open points (SOPs), as power electronic devices connecting feeders and buses in DNs, can boost network efficiency by leveraging their ability to control active power flow and provide reactive power compensation¹⁸. The flexibility offered by SOPs can also be employed in P2P energy trading. The study of Zhao et al.¹⁹ highlights the utility of back-to-back voltage source converter (B2B VSC)-based SOPs in creating flexible connections for regions trading energy via P2P. The proposed methodology involves a two-stage process: in lower stage, a non-cooperative game establishes profit-maximizing P2P transactions between regions. Subsequently, in upper-stage the SOP optimizes its own profitability by adjusting power injections at its terminals, constrained by its technical specifications, through a second order conic programing (SOCP) problem. The initial determination of regional P2P transactions relies on linear power flow equations for balanced DNs, which do not account for lines capacities or losses. Notably, the presented framework can handle only one SOP connecting the regions in the DN.

Researches exploring the flexibility gained from simultaneous application of SOPs and network reconfiguration in DNs are limited. To address loss minimization in balanced DNs, Nguyen et al.²⁰ proposed the combined use of SOPs and reconfiguration, although it neglected SOP losses and employed a metaheuristic algorithm due to the nonlinear problem structure. Similarly, Azizi et al.²¹ presented a mixed-integer nonlinear programing (MINLP) model for balanced DNs, incorporating constraints to maintain protection coordination despite topological changes and SOP existence. Yin et al.²² introduced a deep reinforcement learning framework for reconfiguration in unbalanced DNs with SOP presence. This framework integrates two distinct AI methods: one for learning reconfiguration strategies and another for determining SOP variables. However, this AI-based approach requires significant training time and overlooks both SOP losses and mutual impedances of network lines. Furthermore, its reliance on DGs generations as input parameter, limits the applicability in scenarios requiring DGs production adjustments.

In this study for enhancing the efficiency in optimal operation of three-phase unbalanced DN, it is proposed to utilize the flexibility caused by simultaneous application of SOP and reconfiguration. The suggested model establishes an AC optimal power flow (ACOPF) that considers the mutual impedances of lines and uses more-detailed equations for relation of voltage and flow of lines, moreover, the difference of voltages magnitude and angle with the rated values is not ignored. Given the existence of single and two-phase lines in unbalanced DNs, the independent switching operation for each phase of line is assumed as an available feature. Consequently, in reconfiguration process radiality can be maintained independently for different phases. For visualization, Fig. 1 depicts a simple unbalanced network with a possible radial structure, decomposed by phase after reconfiguration.

Fig. 1

Single-phase reconfiguration in an unbalanced network, with separate radial structures across phases.

The proposed framework for unbalanced DNs determines the optimal network topology and the injected power of SOPs using a linear ACOPF model. This model is formulated as an MILP problem and solved by the DSO through a mathematical multi-objective optimization approach. Its objectives are to minimize power losses, mitigate voltage magnitude and angle imbalances, and facilitate P2P energy transactions. Notably, this study focuses exclusively on the physical layer of P2P transactions, without considering the virtual layer, including market mechanisms. The general overview for the proposed framework of the paper is presented in Fig. 2. To the best of the authors’ knowledge, this study is the first to address the optimal operation of unbalanced DNs by concurrently employing SOPs and network reconfiguration through a linearized mathematical optimization framework. Table 1 provides a comparative analysis of this study’s contributions relative to existing research, highlighting gaps in the literature. The specific contributions of this work are summarized as follows:

• By leveraging the combined flexibility of SOP employment and network reconfiguration, an optimal operation model for unbalanced DNs is proposed. Regarding the challenges of P2P energy trading for physical structure of DNs, the benefits of this joint optimization model for P2P trading facilitation along with voltage imbalance mitigation and loss reduction is investigated.

• The proposed co-optimization model incorporates the mutual impedances of network lines while accounting for the existence of single-, double-, and three-phase buses and laterals. Moreover, the deviations of voltages from their nominal values, both for magnitude and angle, are considered, leading to a more accurate ACOPF formulation. With all these considerations taken into account, a two-stage MILP optimization model is proposed, ensuring tractability and global solvability.

• When single- and double-phase branches exist in the unbalanced network, the framework introduces the capability for single-phase reconfiguration, enabling the formation of separate radial topologies across different phases of the unbalanced DN.

Fig. 2

General overview for the proposed scheme.

Table 1 Contributions comparison to existing researches.

The remainder of the paper is organized as follows. The formulations for unbalanced DN accommodated for reconfiguration operation and comprising SOP modeling are presented in “Unbalanced distribution network modeling” section. In “Optimization problem and solution method” section the solving approach is presented and the objectives are defined. The “Numerical simulation analysis” section outlines the results for numerical simulations, indicating the efficiency of proposed framework, and finally paper’s general findings are summarized in “Conclusion” section.

Unbalanced distribution network modeling

Modeling the unbalanced DN, here comprises different aspects associated with SOP presence, reconfiguration application and physical characteristics of network. Due to the SOP installation, considering and formulating the related operational constraints are necessary. The DNs are mostly designed with meshed structure but operated in a radial way to make the protection and the maintenance easier; thus, constraints for assuring the radial topology are also required. Another important part is to consider the physical rules of network which are manifested as power flow constraints in the presence of switching operations.

In the proposed model, the focus is on near-real-time optimization. Consequently, a time delay exists between the determination of the optimal solution and its implementation. As illustrated in Fig. 2, the DSO gathers data derived from short-term load forecasts, DER generation predictions, and the present condition of the network. This encompasses mutually agreed P2P transactions, technical data pertaining to DERs, network information including bus and line configurations, the locations and operational status of switches, as well as SOP technical data and their respective connection points. The DSO then executes the proposed two-stage optimization algorithm, and the resulting optimal solution provides the setpoints that are transmitted to the controllable entities. The optimal control setpoints are represented by straightforward numerical values: continuous real numbers for SOP injections, P2P curtailments, and dispatchable DER generations, as well as discrete integer values for switch statuses. Consequently, these data can be readily transmitted through existing communication infrastructures. Furthermore, considering the time delay between determining the optimal control setpoints and its subsequent execution, the demands for computational and communication speed are not stringent.

SOP modeling

When the injected powers by the converters are regarded as control variables, the active power transferring capability of SOP can be represented by following equations:

$$P_{i,\varphi }^{SOP} + Ploss_{i,\varphi }^{SOP} + P_{j,\varphi }^{SOP} + Ploss_{j,\varphi }^{SOP} = 0\quad \forall i,j \in B_{sop} ,\forall \varphi \in \{ a,b,c\}$$

(1)

$$Ploss_{i,\varphi }^{SOP} = A_{sop} \cdot \left| {P_{i,\varphi }^{SOP} } \right|\quad \forall i \in B_{sop} ,\forall \varphi \in \{ a,b,c\}$$

(2)

The Eq. (1) denotes the active power balance for SOP converters. While (2) calculates the SOP loss for each terminal with the loss coefficient, $A_{sop}$, is taken here as 0.02²³. Moreover, the capacity constraint for the converters of the SOP is defined by (3):

$$(P_{i,\varphi }^{SOP} )^{2} + (Q_{i,\varphi }^{SOP} )^{2} \le (S_{i,\varphi }^{SOP} {)}^{2} \quad \forall i \in B_{sop} ,\forall \varphi \in \{ a,b,c\} .$$

(3)

The optimal values of injected powers ($P_{i,\varphi }^{SOP}$ and $Q_{i,\varphi }^{SOP}$) are transmitted to the device local controllers as reference setpoints, and the controllers then regulate the injected powers accordingly. Considering the conventional B2B-VSC structure for SOPs, the general control strategy can follow the method proposed in Cao et al.²⁴. In this approach, one VSC operates under a P–Q control scheme to regulate the output powers, while the other VSC operates under a V_dc–Q control scheme to maintain a constant DC-link voltage and adjust the reactive power. The P–Q control scheme is based on a current-control method, which derives reference currents from the reference powers and regulates the output currents to track these values. Furthermore, Wang et al.²⁵ presents a modification of the control strategy in Cao et al.²⁴ to enable unbalanced SOP operation.

Radiality preserving constraints

In DNR problem, the optimal radial configuration is determined through altering the position of switches. Here, the constraints necessary for preserving the radiality of DN during switching operations²⁶ are extended to comply with multi-phase networks. Regardless of the direction of power flow, the formulation introduces two binary variables, $\beta p_{ij}^{\varphi }$ and $\beta p_{ji}^{\varphi }$, associated with each line ij on phase $\varphi$. Moreover, the connection status of the line, is denoted by the variable $\alpha_{ij}^{\varphi }$. The corresponding constraints are expressed accordingly to ensure proper network configuration and operational feasibility.

$$\beta p_{ij}^{\varphi } + \beta p_{ji}^{\varphi } = \alpha_{ij}^{\varphi } \quad \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(4)

$$\sum\limits_{\forall j \in N(i)} {\beta p_{ij}^{\varphi } } = 1\quad \forall i \in B/1,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(5)

$$\beta p_{ij}^{\varphi } = 0\quad \forall i \in slack,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(6)

$$\alpha_{ij}^{\varphi } ,\beta p_{ij}^{\varphi } \in \left\{ {0,1} \right\}\quad \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(7)

$$\sum\limits_{i} {\sum\limits_{j \in N(i)} {\alpha_{ij}^{\varphi } } } = 2(N^{\varphi } - {1)}\quad i \in B,\;\varphi \in \{ a,b,c\}$$

(8)

Equation (4) indicates that for the line ij on phase $\varphi$ either the bus i is the parent of j, or bus j is parent of i. It is noteworthy that for the line ij on phase $\varphi^{\prime }$ the $\beta p_{ij}^{{\varphi^{\prime } }}$ can acquire a different value from $\beta p_{ij}^{\varphi }$ showing the independence of radiality for different phases of unbalanced DN. The (5) signifies that each bus i (except the root bus) on phase $\varphi$ has exactly one parent and (6) tells that the root bus has no parent. Equation (7) confirms the binary entity of variables. Another necessary constraint for radiality is presented by (8) where $N^{\varphi }$ is number of buses in network that comprise the phase $\varphi$. Given the tree-graph structure of connected radial networks, (8) indicates that the number of lines per phase equals the number of buses comprising that phase minus one. The factor 2 in right hand of (8) implies that both ij and ji are considered as available lines in the model. Moreover, it is evident that $\alpha_{ij}^{\varphi } = \alpha_{ji}^{\varphi }$.

Unbalanced DN power flow modeling

In unbalanced DNs, the voltage drop across the line connecting buses i and j is determined by the line impedance matrix ${\vec{\mathbf{Z}}}_{ij}^{{}} \in {\mathbb{C}}^{3 \times 3}$, which accounts for the mutual impedances among the phases, as outlined in Eq. (9). In ${\vec{\mathbf{Z}}}_{ij}$ the diagonal arrays are self-impedances and non-diagonal arrays represent mutual-impedances between phases. For a line with missing phase (or phases) the arrays in column and row of the corresponding missing phase (or phases) are replaced by zero. In (9), $\overrightarrow {{{\mathbf{V}}_{i} }} = \left[ {\begin{array}{*{20}c} {\overrightarrow {{V_{ia} }} } & {\overrightarrow {{V_{ib} }} } & {\overrightarrow {{V_{ic} }} } \\ \end{array} } \right]^{T}$ and $\overrightarrow {{{\mathbf{I}}_{ij} }} = \left[ {\begin{array}{*{20}c} {\overrightarrow {{I_{ij}^{a} }} } & {\overrightarrow {{I_{ij}^{b} }} } & {\overrightarrow {{I_{ij}^{c} }} } \\ \end{array} } \right]^{T}$ are vectors for voltage and current. The symbol $( \bullet )^{T}$ denotes the transposing operation and ($\mathop \bullet \limits^{ \to }$) indicates complex values.

$$\overrightarrow {{\mathbf{V}}}_{i} - \overrightarrow {{\mathbf{V}}}_{j} = {\vec{\mathbf{Z}}}_{ij} .\overrightarrow {{\mathbf{I}}}_{ij} \mathop{\longrightarrow}\limits^{{\mathop {{\mathbf{Y}}_{ij} }\limits^{ \to } = \mathop {{\mathbf{Z}}_{ij}^{ - 1} }\limits^{ \to } }}\overrightarrow {{\mathbf{I}}}_{ij} = {\vec{\mathbf{Y}}}_{ij} ({\vec{\mathbf{V}}}_{i} - {\vec{\mathbf{V}}}_{j} )$$

(9)

Based on (9) the current for each phase of line ij can be articulated as follows:

$$\vec{I}_{ij}^{\varphi } = \sum\limits_{{\varphi^{\prime } \in \Omega_{ij} }} {\vec{Y}_{ij}^{{\varphi \varphi^{\prime } }} \cdot \left( {\vec{V}_{{i\varphi^{\prime } }} - \vec{V}_{{j\varphi^{\prime } }} } \right)}$$

(10)

In (10), the set $\Omega_{ij}$ shows the existing phases for line ij and $\mathop {Y_{ij}^{{\varphi \varphi^{\prime}}} }\limits^{ \to }$ are arrays of line admittance matrix ${\vec{\mathbf{Y}}}_{ij}$ derived by inversion from ${\vec{\mathbf{Z}}}_{ij}$. The apparent power flowing on phase $\varphi$ of the line ij, is presented in (11) which is further elaborated upon in (12). The symbol (*) denotes the conjugate for complex values.

$$\vec{S}_{ij\varphi }^{L} = \vec{V}_{i\varphi } \left( {\sum\limits_{{\varphi^{\prime } \in \Omega_{ij} }} {\vec{Y}_{ij}^{{\varphi \varphi^{\prime } }} (\vec{V}_{{i\varphi^{\prime } }} - \vec{V}_{{j\varphi^{\prime } }} )} } \right)^{*} \quad \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(11)

$$\begin{aligned} \vec{S}_{ij\varphi }^{L} & = V_{i\varphi }^{2} (g_{ij}^{\varphi \varphi } - jb_{ij}^{\varphi \varphi } ) + \sum\limits_{{\varphi^{\prime } \in \Omega_{ij} /\varphi }} {V_{i\varphi } V_{{i\varphi^{\prime } }} } \left( {\cos (\theta_{i\varphi } - \theta_{{i\varphi^{\prime } }} ) + j\sin (\theta_{i\varphi } - \theta_{{i\varphi^{\prime } }} )} \right)\left( {g_{ij}^{{\varphi \varphi^{\prime } }} - jb_{ij}^{{\varphi \varphi^{\prime } }} } \right) \\ & \quad - \sum\limits_{{\varphi^{\prime } \in \Omega_{ij} }} {V_{i\varphi } V_{{j\varphi^{\prime } }} } \left( {\cos (\theta_{i\varphi } - \theta_{{j\varphi^{\prime } }} ) + j\sin (\theta_{i\varphi } - \theta_{{j\varphi^{\prime } }} )} \right)\left( {g_{ij}^{{\varphi \varphi^{\prime } }} - jb_{ij}^{{\varphi \varphi^{\prime } }} } \right) \\ \end{aligned}$$

(12)

By extending the approach of Jabr²⁷ to accommodate with unbalanced DNs, the new variables $U_{i\varphi } = (V_{i\varphi } )^{2}$, $c_{ij}^{{\varphi \varphi^{\prime } }} = \sqrt {U_{i\varphi } } \sqrt {U_{{j\varphi^{\prime } }} } \cos \theta_{ij}^{{\varphi \varphi^{\prime } }}$ and $e_{ij}^{{\varphi \varphi^{\prime } }} = \sqrt {U_{i\varphi } } \sqrt {U_{{j\varphi^{\prime } }} } \sin \theta_{ij}^{{\varphi \varphi^{\prime } }}$ are introduced with $\theta_{ij}^{{\varphi \varphi^{\prime } }}$ denotes the $(\theta_{i\varphi } - \theta_{{j\varphi^{\prime } }} )$. Thus, by decomposing $\vec{S}_{ij\varphi }^{L}$ of (12) into its $P_{ij\varphi }^{L}$ and $Q_{ij\varphi }^{L}$ components and integrating binary variables that represent the connection status of the line, the subsequent equations are derived.

$$\begin{aligned} P_{ij\varphi }^{L} & = \alpha_{ij}^{\varphi } g_{ij}^{\varphi \varphi } U_{i\varphi } + \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} /\varphi }} {\alpha_{ij}^{\varphi } \alpha_{ij}^{{\varphi^{\prime } }} \left( {g_{ij}^{{\varphi \varphi^{\prime } }} c_{ii}^{{\varphi \varphi^{\prime } }} + b_{ij}^{{\varphi \varphi^{\prime } }} e_{ii}^{{\varphi \varphi^{\prime } }} } \right)} \\ & \quad - \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} }} {\alpha_{ij}^{\varphi } \alpha_{ij}^{{\varphi^{\prime } }} \left( {g_{ij}^{{\varphi \varphi^{\prime } }} c_{ij}^{{\varphi \varphi^{\prime } }} + b_{ij}^{{\varphi \varphi^{\prime } }} e_{ij}^{{\varphi \varphi^{\prime } }} } \right)} \quad \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij} \\ \end{aligned}$$

(13)

$$\begin{aligned} Q_{ij\varphi }^{L} & = - \alpha_{ij}^{\varphi } b_{ij}^{\varphi \varphi } U_{i\varphi } + \sum\limits_{{\varphi^{\prime } \in \Omega_{ij} /\varphi }} {\alpha_{ij}^{\varphi } \alpha_{ij}^{{\varphi^{\prime } }} \left( {g_{ij}^{{\varphi \varphi^{\prime } }} e_{ii}^{{\varphi \varphi^{\prime } }} - b_{ij}^{{\varphi \varphi^{\prime } }} c_{ii}^{{\varphi \varphi^{\prime } }} } \right)} \\ & \quad - \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} }} {\alpha_{ij}^{\varphi } \alpha_{ij}^{{\varphi^{\prime } }} \left( {g_{ij}^{{\varphi \varphi^{\prime } }} e_{ij}^{{\varphi \varphi^{\prime } }} - b_{ij}^{{\varphi \varphi^{\prime } }} c_{ij}^{{\varphi \varphi^{\prime } }} } \right)} \quad \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij} \\ \end{aligned}$$

(14)

Equations (13) and (14) show the active and reactive flowing powers on phase $\varphi$ of line ij. For integrating the switching capability, binary variables referring to open/close state of switch are also regarded. In these equations if the switch for phase $\varphi$ of line ij is open ($\alpha_{ij}^{\varphi }$ = 0) then $P_{ij\varphi }^{L}$ and $Q_{ij\varphi }^{L}$ will be zero. Presence of $\alpha_{ij}^{{\varphi^{\prime } }}$ in second and third terms of (13) and (14) signifies the impact of switch condition in the phase $\varphi^{\prime }$ of line ij on the flow at phase $\varphi$.

Based on the structure of (13) and (14) two set of available indices can be defined for the variables c and e, referred to as type-1 and type-2 indices. The benefit of this division of indices is for avoiding the presence of bilinear variables in conic constraint that will be pointed in paper. Thus, all indices for variables c and e fall into two types:

• Type-1 index: In second term of (13) and (14) with general form of $c_{ii}^{{\varphi \varphi^{\prime } }}$ and $e_{ii}^{{\varphi \varphi^{\prime } }}$ that are defined for the condition $\{ \forall i \in B,\forall \varphi ,\varphi^{\prime } \in \Omega_{i} ,\varphi \ne \varphi^{\prime } \}$. In fact, variables c and e on type-1 index are available for each bus i of network and for each pair of dissimilar phases present on this bus.

• Type-2 index: In third term of (13) and (14) with general form of $c_{ij}^{{\varphi \varphi^{\prime } }}$ and $e_{ij}^{{\varphi \varphi^{\prime } }}$ that are defined for the condition $\{ \forall i \in B,\forall j \in N(i),\forall \varphi ,\varphi^{\prime } \in \Omega_{ij} \}$. The variables c and e on type-2 index are available for each line ij of network and for each pair of phases present on this line. Moreover, phase indices $\varphi$ and $\varphi^{\prime }$ can be identical.

During reconfiguration, opening a line switch renders the indices of c and e associated with that line invalid, specifically affecting the type-2 indices. Conversely, the type-1 indices for c and e, which are linked to buses, remain unaffected by the switch operation.

A new binary variable $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }}$ is defined to replace the product of two binary variables $\alpha_{ij}^{\varphi } \cdot \alpha_{ij}^{{\varphi^{\prime } }}$ with the following constraint:

$$\left\{ {\begin{array}{*{20}l} {\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \le \alpha_{ij}^{\varphi } \;{\text{and}}\;\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \le \alpha_{ij}^{{\varphi^{\prime } }} } \hfill \\ {\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \ge \alpha_{ij}^{\varphi } + \alpha_{ij}^{{\varphi^{\prime } }} - 1} \hfill \\ \end{array} } \right.$$

(15)

For the multiplications $\alpha_{ij}^{\varphi } .U_{i\varphi }$ and $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot c_{ii}^{{\varphi \varphi^{\prime } }}$, the new variables $U\alpha_{ij}^{\varphi }$ and $\alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }}$ are defined with the constraints formulated in (16) and (17) respectively. Similar to (17), the variable $\alpha \alpha e1_{ij}^{{\varphi \varphi^{\prime } }}$ is also defined to substitute the product $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot e_{ii}^{{\varphi \varphi^{\prime } }}$. It is notable that here (17) only determines $\alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }}$ (or $\alpha \alpha e1_{ij}^{{\varphi \varphi^{\prime } }}$) based on the c (or e) of type-1 index.

$$\left\{ {\begin{array}{*{20}l} { - \alpha_{ij}^{\varphi } \cdot M \le U\alpha_{ij}^{\varphi } \le \alpha_{ij}^{\varphi } \cdot M} \hfill \\ { - (1 - \alpha_{ij}^{\varphi } ) \cdot M \le U\alpha_{ij}^{\varphi } - U_{i\varphi } \le (1 - \alpha_{ij}^{\varphi } ) \cdot M} \hfill \\ \end{array} } \right.$$

(16)

$$\left\{ {\begin{array}{*{20}l} { - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot M \le \alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }} \le \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot M} \hfill \\ { - \left( {1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} } \right) \cdot M \le \alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }} - c_{ii}^{{\varphi \varphi^{\prime } }} \le \left( {1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} } \right) \cdot M} \hfill \\ \end{array} } \right.\quad \forall i \in B,\forall \varphi ,\varphi^{\prime } \in \Omega_{i} ,\varphi \ne \varphi^{\prime } ,\forall j \in N(i)$$

(17)

Additionally, for the multiplication of $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot c_{ij}^{{\varphi \varphi^{\prime } }}$ the new variable $\alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime } }}$ with constraint formulated in (18) is defined. In a similar way, $\alpha \alpha e2_{ij}^{{\varphi \varphi^{\prime } }}$ is defined to replace the product $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} \cdot e_{ij}^{{\varphi \varphi^{\prime } }}$. Equation (18) only effects $\alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime } }}$ (or $\alpha \alpha e2_{ij}^{{\varphi \varphi^{\prime } }}$) based on the c (or e) of type-2 index.

$$\left\{ \begin{gathered} - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} .M \le \alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime}}} \le \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} .M \hfill \\ - (1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} ).M \le \alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime}}} - c_{ij}^{{\varphi \varphi^{\prime}}} \le (1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} ).M \hfill \\ \end{gathered} \right.\begin{array}{*{20}c} {} & {} & {\forall i \in B,\forall j \in N(i),\forall \varphi ,\varphi^{\prime} \in \Omega_{ij} } \\ \end{array}$$

(18)

It is noteworthy that constraints (17) and (18) do not impose any limitation on value of c and e regardless of whether they correspond to type-1 or type-2 indices. The only use of (17) and (18) is to adjust the value of $\alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }}$ (or $\alpha \alpha e1_{ij}^{{\varphi \varphi^{\prime } }}$) and $\alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime } }}$ (or $\alpha \alpha e2_{ij}^{{\varphi \varphi^{\prime } }}$) to represent the zero or get the value of $c_{ii}^{{\varphi \varphi^{\prime } }}$ (or $e_{ii}^{{\varphi \varphi^{\prime } }}$) and $c_{ij}^{{\varphi \varphi^{\prime } }}$ (or $e_{ij}^{{\varphi \varphi^{\prime } }}$) based on switch status in Eqs. (19) and (20). As a result, incorporating the newly defined variables allows for the reformulation of the active and reactive power flows in Eqs. (13) and (14) as follows:

$$P_{ij\varphi }^{L} = g_{ij}^{\varphi \varphi } U\alpha_{ij}^{\varphi } + \sum\limits_{{\varphi^{\prime } \in \Omega_{ij} /\varphi }} {\left( {g_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }} + b_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha e1_{ij}^{{\varphi \varphi^{\prime } }} } \right)} - \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} }} {\left( {g_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime } }} + b_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha e2_{ij}^{{\varphi \varphi^{\prime } }} } \right)}$$

(19)

$$Q_{ij\varphi }^{L} = - b_{ij}^{\varphi \varphi } U\alpha_{ij}^{\varphi } + \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} /\varphi }} {\left( {g_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha e1_{ij}^{{\varphi \varphi^{\prime } }} - b_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha c1_{ij}^{{\varphi \varphi^{\prime } }} } \right)} - \sum\limits_{{\varphi^{\prime} \in \Omega_{ij} }} {\left( {g_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha e2_{ij}^{{\varphi \varphi^{\prime } }} - b_{ij}^{{\varphi \varphi^{\prime } }} \alpha \alpha c2_{ij}^{{\varphi \varphi^{\prime } }} } \right)}$$

(20)

Mutual couplings in lines create virtual connections between phases, causing the network inherently non-radial and making it harder to determine precise solutions. Therefore, implementing new constraints is of importance to refine the possible solution space and enabling the achievement of precise results. In this context, considering the defined variables c and e, the relation $e_{ij}^{{\varphi \varphi^{\prime } }} = c_{ij}^{{\varphi \varphi^{\prime } }} \tan \theta_{ij}^{{\varphi \varphi^{\prime } }}$ can be incorporated²⁷ as an additional constraint. To address the nonlinearity of $c_{ij}^{{\varphi \varphi^{\prime } }} = \sqrt {U_{i\varphi } } \sqrt {U_{{j\varphi^{\prime}}} } \cos \theta_{ij}^{{\varphi \varphi^{\prime } }}$ and $e_{ij}^{{\varphi \varphi^{\prime } }} = c_{ij}^{{\varphi \varphi^{\prime } }} \tan \theta_{ij}^{{\varphi \varphi^{\prime } }}$, the linearization is achieved through the application of Taylor expansion around the operating points represented by w in Eqs. (21) and (22):

$$e_{ij}^{{\varphi \varphi^{\prime}}} = c_{ij}^{{\varphi \varphi^{\prime}}} \frac{{\sin (\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime}}}^{w} )}}{{\cos (\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime}}}^{w} )}} + c_{ij}^{{\varphi \varphi^{\prime}(w)}} \frac{{\theta_{i\varphi } - \theta_{i\varphi }^{w} }}{{\cos^{2} (\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime}}}^{w} )}} - c_{ij}^{{\varphi \varphi^{\prime}(w)}} \frac{{\theta_{{j\varphi^{\prime}}} - \theta_{{j\varphi^{\prime}}}^{w} }}{{\cos^{2} (\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime}}}^{w} )}}$$

(21)

$$\begin{aligned} c_{ij}^{{\varphi \varphi^{\prime } }} & = V_{i\varphi }^{w} V_{{j\varphi^{\prime } }}^{w} \left( {\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime } }}^{w} } \right)\sin \left( {\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime } }}^{w} } \right) + \frac{1}{2}\frac{{V_{{j\varphi^{\prime } }}^{w} }}{{V_{i\varphi }^{w} }}\cos \left( {\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime } }}^{w} } \right)U_{i\varphi } + \frac{1}{2}\frac{{V_{i\varphi }^{w} }}{{V_{{j\varphi^{\prime } }}^{w} }}\cos \left( {\theta_{i\varphi }^{w} - \theta v} \right)U_{{j\varphi^{\prime } }} \\ & \quad - V_{i\varphi }^{w} V_{{j\varphi^{\prime } }}^{w} \sin \left( {\theta_{i\varphi }^{w} - \theta_{{j\varphi^{\prime } }}^{w} } \right)\left( {\theta_{i\varphi } - \theta_{{j\varphi^{\prime } }} } \right). \\ \end{aligned}$$

(22)

Equations (21) and (22) do not account for line switching operation; however, as previously mentioned, they remain valid for type-1 indices of c and e. For type-2 indices the switching status of lines must be incorporated, as represented in Eqs. (23) and (24). Defining the right hand side of (21) and (22) as Taylor1 and Taylor2, respectively, we obtain:

$$- M\left( {1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} } \right) \le e_{ij}^{{\varphi \varphi^{\prime } }} - Taylor1 \le M\left( {1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} } \right)$$

(23)

$$- M(1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} ) \le c_{ij}^{{\varphi \varphi^{\prime}}} - Taylor2 \le M(1 - \alpha \alpha_{ij}^{{\varphi \varphi^{\prime}}} )$$

(24)

In (23) and (24) for type-2 indices if $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }}$ gets equal to one the e and c get equal to Taylor1 and Taylor2, respectively, aligning with Eqs. (21) and (22). If $\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }}$ gets equal to zero the (23) and (24) don’t impose any limitation on value of e and c, thus, can take any value in a sufficiently large interval.

Based on the definitions of U, c and e, constraint (25) is introduced. To enhance tractability, this constraint is relaxed into a SOC form, as represented in Eq. (26)²⁸:

$$\left( {c_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} + \left( {e_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} = U_{i\varphi } U_{{j\varphi^{\prime } }} \quad i,j,\varphi ,\varphi^{\prime } \in Type - 1\;{\text{or}}\;Type - 2$$

(25)

$$\left( {c_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} + \left( {e_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} \le U_{i\varphi } U_{{j\varphi^{\prime } }} .$$

(26)

Constraint (26) is derived without accounting for the switching operations involved in the DNR. Incorporating the bilinear variables related to switch statuses into the SOC constraint presents significant computational challenges, potentially leading to a nonlinear programming formulation. To avoid these complexities, this work circumvents that approach. For e and c of type-1 index dependent on buses, the opening or closing of line switches is not the matter and (26) will be valid for type-1 without need to any modification. c and e of type-2 index are reliant on lines open/close statues. For this type of index if the switches do not have operation $(\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} = 1)$ there will be no need to modify the (26). In case of switch operation $(\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} = 0)$, an examination of the constraints affecting c and e of type-2 reveals that these variables don’t impact the problem. In detail, in the case of $(\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} = 0)$, Eqs. (13)-(14) and their reformulated form (19)–(20) show that the zero value of the binary variable causes c and e to be omitted, having no influence in flowing power. Additionally, as discussed earlier, (23)–(24) also do not impose any restrictions on c and e. As a result, with $(\alpha \alpha_{ij}^{{\varphi \varphi^{\prime } }} = 0)$, c and e for the type-2 index are free variables, solely bounded by constraint (26). This is based on the fact that if a variable appears only in a single constraint and not in the objective, that variable becomes a free variable and can readily take any value dictated solely by that constraint without impacting the problem^29,30.

The model considers load demands to be composed of two parts: a fixed portion, which the DSO provides, and a flexible portion that can be met with P2P energy trading. This structure provides the DSO with the ability to limit P2P transactions as required. For secure and optimal network operation, the DSO must validate all P2P transactions among seller-buyer pairs. Consequently, the model introduces variables, $P_{i,\varphi }^{curt\_sell}$ and $P_{i,\varphi }^{curt\_buy}$, to enable the DSO’s regulation of these transactions. The detailed mechanisms of buyer–seller negotiations in P2P trading fall beyond the scope of this paper. Instead, it is assumed that the DSO has access to the final outcomes of these negotiations, allowing it to incorporate the resulting trade agreements into its operational framework. At any bus i, each phase $\varphi$ can take one of three roles in the P2P market: seller, buyer, or inactive participant. Consequently, the power balance for buyers in P2P market is formulated by Eq. (27), and the corresponding balance for sellers is given in Eq. (28).

$$P_{i,\varphi }^{g} - P_{i,\varphi }^{d} - P_{i,\varphi }^{buy} + P_{i,\varphi }^{curt\_buy} + P_{i,\varphi }^{SOP} = \sum\limits_{j \in N(i)} {P_{ij\varphi }^{L} }$$

(27)

$$P_{i,\varphi }^{g} - P_{i,\varphi }^{d} + P_{i,\varphi }^{sell} - P_{i,\varphi }^{curt\_sell} + P_{i,\varphi }^{SOP} = \sum\limits_{j \in N(i)} {P_{ij\varphi }^{L} }$$

(28)

Within any single P2P transaction, the amount of power curtailed is the same for both the seller and the buyer, meaning that for each transaction, $P_{{}}^{curt\_buy} = P_{{}}^{crut\_sell}$. For any phase of bus i that is not involved in P2P trading, the active balance is presented with Eq. (29). Additionally, Eq. (30) shows the reactive balance for each phase at its corresponding bus.

$$P_{i,\varphi }^{g} - P_{i,\varphi }^{d} + P_{i,\varphi }^{SOP} = \sum\limits_{j \in N(i)} {P_{ij\varphi }^{L} }$$

(29)

$$Q_{i,\varphi }^{g} - Q_{i,\varphi }^{d} + Q_{i,\varphi }^{SOP} = \sum\limits_{j \in N(i)} {Q_{ij\varphi }^{L} }$$

(30)

Additionally, the limitations on voltage magnitude of buses and capacity of line are specified with Eqs. (31) and (32), respectively. Here, $S_{ij\varphi }^{cap}$ represents the capacity for phase $\varphi$ of line ij. It is worth noting that constraint (32) is a quadratic circular constraint, analogous to (3), and is linearized by approximating it with a convex n-sided regular polygon³¹.

$$\left( {V_{i\varphi }^{\min } } \right)^{2} \le U_{i\varphi } \le \left( {V_{i\varphi }^{\max } } \right)^{2} \, \forall i \in B{,}\forall \varphi \in \Omega_{i}$$

(31)

$$\left( {P_{ij\varphi }^{L} } \right)^{2} + \left( {Q_{ij\varphi }^{L} } \right)^{2} \le \left( {S_{ij\varphi }^{cap} } \right)^{2} \, \forall i \in B,\forall j \in N(i),\forall \varphi \in \Omega_{ij}$$

(32)

Optimization problem and solution method

In this study, the DSO solves a multi-objective problem. The goals are to minimize losses, decrease imbalances in voltage magnitude and angle, and to facilitate P2P energy trading. This is accomplished by leveraging the flexibility offered by the simultaneous adjustment of network topology and SOP injections. Accordingly, the objective function is constructed by linearly combining several terms (33), each with a specific weight, to improve operational efficiency and tackle the network imbalances.

$$\min \;Obj = w_{1} \rho^{Loss} + w_{2} \rho^{mag} + w_{3} \rho^{ang} + w_{4} \rho^{P2P}$$

(33)

$$\rho^{Loss} = \sum\limits_{\begin{subarray}{l} i \in B \\ j \in N(i) \\ \varphi \in \Omega_{ij} \end{subarray} } {P_{ij\varphi }^{L} } + \sum\limits_{\begin{subarray}{l} i \in Bsop \\ \varphi \in \Omega_{i} \end{subarray} } {Ploss_{i,\varphi }^{SOP} }$$

(34)

$$\rho^{mag} = \sum\limits_{\begin{subarray}{l} i \in B \\ \varphi \in \Omega_{i} \end{subarray} } {\left| {V_{i\varphi } - V_{i\varphi }^{nom} } \right|}$$

(35)

$$\rho^{ang} = \sum\limits_{\begin{subarray}{l} i \, \in {\text{B}} \\ a,b \subseteq \Omega_{i} \end{subarray} } {\left| {\theta_{i}^{a} - \theta_{i}^{b} - \frac{2\pi }{3}} \right| + } \sum\limits_{\begin{subarray}{l} i \, \in {\text{B}} \\ b,c \subseteq \Omega_{i} \end{subarray} } {\left| {\theta_{i}^{b} - \theta_{i}^{c} + \frac{4\pi }{3}} \right| + } \sum\limits_{\begin{subarray}{l} i \, \in {\text{B}} \\ c,a \subseteq \Omega_{i} \end{subarray} } {\left| {\theta_{i}^{c} - \theta_{i}^{a} - \frac{2\pi }{3}} \right|}$$

(36)

$$\rho^{P2P} = \sum\limits_{k \in trans} {P_{k}^{curt\_buy} }$$

(37)

In Eq. (34), the first term aims to compute the active losses in the DN lines, and the second term considers the losses induced by the SOP operation. The (35) measures the total difference between the actual voltage magnitudes and their desired values across all existing phases at the network buses. In this paper the nominal value for voltage magnitude is considered 1pu. Equation (36) is designed to minimize deviations in phase angles compared to ideal balanced three-phase state. The first term of this equation quantifies angle deviations at buses having phases a and b, the second term is for buses with phases b and c, and the third term for buses with phases a and c. The objective ρ^P2P in (37), minimizes the curtailment of P2P transactions by the DSO. To address the multi-objective optimization problem, the weighted sum method³² is applied. This requires the use of positive weight coefficients (w₁–w₄) that add up to one. The analytic hierarchy process³³ determines the specific values of these coefficients, reflecting the DSO’s priorities. Henceforth, the DSO formulates the optimal operation model comprising the joint optimization for SOP and DN’s topology while preserving the radiality separately on each phase of unbalanced DN as presented in (38):

$$\left\{ \begin{gathered} Min\;(33) \hfill \\ s.t.\;\;\;\underbrace {\left( 1 \right) - \left( 3 \right) }_{{{\text{SOP}}}}, \, \underbrace {\left( 4 \right) - \left( 8 \right)}_{{{\text{Radiality}}}}, \, \underbrace {{\left( {15} \right) - \left( {20} \right)}}_{{\text{Line flow}}}{,}\underbrace {{\left( {21} \right) - \left( {22} \right)}}_{{\text{Taylor for type - 1 index}}},\underbrace {{\left( {23} \right) - \left( {24} \right)}}_{{\text{Taylor for type - 2 index}}}, \hfill \\ \;\;\;\;\;\;\underbrace {{ \, \left( {26} \right) \, }}_{{{\text{Conic}}}} \, , \, \underbrace {{\left( {27} \right) - \left( {30} \right)}}_{{\text{Power balance}}}, \, \underbrace {{ \, \left( {31} \right) \, }}_{{\text{Voltage limits}}}, \, \underbrace {{ \, \left( {32} \right) \, }}_{{\text{Line capacity}}} \hfill \\ \end{gathered} \right.$$

(38)

Single-stage solution methodology is not applicable here because Taylor-based constraints, derived from initial working point voltages, may not accurately represent the feasible solution space. Furthermore, the interdependence of phase voltages caused by mutual impedances and the presence of SOC constraint reliant on to these interdependent voltages adds the complexity. Therefore, a two-stage solution methodology, similar to successive programming³⁴ for large-scale OPF problems, is employed. The first-stage initially ignores the complex SOC constraints, and the second-stage integrates them while using the resultant solution of the first-stage. Due to presence of Taylor based constraints (21)–(24), the base points $V_{i,\varphi }^{w}$, $\theta_{i,\varphi }^{w}$ are initialized by using a flat voltage $[1\angle 0,1\angle ( - 2\pi /3),1\angle (2\pi /3)]$ on the existing phases across all system buses. Also, the initial value of $c_{ij}^{{\varphi \varphi^{\prime } (w)}}$ is easily obtained by applying the flat voltage and the definition $c_{ij}^{{\varphi \varphi^{\prime } }} = V_{i\varphi } V_{{j\varphi^{\prime } }} \cos \theta_{ij}^{{\varphi \varphi^{\prime } }}$ for any available index of type-1 and type-2. By solving the optimization problem iteratively, the base points for Taylor-approximated constraints are updated after each iteration.

Code availability

In the proposed approach the first-stage iteratively solves the problem (38) by removing the complex SOC constraint (26) until the voltage difference between two successive iterations falls below a predefined threshold. This threshold is set to 0.01 pu for buses’ voltage magnitude and 1 degree for the phase angles. The goal of the first-stage is to improve precision for the operating points. Upon convergence of the first-stage, the SOC constraint is added, and the problem (38) for second-stage is solved iteratively until the SOC relaxation error is less than a predefined threshold of 0.0005. In this approach, the first-stage involves an MILP problem. The second-stage problem, by employing a polyhedral approximation for the SOC constraint (26) addressed in Jabr et al.²⁶, can also be converted into an MILP, allowing it to be solved using a wider range of mixed-integer programming solvers. Additionally, the SOC relaxation error is defined by Eq. (39) and the procedure for solving the problem is demonstrated by the Algorithm 1.

$$e_{soc} = \left( {c_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} + \left( {e_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} - U_{i\varphi } U_{{j\varphi^{\prime } }}$$

(39)

In Stage 1, flag_mag and flag_ang act as convergence indicators for voltage magnitude and angle. In each solving iteration, their values are updated using conditional checks that compare the current magnitude and angle with those from the previous iteration. If the differences fall below predefined thresholds, the corresponding flag is set to zero, indicating convergence. Stage 1 is considered converged only when both flag_mag and flag_ang reach zero; otherwise, the iterative process continues. After Stage 1 completes, Stage 2 begins. In Stage 2, a third indicator, flag_soc, is added to monitor the SOC relaxation error. When this error becomes smaller than its specified threshold, flag_soc is set to zero. Stage 2 is deemed converged once all three flags—flag_mag, flag_ang, and flag_soc—have been set to zero. The paper’s mathematical algorithm is implemented in GAMS environment using CPLEX solver. It is also notable that the solution procedure is presented in a more comprehensive manner through a GAMS-like pseudocode, which is provided as a supplementary material.

Algorithm 1

Two-stage MILP approach for solving the problem

Numerical simulation analysis

The aim of this section is to evaluate the effectiveness of suggested joint optimization scheme for DN topology refinement and SOP injection control. Accordingly, two test systems exhibiting unbalanced conditions are examined. The optimal operation problem is solved and the improvements in objective values are analyzed in IEEE 13-bus DN. Additionally, the scalability for the proposed scheme is confirmed through the implementation on the IEEE 123-bus DN. Here the objective weights w₁–w₄, are respectively taken as (0.25, 0.125, 0.125, 0.5) with the higher weight assigned to ρ^P2P showing the DSO’s preference to curtail smaller amount of P2P transactions. To solve the optimization problem, CPLEX solver is employed within the GAMS environment. The simulations and computations were performed on a personal computer with an Intel Core® i5 CPU and 16 GB of RAM.

IEEE 13-bus network

The IEEE 13-bus DN³⁵ is characterized as a compact and highly loaded distribution network, exhibiting a constant load of 3466 kW and 2102 kVAr. Owing to its highly unbalanced features, it is suggested in Schneider et al.³⁶ as a benchmark for assessing the convergence properties of power flow analysis methodologies. The per-unit base for each phase power is set to 5/3 MVA. With the line voltage of 4.16 kV, the base voltage for individual phases is set to 2.401 kV and the voltage magnitude is constrained to lie within the interval of 0.95 pu to 1.05 pu. The diagram presented in Fig. 3 displays the 13-bus DN, the lines with switching operation capability are depicted by dash-line and it is assumed that all these lines are capable of single phase switching. The switchable lines 3–6, 5–11, 8–12 and 12–13 are added to the original network with their corresponding impedances same as those of line 2–7. Existing phases of double and single-phase lines are also labeled next to each one. For the study cases that comprise SOP, the terminals are also located in Fig. 3. The SOP rating capacity per phase at each terminals is considered 0.75 pu. A total of 5500 kW of PV generation is integrated into the system, and Table 2 contains information for their rating and placement. These PV units operate at power factor of one. Moreover, the DSO controls single-phase DGs, each rated at 333.3 kVA and functioning at a 0.9 power factor, located on each phase of bus 7.

Fig. 3

Diagram for the IEEE 13-bus DN.

Table 2 Capacity and place for PV units in the IEEE 13-bus DN.

This analysis considers the PV generation units as sellers and flexible network loads as buyers in the P2P energy market. The arranged P2P transactions between seller-buyer pairs in the 13-bus DN are outlined in Table 3. These transactions have the total amount 5100 kW, with 4200 kW taking place between seller-buyer pairs located on dissimilar phases. As previously stated, this study focuses on the physical layer of P2P trading, assuming negotiation results are available to the DSO. The DSO then optimizes network operation following the operational constraints and maximizes the volume for executable P2P transactions.

Table 3 P2P powers agreed between seller-buyer pairs for the IEEE 13-bus DN.

To assess the performance of the joint DNR and SOP utilization in the DN optimal operation when addressing the objective improvement, the following three cases are simulated and analyzed within the IEEE 13-bus DN model.

• Case 1: The basic system without reconfiguration operation and SOP presence.

• Case 2: The DN with reconfiguration operation capability.

• Case 3: The DN with simultaneous utilization of SOP and reconfiguration.

For radial operation of basic DN in Case 1, the system configuration regarding the connected/disconnected status of switchable lines are given in Table 4.

Table 4 Line connection status for the IEEE 13-bus system in Case 1.

The results for studied cases are presented in Table 5, demonstrating the consistent decrease in objective value from Case 1 to 3. In Case 2 the objective is decreased 15.83% in comparison to Case 1. Case 3 showing the considerable efficiency of concurrent SOP and DNR utilization, has the significant decrees of 85.51% and 82.87% in objective compared to Cases 1 and 2 respectively. Additionally, both voltage magnitude and angle imbalances have declined from Case 1 to Case 2, and again from Case 2 to Case 3. For loss in Case 3 while the line related losses are decreased in comparison of Cases 1 and 2, the presence of SOP and its loss component has increased the total system losses. For P2P transactions in Case 2, curtailments have decreased by 259.67 kW compared to Case 1. Almost whole of this curtailment reduction is occurred for seller-buyer pairs operating on dissimilar phases. The remarkable 97.64% reduction in volume of P2P curtailments achieved in Case 3 compared to Case 2, highlighting the flexibility that the concurrent operation of SOP and DNR introduces for unbalanced DNs. The computation times for the cases are also presented in Table 5. It is evident that Case 3 exhibits the longest runtime; however, compared to Case 2, the simultaneous implementation of SOP and reconfiguration results in an approximate 16.6% increase in computation time. This increase is considered acceptable given that the objective function improved by approximately 82.87%.

Table 5 Simulation results for IEEE 13-bus system cases.

Figure 4 illustrates the optimal radial topology for the 13-bus DN in Case 2, detailing phase-specific configurations. The lines 7–8 and 8–12 function as three-phase switchable lines, demonstrating selective connectivity. Specifically, when line 8–12 is engaged on phase a, it remains inactive on phases b and c. Conversely, line 7–8 is disconnected on phase a while maintaining connectivity for phases b and c. Additionally, line 7–10, a double-phase connection spanning phases a and c, is activated for phase a but is disconnected on phase c.

Fig. 4

Optimal radial configuration of 13-bus DN, decomposed by phases (Case 2).

Figure 5 presents the optimal radial topology obtained for Case 3, illustrating phase-specific configurations. Across all phases, the three-phase line 7–8 ensures network connectivity, while the line 8–12 is disconnected. The double-phase line 2–5 is active on phase b but deactivated on phase c. Consequently, the disconnection of line 2–5 on phase c is compensated by the connectivity provided through phase c of line 7–10, maintaining overall network integrity.

Fig. 5

Optimal radial configuration of 13-bus DN, decomposed by phases (Case 3).

Figure 6 illustrates the voltage magnitude profiles across phases for the analyzed cases. By adjusting its active and reactive power injections at terminal points, the SOP can alter both the quantity and direction of power flow in the lines. These adjustments, in turn, influence the voltage magnitude and phase angles of the buses.

Fig. 6

Voltage magnitude in the 13-bus DN for: (a) Phase a. (b) Phase b. (c) Phase c.

Figure 7 illustrates the angle deviations associated with the three terms of Eq. (36). The data is presented for buses where at least two relevant phases exist, enabling the calculation of voltage angle differences. Among the analyzed cases, Case 3 exhibits lower angle deviations indicating improved voltage characteristic compared to other scenarios.

Fig. 7

Angle deviations in the 13-bus DN for: (a) Phases a–b. (b) Phases b–c. (c) Phases c–a.

Figure 8 illustrates the initially arranged P2P transactions alongside the curtailments applied for each case study. In Case 2, where only the DNR was utilized, the curtailments were reduced by 259.67 kW in comparison to Case 1. Furthermore, in Case 3, which employed both DNR and SOP, a minimal level of P2P curtailment was observed. Specifically, only 1% of the initially arranged P2P transactions were curtailed in Case 3. As the power transfer between different phases of the DN is not possible, the load of P2P buyer’s must be exclusively supplied from resources on its specific phase, and similarly the seller’s generated power must be consumed within its own phase. Thus, the P2P transactions associated with seller and buyer on dissimilar phases, have the inherent potential to exacerbate imbalances within the DN. The power curtailments for P2P transactions associated with seller and buyer on dissimilar phases were 1817.03 kW, 1557.40 kW, and 51.27 kW for Cases 1, 2, and 3, respectively. Evidently, in all analyzed cases, a substantial portion of the curtailed P2P transactions were associated with these dissimilar-phase interactions: 74.66% in Case 1, 71.63% in Case 2, and all of the curtailments in Case 3. For Case 3 in same time of fulfilling almost all of the initially agreed P2P transactions, the voltage imbalances are effectively reduced. This demonstrates the efficiency of combining SOP and DNR, leading to a significantly improved objective value.

Fig. 8

Initially arranged P2P transactions and the curtailments for the 13-bus DN.

Figure 9 demonstrates the active and reactive powers injected by SOP in each phase of its connected terminals for Case 3, showing that in all phases the active power is transferred from bus 4 to bus 11. The total apparent injected power on terminals of SOP sums up to 4652.57 kVA.

Fig. 9

Injected active and reactive powers by SOP in Case 3 of the 13-bus network.

IEEE 123-bus network

The IEEE 123-bus test system serves as a benchmark for evaluating the scalability of the proposed approach within a large-scale DN. This system consists of a total non-flexible load of 3490 kW and 1920 kVAr, with comprehensive parameter data available in IEEE PES Test Feeders³⁵. Operating with the base power of 5/3 MVA and a base voltage of 2.401 kV per phase, the system is connected to the upstream network at bus 1. Figure 10 illustrates the network’s layout, highlighting the switchable lines (marked in red) and the integration of SOP devices. For the case study incorporating SOP functionality, two SOP units are included: one linking buses 7 and 151, and another positioned between buses 64 and 100. Each terminal maintains a per-phase capacity of 0.75 pu. Additionally, the DSO manages six single-phase DGs, each rated at 111.1 kVA with a power factor of 0.9. These DG units are placed on different phases of buses 60 and 135 to enhance operational flexibility. For the case studies involving the 123-bus DN, Table 6 outlines 20 distinct P2P energy trading agreements between seller-buyer pairs, totaling 5600 kW, with 3000 kW exchanged between pairs operating on dissimilar phases. Notably, the switchable lines 13–30, 36–64, 52–87, and 151–300 are integrated into the original network, each possessing impedances identical to those of line 13–18. All the switchable lines are three-phase, with the exception of the double-phase line 36–64, which includes phases a and b, and the single-phase line 54–94, which contains only phase a. Considering the established network set up, the IEEE 123-bus DN is employed to simulate three case studies that follow a similar structural design to those conducted on the IEEE 13-bus DN. For Case 1, where no reconfiguration is applied, the predefined operational states of switchable lines are specified in Table 7.

Fig. 10

Diagram of the IEEE 123-bus DN.

Table 6 Arranged P2P powers between seller-buyer pairs for the IEEE 123-bus DN.

Table 7 Predefined status of switchable lines in Case 1 of the IEEE 123-bus DN.

The resulted objective and its components for the studied cases related to the IEEE 123-bus DN are depicted in Fig. 11, where the smaller area of the polygon corresponding to each simulated case, exhibits the better performance in objective reduction.

Fig. 11

The objective value and its components for the 123-bus DN.

Case 3, which integrates both SOP and DNR, achieves the lowest objective value and the least P2P curtailments among the studied cases. It demonstrates a 33.07% reduction in the objective value and a 49.46% decrease in P2P curtailments compared to Case 2, which solely employs unbalanced reconfiguration. Figure 12 illustrates power curtailments for P2P arrangements across the cases of the 123-bus DN. The P2P transactions between seller-buyer pairs of dissimilar phases are potentially challenging, as they can introduce significant network imbalances. Consequently, in Cases 2 and 3, all the transaction curtailments are associated with this type of exchanges, and in Case 1, such curtailments account for 89.85% of all P2P reductions. Case 3 exhibits higher system losses compared to the other cases due to SOP-induced losses and the increased volume of P2P transactions. However, the objective function prioritizes maximizing P2P transaction fulfillment, thereby reducing the impact of increased losses on the overall objective value. Ultimately, this optimization leads to a net decrease in the objective function result. In terms of voltage magnitude imbalance improvements, Cases 2 and 3 achieve 19.11% and 44.75% reductions, respectively, when compared to Case 1. Additionally, Case 3 demonstrates a 37.30% improvement in angle imbalance relative to Case 2, highlighting its effectiveness in enhancing network stability.

Fig. 12

The curtailments on initially agreed P2P exchanges in cases of the 123-bus DN.

Table 8 provides details on the connected switchable lines for each phase in Cases 2 and 3, where reconfiguration is applied. As an example of single-phase switch operation in Case 2, line 13–30 is exclusively connected on phase c, while line 151–300 is disconnected on phase a and is connected on phases b and c. These configurations illustrate the selective phase-wise connectivity adjustments made during the reconfiguration process.

Table 8 Connected switchable lines from optimization results for Cases 2 and 3 of the IEEE 123-bus DN.

Equation (40) establishes a three phase imbalance index for voltage¹⁰, incorporating both magnitude and phase angle by expressing the proportion of negative to positive sequence voltage at bus i. This index for three phase buses across the case studies is displayed in Fig. 13. The numbers along the outer circumference indicate the DN’s three-phase buses, while the radial distance from the center represents the index value. A higher concentration of index values near the center signifies improved voltage imbalance mitigation, demonstrating the effectiveness of Case 3.

$$VI_{i} = \frac{{\left| {\vec{V}_{i}^{ - } } \right|}}{{\left| {\vec{V}_{i}^{ + } } \right|}} = \frac{{\left| {\vec{V}_{ia}^{{}} + \vec{V}_{ib}^{{}} e^{j4\pi /3} + \vec{V}_{ic}^{{}} e^{j2\pi /3} } \right|}}{{\left| {\vec{V}_{ia}^{{}} + \vec{V}_{ib}^{{}} e^{j2\pi /3} + \vec{V}_{ic}^{{}} e^{j4\pi /3} } \right|}}$$

(40)

Fig. 13

Three-phase voltage imbalance index for the 123-bus DN.

While the SOC relaxation may lead to deviations from the exact solution of the original non-convex problem, existing literature^8,37 support the fact that minor differences are generally acceptable due to the significant improvements in computational efficiency. Thus, the SOC relaxation error, defined as $e_{soc} = \left( {c_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} + \left( {e_{ij}^{{\varphi \varphi^{\prime } }} } \right)^{2} - U_{i\varphi } U_{{j\varphi^{\prime } }}$, is calculated for three cases studies of 123-bus DN. In Table 9, the average and maximum values of the errors for each case study are presented, showing convincingly small values that demonstrate the proposed model’s accuracy and reliability.

Table 9 SOC relaxation errors in the IEEE 123-bus DN case studies.

Figure 14 presents the injected active and reactive powers by the SOPs in Case 3 of the IEEE 123-bus system. SOP1 facilitates active power transfer, with phases a and c directing power from bus 151 to bus 7, while phase b operates in the reverse direction, transferring power from bus 7 to bus 151. Additionally, SOP1 exhibits substantial reactive power injection and absorption, where positive values represent injection and negative values indicate absorption. SOP2 follows a similar operational pattern but at lower injection levels compared to SOP1.

Fig. 14

Powers injected by SOP1 and SOP2 in Case 3 of the 123-bus network.

The proposed approach is also compared with the unbalanced DNR method presented in Zhou et al.¹³, and the results, including computation time, are shown in Table 10. Since the method in Zhou et al.¹³ does not support single-phase switching and cannot account for voltage angle imbalances, certain adjustments were made to ensure a fair comparison under equivalent conditions. Specifically, the reconfiguration was performed without single-phase switching, the angle imbalance term addressed in Eq. (36) was excluded from the objective function, and SOP operation was omitted in both approaches. The objective function’s weight factors were set to 0.25 for both power loss and voltage magnitude imbalance, and 0.5 for P2P transaction curtailments. The switchable lines and all other conditions used in this comparison are consistent with those previously described and shown in Fig. 10. The higher losses observed in the proposed method are due to its more accurate power flow modeling, which avoids simplifications. In contrast, the model in Zhou et al.¹³ neglects line losses in the power balance equations and does not account for voltage magnitude deviations from the nominal 1 per unit value when calculating losses. Additionally, it simplifies the relationships between bus voltage magnitudes and line power flows. These simplifications lead to underestimated losses and reduced computational time. With the reconfiguration operation the proposed approach achieves a 15.58% improvement in the objective, while the method of Zhou et al.¹³ results in a 3.52% improvement.

Table 10 Comparison of the proposed method with the approach of reference¹³.

Conclusion

This paper introduces a two-stage MILP framework for optimizing the operation of unbalanced DNs, integrating both network reconfiguration and SOP functionalities. The optimization is executed using a successive linear programming approach, ensuring computational efficiency. The OPF model considers single, double, and three-phase buses and lines, explicitly accounting for mutual line impedances. Additionally, the reconfiguration model enables single-phase switching while preserving phase-wise radiality. To address the complexities of P2P energy trading in unbalanced networks, the proposed framework enhances operational flexibility and efficiency by facilitating P2P transactions and mitigating voltage imbalances. Case studies validate the model’s effectiveness: in 13-bus DN, network reconfiguration alone improved the objective function by 15.38%, whereas the combined use of DNR and SOP achieved an 85.51% improvement, along with significant reductions in P2P transaction curtailments. Similarly, in 123-bus DN, reconfiguration alone lowered P2P curtailments by 16.88%, while integrating both SOP and DNR reduced curtailments by 57.99%. Future research could enhance the proposed framework by incorporating uncertainties in load variations and DERs injections, using approaches such as robust optimization techniques³⁸ or chance-constrained formulations³⁹. Additionally, optimizing SOP placement and capacity, along with developing a mechanism to compensate SOP investment costs related to P2P facilitation, would further improve the comprehensiveness of the proposed approach.