Optimal Power Flow of Unbalanced Distribution Networks Using a Novel Shrinking Net Algorithm

Abstract

The increasing penetration of distributed energy resources (DERs) in unbalanced distribution networks presents significant challenges for optimal operation, particularly concerning power loss minimization and voltage regulation. This paper proposes a comprehensive Optimal Power Flow (OPF) model that coordinates various assets, including on-load tap changers (OLTCs), reactive power compensators, and controllable electric vehicles (EVs). To solve this complex and non-convex optimization problem, we developed the Shrinking Net Algorithm (SNA), a novel metaheuristic with mathematically proven convergence. The proposed framework was validated using the standard IEEE 123-bus test system. The results demonstrate significant operational improvements: total active power loss was reduced by 32.1%, from 96.103 kW to 65.208 kW. Furthermore, all node voltage violations were eliminated, with the minimum system voltage improving from 0.937 p.u. to a compliant 0.973 p.u. The findings confirm that the proposed SNA is an effective and robust tool for this application, highlighting the substantial economic and technical benefits of coordinated asset control for modern distribution system operators.

1. Introduction

The Optimal Power Flow (OPF) of distribution networks is effective technology to ensure safe and economic operation and overall voltage level. In the distribution network, the on-load tap changer (OLTC) transformers, reactive power compensators, controllable distributed generators, and electric vehicles (EVs) can be used as control devices to improve network performance.

In response to the complex challenges of OPF in modern distribution networks, researchers have proposed a variety of solution methodologies [1,2]. These can be broadly categorized into two main groups: traditional mathematical optimization methods and modern metaheuristic algorithms. Traditional approaches, such as linear programming [3], nonlinear programming [4], and mixed-integer programming [5], can be highly efficient and guarantee optimality for convex problems. However, their effectiveness is often limited when applied to the non-convex and non-differentiable OPF problem characteristic of unbalanced distribution systems. For non-convex problems like OPF, the performance of these solvers is particularly sensitive to the choice of initial values, and they are prone to converging to local optima rather than the global solution. Furthermore, many of these methods require the objective functions and constraints to be differentiable and continuous.

To overcome these limitations, modern metaheuristic algorithms have gained significant popularity. Algorithms such as genetic algorithms (GAs) [6], Particle Swarm Optimization (PSO) [7,8], and others like the crow search optimizer have been successfully applied to various power system optimization problems [9,10], including reactive power planning [7] and economic load dispatch [8]. These methods do not require gradient information, making them well-suited for complex, non-differentiable problems. While powerful, many of these metaheuristics introduce their own set of challenges. Their performance often depends on the careful tuning of several algorithm-specific parameters (e.g., inertia weights in PSO, crossover and mutation rates in GA), and they can be susceptible to premature convergence, especially in the high-dimensional search space of the distribution system OPF problem. This analysis reveals a distinct research gap, highlighting the need for a novel optimization algorithm that combines the robustness of metaheuristics with algorithmic simplicity, requiring minimal parameter tuning while effectively avoiding premature convergence in complex OPF problems. To address this gap, this paper introduces the Shrinking Net Algorithm (SNA). The novelty of SNA lies in two key areas: (1) a unique ‘boundary-in’ search mechanism that initializes the population on the edge of the feasible space to promote a wide initial search; and (2) a simple structure governed by a single primary control parameter: the extension coefficient. These features are specifically designed to enhance robustness while significantly reducing the tuning effort compared to many existing algorithms.

In this paper, with the minimum loss and the reasonable voltage distribution as the objective, the optimization model of modern low-voltage distribution system is established and a novel heuristic intelligent optimization algorithm simulating fishermen fishing, entitled SNA, is proposed. This algorithm, starting from the boundary of the search space and gradually conducting a compressed search toward the interior space, has fewer parameters, stronger global convergence, and is more effective at pursuing the optimal solution. The convergence of the algorithm is mathematically guaranteed, and it is applied to solve the optimization problem of three-phase asymmetrical distribution systems. This paper is arranged as following: Section 2 presents the Optimal Power Flow model; Section 3 discusses the calculation method for power flow; Section 4 proposes a novel heuristic intelligent optimization algorithm—SNA; a case study is implemented in Section 5; and conclusions are drawn in Section 6.

2. Optimal Power Flow Model

In general, there are only two types of nodes in traditional low-voltage distribution networks: balance nodes and PQ nodes. With the penetration of distributed power resources and new types of loads, more types of nodes should be defined. Moreover, there are asymmetrical three-phase line impedance and unbalanced three-phase loads that usually exist, and their impedance ratios are relatively larger than those of transmission networks.

The OPF problem in this study is formulated as a static optimization to determine the best operating state of the network at a specific moment in time, typically corresponding to a critical condition such as peak load. The primary goal is to minimize total active power loss while maintaining all node voltages within acceptable limits. To ensure voltage security, any violation of these limits is incorporated as a penalty term in the objective function. This optimization is subject to constraints imposed by the power flow equations and the operational limits of all controllable assets, including reactive power compensators, OLTC transformers, distributed resources, and EVs. The objective function is

$$\begin{array}{cc}\min & f=\end{array}{\displaystyle \sum _{(i,j)\in \mathcal{B}}{\displaystyle \sum _{\nu \in \mathrm{\Phi }}{I}_{ij(\nu )}^2\cdot Z_{ij(\nu )}+\eta {\displaystyle \sum _{i\in N}{\displaystyle \sum _{\nu \in \mathrm{\Phi }}\frac{\Delta V_{i(\nu )}}{V_{i(\nu )}^{\max }-V_{i(\nu )}^{\min }}}}}}$$

(1)

where the indices i and j represent system nodes from the set of all nodes $N=\{1,2,\dots ,N_B\}$, where N_B is the total number of nodes; the set of all branches is denoted by $\mathcal{B}\subseteq N\times N$, where each branch is an ordered pair (i,j) indicating an upstream–downstream connection; the index $\nu$ represents the phase from the set of phases Φ = {A,B,C}; key electrical variables include the voltage $V_{i(\nu )}$ at any node i, the current $I_{ij(\nu )}$. in a branch, and the corresponding branch impedance $Z_{ij(\nu )}$; $\eta$ is the coefficient of the penalty factor; and $\Delta V_{i(\nu )}$ is the voltage violation at node i, which is determined by the maximum and minimum allowable voltage limits, $V_{i(\nu )}^{\max }$ and $V_{i(\nu )}^{\min }$.

The power balance for the entire network is expressed in a compact vector form, where the specified net power injection must equal the power calculated from the system’s voltage state:

$${\mathit{P}}_g+{\mathit{P}}_{ev}-{\mathit{P}}_l={\mathit{P}}_{calc}(\mathit{e},\mathit{f})$$

(2)

$${\mathit{Q}}_g+{\mathit{Q}}_c-{\mathit{Q}}_l={\mathit{Q}}_{calc}(\mathit{e},\mathit{f})$$

(3)

where ${\mathit{P}}_g$, ${\mathit{P}}_p$, ${\mathit{P}}_l$ are vectors representing the active power from generators, EV, and loads for all nodes and phases in the system; ${\mathit{Q}}_g$, ${\mathit{Q}}_c$, ${\mathit{Q}}_l$. are vectors representing the reactive power from generators, compensation devices, and loads for all nodes and phases; ${\mathit{P}}_{calc}(\mathit{e},\mathit{f})$ and ${\mathit{Q}}_{calc}(\mathit{e},\mathit{f})$ are vector functions that compute the resulting active and reactive power injections, respectively, based on the system’s voltage state; and e, f are vectors containing the real and imaginary parts of all node voltages, respectively.

The voltage at a downstream node is determined by the voltage at the upstream node and the voltage drop across the connecting branch:

$$\begin{array}{cc}{V}_{j(\nu )}=V_{i(\nu )}-Z_{ij(\nu )}\cdot I_{ij(\nu )}& \forall (i,j)\in \mathcal{B},\nu \in \mathrm{\Phi }\hfill \end{array}$$

(4)

The voltage violation term used in the objective function is defined as

$$\Delta V_{i(\nu )}=\left\{\begin{array}{rr}{V}_{i(\nu )}-V_{i(\nu )}^{\max }\hfill & V_{i(\nu )}>V_{i(\nu )}^{\max }\hfill \\ 0\hfill & V_{i(\nu )}^{\min }\le V_{i(\nu )}\le V_{i(\nu )}^{\max }\hfill \\ V_{i(\nu )}^{\min }-V_{i(\nu )}\hfill & V_{i(\nu )}<V_{i(\nu )}^{\min }\hfill \end{array}\right.$$

(5)

The branch currents are calculated based on the sum of all downstream currents, as defined by the following relationship:

$$I_{ij(\nu )}=I_{j(\nu )}+{\displaystyle \sum _{m\in j}{I}_{jm(\nu )}}$$

(6)

where $I_{j(\nu )}$ is the load current of node j of phase $\nu$.

All controllable assets and system voltages must operate within their specified physical and operational limits:

$$\begin{array}{cc}{Q}_{c(\nu )}^{\min }\le Q_{c(\nu )}\le Q_{c(\nu )}^{\max }& \forall c\in \{1,\cdot \cdot \cdot ,N_C\},\nu \in \mathrm{\Phi }\end{array}$$

(7)

$$\begin{array}{cc}{T}_{t(\nu )}^{\min }\le T_{t(\nu )}\le T_{t(\nu )}^{\max }& \forall t\in \{1,\cdot \cdot \cdot ,N_T\},\nu \in \mathrm{\Phi }\end{array}$$

(8)

$$\begin{array}{cc}{P}_{g(\nu )}^{\min }\le P_{g(\nu )}\le P_{g(\nu )}^{\max }& \forall g\in \{1,\cdot \cdot \cdot ,N_G\},\nu \in \mathrm{\Phi }\end{array}$$

(9)

$$\begin{array}{cc}{P}_{ev(\nu )}^{\min }\le P_{ev(\nu )}\le P_{ev(\nu )}^{\max }& \forall p\in \{1,\cdot \cdot \cdot ,N_{EV}\},\nu \in \mathrm{\Phi }\end{array}$$

(10)

where the model incorporates several types of controllable devices, including the reactive power $Q_{c(\nu )}$ of the c^th compensation device, the tap position $T_{t(\nu )}$ of the t^th OLTC transformer, the active power output $P_{g(\nu )}$ of the g^th distributed resource, and the charging/discharging power $P_{p(\nu )}$ of the p^th EV; each of these controllable variables is bound by its respective maximum and minimum limits (e.g., $Q_{c(\nu )}^{\max }$, $Q_{c(\nu )}^{\min }$). Finally, the total numbers of these devices in the system are denoted by N_C, N_T, N_G, and N_P, respectively.

Due to the existence of the penalty term $\eta {\displaystyle \sum _i{\displaystyle \sum _{\nu }\frac{\Delta V_{i(\nu )}}{V_{i(\nu )}^{\max }-V_{i(\nu )}^{\min }}}}$, the objective function is the nonlinear and non-differentiable optimization problem, which can be solved by the heuristic intelligent optimization algorithm.

During the optimization process, a key task is to calculate the power flow of the unbalanced three-phase distribution network, which is radial and includes assets such as distributed power resources, EVs, and OLTCs. An effective calculation method is discussed in the next section.

3. Power Flow Calculation Method

To accurately solve the power flow for modern distribution networks, which can be weakly meshed (i.e., containing loop circuits), an advanced solution method is required. Standard forward–backward sweep algorithms are designed for purely radial networks. To handle loops, we employ a common network analysis technique based on Kron’s reduction. The procedure involves notionally opening each loop circuit by removing a single branch within the loop. This converts the network topology into a purely radial one, which can be solved with a standard forward–backward sweep. The effect of the removed branch is then compensated for by calculating and applying a pair of equivalent power injections at the two nodes where the branch was disconnected. This process is iterated until the voltage difference across the “open” points converges to a value near zero, at which point the solution accurately reflects the behavior of the original looped network [11].

3.1. Branch Model

3.1.1. Line

In a low-voltage distribution network, overhead lines and cables coexist. Ignoring the equivalent shunt branch to earth [12], the impedance in matrix is

$${\mathit{Z}}_L=\left[\begin{array}{ccc}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ba}& Z_{bb}& Z_{bc}\\ Z_{ca}& Z_{cb}& Z_{cc}\end{array}\right]$$

(11)

where Z_L is a 3 × 3 impedance matrix, whose diagonal elements have self-impedance and the remaining elements have mutual impedance. If a line or cable is two-phase or single-phase, the corresponding self-impedance and mutual impedance would be set as zero.

3.1.2. Transformer

The tap position can be adjusted to regulate the voltage magnitude of the feeders in the secondary side, and the voltage angle difference in the two sides of the transformer is determined after the connection modes of the windings. The equivalent circuit of two windings transformer is shown in Figure 1.

The positive, negative, and zero sequence components of the currents and the voltages at the two sides satisfy

$$\left[\begin{array}{l}{I}_{H(1)}\\ I_{H(2)}\\ I_{H(0)}\end{array}\right]={\displaystyle \frac{1}{k}}\left[\begin{array}{l}{I}_{n(1)}{e}^{-j\theta }\\ I_{n(2)}{e}^{j\theta }\\ I_{n(0)}\end{array}\right]$$

(12)

$$\left[\begin{array}{l}{V}_{n(1)}\\ V_{n(2)}\\ V_{n(0)}\end{array}\right]={\displaystyle \frac{1}{k}}\left[\begin{array}{l}{V}_{H(1)}{e}^{j\theta }\\ V_{H(2)}{e}^{-j\theta }\\ V_{H(0)}\end{array}\right]$$

(13)

Based on the symmetrical component method, the phase voltage and phase current satisfy

$$\left[\begin{array}{l}{I}_{HA}\\ I_{HB}\\ I_{HC}\end{array}\right]={\displaystyle \frac{1}{3k}}\mathit{A}\left[\begin{array}{l}{I}_{nA}\\ I_{nB}\\ I_{nC}\end{array}\right]$$

(14)

$$\left[\begin{array}{l}{V}_{nA}\\ V_{nB}\\ V_{nC}\end{array}\right]={\displaystyle \frac{1}{3k}}\mathit{B}\left[\begin{array}{l}{V}_{HA}\\ V_{HB}\\ V_{HC}\end{array}\right]$$

(15)

$$\mathit{A}=\left[\begin{array}{ccc}{e}^{-j\theta }+e^{j\theta }+\lambda & a{e}^{-j\theta }+a^2{e}^{j\theta }+\lambda & a^2{e}^{-j\theta }+a{e}^{j\theta }+\lambda \\ a^2{e}^{-j\theta }+a{e}^{j\theta }+\lambda & e^{-j\theta }+e^{j\theta }+\lambda & a{e}^{-j\theta }+a^2{e}^{j\theta }+\lambda \\ a{e}^{-j\theta }+a^2{e}^{j\theta }+\lambda & a^2{e}^{-j\theta }+a{e}^{j\theta }+\lambda & e^{-j\theta }+e^{j\theta }+\lambda \end{array}\right]$$

$$\mathit{B}=\left[\begin{array}{ccc}{e}^{j\theta }+e^{-j\theta }+\lambda & a{e}^{j\theta }+a^2{e}^{-j\theta }+\lambda & a^2{e}^{j\theta }+a{e}^{-j\theta }+\lambda \\ a^2{e}^{j\theta }+a{e}^{-j\theta }+\lambda & e^{j\theta }+e^{-j\theta }+\lambda & a{e}^{j\theta }+a^2{e}^{-j\theta }+\lambda \\ a{e}^{j\theta }+a^2{e}^{-j\theta }+\lambda & a^2{e}^{j\theta }+a{e}^{-j\theta }+\lambda & e^{j\theta }+e^{-j\theta }+\lambda \end{array}\right]$$

where $k$ is the transformer voltage ratio; $\theta$ is the angle difference between the two sides; $a=e^{j120°}$ is the rotation factor; and $\lambda$ is the mode factor of the zero sequence current. $\lambda$ is set to 0 if the zero sequence current cannot flow back; otherwise, it is set to 1.

Suppose that $Z_1$, $Z_2$, and $Z_0$ are the positive, negative, and zero sequence impedance of the OLTC transformer, respectively. In general, $Z_1=Z_2$. The impedance of the transformer is

$${\mathit{Z}}_{\mathit{T}}=\left(\begin{array}{ccc}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ba}& Z_{bb}& Z_{bc}\\ Z_{ca}& Z_{cb}& Z_{cc}\end{array}\right)={\displaystyle \frac{1}{3}}\left(\begin{array}{ccc}{Z}_0+2Z_1& Z_0-Z_1& Z_0-Z_1\\ Z_0-Z_1& Z_0+2Z_1& Z_0-Z_1\\ Z_0-Z_1& Z_0-Z_1& Z_0+2Z_1\end{array}\right)$$

(16)

3.2. Injection Power Model

3.2.1. The General Load

General loads can be equivalent to three types: constant power (PQ), constant current (I), and constant impedance (Z).

3.2.2. EVs

EVs usually absorb constant charging active power from the power system, and can also be dispatched by the power system under special conditions. Their charging power is changeable, and they can even supply energy for the power system. EVs are seen as PQ nodes in this paper, and their charging and discharging power is assumed to be controllable [13,14].

3.2.3. Distributed Power Resource

There are many kinds of distributed power resources, which can be divided into several types [12] according to their operation characteristics in distribution networks:

• PV nodes, such as micro gas turbines, fuel cells, and other controllable distributed power resources;
• Constant current (I) nodes, such as photovoltaic generation systems, energy storage devices, etc.;
• PQ(V) nodes, such as the combined heat and power asynchronous units of power frequency, whose active power output is constant, while reactive power output is relevant to the node voltage;
• PQ nodes, such as the combined heat and power synchronous units of power frequency, etc.

I nodes and PQ nodes are transformed directly into the injection current to be calculated in the forward backward sweep method. For PV nodes, some improvements should be implemented when calculated in the forward or backward sweep method.

3.3. Improvement Method for Loop Circuit and PV Node

The traditional forward–backward sweep method cannot be directly applied to networks with loop circuits or multiple PV nodes. The improvement method involves treating each loop and PV node connection as an open point [15,16].

The relationship between voltage and current in a power system is fundamentally described by Ohm’s law. For a branch between two nodes, i and j, which has been notionally opened, the voltage difference across the opening is given by

$$\Delta V_{ij}=V_i-V_j=Z_{loop}·I_{comp}$$

(17)

where $Z_{loop}$ is the impedance of the looped path and $I_{comp}$ is the compensating current that would flow through the original branch.

The complex power injected at node i can be expressed as $S_i=V_i{I}_i^{*}$. If we assume that the system is close to convergence, the voltage magnitude at all nodes is approximately 1.0 p.u. ($\left|V_i\right|$) and the voltage angles are small. Under these standard linearization assumptions, the compensating current can be approximated from the compensating power $S_{comp}$ as $I_{comp}\approx {(S_{comp}/V_i)}^{*}\approx S_{comp}^{*}$. Substituting this into the voltage difference equation gives

$$\Delta V_{ij}\approx Z_{loop}·S_{comp}^{*}=(R+jX)(P_{comp}-j{Q}_{comp})$$

(18)

By separating the real and imaginary parts of the voltage difference ($\Delta V_{ij}=\Delta V_{re}+j\Delta V_{im}$), we can establish a direct relationship between the voltage difference and the compensating power components:

$$\left[\begin{array}{c}\Delta V_{re}\\ \Delta V_{im}\end{array}\right]=\left[\begin{array}{cc}R& X\\ -X& R\end{array}\right]\left[\begin{array}{c}{P}_{comp}\\ Q_{comp}\end{array}\right]$$

(19)

Inverting this matrix relationship allows us to calculate the required active and reactive compensation power ($P_{comp}$ and $Q_{comp}$) based on the calculated voltage difference across the open points from the previous power flow iteration:

$$\left[\begin{array}{c}{P}_{comp}\\ Q_{comp}\end{array}\right]={\displaystyle \frac{1}{R^2+X^2}}\left[\begin{array}{cc}R& -X\\ X& R\end{array}\right]\left[\begin{array}{c}\Delta V_{re}\\ \Delta V_{im}\end{array}\right]$$

(20)

These calculated compensation powers are then added to the injection power at the relevant nodes in the next iteration of the power flow calculation. The process terminates when the voltage difference across all open points is below a predefined tolerance, indicating convergence.

3.4. Process of the Power Flow Calculation

The complete iterative procedure for calculating the power flow in a weakly meshed, unbalanced distribution network is as follows:

Step 1 (Initialization): Read the network data, including line impedances, load data, and device parameters. Initialize all node voltages, typically to 1.0 p.u.

Step 2 (Topology Modification): Identify all loop circuits and PV nodes. For each loop, select one branch to notionally open. Initialize the compensation power injections ($P_{comp}$ and $Q_{comp}$) at all open points to zero. The network is now treated as a radial system.

Step 3 (Injection Current Calculation): For the current iteration, calculate the total injection current at each node. This includes contributions from constant power PQ loads, constant current I loads, and the previously calculated compensation powers for loops and PV nodes.

Step 4 (Forward Sweep—Branch Currents): Starting from the network extremities and moving toward the substation, calculate the current flowing in each branch by summing the currents of all downstream branches and the injection current at the downstream node.

Step 5 (Backward Sweep—Node Voltages): Starting from the substation and moving toward the extremities, calculate the voltage at each downstream node using the voltage of its upstream node and the calculated voltage drop across the connecting branch.

Step 6 (Convergence Check and Update): Calculate the voltage differences ($\Delta V_{re}$ and $\Delta V_{im}$) across all the notionally opened points from Step 2. Check if the magnitude of these voltage differences is below a predefined convergence tolerance. If converged, the power flow solution is found. Terminate the algorithm and output the results (voltages, currents, power losses); otherwise, use (20) to update the compensation power injections ($P_{comp}$ and $Q_{comp}$) for each open point based on the calculated voltage differences. Return to Step 3 for the next iteration.

4. Shrinking Net Algorithm

For solving the optimization model in Section 2, a novel heuristic intelligent optimization algorithm simulating fishermen fishing, entitled the Shrinking Net Algorithm, is proposed. The algorithm begins its compressed search from the boundary of the search space, moving toward the global optimal solution.

4.1. Terms

Dimension: The number of optimization variables, denoted by D.

Node: A feasible value, ${\mathit{X}}_i=[X_{i1},X_{i2},\dots ,X_{iD}]$, where $X_{id}$ is the $d^{th}$ dimensional component of the $i^{th}$ node, $d=1,2,\dots ,D$.

Vertex: A special kind of node, composed of groups of each variable’s upper and lower limits. For a search space of D dimensions, the number of vertices is $2^D$.

Net Surface: A collection of all the initialized nodes, composed of vertices and a number of random nodes on the boundary surface, which is formed by the hyperplane of the upper and lower limits of each dimension. If $N$ nodes are selected randomly on each hyperplane, the number of nodes in the net surface is $P$, and $P=2^D+2ND$.

Optimal Node: An optimal solution of the current iterative round.

Shrink: The operation of updating the iteration of all nodes.

Extension Coefficient: A key parameter to influence the change in all nodes in the shrinking operation.

Violation: The value of a node out of its upper and lower limits in the process of shrinking.

4.2. Mathematical Analysis of the Update Mechanism

The core of the SNA’s search behavior is encapsulated in the node update equation:

$$\begin{array}{c}\hfill {\mathit{X}}_i^{(m+1)}={\mathit{X}}_i^{\left(m\right)}+C{\displaystyle \frac{m}{M}}{\tilde{\xi }}^{\left(m\right)}({\mathit{X}}^{best}-{\mathit{X}}_i^{\left(m\right)})\\ -(C-1){\displaystyle \frac{m}{M}}{\tilde{\varsigma }}^{\left(m\right)}({\mathit{X}}^{best}-{\mathit{X}}_i^{\left(m\right)})\end{array}$$

(21)

where $i=1,2,\dots ,P$, $m=1,2,\dots ,M-1$; P is the total number of nodes on the net surface; M is the maximum iteration time; ${\mathit{X}}_i^{\left(m\right)}$ is $i^{th}$ node in the $m^{th}$ iteration; ${\mathit{X}}^{best}$ is the current optimal node; C is the extension coefficient, whose value influences the global and local searching ability of SNA; and ${\tilde{\mathit{\xi }}}^{\left(m\right)}=[{\tilde{\xi }}_1^{\left(m\right)},{\tilde{\xi }}_2^{\left(m\right)},\dots ,{\tilde{\xi }}_D^{\left(m\right)}]$ and ${\tilde{\mathit{\varsigma }}}^{\left(m\right)}=[{\tilde{\varsigma }}_1^{\left(m\right)},{\tilde{\varsigma }}_2^{\left(m\right)},\dots ,{\tilde{\varsigma }}_D^{\left(m\right)}]$ are the random vectors that evenly distribute between 0 and 1.

The mechanism of (21) is driven by the interplay of a primary convergence vector and an opposing stochastic vector, which are both oriented along the directional axis from the current node ${\mathit{X}}_i^{\left(m\right)}$ to the best-known solution ${\mathit{X}}^{best}$. The influence of these competing terms is scaled by a linearly increasing adaptive weight ${\displaystyle \frac{m}{M}}$, which ensures that node movements are initially fine-grained and become progressively larger to accelerate convergence in later stages. The use of an iteration-dependent adaptive weight to balance exploration and exploitation is a well-established technique in the field of metaheuristics, notably popularized in variants of PSO [17]. The critical element of this dynamic is the extension coefficient C. It arbitrates the balance between the two randomized terms, directly tuning the algorithm’s exploratory-exploitative character. This strategy of creating a search vector from the interplay of multiple population members shares conceptual similarities with other powerful optimizers, such as Differential Evolution, which uses differences between population vectors to guide its search [18].

4.3. Basic Procedure of SNA

Step 1: Initialize net surface. When the dimension D of optimization space is relatively low, the net surface is composed of the vertices and random nodes on the dimensional hyperplane; when the dimension is relatively high, in order to avoid the dimensional disaster, simplifying the net surface and ignoring all the vertices, the net surface is composed only by random nodes on each dimensional hyperplane, whose number is $2ND$.

Step 2: Calculate the objective function values of each node on the net surface and obtain the global optimal value $X^{best}$.

Step 3: Shrink. Each node is updated according to (21).

Step 4: Judge violation if any variable’s value after an update exceeds its upper or lower limits.

Step 5: Return to Step 2 until $m=M-1$, and then output the optimization results.

The optimal node of each iteration after being updated according to (21) is still itself, and it is replaced only when a more optimal node appears, which ensures the non-inferiority of the nodes. The complete step-by-step process of the proposed SNA is illustrated in the flowchart shown in Figure 2.

4.4. Analysis of Convergence

Equation (21) can be transformed as

$${\displaystyle \frac{{\mathit{X}}^{best}-{\mathit{X}}_i^{(m+1)}}{{\mathit{X}}^{best}-{\mathit{X}}_i^{(m)}}}=1-C{\displaystyle \frac{m}{M}}{\overline{\xi }}^{(m)}+(C-1){\displaystyle \frac{m}{M}}{\overline{\zeta }}^{(m)}$$

(22)

If ${\mathit{X}}^{best}$ is constant in the process of iteration, suppose that

$${\mathit{Q}}^{\left(m\right)}=1-C{\displaystyle \frac{m}{M}}{\tilde{\mathit{\xi }}}^{\left(m\right)}+(C-1){\displaystyle \frac{m}{M}}{\tilde{\mathit{\varsigma }}}^{\left(m\right)}$$

(23)

Then

$$\begin{array}{c}{\mathit{X}}_i^{(m+1)}={\mathit{X}}^{best}(1-{\mathit{Q}}^{\left(m\right)})+{\mathit{X}}_i^{\left(m\right)}{\mathit{Q}}^{\left(m\right)}\hfill \\ ={\mathit{X}}^{best}(1-{\mathit{Q}}^{\left(m\right)}{\mathit{Q}}^{(m-1)})+{\mathit{X}}_i^{(m-1)}{\mathit{Q}}^{\left(m\right)}{\mathit{Q}}^{(m-1)}\hfill \\ =\dots \hfill \\ ={\mathit{X}}^{best}(1-{\displaystyle \mathbf{\prod }_{j=1}^m{\mathit{Q}}^{\left(j\right)}})+{\mathit{X}}_i^{\left(1\right)}{\displaystyle \mathbf{\prod }_{j=1}^m{\mathit{Q}}^{\left(j\right)}}\hfill \end{array}$$

(24)

where ${\mathit{Q}}^{\left(m\right)}$ is the ratio of the distance between current node and the optimal node in the recent two iterations. If $\left|{\mathit{Q}}^{\left(m\right)}\right|>1$, as the iteration proceeds, the current node moves further and further away from the optimal one; if $\left|{\mathit{Q}}^{\left(m\right)}\right|<1$, it means that the current node and the optimal node are getting closer.

According to (23), $Q_d^{\left(m\right)}$, the $d^{th}$ element in ${\mathit{Q}}^{\left(m\right)}=[Q_1^{\left(m\right)},Q_2^{\left(m\right)},\dots ,Q_D^{\left(m\right)}]$, where $d=1,2,\dots ,D$, can be expressed as

$$Q_d^{\left(m\right)}=1-C{\displaystyle \frac{m}{M}}{\tilde{\xi }}_d^{\left(m\right)}+(C-1){\displaystyle \frac{m}{M}}{\tilde{\varsigma }}_d^{\left(m\right)}$$

(25)

Because ${\tilde{\xi }}_d^{\left(m\right)}$ and ${\tilde{\varsigma }}_d^{\left(m\right)}$ are the random numbers mutually independent and uniformly distributed in [0, 1], the expectation value of $Q_d^{(m)}$ is

$$\begin{array}{c}E\left(Q_d^{\left(m\right)}\right)\hfill \\ ={\displaystyle \frac{1}{M-1}}{\displaystyle \sum _{m=1}^{M-1}{Q}_d^{\left(m\right)}}\hfill \\ =1-C({\displaystyle \frac{1}{M-1}}{\displaystyle \sum _{m=1}^{M-1}\frac{m}{M})(\frac{1}{M-1}{\displaystyle \sum _{m=1}^{M-1}{\tilde{\xi }}_d^{\left(m\right)}})}\hfill \\ \hspace{0.33em}+(C-1)({\displaystyle \frac{1}{M-1}}{\displaystyle \sum _{m=1}^{M-1}\frac{m}{M})(\frac{1}{M-1}{\displaystyle \sum _{m=1}^{M-1}{\tilde{\varsigma }}_d^{\left(m\right)}})}\hfill \\ =1-{\displaystyle \frac{1}{2}}C({\displaystyle \frac{1}{M-1}}{\displaystyle \sum _{m=1}^{M-1}\frac{m}{M})}\hspace{0.33em}+{\displaystyle \frac{1}{2}}(C-1)({\displaystyle \frac{1}{M-1}}{\displaystyle \sum _{m=1}^{M-1}\frac{m}{M})}\hfill \\ =1-{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{M-1}}{\displaystyle \frac{1}{M}}{\displaystyle \frac{M(M-1)}{2}}\hfill \\ =0.75\hfill \end{array}$$

(26)

Thus, it can be seen that for a sufficiently large M, ${\displaystyle \mathbf{\prod }_{j=1}^m{\mathit{Q}}^{(j)}}$ equals to 0, which is not influenced by the value of extension coefficient C. Equation (24) shows that ${\mathit{X}}_i^{(m+1)}$ finally tends to ${\mathit{X}}^{best}$ and is irrelevant to ${\mathit{X}}_i^{\left(1\right)}$.

An expectation value of $Q_d^{(m)}$ equaling to 0.75 shows that the nodes on the net surface become closer to the optimal node. In fact, the value of $Q_d^{(m)}$ depends on the two random numbers ${\tilde{\xi }}_d^{\left(m\right)}$ and ${\tilde{\varsigma }}_d^{\left(m\right)}$ and extension coefficient C, so a suitable extension coefficient can accelerate the convergence of SNA.

4.5. Selection of Extension Coefficient

The extension coefficient is the only parameter needed to be set in SNA, and it effects the value of $Q_d^{\left(m\right)}$, according to (25),

$$Q_d^{\left(m\right)}\in \left\{\begin{array}{ll}[1+(C-1){\displaystyle \frac{m}{M}},1-C{\displaystyle \frac{m}{M}}]\hfill & C<0\hfill \\ [1-{\displaystyle \frac{m}{M}},1]\hfill & 0\le C\le 1\hfill \\ [1-C{\displaystyle \frac{m}{M}},1+(C-1){\displaystyle \frac{m}{M}}]\hfill & C>1\hfill \end{array}\right.$$

(27)

When $0\le C\le 1$, $Q_d^{\left(m\right)}$ is in the (0, 1], which means that ${\mathit{X}}_i^{(m+1)}$ is gradually getting closer to ${\mathit{X}}^{best}$ in one direction, which reflects the local convergence of SNA; when C is not in (0, 1], $Q_d^{\left(m\right)}$ may be less than 0 or more than 1, which reflects the global optimization ability of SNA.

Making further analysis on the final several rounds of iterations, when m is large enough, and $C\in [0,1]$, $Q_d^{(m)}$ belongs to (0, 1] and is uniformly distributed in the range of its value because of random numbers ${\tilde{\xi }}_d^{\left(m\right)}$ and ${\tilde{\varsigma }}_d^{\left(m\right)}$. If ${\mathit{X}}^{best}$ does not change in the last several rounds of iterations, ${\tilde{\mathit{\alpha }}}^{(j)}=[{\tilde{\alpha }}_1^{(j)},{\tilde{\alpha }}_2^{(j)},\dots ,{\tilde{\alpha }}_D^{(j)}]$ is adopted to present Q^(j), then

$${\mathit{X}}_i^{(m+1)}={\mathit{X}}^{best}(1-{\displaystyle \mathbf{\prod }_{j=1}^m{\tilde{\mathit{\alpha }}}^{(j)}})+{\mathit{X}}_i^{\left(p\right)}{\displaystyle \mathbf{\prod }_{j=p}^m{\tilde{\mathit{\alpha }}}^{(j)}}$$

(28)

which shows that, if m is large enough, ${\displaystyle \mathbf{\prod }_{j=1}^m{\tilde{\mathit{\alpha }}}^{(j)}}$ and ${\displaystyle \mathbf{\prod }_{j=p}^m{\tilde{\mathit{\alpha }}}^{(j)}}$ tend to be 0, so no matter how the optimal node changes in early iterations, as long as the optimal node tends to be stable in the final several iterations, all nodes will converge to the optimal node.

Based on the analysis mentioned above, the selecting principles of extension coefficient C are as follows: C is set to be a larger number in the early iterations, such as C > 1, to improve the global optimization ability, and C is set to belong to [0, 1] in the later iterations to guarantee the local convergence.

4.6. Performance Test of the Algorithm

In order to verify the analysis results, “Generalized Schwefel’s Problem” function is selected to test the global search performance of the optimization method. The mathematical function expression is

$$f(x,y)=-x\sin (\sqrt{\left|x\right|})-y\sin (\sqrt{\left|y\right|})$$

(29)

where $x,y\in \left[-500, 500\right]$. Its search space is shown in Figure 3.

The SNA and standard particle swarm algorithm (PSO) are run 200 times, respectively, for comparison test. The maximum number of iterations is set to be 50, that is, M = 50. In SNA, the two-dimensional space has four hyperplanes, the four sides of the quadrilateral shown in Figure 3, and four vertices, and ten nodes are selected randomly on each hyperplane, that is, N = 10, and then the number of initial nodes is $P=2^D+2ND=2^2+2\times 10\times 2=44$, and four constant extension coefficients and a variable extension coefficient $C=f(m)=20-19.5 m/M$ are set. In PSO, the group size is also 44, and the value range of the inertia weight $\omega$ is [0.5, 1.4], and $c_1$ = $c_2$ = 2 [19]. The comparison results of the two algorithms are listed in

Table 1. Statistics of optimization results.

Algorithm		Minimum	Average	Standard Deviation	Convergence Rate
SNA	C = 0.5	−837.966	−812.056	45.927	77.5%
	C = 5	−837.966	−822.569	39.931	87.0%
	C = 15	−837.966	−833.886	20.969	98.5%
	C = 30	−837.966	−804.497	62.293	80.0%
	C = f(m)	−837.966	−835.005	18.538	97.5%
PSO		−837.966	−753.529	89.557	46.5%

, and only the result with highest success rate of converge of PSO is shown from when the inertia weight coefficient $\omega$ equals 0.7.

The following conclusions can be drawn based on Table 1:

• Both SNA and PSO have the ability of global optimization, and may have the situation of local optimum with a specified probability.
• Taken as a whole, SNA has stronger global optimization ability, better stability, and higher accuracy.
• When the extension coefficient is constant and is less than 1, SNA loses a certain ability of global optimization in the process of each node strictly moving closer to the optimal node. As the extension coefficient increases, the ability of global optimization is significantly improved, and when the extension coefficient is too large, the optimization accuracy is reduced in the later iterations, and each node may even present the state of dispersing in the optimization process, all of which reduce the optimization success rate.
• By selecting the extension coefficient that automatically adapts to the varying iteration times, the ability of global optimization of SNA is improved and the optimization accuracy is higher, which verifies the selection principles of the extension coefficient mentioned above.

4.7. Application of SNA to the Optimal Power Flow Problem

To solve the OPF model formulated in Section 2, the SNA is specifically tailored as outlined below. Each component of the SNA is mapped directly to a feature of the OPF problem, ensuring a targeted and efficient search for the optimal operating point.

• Defining a Node: In the context of this OPF problem, a single node or candidate solution X_i in the SNA population is a multi-dimensional vector. Each element of this vector corresponds to a controllable variable in the distribution network. The dimension of the search space, D, is therefore the total number of controllable assets: D = N_T + N_C + N_G + N_EV.
• Initialization of the Net Surface: The initial population of candidate solutions is generated by creating vectors where the values for each controllable variable are set at or near their operational limits. This aligns with SNA’s boundary-in-search mechanism, ensuring that the initial search covers the entire feasible operating range of the controllable devices.
• Evaluating the Objective Function: For each candidate solution vector X_i in the population, its fitness is evaluated by solving the full three-phase unbalanced power flow, as described in Section 3. The control variables from the vector X_i are used as inputs to the power flow calculation. The resulting system state is then used to calculate the objective function value f(X_i), which includes both the total active power loss and the penalty for any voltage violations.
• The Shrinking Process: During each iteration, the SNA’s update equation is applied to every solution vector X_i. The algorithm shrinks the population towards the current best-known solution X_best. If any variable in a vector is updated to a value outside its operational limits, it is reset to the corresponding boundary value, thus enforcing the inequality constraints of the OPF model. This iterative process continues until the termination criterion is met, yielding the final optimal set of control parameters.

5. Case Study

In this paper, the IEEE 123-node distribution system is adopted as the test [14,20]. To ensure the reproducibility of our results, all line, transformer, and load parameters are used exactly as specified in the official documentation provided by the IEEE PES Working Group, which is a widely accepted benchmark. The system’s network topology is shown in Figure 4. For this study, the base system was modified by adding a number of controllable distributed power supplies and EVs at eight specific nodes. In addition to these, other controllable devices, such as OLTC transformers and reactive power compensation devices (Cap), were also included in the optimization. The specific parameters and operational limits of all controllable devices are listed in

Table 2. The parameters of controllable devices and the optimization results.

Node	Phase	Type	Limit	Initial Value	Optimal Value	Unit
26	C	DG	[0, 280]	0	226.337	kW
32	A	DG	[0, 100]	0	100	kW
60	A	DG	[0, 40]	0	40	kW
67	ABC	DG	[0, 100]	0	100	kW
76	ABC	DG	[0, 100]	0	100	kW
95	ABC	DG	[0, 100]	0	100	kW
36	B	EV	[−40, 0]	−40	0	kW
109	A	EV	[−4.8, 0]	−4.8	0	kW
83	ABC	Cap	[0, 600]	600	600	kvar
88	B	Cap	[0, 50]	50	50	kvar
90	B	Cap	[0, 50]	50	0	kvar
92	C	Cap	[0, 500]	50	50	kvar
150–149	ABC	OLTC	[0.9, 1.1]	1	1.0618	p.u.
61–610	ABC	OLTC	[0.9, 1.1]	1	1	p.u.

. The upper and lower voltage limits for each phase of each node are set to 1.05 p.u. and 0.95 p.u., respectively.

The established optimization model and the proposed SNA in this paper are used for the optimization calculation. The maximum number of iterations is set to be 50, that is, $M=50$, and the extension coefficient is selected as $C=f(m)=20-19.5 m/M$. Corresponding to the 14 controllable devices, two nodes are selected on each boundary randomly, then the scale of initial nodes is $P=2^D+2ND=2^{14}+2\times 2\times 14=\mathrm{16,440}$, which leads to over-computation. Adopt the method of reducing the initial node scale mentioned above and ignore all vertices; then, the number of initial nodes is 56. The PSO was implemented with a linearly decreasing inertia weight from 0.9 to 0.4 and cognitive and social coefficients both set to 2, which are common settings. Both algorithms were executed for 30 independent trial runs to assess their statistical robustness and consistency. The statistical results of the 30 trial runs for both SNA and PSO are summarized in

Table 3. Statistical comparison of SNA and PSO for the OPF problem.

Algorithm	Best Result (kW)	Worst Result (kW)	Mean (kW)	Std. Deviation (kW)
SNA	65.208	65.812	65.431	0.197
PSO	66.953	71.245	68.816	1.324

. The objective function value is used as the primary metric for comparison.

The results in Table 3 clearly demonstrate the superior performance and reliability of the proposed SNA. The SNA consistently found a more optimal solution, with its best result, 65.208 kW, significantly outperforming PSO’s best result, 66.953 kW. More importantly, the SNA exhibits exceptional robustness, as indicated by a standard deviation of only 0.197 kW, compared to PSO’s much larger 1.324 kW. This implies that the SNA is far less susceptible to premature convergence and can reliably find a near-optimal solution on every run, a crucial advantage for real-world operational planning. The convergence curve for the best SNA run is shown in Figure 5.

When the IEEE 123-node system operates at full load status, the active power loss of the system is 96.103 kW, the voltage of each node is lower, and the average voltage is 0.969 p.u. The voltage of phase C of node 104 is the minimum, which is 0.937 p.u. After optimization, the active power loss is 65.208 kW, the distribution of node voltage is more reasonable, all nodes meet the voltage upper and lower limits, and the average voltage is 1.017 p.u. The voltage of phase A of node 114 is the minimum, which is 0.973 p.u. The optimization results of all the controllable parameters are shown in Table 2. The results reveal an intelligent coordination of assets. For example, the capacitor at node 90 was switched off (0 kvar) because its reactive power injection, while locally beneficial, was found to increase total system losses and cause overvoltage at nearby nodes under this specific load condition. Similarly, the charging of the two EVs was curtailed to 0 kW. While EV charging is a primary function, from a grid optimization perspective, reducing this significant load at a critical peak moment provides substantial benefits in both loss reduction and voltage support for adjacent nodes. The main OLTC tap was adjusted to 1.0618 p.u., proactively boosting the voltage profile across the entire feeder to counteract voltage drop.

While the proposed SNA has demonstrated strong performance, it is important to acknowledge its potential limitations, which also suggest avenues for future research. Firstly, while the simplified boundary initialization strategy is effective, the algorithm’s performance may face challenges in extremely high-dimensional optimization problems, a phenomenon often referred to as the curse of dimensionality. Future work could investigate new initialization strategies to maintain search efficiency in such large-scale problems. Secondly, although SNA benefits from having only a single primary tuning parameter, the extension coefficient C, the results indicate that the algorithm’s performance is sensitive to its value. Developing a more robust, problem-independent adaptive strategy for C would be a valuable enhancement. Finally, the current formulation of SNA is designed for continuous variables. To handle the discrete OLTC tap positions in this study, a post-optimization rounding step was applied. Future research could focus on developing a native mixed-integer or discrete version of SNA to handle such variables more efficiently.

6. Conclusions

This paper introduces the SNA, a novel and conceptually simple metaheuristic designed to solve the complex OPF problem in unbalanced distribution networks. When validated on the IEEE 123-bus system, the proposed SNA framework proved to be highly effective, reducing active power loss by a significant 32.1% while eliminating all voltage violations. A comparative analysis against standard Particle Swarm Optimization further confirmed SNA’s superior solution quality and robustness, establishing it as a valuable new tool for enhancing grid efficiency and reliability. Despite its strong performance, the SNA presents limitations that also highlight clear avenues for future research. A key limitation is the algorithm’s sensitivity to its single control parameter, the extension coefficient C, which suggests that developing a more robust adaptive strategy would be a valuable enhancement. Additionally, the current continuous version of SNA requires post-processing for discrete variables. A primary direction for future work is therefore the development of a native mixed-integer SNA to handle such optimization tasks more efficiently and expand its applicability.