An Optimized Method for Setting Relay Protection in Distributed PV Distribution Networks Based on an Improved Osprey Algorithm

Abstract

The high penetration of distributed photovoltaics (PV) into distribution networks alters the system’s short-circuit current characteristics, posing risks of maloperation and failure-to-operate to conventional inverse-time overcurrent protection. Based on an equivalent model of distributed PV during faults, this paper analyzes its impact on the protection characteristics of traditional distribution networks. With protection selectivity and the physical constraints of protection devices as conditions, an optimization model for inverse-time overcurrent protection is established, aiming to minimize the total operation time. To enhance the solution capability for this complex optimization problem, the standard Osprey Optimization Algorithm (OOA) is improved through the incorporation of three strategies: arccosine chaotic mapping for population initialization, a nonlinear convergence factor to balance global and local search, and a dynamic spiral search strategy combining mechanisms from the Whale and Marine Predators algorithms. Based on this improved algorithm, an optimized protection scheme for distribution networks with distributed PV is proposed. Simulations conducted in PSCAD/EMTDC (V4.6.2) and MATLAB (R2023b) verify that the proposed method effectively prevents protection maloperation and failure-to-operate under both fault current contribution and extraction scenarios of PV, while also reducing the overall relay operation time.

1. Introduction

Conventional power systems are predicated upon the utilization of primary energy sources, including coal and oil. With ongoing societal advancement, this paradigm is increasingly constrained by energy and environmental factors. Distributed photovoltaic (PV) power generation has thus gained prominence and is being integrated into distribution networks, owing to its merits of flexibility, efficiency, and minimal environmental impact [1,2,3]. The integration, however, introduces complex fault behaviors and power stochasticity, thereby altering network topology and power flow patterns. These alterations challenge the adaptability of conventional relay protection schemes [4]. Consequently, it is imperative to modify and optimize protection settings to fulfill the requirements imposed by the proliferation of distributed energy resources.

The existing literature on inverse-time overcurrent relay coordination in distribution networks encompasses several methodological approaches. Research efforts range from fusion algorithms that enhance convergence properties [5] and neural network-based characteristic optimization [6], to the combined use of linear programming and genetic algorithms for optimizing operating time [7]. In addressing protection zone overlap, multi-objective particle swarm optimization has been utilized to improve response speed [8]. Another study applied a harmony search algorithm to refine coordination margins [9]. A common limitation among these works, however, was the lack of quantified analysis regarding PV-induced changes in short-circuit current characteristics, with improvements often restricted to parametric adjustments. Therefore, while these methods form a reliable foundation for traditional network protection, they do not adequately address the distinct challenges introduced by distributed PV integration, indicating a clear need for more context-aware solutions.

To address this issue, scholars have proposed protection configuration and setting methods that include distributed PV scenarios. Reference [10] considers the impact of distributed PV on the current contribution and extraction effects in multi-level protection and proposes a protection scheme based on distance protection principles. In distribution networks with DG, Reference [11] uses the total operating time of inverse-time overcurrent relays as the objective function and the “four characteristics” principle as constraints, applying a particle swarm algorithm to optimize the settings. Reference [12] considers different operational states of DG-integrated distribution networks and proposes a multi-agent distributed protection coordination framework to achieve temporal optimization of protection devices and avoid coordination mismatch. Reference [13] minimizes the total relay coordination time while ensuring reliable coordination between protections and optimizes the starting current and time setting coefficients using an improved differential evolution algorithm. Reference [14] introduces an improved gravitational search algorithm with a hybrid particle swarm, considers coordination constraints between primary protections, and optimizes the operating parameters of inverse-time over-current protection. Although these works considered the impact of distributed generation on protection devices, they often lacked rigorous quantitative analysis of the PV fault behavior. As a result, the defined objective functions and constraints were prone to local optima, and the computational efficiency of the optimization schemes remained inadequate.

Against this background, the contributions of this work are presented in the following three aspects. First, a precise fault equivalent model for distributed PV is established, incorporating its low-voltage ride-through (LVRT) control strategy, which provides a quantitative basis for analyzing its impact on protection. Second, a comprehensive optimization model for inverse-time overcurrent protection is constructed, explicitly considering the conflicting requirements of selectivity, sensitivity, and speed under both fault current contribution and extraction scenarios. Furthermore, an Improved Osprey Optimization Algorithm (IOOA) is developed for this specific problem. The IOOA synergistically integrates an arccosine chaotic mapping to enhance population diversity during initialization, a nonlinear convergence factor to adaptively balance global exploration and local exploitation across iterations, and a dynamic spiral search strategy that leverages mechanisms from other algorithms to prevent premature convergence and strengthen local refinement. This coordinated improvement approach is designed to effectively overcome the limitations of the standard OOA and other metaheuristics in handling the high-dimensional, nonlinear, and constrained nature of the protection coordination problem in PV-integrated distribution networks.

2. Impact of Distributed PV Integration on Protection in Distribution Networks

2.1. Fault Equivalent Model of Distributed PV

When a fault occurs in the distribution network, a voltage sag arises at the Point of Common Coupling (PCC) of the distributed photovoltaic (PV) system. In accordance with the Technical Requirements for Connecting PV Power Systems to Distribution Networks [15], PV units must possess Low Voltage Ride Through (LVRT) capability during grid faults. Consequently, by detecting the severity of the voltage sag at the PCC in real-time, the distributed PV system can provide corresponding reactive power support to the grid. During the fault ride-through period, the relationship between the current reference value in the LVRT strategy and the severity of the voltage sag is given as follows:

$$i_d^{ref}=\left\{\begin{array}{c}\begin{array}{l}{P}^{ref}/U_{\mathrm{PCC}}\begin{array}{c}\begin{array}{c} 0.9<U_{\mathrm{PCC}}\end{array}\end{array}\\ P^{ref}/k_1{U}_{\mathrm{PCC}}\begin{array}{cc}& \end{array} 0.84<U_{\mathrm{PCC}}\le 0.9\end{array}\\ \sqrt{{\left(I_{\max }^2-i_q^{ref}\right)}^2} U_{\mathrm{PCC}}\le 0.84\hfill \end{array}\right.$$

(1)

$$i_q^{ref}=\left\{\begin{array}{c}\begin{array}{l}\begin{array}{cc}\begin{array}{ccc} 0& & \end{array}\hfill & 0.9<U_{\mathrm{PCC}}\end{array}\\ k_1(0.9-U_{\mathrm{PCC}})I_{\mathrm{N}}\begin{array}{cc}& \end{array}0.2<U_{\mathrm{PCC}}\le 0.9\end{array}\\ k_2{I}_{\mathrm{N}} U_{\mathrm{PCC}}\le 0.2\hfill \end{array}\right.$$

(2)

where $i_d^{ref}$ and $i_q^{ref}$ are the reference values for the active and reactive currents of the inverter, respectively; P^ref is the reference value for active power; I_max is the current limiting amplitude of the distributed PV inverter; I_N is the rated grid-connected current on the PV side; k₁ shall be no less than 1.5; k₂ shall be no less than 1.05.

As indicated by Equations (1) and (2), during fault ride-through, the grid voltage support requirements and the short-circuit current limiting strategy cause the reactive current output from the inverter to increase progressively with the depth of the voltage sag at the PCC. The control parameters k₁ and k₂ are determined according to the technical requirements specified in the standard [16], which stipulates the reactive power support capabilities for photovoltaic power stations during low-voltage ride-through conditions. Specifically, parameter k₁ shall be no less than 1.5 and k₂ shall be no less than 1.05, ensuring adequate reactive current injection during grid faults. The reactive power output by the inverter under these conditions can be specifically expressed as follows [17]:

$$\left\{\begin{array}{ll}{Q}_0=0& U_{\mathrm{PCC}}>0.9\\ Q_0⩾k_1{E}_{\mathrm{N}}{I}_{\mathrm{N}}\left(0.9-U_{\mathrm{PCC}}\right)& 0.2⩽U_{\mathrm{PCC}}⩽0.9\\ Q_0⩾k_2{E}_{\mathrm{N}}{I}_{\mathrm{N}}& U_{\mathrm{PCC}}<0.2\end{array}\right.$$

(3)

where E_N is the rated voltage on the system side; Q₀ is the reactive power output from the PV system.

To validate the correctness of the aforementioned fault equivalent model for distributed photovoltaics (PV), a simulation model illustrated in Figure 1 was developed in PSCAD. The model parameters are configured as follows: system base capacity of 100 MVA, voltage level of 10.5 kV, and distributed PV capacity of 15 MW. The PV inverter employs a grid-following control architecture, featuring a dual-loop control structure comprising a power outer loop and a current inner loop, and incorporates Low Voltage Ride-Through (LVRT) capability compliant with the standard [16]. The current output limiting strategy is set to 1.5 times the rated current to ensure device safety during faults. In the simulation, a three-phase bolted short-circuit fault, with a fault resistance set to 0.001 Ω, is applied at the 50% point of branch line Br₁. The fault is initiated at 2.0 s and lasts for 0.5 s.

2.2. Analysis of Inverse-Time Overcurrent Protection Characteristics

2.2.1. Impact of PV Fault Current Contribution

A single-line diagram of a 10 kV distribution network with distributed PV is shown in Figure 2. In the diagram, the upstream system is connected to the distribution network via a 66/10 kV transformer. The distributed PV system is connected to the distribution bus M through a 0.69/10.5 kV step-up transformer. An inverse-time overcurrent protection relay is configured at the head end of each feeder section.

The operating characteristic of the inverse-time overcurrent protection relay shown in the figure complies with the time-current relationship specified for standard inverse-time overcurrent protection in the standard [18]. The corresponding operating characteristic equation is given as follows:

$$t=\left[{\displaystyle \frac{k}{{(I/I_{\mathrm{B}})}^a-1}}+c\right]\cdot T_{\mathrm{DS}}$$

(4)

where I is the measured value of the input current; I_B is the base current, in amperes (A); a is the inverse-time characteristic constant, dimensionless; k and c are inverse-time constants, in seconds (s); T_DS is the time dial setting, dimensionless; t is the theoretical operating time, in seconds (s).

The international standard IEC 60255-3 also specifies recommended values for parameters a, k, and c for several standard inverse-time characteristic curves. In the subsequent model established in this paper, calculations are performed using the general inverse-time parameters defined in this standard, namely a = 0.02, k = 0.14, and c = 0. Within Equation (4), both the Time Dial Setting (T_DS) and the Base Current (I_B) are set as the protection values to be optimized.

As shown in Figure 2, when a three-phase short-circuit fault occurs at point f on the branch line, the output current of the distributed generation is controlled by power electronic devices and is assumed to remain governed by the superposition theorem. It is equivalently represented as a current source in the fault additional network. Figure 3 shows the positive-sequence equivalent network of the distribution network with distributed PV. In the diagram, parameters x and y represent the grid integration locations along line segments MP and PN, respectively; k indicates the location of the fault point f on branch Br₁; Z_S is the equivalent impedance of the system side; Z_F is the fault resistance; Z_Load is the load impedance; ${\dot{I}}_{\mathrm{s}}$ and ${\dot{I}}_{\mathrm{PV}}$ are the short-circuit currents supplied to the fault point from the system and the distributed PV, respectively.

As observed from Figure 2, the short-circuit current flowing through R1 can be expressed as the sum of the current supplied by the system source and the component of the photovoltaic (PV) output current that flows toward the fault point:

$$\begin{array}{l}\begin{array}{c}\begin{array}{l}{\dot{I}}_1={\dot{I}}_{\mathrm{s}}+{\dot{I}}_{\mathrm{PV}.f}=\\ {\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+\beta Z_{\mathrm{MP}}+k{Z}_{{\mathrm{BR}}_1}+Z_{\mathrm{F}}}}+{\dot{I}}_{\mathrm{PV}}{\displaystyle \frac{Z_s}{Z_{\mathrm{PV}}+\beta Z_{\mathrm{MP}}+k{Z}_{{\mathrm{BR}}_1}+Z_{\mathrm{F}}}}\end{array}\end{array}\\ ={\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+Z_{\mathrm{r}}}}+{\dot{I}}_{\mathrm{PV}}{\displaystyle \frac{Z_s}{Z_{\mathrm{PV}}+Z_{\mathrm{r}}}}\end{array}$$

(5)

where U_s is the rated voltage on the system side; U_PCC is the voltage at the distributed PV point of common coupling. ${\dot{I}}_{\mathrm{PV}.f}$ denotes the portion of the total PV output current ${\dot{I}}_{\mathrm{PV}}$ that flows through R1 toward the fault point, determined by the current division between the system impedance and the fault loop impedance. Furthermore, the impedance Z_r (denoting the sum of impedances from bus M to the fault point) satisfies Z_r = βZ_MP + kZ_Br1 + Z_F.

Assuming the rated capacity of the distributed PV does not exceed 100% of the upstream transformer’s capacity, and the current limiting strategy of the distributed PV inverter restricts the output short-circuit current to no more than 1.5 times the rated current, then the magnitude of ${\dot{I}}_1$ should satisfy the following inequality:

$$\left|{\dot{I}}_1\right|⩽\left|{\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+Z_{\mathrm{r}}}}\right|+\left|{\dot{I}}_{PV}\cdot {\displaystyle \frac{Z_{\mathrm{s}}}{Z_{\mathrm{PV}}+Z_{\mathrm{r}}}}\right|⩽\left|{\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+Z_{\mathrm{r}}}}\right|+\left|{\displaystyle \frac{1.5S_{\mathrm{T}}}{\sqrt{3}{U}_{\mathrm{s}}}}\cdot {\displaystyle \frac{Z_{\mathrm{s}}}{Z_{\mathrm{PV}}+Z_{\mathrm{r}}}}\right|$$

(6)

where S_PV is the rated capacity of the distributed PV system; S_T is the capacity of the main transformer in the system; $\left|•\right|$ represents the magnitude of the corresponding variable.

Based on Equations (5) and (6), the fault current amplification factor φ for R1 is defined as the ratio of the short-circuit current magnitudes flowing through it before and after PV integration. φ reaches its maximum under the critical condition where the phase of the PV short-circuit current aligns with that of the system short-circuit current, and the PV current attains its maximum rated value. When the equivalent impedance on the PV side is much smaller than the fault loop impedance (i.e., $Z_{\mathrm{pv}}\ll Z_{\mathrm{r}}$), the maximum value of φ can be expressed as:

$$\begin{array}{c}\phi ={\displaystyle \frac{\left|{\dot{I}}_1\right|}{\left|{\dot{I}}_{\mathrm{s}}\right|}}={\displaystyle \frac{\left|\frac{{\dot{U}}_{\mathrm{s}}}{Z_{\mathrm{s}}+Z_{\mathrm{r}}}+{\dot{I}}_{\mathrm{PV}}\frac{Z_{\mathrm{s}}}{Z_{\mathrm{pv}}+Z_{\mathrm{r}}}\right|}{\left|\frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+Z_{\mathrm{r}}}\right|}}\le 1+{\displaystyle \frac{1.5\cdot S_{\mathrm{PV}}}{\sqrt{3}{U}_n}}\cdot {\displaystyle \frac{\left|Z_s\right|}{\left|\frac{{\dot{U}}_s}{Z_s+Z_r}\right|}}\\ \le 1+1.5\cdot S_{\mathrm{PV}}^{\ast }\cdot |Z_{\mathrm{T}}^{\ast }|\cdot {\displaystyle \frac{|Z_{\mathrm{T}}^{\ast }+Z_r^{\ast }|}{\left|Z_r^{\ast }\right|}}\end{array}$$

(7)

where $Z_{\mathrm{T}}^{\ast }$ is the per-unit impedance of the main system transformer, calculable from its short-circuit voltage percentage U_k%; $S_{\mathrm{PV}}^{\ast }=S_{\mathrm{PV}}/S_T$ is the per-unit value of the installed distributed PV capacity; and $Z_{\mathrm{r}}^{\ast }$ is the total per-unit impedance from bus M to the fault location. From Equation (7), it can be seen that under this scenario, the augmentation effect of the PV is related to its penetration ratio. It should also be pointed out that, in this topology, φ will always be greater than 1. This indicates that the distributed PV continuously feeds fault current into the fault point, causing the short-circuit current flowing through R1 to increase significantly compared to the case without PV integration, thereby affecting the operating selectivity of the inverse-time overcurrent protection.

To assess the impact of this fault current contribution on the selectivity, speed, and sensitivity of inverse-time overcurrent protection, the operating time difference ΔT and the sensitivity coefficient K_seni for R1 and R3 can be expressed as follows, with reference to Figure 2 and Equation (7):

$$\left\{\begin{array}{l}\mathsf{\Delta }T=t_3-t_1\\ K_{\mathrm{sen}3}=I_{\mathrm{B}3}/I_{3\min }^{(2)},K_{\mathrm{sen}1}=I_{\mathrm{B}1}/I_{1\min }^{(2)}\\ {\dot{I}}_1={\dot{I}}_3={\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+\beta Z_{\mathrm{MP}}+k{Z}_{{\mathrm{Br}}_1}+Z_{\mathrm{F}}}}+{\displaystyle \frac{U_{\mathrm{PCC}}}{Z_{\mathrm{PV}}+\beta Z_{\mathrm{MP}}+k{Z}_{{\mathrm{Br}}_1}+Z_{\mathrm{F}}}}\end{array}\right.$$

(8)

where t_i is the operating time of protection i; I_i is the magnitude of the short-circuit current flowing through protection i; $I_{i\min }^{(2)}$ is the short-circuit current through protection i when a line-to-line fault occurs at the end of its protected line under the system’s minimum operating condition; K_seni is the sensitivity coefficient of protection i; and I_Bi is the starting current of protection i.

According to Equations (7) and (8), both the protection operating time difference ΔT and the sensitivity coefficient K_seni are influenced by the fault current contribution from the distributed PV. As shown in Figure 4, when a fault occurs at point f on branch line Br₁, the distributed PV remains grid-connected during the fault via its LVRT control strategy and injects fault current into the system. This current superimposes with the short-circuit current supplied from the system side, resulting in an increase in the magnitude of the short-circuit current flowing through both R1 and R3. Consequently, the operating time difference ΔT between R1 and R3 is reduced, which undermines the time-graded coordination between the protective devices and increases the risk of protection mismatch [19].

2.2.2. Impact of PV Fault Current Extraction

When a distributed PV system is connected between a protection device and a fault point, the short-circuit current it injects can partially offset the fault current supplied from the main grid.

As illustrated in Figure 5, when the PV source is connected to Bus P and a three-phase short-circuit fault occurs at point f on branch line Br₁, the short-circuit current output from the PV introduces a fault current extraction effect. This reduces the magnitude of the short-circuit current flowing through R1, thereby impairing its operating speed and sensitivity. A detailed analysis is provided below:

Based on the equivalent circuit shown in the Figure 6, the current flowing through R1 is analyzed using the superposition theorem.

When only the system side is considered, the PV source is treated as an open circuit, and the current $I_1^{\prime }$ flowing through R1 is given by:

$$I_1^{\prime }={\displaystyle \frac{U_{\mathrm{s}}}{Z_{\mathrm{s}}+x{Z}_{\mathrm{MP}}+k{Z}_{{\mathrm{Br}}_1}+Z_{\mathrm{F}}}}$$

(9)

When only the PV is active, the short-circuit current fed by the PV undergoes current division at the grid-connection point on branch line Br₁. The component of this current flowing through R1 is given by:

$$I_{\mathrm{PV}.1}=-I_{\mathrm{PV}}\cdot {\displaystyle \frac{Z_{\mathrm{F}}}{k{Z}_{{\mathrm{Br}}_1}+x{Z}_{\mathrm{MP}}+Z_{\mathrm{s}}+Z_{\mathrm{F}}}}$$

(10)

According to the superposition theorem, the total current flowing through R1 is expressed as:

$$I_1=I_1^{\prime }+I_{\mathrm{PV}.1}={\displaystyle \frac{U_{\mathrm{s}}-I_{\mathrm{PV}}\cdot Z_{\mathrm{F}}}{Z_{\mathrm{s}}+x{Z}_{\mathrm{MP}}+k{Z}_{{\mathrm{Br}}_1}+Z_{\mathrm{F}}}}$$

(11)

As shown in Equation (11), the integration of distributed PV leads to a reduction in the short-circuit current I₁ flowing through R1. According to the inverse-time overcurrent protection characteristic in Equation (4), a decrease in the measured current I results in an increase in its operating time t₁. Meanwhile, the sensitivity K_sen1 of R1 also decreases due to the reduction in the minimum short-circuit current I_min [20].

R1 serves as the backup for R3. In this scenario, the fault current extraction effect reduces the backup capability of R1 to some extent. The prolonged operating time implies an overall delay in fault clearance, thereby exacerbating the risks of equipment damage and system instability.

The above analysis demonstrates that the fault current contribution and extraction effects of distributed PV alter the short-circuit current distribution in the distribution network, posing a core challenge to the global coordination optimization of inverse-time overcurrent protection: under the contribution effect, the selectivity of upstream protection must be ensured to prevent unintended maloperation; whereas under the extraction effect, the sensitivity and speed of protection need to be enhanced to avoid failure-to-operate risks. Conventional optimization models struggle to simultaneously balance the performance requirements of selectivity, sensitivity, and speed after distributed PV integration. Therefore, further research is needed to develop a coordination optimization model capable of accounting for these three performance metrics.

3. Design of a Coordination Optimization Model for Distribution Networks with Distributed PV Integration

3.1. Formulation of the Objective Function

Based on the protection coordination principles for distribution networks with distributed PV integration and the operating characteristics of inverse-time overcurrent relays, the optimization objective function derived according to this standard is formulated as follows:

$$f=\sum _{i=1}^I\sum _{j=1}^J{\displaystyle \frac{0.14T_{{\mathrm{DS}}_{ij}}}{{\left(\frac{I_i}{I_{B_{ij}}}\right)}^{0.02}-1}}+P$$

(12)

where i denotes the relay index; j denotes the fault point index; I_i is the fault current; P is the penalty term for constraint violation. Thus, the algorithm aims to determine the values of the time dial setting $T_{{\mathrm{DS}}_{ij}}$ and the starting current I_B that minimize the objective function f.

3.2. Constraint Setting

The coordination relationships between inverse-time overcurrent relays serve as the constraints of the proposed algorithm. These include the boundaries of the time dial settings, the physical limits of the starting currents, the coordination constraints between relays, and the inherent physical limitations of the relays themselves. For a system with n relays and m fault points, the specific mathematical formulations are described as follows:

$$T_{{\mathrm{DS}}_j}^{\min }\le T_{{\mathrm{DS}}_j}\le T_{{\mathrm{DS}}_j}^{\max }, j=1,\dots ,n.$$

(13)

$$k_{1}I{L}_{{\max }_j}\le I_{pj}\le k_{2}I{L}_{{\min }_j}, j=1,\dots ,n.$$

(14)

$$t_{qk}-t_{pk}\ge \mathsf{\Delta }T, \forall k=1,\dots ,m.$$

(15)

$$I_{\mathrm{B}i\min }⩽I_{\mathrm{B}i}⩽I_{\mathrm{B}i\max }$$

(16)

where $T_{{\mathrm{DS}}_j}^{\min }$ and $T_{{\mathrm{DS}}_j}^{\max }$ denote the lower and upper bounds of the time dial setting for the j-th relay, respectively; k₁ and k₂ represent the reliability coefficient and sensitivity coefficient, respectively; $L_{{\max }_j}$ and $L_{{\min }_j}$ are the maximum load current and minimum short-circuit current corresponding to the j-th relay, respectively; t_qk and t_pk are the operating times of two relays that need to be coordinated under the same fault; I_Bimin and I_Bimax are the minimum and maximum currents allowed to flow through the relay, respectively, preventing the fault current from exceeding the relay’s withstand limits. The bounds for these parameters, including $T_{{\mathrm{DS}}_j}$, $I_{\mathrm{B}i}$ and the coordination time margin, are established based on industry standards [18] and extensive simulation experiments and engineering practices [18,21].

3.3. Penalty Terms Configuration

Penalty functions are established for the time dial setting boundary constraints, starting current physical boundary constraints, and protection coordination constraints, respectively, to guide the optimization process toward the feasible region. The specific expressions are as follows:

$$P=P_1+P_2+P_3$$

(17)

$$\begin{array}{l}{P}_1={10}^5{\displaystyle \sum _{i=1}^n}\left[\max \{0,T_{\mathrm{DS},j}^{\min }-T_{\mathrm{DS},j}\}+\right.\\ \begin{array}{c} \left.\max \{0,T_{\mathrm{DS},j}-T_{\mathrm{DS},j}^{\max }\}\right]\end{array}\end{array}$$

(18)

$$\begin{array}{l}{P}_2={10}^5{\displaystyle \sum _{j=1}^n}\left[\max \{0,I_{p,j}^{\min }-I_{p,j}\}+\right.\\ \begin{array}{c}\left.\max \{0,I_{p,j}-I_{p,j}^{\max }\}\right]\end{array}\end{array}$$

(19)

$$P_3={10}^5\sum _{(p,q)\in S}\sum _{k=1}^m\max \{0,\mathsf{\Delta }T-(t_{qk}-t_{pk})\}$$

(20)

where P₁, P₂ and P₃ are the penalty terms for the time dial setting boundary constraints, starting current boundary constraints, and protection coordination constraints, respectively; $\mathsf{\Delta }T$ is the coordination time margin, set here to 0.3 s; S is the set of all relay pairs that require coordination. The values of the penalty coefficients P₁, P₂ and P₃ were determined through extensive simulation experiments and sensitivity analysis, which confirmed their capability to ensure stable convergence and effective constraint enforcement across various fault scenarios, thereby demonstrating strong robustness.

4. Model Solution Based on Improved Osprey Optimization Algorithm

4.1. Algorithm Improvement Strategies

Based on the optimization model established in Section 3, which takes the starting current IB and the time dial setting T_DS as optimization variables, the model comprehensively considers the principles of reliability, selectivity, speed, and sensitivity of protection devices under distributed PV integration. It exhibits characteristics of being multidimensional, highly nonlinear, and subject to complex constraints. Due to the stochastic nature of distributed PV output and the complexity of its control strategies, the system presents multidimensional coupling issues, making it necessary to introduce efficient algorithms with corresponding solving capabilities. The Osprey Optimization Algorithm (OOA) [22], known for its strong convergence accuracy and global search capability, is well-suited for solving such models. However, the traditional OOA suffers from limitations such as inefficient global search and a tendency to fall into local optima. To address these issues, this paper designs an improved method that combines strong global exploration capability with efficient convergence characteristics, aiming to enhance the solution performance of the coordination optimization model and meet the requirements for adaptive protection coordination in distribution networks with distributed PV integration.

4.1.1. Nonlinear Chaotic Mapping

In the population initialization phase of the standard Osprey Optimization Algorithm (OOA), the algorithm employs random numbers rand(0,1) for generating the initial population. In practice, such random numbers are pseudorandom, characterized by inherent periodicity and non-uniform distribution. These characteristics can easily lead to insufficient population diversity, thereby compromising global search efficiency. To address this limitation, this paper introduces an arccosine chaotic mapping to reconstruct the initialization process.

Chaotic mapping is a typical nonlinear theory. In the initialization phases of many optimization algorithms, chaotic maps are employed as random number generators to leverage their beneficial properties, including randomness, ergodicity, periodicity, and sensitivity to initial conditions, for algorithmic improvements [23,24,25]. The iterative formula for defining the chaotic sequence {r_j} is as follows:

$$\left\{\begin{array}{c}{r}_1=\mathrm{rand}(0,1)\\ r_{i+1}=\left|\arccos (\pi \cdot r_i)\right|, i=1,2,\dots ,N-1\end{array}\right.$$

(21)

where N is the population size; rand(0,1) generates a random number uniformly distributed in [0, 1]; the pseudorandom initial value rand(0,1) is replaced by extracting an N-1-dimensional chaotic vector $\mathit{R}=\{r_2,r_3,\dots ,r_N\}$, which is then used to generate the initial position of the i-th individual in the D-dimensional solution space. The mathematical expression is given by:

$$x_{ij}=b_j^{\mathrm{L}}+R(i)\cdot (b_j^{\mathrm{L}}-b_j^U)$$

(22)

where $b_j^{\mathrm{L}}$ and $b_j^U$ are the lower bound and upper bound of the j-th variable, respectively.

4.1.2. Nonlinear Convergence Factor

In the standard Osprey Optimization Algorithm (OOA), individual position updates primarily rely on random perturbations to guide the search direction. While this mechanism provides strong global exploration capability, it lacks adaptive control over the convergence speed. To balance the algorithm’s global exploration and local exploitation abilities, this paper introduces a nonlinear convergence factor, denoted as Φ, whose mathematical expression is given by:

$$\Phi =\exp \left(-{\left({\displaystyle \frac{t}{T_{\max }}}+1\right)}^2\right)$$

(23)

where t is the current iteration count; T_max is the maximum number of iterations.

As the expression shows, this factor decays exponentially with the number of iterations, and the decay rate accelerates as t increases. This behavior aligns with the search strategy of “emphasizing exploration in early stages and exploitation in later stages” during the optimization process. Specifically, when t = 0, $\Phi \approx e^{-1}\approx 0.367$, maintaining a relatively large value to preserve population diversity. When $t\to T_{\max }$, $\Phi \to e^{-4}\approx 0.018$, and its sharp decay accelerates local convergence. Specifically, when incorporated into the standard algorithm, it takes the form shown in Equation (24):

$$x_{i,j}^{\mathrm{P}1}=\Phi x_{i,j}+r_{i,j}(S{F}_{i,j}-I_i{x}_{i,j})$$

(24)

During the early stages of iteration, when the value of t is small and Φ is relatively large, the current position information is largely preserved, mitigating drastic fluctuations in the search direction caused by random perturbations. In later iterations, as t increases and Φ approaches zero, the individual’s position rapidly converges toward the region of the optimal solution, thereby enhancing the efficiency of local search.

4.1.3. Improved Prey Processing Phase

To further mitigate the tendency of OOA to converge to local optima, this paper integrates the spiral search mechanism from the Whale Optimization Algorithm (WOA) [26] and the dynamic adjustment strategy of the Marine Predators Algorithm (MPA) [27], proposing a dynamic spiral position update method. The corresponding mathematical model is defined as follows:

$$\begin{array}{c}\alpha =-1+{\displaystyle \frac{t}{T_{\max }}}\cdot (-1)\\ \beta =(\alpha -1)\cdot \delta +1\begin{array}{cc}& \end{array}\delta \sim \mathrm{U}[0,1]\\ \mathcal{L}=|\exp (\beta )\cdot \cos (2\pi \beta )|\end{array}$$

(25)

where $\alpha$ is the shape control parameter, whose value linearly decreases within the range [−2, −1]; $\beta$ is the phase parameter for generating random perturbations; $\delta$ is a random number uniformly distributed in [0, 1]; $\mathcal{L}$ is the dynamic spiral factor, which exhibits non-monotonic oscillations over the interval $\beta \in [\alpha ,1]$. Specifically, when incorporated into the standard algorithm, it takes the form shown in Equation (26):

$$x_{i,j}^{\mathrm{P}2}=x_{i,j}\cdot \mathcal{L}+{\displaystyle \frac{b_j^L+{\delta }^{\prime }\cdot (b_j^U-b_j^L)}{t}}$$

(26)

where ${\delta }^{\prime }~\mathrm{U}[0,1]$. By integrating the dynamic spiral factor $\mathcal{L}$ as a spiral-driven component in the position update equation and introducing the random perturbation term $\delta$ to construct a boundary perturbation mechanism, its periodic oscillatory behavior—alternating between positive and negative phases—enables individuals to perform bidirectional search within the solution space. Specifically, the forward direction inherits historical position information to accelerate convergence, while the reverse direction breaks through local optima to escape gradient-descent traps. This mechanism further strengthens the algorithm’s convergence capability and local search performance.

4.2. Convergence and Computational Complexity Analysis

The improved Osprey Optimization Algorithm is modeled as a Markov chain:

$$P_{t+1}=T(P_t)=S\circ V_1(P_t)\circ V_2(P_t)$$

(27)

where the operator $\circ$ denotes the composition of functions, S represents the selection operator, and V₁ and V₂ denote the improved global search operator and the dynamic spiral local search operator, respectively.

Three improvement strategies collectively ensure the global convergence characteristics of the algorithm. The chaotic mapping initialization employs the arccosine mapping $r_{i+1}=\left|\arccos (\pi \cdot r_i)\right|$ to generate an initial population with ergodicity, which mitigates the periodicity and non-uniform distribution issues of traditional pseudo-random numbers, thereby providing favorable initial diversity for global search. The nonlinear convergence factor Φ maintains relatively large values during the early iterations (when t is small), enhancing global exploration capability, and decays rapidly in the later stages (when t approaches T_max), accelerating local convergence. This adaptive balancing mechanism contributes to improved convergence speed. The dynamic spiral search utilizes the periodic oscillatory property of $\mathcal{L}$ to perform bidirectional search within the solution space, effectively avoiding local optima.

In summary, the improved algorithm satisfies the convergence criteria for stochastic optimization algorithms: a compact search space, continuous fitness function, strict elite retention strategy, and probabilistic positivity in the exploration mechanism. Consequently, global convergence is guaranteed.

From the perspective of computational complexity analysis, the improved OOA incurs a marginally higher computational burden compared to the original algorithm. The initialization phase, which includes chaotic sequence generation and population initialization, possesses a complexity of:

$$O(N\times D)$$

(28)

where N represents the population size and D denotes the problem dimensionality. Each iteration involves fitness evaluation, global search operation, local search operation, and boundary handling. The complexity of the fitness evaluation is $O(N\times C_f)$ (where $C_f$ signifies the computational complexity of the fitness function itself), while the complexity of all other operations is $O(N\times D)$.

Therefore, the complexities per iteration and the overall complexity are, respectively:

$$O(N\times D+N\times C_f)$$

(29)

$$T_{\mathrm{total}}=O(N\times D)+T_{\max }\times O(N\times D+N\times C_f)$$

(30)

For the inverse-time overcurrent protection coordination problem addressed in this paper, the parameters are set as T_max = 100, N = 50, D = 8, resulting in a total computational complexity of $O(80000)$ basic operations. Compared to the original OOA algorithm, the improvement strategies introduce additional computational load; however, they maintain the same order of time complexity. The computational efficiency thus satisfies the requirements for practical engineering applications.

To further verify the computational efficiency of the proposed Improved Osprey Optimization Algorithm (IOOA), the inverse-time overcurrent protection setting optimization model established in Section 3 was solved under a typical configuration (population size = 100, maximum iterations = 100). The average computation time per single run of IOOA was approximately 0.060 s (with a standard deviation of ±0.002 s), while that of the standard OOA was 0.033 ± 0.001 s. Although the introduced enhancement strategies slightly increase the computational burden per iteration, the overall solution time remains within the millisecond range, fully meeting the real-time requirements for relay protection setting calculations in practical engineering applications.

4.3. Flowchart of the Improved Osprey Algorithm

Based on the protection setting optimization model, constraint conditions, and the improved osprey optimization algorithm constructed in the preceding sections, the overall optimization process is illustrated in the Figure 7:

5. Simulation Analysis

5.1. Performance of Inverse-Time Overcurrent Relay Setting in Distribution Networks

To systematically evaluate the setting performance of the proposed Improved Osprey Optimization Algorithm (IOOA) and its adaptability to different photovoltaic (PV) penetration levels, this section conducts simulation tests with varying PV integration capacities: with the PV current contribution effect representing a medium-to-low penetration scenario, and the PV current extraction effect representing a high penetration scenario. Output fluctuation tests are further superimposed on these two fundamental scenarios. The selected PV capacities are set slightly higher than the typical ranges specified in general engineering guidelines, in order to thoroughly evaluate the performance of the protection scheme under the extreme effects of both current contribution and extraction. The objective of this section is to verify whether the inverse-time overcurrent protection settings obtained through the improved algorithm can satisfy the core requirements of selectivity, speed, sensitivity, and reliability across a series of practical conditions formed by different penetration levels and their inherent output fluctuations.

5.1.1. Adaptability of the Setting Method to the PV Current Contribution Effect

To validate the adaptability of the proposed Improved Osprey Optimization Algorithm (IOOA) under the PV current contribution effect, the simulation model presented in Figure 2 is implemented in PSCAD. The system configuration includes a 100 MVA main transformer with a voltage ratio of 66 kV/10.5 kV. A PV station with an output of 15 MW is integrated, employing PQ control and featuring low-voltage ride-through capability. Line parameters follow the specifications in

Table 1. Line parameters of 10 kV distribution network.

Line	Length (km)	Conductor Specification	Positive-Sequence Impedance (Ω/km)	Zero-Sequence Impedance (Ω/km)
MP	3.91	YJV_240	0.069 + j0.099	0.110 + j0.159
PN	3.17	YJV_240	0.069 + j0.099	0.110 + j0.159
Br1	1.42	YJV_120	0.106 + j0.153	0.169 + 0.245
Br2	1.55	YJV_120	0.106 + j0.153	0.169 + 0.245

, and Load 1, Load 2, and Load 3 are modeled as purely active loads, each rated at 5 MW.

As illustrated in Figure 3, three-phase bolted short-circuit faults, each with a fault resistance of 0.001 Ω, are applied under normal system operating conditions at the beginning, middle, and end sections of the main lines MP and PN, as well as the branch lines Br₁ and Br₂. The short-circuit currents flowing through each protection device (in amperes, A) are recorded, resulting in a 12 × 3 fault current matrix, as shown in the

Table 2. Three-phase fault current matrix.

	Protection	R1	R2	R3	R4
Fault Point
1: MP Start Section		8124	18	5	9
2: MP Middle Section		7814	17	5	9
3: MP End Section		6523	15	41	7
4: PN Start Section		6524	6416	42	0
5: PN Middle Section		5648	5664	58	6
6: PN End Section		4809	4811	70	36
7: Br1 Start Section		7801	18	7799	9
8: Br1 Middle Section		6992	75	6989	38
9: Br1 End Section		6393	132	6391	66
10: Br2 Start Section		5623	5601	58	5603
11: Br2 Middle Section		5611	5599	67	5600
12: Br2 End Section		5201	5119	75	5117

.

The model was executed in a Windows 11 environment using algorithms written in the MATLAB programming language. The PC configuration was as follows: Intel(R) Core (TM) i9-13900HX processor, 32.00 GB RAM, and an NVIDIA GeForce RTX 4070 GPU. The algorithm population size was set to 150, with the maximum number of iterations set to 50. Under this configuration, the resulting inverse-time overcurrent relay setting values for each protection device are listed in

Table 3. Inverse-time overcurrent relay setting results under current contribution effect.

Setting Results	R1	R2	R3	R4
TDS	0.14	0.11	0.07	0.07
IB	1273.7	961.5	762.2	692.4

, and the heatmap of the overall operating time is shown in Figure 8.

As shown in Figure 8, the relay operating times exhibit a distinct hierarchical staircase pattern. Under this operational scenario with PV integration, the sequential distribution of protection operating times ensures coordination selectivity. Specifically, at fault points 7–9, where the distributed PV current contribution effect is most pronounced, R3 reliably clears the fault within 0.23–0.26 s while ensuring R1 remains secure against maloperation, thereby avoiding the over-tripping risk inherent in traditional instantaneous overcurrent protection schemes.

5.1.2. Adaptability of the Setting Method to the PV Current Extraction Effect

A simulation model shown in Figure 5 was built in the PSCAD platform. Based on the analysis in Section 2.2.2, the PV station was connected to Bus P with an output of 36 MW, while other parameters remained consistent with those described in Section 5.1.1. Fault locations were the same as in Table 2, with the fault type set to a BC-phase bolted short-circuit fault with a fault resistance of 0.001 Ω. The resulting fault current matrix is shown in

Table 4. BC-phase fault current matrix.

	Protection	R1	R2	R3	R4
Fault Point
1: MP Start Section		8138	366	101	195
2: MP Middle Section		5640	354	96	177
3: MP End Section		3813	332	97	166
4: PN Start Section		3819	8018	105	158
5: PN Middle Section		3354	6263	110	6
6: PN End Section		3670	4392	111	173
7: Br1 Start Section		5632	351	9076	177
8: Br1 Middle Section		5100	371	7788	199
9: Br1 End Section		4874	380	5419	190
10: Br2 Start Section		3346	6260	110	6056
11: Br2 Middle Section		3193	5186	117	4831
12: Br2 End Section		3577	4007	111	4107

(in amperes, A), serving to validate the method’s adaptability to the PV current extraction effect.

Through iterative calculations of the optimization algorithm, the protection setting results under the PV current extraction effect were obtained (

Table 5. Inverse-time overcurrent relay setting results under current extraction effect.

Setting Results	R1	R2	R3	R4
TDS	0.11	0.09	0.06	0.06
IB	1371.7	1443.3	1829.3	1412.5

), and a heatmap of the operating times of each protection device at the aforementioned fault points was plotted (Figure 9), providing a visual representation of their operating times and coordination relationships.

As shown in Figure 9, under operating conditions where distributed PV causes significant fault current extraction on R1, the protection device can still reliably respond to faults occurring within its local line zone. Specifically, in fault zones 1–3, although PV integration reduces the short-circuit current detected by R1, its operating time remains stable between 0.42 and 0.75 s. Compared to the potential failure-to-trip risks associated with conventional coordination methods, the proposed optimization approach ensures reliable clearance of faults along the entire line, validating the improved algorithm’s adaptability to the fault current extraction effect.

5.1.3. Adaptability of the Coordination Method to PV Output Fluctuation Scenarios

For the aforementioned scenarios involving both PV fault current contribution and extraction, the output of the PV power station was set to fluctuate randomly within the range of 80% to 110%. Keeping the protection settings for these two typical scenarios unchanged, the method’s adaptability was evaluated by simulating three-phase and line-to-line short-circuit faults across the twelve designated zones.

As shown in the operating time heatmaps of Figure 10 and Figure 11, under conditions of fluctuating PV output, the protection settings generated by the proposed improved optimization algorithm are capable of ensuring reliable operation. Compared with traditional coordination methods, the proposed algorithm effectively addresses both the fault current contribution and extraction effects introduced by PV integration, thereby preventing maloperation or failure-to-trip.

5.2. Adaptability Verification Under Different Fault Scenarios

5.2.1. Adaptability Analysis Under Different PV Integration Penetration Ratios

A 10 kV distribution network simulation model, as shown in Figure 2, was built in PSCAD/EMTDC. To verify the adaptability of the proposed optimization algorithm to different PV integration capacities, a bolted three-phase short-circuit fault with a fault resistance of 0.001 Ω was set at fault point f on branch Br₁. The capacity of the integrated PV station was successively varied: 0 MW (scenario without PV), 6 MW, and 12 MW, in order to analyze the selectivity of Protection R₁ and Protection R₃ under different PV penetration levels. The simulation results are presented in the

Table 6. Simulation results under current contribution effect with different PV penetration ratios.

PV Integration Capacity	R1 Operating Time (t/s)	R3 Operating Time (t/s)	$\mathbf{\Delta }\mathit{t}$
12 MW	0.56	0.24	0.34
6 MW	0.63	0.28	0.35
0 MW	0.67	0.31	0.36

.

As the PV integration capacity increases, the short-circuit currents flowing through protection devices R₁ and R₃ are augmented by the PV current contribution effect, leading to an overall upward trend in their operating times. In the high-penetration-ratio scenario, although the operating time of R₁ decreases to some extent compared to the scenario without PV, its operating time difference ($\Delta t$) remains within a reasonable range. This demonstrates that the algorithm maintains good selectivity even under high penetration conditions, thereby addressing the impact of the PV current contribution effect on the coordination between upstream and downstream protections.

Similarly, to visually verify the impact of the PV current extraction effect on the sensitivity of protection device R₁ under different penetration ratios, a 10 kV distribution network simulation model shown in Figure 5 was built in PSCAD/EMTDC. A bolted BC two-phase short-circuit fault with a fault resistance of 0.001 Ω was configured at the end of line MP. The PV integration capacity was successively changed to 20 MW, 24 MW, and 30 MW to verify whether protection R₁ can operate reliably under extreme conditions. The simulation results are presented in the

Table 7. Simulation results under current extraction effect with different PV penetration ratios.

PV Integration Capacity	R1 Operating Time (t/s)	Ksen1
30 MW	0.70	2.44
24 MW	0.64	2.62
20 MW	0.59	2.71

.

As the penetration ratio of the integrated PV increases, the extraction effect intensifies progressively. This leads to a further reduction in the short-circuit current detected by protection device R₁, resulting in a gradual extension of its operating time. Concurrently, the sensitivity coefficient K_sen1 decreases from 2.71 to 2.44. Despite the high penetration environment of 30 MW, protection R₁ still maintains the capability for reliable operation, indicating that the proposed optimization method can sustain the fundamental sensitivity requirements of the protection under varying penetration levels in current extraction scenarios.

5.2.2. Adaptability Analysis Under Different Fault Scenarios

In the 10 kV distribution network simulation models shown in Figure 2 and Figure 5, various types of short-circuit faults were applied at typical locations, namely the 75% point of branch line Br₁ and the end of the main line MP, to validate the adaptability of the proposed algorithm under the PV current contribution and extraction effects, respectively. The PV capacities were set to 12 MW and 30 MW accordingly, while all other line parameters, system parameters, and load configurations remained consistent with those specified in Section 5.1.1 and Section 5.1.2.

As can be seen from the

Table 8. Relay operating times under different fault scenarios.

Operating Scenario	Fault Type	R1	R3	$\mathbf{\Delta }\mathit{t}$	Ksen1
PV Current Contribution Effect	ABC (0.001 Ω)	0.66	0.27	0.39	-
	BC (0.001 Ω)	0.55	0.21	0.34	-
	BC-G (1 Ω)	0.51	0.19	0.32	-
	BC-G (5 Ω)	0.49	0.18	0.31	-
	BC-G (10 Ω)	0.42	0.15	0.27	-
PV Current Extraction Effect	BC (0.001 Ω)	0.73	-	-	2.67
	BC-G (1 Ω)	0.86	-	-	1.87
	BC-G (10 Ω)	0.95	-	-	1.51
	AC-G (1 Ω)	0.86	-	-	1.87
	AC-G (10 Ω)	0.95	-	-	1.51

, in the PV current contribution scenario, for bolted three-phase/two-phase short-circuits and two-phase-to-ground short-circuits with different fault resistances, the upstream protection R1 and the downstream protection R3 maintain proper time-graded coordination, with no occurrence of unintended maloperation (over-tripping). In the PV current extraction scenario, as the fault resistance increases from 0 Ω to 10 Ω, the fault current gradually decays; nevertheless, the sensitivity of protection R1 remains reliable. In summary, across different fault types and fault resistance scenarios, the proposed optimization method can effectively coordinate the protection operation.

5.2.3. Adaptability Analysis Under Different Network Topologies

To verify the adaptability of the proposed optimization algorithm to different network topologies, this section introduces a dual-feeder distribution network, as illustrated in the figure below, for comparative analysis against the single-feeder with T-connected branch structures shown earlier in Figure 2 and Figure 5. In Figure 12, the PV integration points PV₁ and PV₂ are used to simulate the two typical scenarios of PV current contribution and extraction, respectively, in order to verify the algorithm’s versatility across different network topologies.

To verify the adaptability of the proposed optimization algorithm under different network topologies, this section conducts simulation tests on the dual-feeder distribution network shown in the figure above. In the diagram, the PV units PV₁ and PV₂ are integrated at different locations to simulate the typical scenarios of current contribution and extraction, respectively. In the simulation model, the system base capacity is set to 100 MVA, with PV₁ capacity at 12 MW and PV₂ capacity at 30 MW. The loads L₁ and L₂ are both 5 MW. The line parameters are presented in the

Table 9. Line parameters of the dual-feeder distribution network.

Line	Length (km)	Conductor Specification	Positive-Sequence Impedance (Ω/km)	Zero-Sequence Impedance (Ω/km)
AB	3.12	YJV_240	0.069 + j0.099	0.110 + j0.159
BC	3.24	YJV_240	0.069 + j0.099	0.110 + j0.159
CD	2.43	YJV_120	0.106 + j0.153	0.169 + 0.245
AE	3	YJV_240	0.069 + j0.099	0.110 + j0.159
EF	3.36	YJV_240	0.069 + j0.099	0.110 + j0.159
FG	2.82	YJV_120	0.106 + j0.153	0.169 + 0.245

.

Based on the parameters in the above table, a simulation model was built in PSCAD. To separately verify the protection performance under the PV current contribution and extraction effects, the following fault scenarios were set: when PV₁ is individually put into/out of operation, a three-phase bolted short-circuit with a fault resistance of 0.001 Ω is applied at k₂ (50% of line BC) and k₅ (50% of line EF); when PV₂ is individually put into/out of operation, a BC two-phase bolted short-circuit with a fault resistance of 0.001 Ω is applied at k₁ (50% of line AB) and k₂ (50% of line BC). The simulation results are shown in the

Table 10. Simulation results of protection performance for the dual-feeder distribution network under different PV operating modes.

Operating Scenario	Fault Point	R1 (t/s)	R2 (t/s)	R4 (t/s)	R5 (t/s)	$\mathbf{\Delta }\mathit{t}$	Ksen1
PV Current Contribution Effect	K2	0.64	0.43	-	-	0.21	-
	K5	-	-	0.64	0.43	0.21	-
PV de-energized	K2	0.74	0.51	-	-	0.23	-
	K5	-	-	0.74	0.51	0.23	-
PV Current Extraction Effect	K1	0.47	1.02	-	-	-	2.04
	K2	1.01	0.69	-	-	-	2.04
PV de-energized	K1	0.44	-	-	-	-	2.03
	K2	0.79	0.58	-	-	-	2.13

, which are used to evaluate the adaptability of the proposed method in the dual-feeder structure.

The above simulation results indicate that the protection setting optimization method based on the improved OOA proposed in this paper also demonstrates good adaptability in the dual-feeder topological scenario. In this configuration, PV₁ induces a current contribution effect while PV₂ induces a current extraction effect. After optimization, all protection devices maintain correct operating sequences and sufficient sensitivity. In the current contribution scenario, the protection operating time curves remain parallel regardless of whether the PV is in operation or disconnected. Protections R1–R2 and R4–R5, which are significantly affected by the contribution effect, still achieve reliable coordination without maloperation. In the current extraction scenario, protection R2 is notably influenced by PV₂, resulting in prolonged operating time due to the extraction current. Nevertheless, its setting values ensure adequate sensitivity, effectively preventing the risk of failure to operate.

5.3. Comparative Analysis with the Original Algorithm

To validate the effectiveness of the proposed improvement strategies in enhancing the adaptability and performance of the Osprey Optimization Algorithm (OOA), this section presents a comparative analysis between the improved OOA and the original OOA under identical simulation environments and tuned parameters. The comparison encompasses convergence speed, optimization accuracy, and adaptability to scenarios involving PV integration.

5.3.1. Comparison of Setting Results Under PV Current Contribution Scenario

To validate the effectiveness of the proposed improved method under the PV current contribution scenario, a simulation model shown in Figure 2 was built in the PSCAD platform, with the PV station connected to Bus M at an output of 15 MW. A three-phase short-circuit fault was applied, and the resulting fault current matrix is given in Table 2. Line parameters and protection configurations remain consistent with those in Section 5.1.1.

Under this scenario, both the original OOA and the improved OOA (IOOA) were used to optimize the inverse-time overcurrent relay settings. The algorithm population size was set to 100, and the maximum number of iterations was set to 50. The resulting optimal settings for each relay are listed in

Table 11. Comparison of Inverse-Time Overcurrent Relay Settings between Original OOA and Improved OOA under Current Contribution Effect.

Setting Results	Algorithm Type	R1	R2	R3	R4
TDS	Original OOA	0.21	0.12	0.13	0.07
	Improved OOA	0.14	0.11	0.07	0.07
IB	Original OOA	1054.1	1344.7	2624.8	943.7
	Improved OOA	1273.7	961.5	762.2	692.4

, and the overall operating time heatmap is shown in Figure 13.

As shown in Figure 14, the simulation results indicate that the original algorithm’s insufficient adaptability to PV fault current contribution scenarios leads to protection miscoordination in its setting results, potentially expanding the outage scope caused by faults. In contrast, the improved algorithm not only effectively adapts to the PV fault current contribution scenario in its setting calculations but also outperforms the original algorithm across multiple performance metrics, including Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and overall average operating time.

5.3.2. Comparison of Setting Results Under PV Current Extraction Scenario

To validate the performance of the Improved Osprey Optimization Algorithm (IOOA) under the PV current extraction scenario, a simulation model shown in Figure 5 was built in PSCAD, with the PV station connected to Bus P at an output of 36 MW. A BC-phase short-circuit fault was applied, and the corresponding fault current matrix is provided in Table 5. Line parameters and protection configurations remain consistent with those described in Section 5.1.2.

Under this scenario, both the original OOA and the improved OOA (IOOA) were utilized to optimize the settings of inverse-time overcurrent relays. The population size and maximum number of iterations were set to 100 and 50, respectively. The resulting relay setting values are summarized in

Table 12. Comparison of inverse-time overcurrent relay settings between Original OOA and improved OOA under current extraction effect.

Setting Results	Algorithm Type	R1	R2	R3	R4
TDS	Original OOA	0.11	0.12	0.06	0.08
	Improved OOA	0.11	0.09	0.06	0.06
IB	Original OOA	1569.9	1837.2	1771.6	582.7
	Improved OOA	1371.7	1443.3	1829.3	1412.5

, and the overall operating time heatmap is presented in Figure 15.

As shown in Figure 16, it can be observed that under the PV current extraction effect, the improved OOA generally achieves shorter operating times across all fault points, with a lower overall average operating time compared to the original OOA, demonstrating superior response speed.

In summary, the improved OOA exhibits better adaptability and comprehensive performance in both PV current contribution and extraction scenarios. By incorporating chaotic initialization, a nonlinear convergence factor, and a dynamic spiral search mechanism, the improved algorithm effectively addresses the original algorithm’s shortcomings—sensitivity to initial values, susceptibility to local optima, and slow convergence in later stages—significantly enhancing the setting accuracy and response speed under complex PV operating conditions.

5.3.3. Ablation Experiment Validation for the Improved OOA

To quantitatively evaluate the specific contributions of the three proposed improvement strategies to the algorithm’s performance enhancement, this subsection designs an ablation experiment. Under the same simulation conditions as in Section 5.2.1, a comparison is made with the original OOA. The simulation results are presented in the

Table 13. Ablation experiment results of improvement strategies under the PV current contribution effect.

Algorithm Variant	Improved OOA Avg. Relay Operating Time (s)	Convergence Iterations	Standard Deviation
Original OOA	0.78	97	0.092
OOA with Arccosine Mapping Only	0.69	85	0.073
OOA with Nonlinear Convergence Factor Only	0.64	79	0.045
OOA with Dynamic Spiral Search Only	0.53	72	0.032
Full Improved OOA	0.52	35	0.012

:

The ablation experiment results indicate that each improvement strategy effectively enhances algorithm performance, with distinct optimization emphases. The dynamic spiral search strategy contributes most significantly to reducing the protection operating time; the arccosine chaotic mapping and the nonlinear convergence factor effectively improve the population diversity during the initial phase and the convergence balance during iterations, respectively. The three strategies work synergistically, collectively enabling the Improved OOA to achieve optimal performance in convergence speed, setting accuracy, and robustness.

5.3.4. Sensitivity Analysis of Algorithm Parameters

To comprehensively evaluate the performance of the improved OOA, a sensitivity analysis was further conducted on its population size and iteration count, with comparative tests performed against the original OOA. The testing conditions and parameter settings were identical to those described in Section 5.1.1.

Regarding sensitivity to population size, with the iteration count fixed at 100, the algorithm’s performance was tested under different population sizes. The results are shown in

Table 14. Sensitivity analysis of population size.

Population Size	Improved OOA Avg. Relay Operating Time (s)	Original OOA Avg. Relay Operating Time (s)	Performance Improvement
30	0.4983	0.5892	15.43%
50	0.5116	0.5767	11.2%
100	0.5099	0.5669	10.00%
150	0.5192	0.5701	8.92%

. The improved OOA demonstrated consistent optimization performance across different population sizes, with average relay operating times ranging from 0.4983 to 0.5192 s. In contrast, the results from the original OOA showed comparable fluctuations (ranging from 0.5669 to 0.5892 s) as the population size varied.

Regarding sensitivity to iteration count, with the population size fixed at 100, the algorithm’s performance was tested under different iteration counts. The results are shown in

Table 15. Sensitivity analysis of iteration count.

Iteration Count	Improved OOA Avg. Relay Operating Time (s)	Original OOA Avg. Relay Operating Time (s)	Performance Improvement
30	0.5225	0.6541	20.11%
50	0.4974	0.5531	10.07%
100	0.5101	0.5970	14.56%
150	0.5329	0.5669	5.97%

. The optimized average relay operating time obtained by the improved OOA remained relatively stable (ranging from 0.4974 to 0.5329 s) across different iteration counts. Particularly noteworthy is that the improved OOA achieved its best performance (0.4974 s) with just 50 iterations, demonstrating its efficient convergence capability. The results from the original OOA exhibited more significant fluctuations (ranging from 0.5531 to 0.6541 s) with increasing iteration counts, indicating poorer stability and convergence characteristics compared to the improved OOA.

5.4. Algorithm Robustness Verification

To validate the robustness of the improved OOA under random initial conditions, this paper designed simulation tests with 30 independent random initializations and compared the results with the original OOA under identical parameter settings. The specific results are illustrated below:

The robustness analysis results are shown in Figure 17.

In conclusion, the random initialization simulations verify that the improved OOA possesses excellent robustness. It can stably converge to optimization solutions with similar performance under different random starting points.

5.5. Comparative Analysis with Peer Algorithms

To comprehensively evaluate the solution performance and optimization capability of the proposed Improved Osprey Optimization Algorithm (IOOA), this study conducts a comparative analysis against various classical metaheuristic algorithms as well as other recently proposed improved versions of OOA. The algorithms used for comparison include the Butterfly Optimization Algorithm (BOA), the Whale Optimization Algorithm (WOA), the Particle Swarm Optimization (PSO), and an improved version of OOA that incorporates Cauchy mutation and opposition-based learning strategies (denoted as IOOA-B) [28]. To ensure a fair comparison, all algorithms are configured using the default parameters recommended in their respective original literature and are tested under identical conditions. The standard deviation of the obtained objective function values and the number of iterations required to reach the optimal solution are used as metrics to evaluate precision and speed, respectively. The model evaluation is based on the photovoltaic (PV) current contribution and extraction scenarios described in the preceding sections. The comparative results are summarized in

Table 16. Performance metrics of different optimization algorithms under PV integration scenarios.

Operating Scenario	Model	Convergence Iterations	Standard Deviation	Overall Average Operating Time (t/s)
PV Current Contribution Effect	Improved OOA	30	0.012	0.522
	WOA	94	0.042	0.632
	BOA	79	0.094	0.784
	PSO	97	0.237	0.617
	IOOA-B	49	0.029	0.536
PV Current Extraction Effect	Improved OOA	30	0.007	0.454
	WOA	94	0.261	0.783
	BOA	79	0.394	0.588
	PSO	97	0.478	1.011
	IOOA-B	44	0.024	0.628

.

As shown in Table 16, the improved Osprey Optimization Algorithm exhibits higher solution accuracy and faster convergence speed compared to traditional optimization algorithms. Its overall performance is superior, making it more suitable for application in distribution network relay protection.

5.6. Economic Benefit Analysis

Based on the simulation results presented in Section 5, the proposed protection setting optimization method employing the Improved Osprey Optimization Algorithm (IOOA) demonstrates significant advantages in preventing maloperation and failure-to-trip, as well as in reducing the overall average operating time of the relays. To quantify the practical value of these technical improvements for distribution system operation, a brief benefit estimation is conducted below based on the simulation data.

In terms of reliability, the method maximizes the avoidance of unnecessary expansion of the outage area by ensuring faults are accurately and reliably cleared. Taking the 10 kV feeder from the simulation as an example, its average load current is approximately 556 A (see Table 3), corresponding to a load power of about 9.1 MW. Assuming this feeder experiences one permanent fault per year requiring protection intervention, the proposed method can reduce the average affected outage area per fault by approximately 25% by preventing maloperation. Simultaneously, the reduction in relay operating time (an average decrease of about 0.31 s), as shown in the heatmaps of Figure 8, Figure 9, Figure 10 and Figure 11, further decreases the fault duration. Combining these two aspects of improvement, a preliminary estimation indicates that this method can reduce the annual Energy Not Supplied (ENS) for a typical feeder by approximately 4.1 MWh.

Referring to current research on interruption cost in typical distribution scenarios and considering the topology used in this paper’s simulation, a unit interruption cost of 2500 USD/MWh is adopted for estimation [29,30]. Accordingly, the annual economic loss avoided per feeder due to the reduction in ENS is calculated to be about 10,250 USD. Although the specific value is influenced by the actual distribution system’s topology, load characteristics, and local economic conditions, this estimation still reflects a positive trend in economic benefits.

Regarding implementation feasibility, as described in Section 4.2, a single optimization run requires only about 0.06 s on a standard personal computer. This computational burden is negligible within the time scales required for relay protection. The algorithm only requires network topology, line parameters, and fault current data, which are typically available from distribution network measurement devices and dispatch systems. It does not rely on the system’s communication infrastructure for peer-to-peer information transmission, thus holding certain practical value for engineering application.

6. Conclusions

To address the inadequate adaptability of conventional inverse-time overcurrent protection in distribution networks caused by distributed photovoltaic (PV) integration, this paper proposes a protection setting optimization method based on an Improved Osprey Optimization Algorithm (IOOA). The following conclusions are drawn through simulation verification.

(1)

This paper establishes a fault equivalent model for distributed PV and deduces the mechanism by which distributed PV integration leads to protection mismatch risks and reduced sensitivity in inverse-time overcurrent protection. On this basis, an optimization model for protection settings is constructed, with protection selectivity as the constraint and minimization of the total operating time as the objective, to coordinate the contradiction between protection speed and sensitivity caused by PV integration.

(2)

By incorporating arccosine chaotic mapping, a nonlinear convergence factor, and a dynamic spiral search strategy into the standard Osprey Optimization Algorithm (OOA), the convergence accuracy and global search capability of the algorithm in solving the high-dimensional nonlinear optimization model for inverse-time overcurrent protection are significantly enhanced, effectively avoiding local optima.

(3)

Simulation results show that compared with traditional algorithms such as Particle Swarm Optimization (PSO) and Whale Optimization Algorithm (WOA), the optimization scheme of the improved Osprey algorithm proposed in this paper achieves significantly better search performance. Its setting results can correctly coordinate the operating time curves of upstream and downstream protections. Under different scenarios of PV access capacity, location, and output fluctuation, this method can ensure correct and rapid operation of the protection, demonstrating good adaptability and reliability, and enhancing the distribution network’s capacity to accommodate distributed PV.

The proposed protection setting optimization method has achieved satisfactory performance in distribution networks with a single PV source. Future work will focus on its application in networks with multiple PV systems.