Type-2 Fuzzy C-Means-Based Clustering-Decomposed Coordination of Directional Overcurrent Relays

Abstract

Optimal coordination of directional overcurrent relays (DOCRs) in medium-to-large power systems constitutes a computationally demanding, mixed-integer, nonlinear optimisation problem whose complexity escalates rapidly with system size, making the simultaneous minimisation of relay operating time and computational cost a critical open challenge. This study presents a two-level hierarchical framework in which Type-2 Fuzzy C-Means (T2FCM) clustering partitions 226 fault scenarios into subproblems at the upper level, while the Hybrid Fractional Entropy Evolution (HFEE) algorithm independently optimises relay settings for each cluster at the lower level. HFEE integrates fractional-order velocity updates—derived from the Grünwald–Letnikov formulation—with a Shannon entropy diversity-control mechanism to prevent premature convergence. T2FCM captures inherent fault-current uncertainty through interval-valued type-2 fuzzy memberships, yielding more robust cluster assignments near protection-zone boundaries than crisp partitioning methods. The framework is validated on the extended IEEE 30-bus system. An ablation study demonstrates that standalone HFEE achieves a 29.19% improvement in $T_{\mathrm{op}}$ over the prior best-reported result; however, a comprehensive parameter sweep over cluster counts $K\in \{2,\dots ,8\}$ and fractional orders $\alpha \in \{0.1,\dots ,0.9\}$ across 50 independent runs per configuration shows that the proposed clustering-decomposed method achieves 3.68–66.67% lower wall-clock computation time while maintaining zero CTI violations across all active relay pairs. The communicationless, entirely offline framework demonstrates scalability for simultaneous sub-transmission and distribution protection coordination and offers a practically deployable strategy for modern power networks.

1. Introduction

In modern power systems, directional overcurrent relays (DOCRs) form the primary line of defence for isolating faulted segments while preserving the continuity of supply in healthy zones. These relays detect fault current magnitude and direction, making their coordinated operation fundamental to network stability and reliability [1,2,3]. DOCR coordination requires the simultaneous determination of optimal time multiplier settings (TMS) and plug settings (PS) so that each primary relay trips before its designated backup by a prescribed coordination time interval (CTI), while minimising total relay operating time.

The problem grows considerably more demanding in large-scale networks. The extended IEEE 30-bus system—a representative sub-transmission/distribution benchmark—comprises 81 DOCRs protecting 41 lines and 4 transformers across both 132 kV and 33 kV voltage levels [4,5]. With 162 decision variables (81 TMS and 81 PS) and 226 primary-–backup coordination constraints, the resulting mixed-integer nonlinear program is intractable for purely analytical methods [6].

Conventional solution strategies span deterministic mathematical programming, including sequential quadratic programming (SQP) [7] and mixed-integer linear programming [8], and population-based metaheuristics such as particle swarm optimisation (PSO) [9], gravitational search algorithm (GSA) [10], differential evolution [11,12], firefly algorithm [13], and cuckoo search [14]. Mathematical programming methods offer convergence guarantees but become computationally prohibitive as problem size increases and are prone to entrapment in local optima when discrete PS values are handled alongside continuous TMS variables [15]. Population-based metaheuristics scale better, yet suffer from premature convergence in high-dimensional search spaces [6].

A critical limitation shared by the majority of prior studies is that they are validated exclusively on the reduced 38-relay configuration of the IEEE 30-bus system or on networks with fewer than 120 coordination constraints [7,12,16]. Methods applied to the full 81-relay configuration with 226 constraints are scarce; Abdelaziz et al. [5] solved a linear programming formulation, and Bayazidi et al. [4] employed HEPSO and TFWO, achieving 35.73 s on the extended system while emphasising the practical importance of discrete relay settings. Neither study, however, employs clustering-based decomposition, and no prior work has validated T2FCM clustering on a system with more than 200 coordination constraints.

A complementary research direction has pursued decomposition strategies, partitioning large coordination problems into tractable subproblems. Ojaghi and Mohammadi [17] demonstrated that clustering can reduce the number of distinct setting groups required for adaptive relay coordination. Zand et al. [18] applied clustering-based zero-miscoordination techniques to group network topologies under N-2 contingencies. However, these approaches rely on crisp clustering assignments, which force fault scenarios near protection-zone boundaries into a single cluster and cannot model the measurement uncertainty inherent in fault current data.

Type-2 Fuzzy C-Means (T2FCM) [19] overcomes this limitation through interval-valued memberships $[{\underline{u}}_{ij},{\overline{u}}_{ij}]$, whose footprint of uncertainty (FOU) captures the ambiguity in scenario-cluster assignment. The authors recently introduced the Hybrid Fractional Entropy Evolution (HFEE) framework. In [20,21], a fractional PSO–GSA with entropy evolution was validated on small IEEE 3-bus and 8-bus systems, achieving approximately 9–12% improvements in operating time over the base PSO–GSA. A subsequent paper [22] extended the framework to IEEE 3-, 8-, and 15-bus networks with 15–26% reductions. Each of those studies, however, optimised the full relay network as a single monolithic problem, which becomes increasingly intractable as system size grows.

This work extends the HFEE framework by incorporating T2FCM as a decomposition stage, creating a two-level hierarchical optimisation architecture for large-scale DOCR coordination. An ablation study against monolithic HFEE without clustering isolates the contribution of the T2FCM decomposition stage and quantifies the resulting trade-off between operational efficiency and computational scalability. The principal contributions are as follows:

• The HFEE framework [20] is implemented and validated on the extended IEEE 30-bus system (81 relays, 226 pairs)—a scale not addressed in prior HFEE studies, which were confined to smaller networks of at most 15 buses [22]. Monolithic HFEE achieves a mean $T_{\mathrm{op}}$ of 25.30 s, a 29.19% improvement over the prior best result of 35.73 s [4].
• To the best of the authors’ knowledge, this is the first application of T2FCM clustering to DOCR coordination. A two-level hierarchical framework is introduced in which T2FCM partitions the fault scenarios at the upper level, and HFEE independently optimises relay settings for each cluster at the lower level.
• An ablation study compares T2FCM+HFEE against monolithic HFEE, demonstrating a 3.68–66.67% reduction in wall-clock computation time through clustering decomposition, at the cost of a higher $T_{\mathrm{op}}$.
• Validation on the extended IEEE 30-bus benchmark confirms zero CTI violations across all active relay pairs, with statistical reliability established over 50 independent runs.

The remainder of this paper is organised as follows. Section 2 reviews the relevant literature. Section 3 states the coordination mathematical model. Section 4 describes the T2FCM and HFEE methodology. Section 5 presents the test system and experimental setup. Section 6 reports simulation results and discussion. Section 7 concludes the study.

2. Literature Review

2.1. Evolution of DOCR Coordination Methods

The coordination of DOCRs has evolved significantly over the past two decades, though most advances remain confined to small-scale benchmark systems. Birla et al. [7] applied SQP to the standard 38-relay IEEE 30-bus system, achieving 116.50 s. While SQP guarantees local optimality, it is sensitive to the initial point and cannot handle discrete PS variables without reformulation, a significant practical limitation. Singh et al. [12] introduced an informative differential evolution algorithm with continuous variable relaxation on the same 38-relay configuration, reducing the operating time to 65.40 s, but the continuous relaxation of PS values yields settings that may not correspond to physically realisable relay tap positions. Korashy et al. [16] applied enhanced equilibrium optimisers on the 38-relay system, achieving 49.30 s and 59.01 s; however, the reduction from 226 to fewer constraints means the coordination problem is significantly less constrained than the full system (detailed in

Table 1. Comparative analysis of DOCR coordination methods on the IEEE 30-bus system.

Reference	Year	${\mathit{T}}_{\mathbf{op}}$ (s)	Relays	Technique	$\ge 120$ Constraints	Clustering
Birla et al. [7]	2006	116.5	38	SQP	×	×
Singh et al. [12]	2014	65.4	38	IDA-Cont.	×	×
Korashy et al. [16]	2024	49.3/59	38	EEO/EO	×	×
Abdelaziz et al. [5]	2002	—	81	LP	√	×
Labrador et al. [23]	2019	150/120	68	ACO/H-ACO	√	×
Bayazidi et al. [4]	2024	35.73	81	HEPSO/TFWO	√	×
Ayvaz [24]	2024	30.1	38	EWSO	×	√
Zand et al. [18]	2023	52.33	38	FA+K-means	×	√
Al-Bhadely and Inan [25]	2024	76.2	38	H-GA-SQP	×	√
This study: Non-Clustered	2026	25.3	81	HFEE	√	×
This study: Clustered	2026	61.54	81	T2FCM+HFEE	√	√

“—” = not reported in original study; √ = satisfied. × = not satisfied.

).

Among studies on the 81-relay extended system, Abdelaziz et al. [5] solved a linear programming (LP) formulation, providing an early demonstration of the problem’s scale. Labrador Rivas et al. [23] extended the problem to 68 relays and applied ant colony optimisation (ACO) and hybrid ACO, obtaining operating times of 150.3 s and 120.8 s respectively; neither method employs decomposition, and neither scales to the full 226-constraint problem. Bayazidi et al. [4] applied HEPSO and TFWO to the full 81-relay system, reporting 35.73 s with careful attention to discrete variable treatment. Although this represents the state of the art for monolithic single-scenario formulations on the extended system, the approach solves one optimisation per relay pair without simultaneously enforcing all 226 constraints, and no clustering-based decomposition is employed.

2.2. Clustering-Based Decomposition in Protection Coordination

Clustering-based decomposition offers a practical path to tackling large coordination problems by dividing them into independently solvable subproblems. Ojaghi and Mohammadi [17] were among the first to demonstrate that K-means clustering could reduce the number of distinct relay-setting groups required for adaptive coordination. Their work, however, targets the setting-group assignment problem rather than fault-scenario decomposition, and K-means imposes hard cluster boundaries that are poorly suited to scenarios near protection-zone boundaries. Zand et al. [18] employed K-means and contingency-based zero-miscoordination techniques under N-2 contingency scenarios for a 38-relay system; the use of crisp K-means partitioning means that ambiguous scenarios near zone boundaries receive forced assignments, potentially disrupting coordination chains. Ghadiri and Mazlumi [26] applied self-organising maps to cluster operational scenarios in microgrids, demonstrating the superior topology-preserving properties of neural-network clustering over conventional partitioning; however, their work does not address DOCR settings optimisation.

A common limitation of all existing clustering-based protection studies is that none have been demonstrated on a system with more than 200 coordination constraints. Furthermore, all prior methods employ crisp or type-1 fuzzy partitioning, which cannot model the measurement uncertainty inherent in fault current data from varying network topologies and generation profiles.

2.3. Type-2 Fuzzy C-Means for DOCRs

Standard Fuzzy C-Means (FCM) [27] assigns each scenario a soft membership grade $u_{ij}\in [0,1]$ to each cluster, partially addressing the hard-boundary limitation of K-means. However, the membership values themselves are crisp real numbers and cannot represent uncertainty about the grade itself, i.e., the imprecision in determining how strongly a boundary scenario belongs to a given cluster.

T2FCM [19] overcomes this by replacing the scalar $u_{ij}$ with an interval $[{\underline{u}}_{ij},{\overline{u}}_{ij}]$, whose width—the FOU—encapsulates the range of plausible membership values. For DOCR coordination, this interval-valued representation offers four specific advantages:

• Measurement uncertainty absorption.
  Fault current magnitudes depend on network topology, generation dispatch, and measurement precision. The FOU absorbs these variations, preventing distorted cluster assignments [28].
• Protection-zone boundary handling. Scenarios near the boundary between two protection zones exhibit characteristics of both zones. T2FCM maintains partial membership in both clusters, preventing the forced misassignment that breaks coordination chains across backup paths [29].
• Noise robustness. The interval-valued memberships dampen the influence of atypical fault scenarios on cluster centroids, unlike K-means, which is highly sensitive to outliers [28].
• Richer cluster confidence information. Wide FOU intervals (${\overline{u}}_{ij}-{\underline{u}}_{ij}$) identify genuinely ambiguous scenarios, flagging potential coordination vulnerabilities for protection engineers.

3. DOCR Coordination Mathematical Model

3.1. Operating Time Formulation

The operating time of a normal inverse (NI) relay according to IEC 60255 [30] is

$$T_i={\mathrm{TMS}}_i\times {\displaystyle \frac{\gamma }{{\left(I_f/I_p\right)}^{\beta }-1}}$$

(1)

where $\gamma =0.14$ and $\beta =0.02$ for the NI characteristic, $I_f$ is the fault current seen by relay i, and $I_p=\mathrm{CTR}\times \mathrm{PS}$ is the pickup current.

3.2. Objective Function

The total relay operating time to be minimised is

$$T_{\mathrm{op}}={\displaystyle \sum _{s=1}^S}\sum _{r\in R_{\mathrm{primary}}}{T}_{r,s}$$

(2)

where S is the number of fault scenarios and $R_{\mathrm{primary}}$ is the set of primary relays.

3.3. Coordination Constraints

For each primary–backup relay pair, the backup relay must operate at least CTI seconds after the primary:

$$T_{b,j}-T_{p,i}\ge \mathrm{CTI},i=1,\dots ,n_r;j=1,\dots ,k_i$$

(3)

A CTI of 0.3 s is adopted for the clustering study (consistent with standard practice [31]), and 0.2 s for the non-clustered ablation baseline.

3.4. Boundary Constraints

Following the MiCOM P127 relay specifications [32]:

$$\begin{array}{cc}\hfill 0.10\le {\mathrm{TMS}}_i& \le 1.00\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill 0.50\le {\mathrm{PS}}_i& \le 2.50\hfill \end{array}$$

(5)

3.5. Entropy-Augmented Fitness Function

To promote balanced relay load distribution and to extend earlier entropy-based formulations [20,21] to the decomposed setting, Shannon entropy is incorporated into the fitness function:

$$F=\underset{\mathrm{total}\mathrm{operating}\mathrm{time}}{\underbrace{\sum _{s=1}^S\sum _{r\in R_{\mathrm{primary}}}{T}_{r,s}}}+\underset{\mathrm{CTI}\mathrm{violation}\mathrm{penalty}}{\underbrace{{\lambda }_{\mathrm{penalty}}\sum _{p=1}^P{\left({\mathrm{CTI}}_p^{\mathrm{viol}}\right)}^2}}-\underset{\mathrm{entropy}\mathrm{reward}}{\underbrace{{\lambda }_{H}H\left(X\right)}}$$

(6)

where ${\mathrm{CTI}}_p^{\mathrm{viol}}=max(0,{\mathsf{\Delta }}_{min}-{\mathrm{CTI}}_p)$ is the shortfall of pair p below the minimum coordination interval, with ${\lambda }_{\mathrm{penalty}}={10}^5$ and ${\lambda }_H=0.05$.

Two distinct entropy quantities appear in this work. The relay-time entropy $H\left(X\right)$ in Equation (6) quantifies the spread of relay operating times and is computed as

$$H\left(X\right)=-{\displaystyle \sum _{i=1}^n}{p}_{i}log{p}_i,p_i={\displaystyle \frac{d_i}{{\sum }_{j=1}^n{d}_j}}$$

(7)

where $d_i$ is the operating time of relay i and n is the total number of primary relays. This is distinct from the swarm-diversity entropy $H_D\left(X\right)$ used within the HFEE velocity update (Section 4.3), where $d_i$ is the Euclidean distance from particle i to the global best position. Maximising $H\left(X\right)$ promotes uniform relay utilisation; maximising $H_D\left(X\right)$ sustains population diversity in the optimiser.

Parameter Selection for ${\lambda }_{\mathrm{penalty}}$ and ${\lambda }_H$

The weight ${\lambda }_{\mathrm{penalty}}={10}^5$ is selected so that a single CTI violation imposes a fitness penalty of ${\lambda }_{\mathrm{penalty}}\times {\mathsf{\Delta }}_{min}^2\approx {10}^5\times {\left(0.3\right)}^2=9000$, which exceeds the maximum feasible $T_{\mathrm{op}}$ by a factor of approximately 150. This effectively converts the CTI inequality into a hard constraint: any solution with a violation is so heavily penalised that it is never selected as a best solution, regardless of its operating time. Reducing ${\lambda }_{\mathrm{penalty}}$ below ${10}^4$ produced CTI violations in preliminary trials, while values above ${10}^6$ slowed convergence by producing near-flat fitness landscapes for feasible solutions.

The entropy reward weight ${\lambda }_H=0.05$ is chosen so that the entropy term contributes approximately 1–3% of the total fitness for a typical feasible solution ($H\left(X\right)\in [1.0,3.5]$, $T_{\mathrm{op}}\approx 60$–70 s), sufficient to bias solutions towards balanced relay load distribution without distorting the primary $T_{\mathrm{op}}$ objective. Values of ${\lambda }_H>0.2$ were found to over-regularise the solution, producing artificially uniform settings at the expense of coordination optimality. These selections are consistent with the entropy-weight choices reported in the authors’ prior studies [20,21].

4. Methodology

4.1. HFEE Without Clustering (Monolithic Baseline)

In the monolithic baseline, HFEE is applied directly to the full 226-scenario coordination problem without any decomposition stage. The input is the complete fault-current matrix $\mathbf{X}\in {\mathbb{R}}^{226\times 81}$, where $X_{i,j}$ denotes the RMS three-phase short-circuit current (A) seen by relay j under scenario i. The objective function of Equation (6) is evaluated over all 226 scenarios and all 226 primary–backup pairs simultaneously. The optimiser seeks a single set of 81 TMS and 81 PS values (162 decision variables in total) that minimises F for the full problem.

This formulation produces one globally consistent relay setting: each relay receives exactly one TMS value and one PS value. No cluster assignment or setting-group selection is required at deployment; the settings are loaded offline into relay non-volatile memory once and remain fixed during normal operation.

The monolithic HFEE baseline is solved for nine fractional orders $\alpha \in \{0.1,0.2,\dots ,0.9\}$, each over 50 independent runs, to characterise the statistical distribution of $T_{\mathrm{op}}$ and to provide a fair comparison with the clustered framework. The CTI threshold for the baseline is 0.2 s, consistent with the reference study [4], enabling direct numerical comparison.

4.2. T2FCM Clustering-Decomposed HFEE

The proposed framework adopts a two-level hierarchical optimisation architecture. A zero-miscoordination criterion is applied: any $(K,\alpha )$ configuration producing CTI violations is excluded from the final ranking, ensuring that only strictly feasible solutions are reported [18]. The overall methodology, of HFEE framework, is illustrated in Figure 1.

4.2.1. Input Data Representation

The input to the clustering stage is the fault-current matrix $\mathbf{X}\in {\mathbb{R}}^{226\times 81}$, where each row represents one fault scenario as an 81-dimensional feature vector of RMS short-circuit currents. The matrix structure is

$$X=\left[\begin{array}{ccccc}11936& 0& 0& \dots & 0\\ 11936& 0& 0& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1972\end{array}\right]$$

(8)

where $X_{i,j}$ is the RMS three-phase short-circuit current (A) seen by relay j during scenario i, computed at 1% of line length from each bus per IEC 60255 [30].

4.2.2. Standard FCM Foundation

Standard FCM [27] assigns each scenario a soft membership $u_{ij}\in [0,1]$ subject to ${\sum }_{j=1}^K{u}_{ij}=1$, minimising

$$J_m={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^K}{u}_{ij}^m{\parallel {\mathbf{x}}_i-{\mathbf{v}}_j\parallel }^2$$

(9)

where $m>1$ is the fuzzification parameter.

4.2.3. T2FCM Extension

T2FCM [19] replaces the scalar $u_{ij}$ with an interval $[{\underline{u}}_{ij},{\overline{u}}_{ij}]$:

$${\overline{u}}_{ij}={\displaystyle \frac{1}{{\displaystyle {\displaystyle \sum _{l=1}^K}{\left(\frac{\parallel {\mathbf{x}}_i-{\mathbf{v}}_j\parallel }{\parallel {\mathbf{x}}_i-{\mathbf{v}}_l\parallel }\right)}^{2/(m_1-1)}}}}$$

(10)

$${\underline{u}}_{ij}={\displaystyle \frac{1}{{\displaystyle {\displaystyle \sum _{l=1}^K}{\left(\frac{\parallel {\mathbf{x}}_i-{\mathbf{v}}_j\parallel }{\parallel {\mathbf{x}}_i-{\mathbf{v}}_l\parallel }\right)}^{2/(m_2-1)}}}}$$

(11)

with fuzzification exponents $m_1=2$ and $m_2=3$. The type-reduced membership for centroid computation is

$$u_{ij}^{*}={\displaystyle \frac{{\underline{u}}_{ij}+{\overline{u}}_{ij}}{2}}$$

(12)

and updated as:

$${\mathbf{v}}_j={\displaystyle \frac{{\sum }_{i=1}^n{\left(u_{ij}^{*}\right)}^m{\mathbf{x}}_i}{{\sum }_{i=1}^n{\left(u_{ij}^{*}\right)}^m}}$$

(13)

The algorithm iterates until the maximum membership change falls below ${10}^{-5}$ or 100 iterations are reached.

4.2.4. Setting Recombination and Deployment

A key distinction from the monolithic baseline is the per-cluster setting structure. T2FCM partitions the 226 fault scenarios into $K=6$ disjoint groups; HFEE is then applied independently to each group, producing $K=6$ complete sets of 81 TMS and 81 PS values—one set per cluster. Consequently, each relay stores $K=6$ (TMS, PS) pairs in its non-volatile memory, one for each cluster. At runtime, the relay independently selects the active pair based on its local fault-current measurement, comparing the measured current against the per-cluster pickup thresholds precomputed offline from the cluster centroids. No inter-relay communication is required: each relay selects its setting group autonomously from a local decision rule, without exchanging data with any other relay. This is consistent with the multi-group capability of commercial numeric relays; the MiCOM P127 supports up to eight pre-loaded setting groups switchable via programmable binary inputs or local logic [32]. In contrast, the monolithic HFEE produces only a single (TMS, PS) pair per relay, requiring no group selection at all.

4.2.5. Cluster Count Restriction Range

The cluster count is swept from $K=2$ to $K=8$. Below $K=6$, per-cluster scenario subsets are too large for the optimiser to reliably resolve all conflicting CTI constraints within the 100-iteration budget. Above $K=6$, wall-clock computation time increases proportionally, and the risk of splitting operationally coupled relay pairs across cluster boundaries rises, leading to CTI violations at $K=8$ for $\alpha \in \{0.1,0.9\}$. This behaviour is also discussed in Section 6, where the computation time–$T_{\mathrm{op}}$ trade-off shows diminishing returns beyond $K=6$. Furthermore, the MiCOM P127 supports a maximum of eight setting groups [32]; $K>8$ would exceed deployable hardware capacity. Detailed results for all $(K,\alpha )$ combinations are available at  https://github.com/Mubashar-3001/fpsogsa-ee-cluster (accessed on 10 May 2026). To complement the operational justification presented above, three established clustering validation indices were computed on the fault-current matrix $\mathbf{X}$ for $K\in \{2,\dots ,8\}$ using the hard cluster assignments derived from K-means (as a proxy for the T2FCM hard labels, following standard practice in fuzzy clustering evaluation [33]).

Table 2. Clustering validation indices for the fault-current matrix ($226\times 81$) over $K=2$–8.

K	Silhouette ↑	Davies–Bouldin ↓	Xie–Beni ↓
2	0.2449	2.3801	1.5364
3	0.2042	2.2191	1.5064
4	0.3115	1.5460	1.2458
5	0.0829	1.8317	1.6424
6	0.0306	1.8556	2.5007
7	0.1744	1.7965	1.5732
8	0.0818	1.8267	2.5558

reports the Silhouette index (higher is better), Davies–Bouldin index (lower is better), and Xie–Beni index (lower is better).

All three indices unanimously identify $K=4$ as the statistically optimal cluster count for the fault-current data, exhibiting the highest Silhouette score (0.3115), the lowest Davies–Bouldin index (1.5460), and the lowest Xie–Beni index (1.2458). The operational parameter sweep (Section 6.3), however, identifies $K=6$ as the configuration that minimises $T_{\mathrm{op}}$ with zero CTI violations. This discrepancy is physically meaningful and expected: $K=4$ produces the most internally cohesive clusters with the sharpest inter-cluster separation, but the coarser decomposition leaves each subproblem with a larger number of conflicting coordination constraints, making it harder for HFEE to find a feasible solution within the 100-iteration budget. $K=6$ achieves finer decomposition, reducing per-cluster constraint complexity at the cost of some cluster purity, and enables HFEE to converge reliably to zero-violation solutions.

A second important observation from Table 2 is that all Silhouette values are below 0.35, indicating that the fault-current scenarios do not form sharply separated natural clusters. This is physically consistent with the meshed network topology of the IEEE 30-bus system, where similar fault-current magnitudes appear across multiple protection zones. Crucially, this finding provides the strongest validation for the choice of T2FCM over K-means or standard FCM: when cluster boundaries are inherently ambiguous, crisp and type-1 fuzzy assignments are unreliable, and the interval-valued memberships of T2FCM are essential to capture the genuine uncertainty in scenario-cluster assignment [28,29].

4.3. Hybrid Fractional Entropy Evolution (HFEE) Optimizer

The HFEE framework integrates fractional calculus and Shannon entropy into the PSO–GSA optimisation scheme [20]. Fractional calculus was first incorporated into PSO–GSA in [22]; Shannon entropy integration (without fractional calculus) was later introduced in [21].

4.3.1. Fractional-Order Velocity Update

Incorporating the Grünwald–Letnikov fractional derivative of order $\alpha$ introduces memory effects into the velocity update [34,35]:

$$\begin{array}{cc}\hfill v_i^{n+1}& =\alpha v_i^n+\frac{1}{2}\alpha (1-\alpha )v_i^{n-1}+\frac{1}{6}\alpha (1-\alpha )(2-\alpha )v_i^{n-2}\hfill \\ \hfill & +\frac{1}{24}\alpha (1-\alpha )(2-\alpha )(3-\alpha )v_i^{n-3}\hfill \\ \hfill & +c_1{r}_1·a{c}_i\left(t\right)+c_2{r}_2·(G_{\mathrm{best}}-X_i\left(t\right))\hfill \end{array}$$

(14)

where $a{c}_i\left(t\right)$ is the gravitational acceleration from GSA [10]. When $\alpha =1$, Equation (14) reduces to standard PSO–GSA [36]. Lower values of $\alpha$ reduce the memory effect, encouraging broader exploration.

4.3.2. Shannon Entropy Integration

The swarm-diversity entropy $H_D\left(X\right)$ is computed at each iteration:

$$p_i={\displaystyle \frac{d_i}{{\sum }_{j=1}^n{d}_j}},H_D\left(X\right)=-{\displaystyle \sum _{i=1}^n}{p}_{i}log{p}_i$$

(15)

where $d_i$ is the Euclidean distance from particle i to $G_{\mathrm{best}}$. $H_D\left(X\right)$ modulates the velocity update:

$$v_i(t+1)=\omega v_i\left(t\right)+c_1{r}_1\left(1+H_D\left(X\right)\right)a{c}_i\left(t\right)+c_2{r}_2\left(G_{\mathrm{best}}-X_i\left(t\right)\right)$$

(16)

When $H_D\left(X\right)$ falls below threshold $H_{min}$, the worst-performing 20% of agents are reinitialised to restore diversity and prevent premature convergence.

4.4. Clustering-Decomposed HFEE Procedure

The complete procedure is summarised in Algorithm 1. Figure 1 illustrates the corresponding framework.

Algorithm 1: T2FCM Clustering-Decomposed HFEE for DOCR Coordination
Input: Fault-current matrix $(226\times 81)$, CTR, relay pairs, TMS/PS bounds, CTI = 0.3 s
Output: Optimised TMS and PS per relay; total $T_{\mathrm{op}}$; CTI violations
	Step 1: Data Preparation
1:	Load fault currents; standardise to zero mean, unit variance
	Step 2: T2FCM Clustering (Upper Level)
3:	for each $K\in \{2,3,\dots ,8\}$ do
4:	Apply T2FCM with $m_1=2$, $m_2=3$
5:	Compute interval memberships $[{\underline{u}}_{ij},{\overline{u}}_{ij}]$
6:	Assign scenarios to clusters via $u_{ij}^{*}$
7:	Merge clusters with fewer than 3 scenarios
8:	end for
	Step 3: Per-Cluster HFEE Optimisation (Lower Level) [20]
9:	for each $\alpha \in \{0.1,0.2,\dots ,0.9\}$ do
10:	for each cluster $C_j$ do
11:	Initialise 50 agents within TMS/PS bounds
12:	for iteration $=1$ to 100 do
13:	Evaluate fitness (Equation (6))
14:	Compute gravitational mass, force, acceleration
15:	Compute $H_D\left(X\right)$ (Equation (15))
16:	Update velocity (Equations (14) and (16))
17:	Update positions; enforce bounds
18:	Update local and global best
19:	if $H_D\left(X\right)<H_{min}$: reinitialise worst 20% of agents
20:	end for
21:	end for
22:	end for
	Step 4: Aggregation and Validation
23:	Combine per-cluster solutions into the full 81-relay setting vectors
24:	Evaluate system-wide $T_{\mathrm{op}}$ and CTI compliance
25:	Record $T_{\mathrm{op}}$, violations, entropy, per-relay times
26:	Identify the best feasible $(K,\alpha )$ configuration

5. Test System and Experimental Setup

5.1. IEEE 30-Bus Extended Network

The extended IEEE 30-bus network is a sub-transmission/ distribution system consisting of 2 generators, 4 synchronous condensers, 13 lines at 132 kV, 4 transformers (132/33 kV), and 23 lines at 33 kV [4,37]. The complete system is protected by 81 DOCRs with 226 primary–backup coordination pairs. The single-line diagram, relay locations, and inter-relay coordination links are presented in [4].The corresponding CTRs are provided in

Table 3. Current transformer ratios for the IEEE 30-bus extended system [4,5].

CT Location	CT Ratio	Step Size (A)
All lines (R1–R70, R71)	1000/5	10
Transformer primary (R72, R74, R77, R79)	500/5	5
Transformer secondary (R73, R75, R76, R78, R80, R81)	2000/5	20

, and the fault currents are in Table 4. The relay pairs and three-phase near-end RMS fault currents follow the IEC 60255 standard, as detailed in [4,5].

5.2. Current Transformer Ratio

The CTR directly affects optimisation outcomes because it determines the discrete pickup current values and consequently shifts the relay operating-time curve. The pickup current $I_p=\mathrm{CTR}\times \mathrm{PS}$ appears in the PSM ratio $I_f/I_p$; any change in CTR therefore alters relay sensitivity and operating time. Accurate CTR values from relay specifications are essential for producing implementable settings [4,6].

The CTRs used follow [4] and are summarised in Table 3. Three CT groups are applied: line relays (R1–R70) use a 1000/5 A ratio; transformer primary relays use 500/5 A; and transformer secondary relays use 2000/5 A. The pickup-current step size on the primary side is $\mathsf{\Delta }{I}_p=0.01\times 5\times \mathrm{CTR}=0.05\times \mathrm{CTR}$ [4]. This discrete quantisation is enforced throughout optimisation to ensure that all optimised PS values correspond to physically realisable tap positions on commercial relays, preventing miscoordination caused by impractical inter-tap settings.

5.3. Scenario Configuration and Excluded Pairs

Network topology changes are considered when determining pickup currents to prevent false tripping under transformer outage conditions. Five scenarios are evaluated: (1) all transformers in service, and (2)–(5) each of the four transformers individually out of service. For each scenario, the maximum load current across all conditions determines the pickup current for each relay. After verification, 226 valid fault scenarios are obtained.

Seven primary/backup pairs in the extended IEEE 30-bus system are structurally constrained such that no TMS or PS assignment can satisfy CTI $\ge 0.2$ s, owing to the physical fault-current distribution. These pairs fall into two categories. Bidirectional pairs (R2→R3, R4→R1, R32→R33, R76→R77) arise from the meshed topology, where the same relay alternates between primary and backup roles depending on fault location. The directional element of the DOCR resolves this in practice by blocking operation for reverse-direction faults, ensuring the conflicting CTI constraints never apply simultaneously. Transformer backup pairs (R15→R74, R75→R79, R78→R75) involve transformer-connected relays where the fault-current distribution inherently favours the backup relay. These are addressed through dedicated current-differential protection, which provides faster and more selective operation than overcurrent time-grading [38]. Accordingly, these seven pairs are excluded from the CTI coordination metric, leaving 219 active coordination pairs, consistent with standard protection engineering practice [39].

5.4. Relay Pairs and Fault Currents

Table 4 presents a representative subset of the 226 primary–backup relay pairs and their three-phase near-end RMS fault currents. The matrix $\mathbf{X}$, used as input to the clustering stage, is structured as in Equation (8), where $X_{i,j}$ is the RMS three-phase short-circuit current (A) seen by relay j during scenario i.

Table 4. IEEE  30-bus extended system relay pairs and fault currents [4,40].

No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$	No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$	No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$
1	R1	R4	11,936	1399	77	R23	R22	7795	1602	153	R46	R52	2880	2880
2	R1	R6	11,936	1052	78	R23	R76	7795	1181	154	R47	R39	8509	5472
3	R2	R3	7857	2090	79	R23	R78	7795	309	155	R47	R43	8509	1402
4	R2	R8	7857	1100	80	R23	R79	7795	10,345	156	R47	R46	8509	1654
5	R2	R10	7857	1104	81	R24	R25	1303	1049	157	R48	R60	2825	2825
6	R2	R12	7857	1178	82	R24	R81	1303	1021	158	R49	R41	4825	4825
7	R3	R2	11,936	1399	83	R25	R21	5660	3545	159	R50	R27	7517	7517
8	R3	R6	11,936	1052	84	R26	R23	3647	3396	160	R51	R45	3845	3845
9	R4	R1	7857	2090	85	R26	R81	3647	1008	161	R52	R54	3970	3970
10	R4	R8	7857	1100	86	R27	R30	12,311	1166	162	R53	R51	2808	2808
11	R4	R10	7857	1104	87	R27	R32	12,311	655	163	R54	R29	4863	4863
12	R4	R12	7857	1178	88	R27	R34	12,311	655	164	R55	R31	10,407	5203
13	R5	R2	12,156	1403	89	R27	R36	12,311	369	165	R55	R33	10,407	5203
14	R5	R4	12,156	1403	90	R27	R75	12,311	748	166	R56	R35	4226	2225
15	R6	R14	3757	3757	91	R27	R77	12,311	876	167	R56	R58	4226	2028
16	R7	R1	8737	2078	92	R27	R79	12,311	9287	168	R57	R35	10,064	2212
17	R7	R3	8737	2078	93	R28	R49	2799	2799	169	R57	R55	10,064	7851
18	R7	R10	8737	1085	94	R29	R28	13,053	2005	170	R58	R59	3750	2123
19	R7	R12	8737	1148	95	R29	R32	13,053	647	171	R58	R62	3750	1628
20	R8	R13	5764	1608	96	R29	R34	13,053	647	172	R59	R47	4012	4012
21	R8	R16	5764	3297	97	R29	R36	13,053	365	173	R60	R57	6036	4419
22	R8	R73	5764	184	98	R29	R75	13,053	741	174	R60	R62	6036	1620
23	R8	R74	5764	10,009	99	R29	R77	13,053	868	175	R61	R57	6514	4407
24	R9	R1	8714	2073	100	R29	R79	13,053	9202	176	R61	R59	6514	2107
25	R9	R3	8714	2073	101	R30	R53	2398	2398	177	R62	R64	2754	2754
26	R9	R8	8714	1078	102	R31	R28	13,666	2036	178	R63	R61	2749	2749
27	R9	R12	8714	1152	103	R31	R30	13,666	1171	179	R64	R80	4219	1055
28	R10	R18	3655	1529	104	R31	R34	13,666	597	180	R65	R63	5987	1944
29	R11	R1	8661	2079	105	R31	R36	13,666	358	181	R65	R68	5987	17
30	R11	R3	8661	2079	106	R31	R75	13,666	750	182	R65	R80	5987	1045
31	R11	R8	8661	10,711	107	R31	R77	13,666	879	183	R66	R70	836	836
32	R11	R10	8661	1082	108	R31	R79	13,666	9313	184	R67	R63	5942	1932
33	R12	R15	6585	2454	109	R32	R33	7784	5157	185	R67	R66	5942	24
34	R12	R20	6585	1150	110	R32	R56	7784	2651	186	R67	R80	5942	1039
35	R12	R22	6585	1590	111	R33	R28	13,666	2036	187	R68	R69	1102	1102
36	R12	R24	6585	340	112	R33	R30	13,666	1171	188	R69	R65	2124	2124
37	R12	R76	6585	1165	113	R33	R32	13,666	597	189	R70	R67	1589	1589
38	R12	R78	6585	301	114	R33	R36	13,666	358	190	R71	R61	5415	2725
39	R12	R79	6585	10,185	115	R33	R75	13,666	750	191	R71	R64	5415	2736
40	R13	R5	2085	2085	116	R33	R77	13,666	879	192	R72	R7	6557	1532
41	R14	R7	5761	1524	117	R33	R79	13,666	9313	193	R72	R13	6557	1648
42	R14	R16	5761	3364	118	R34	R31	7784	5157	194	R72	R16	6557	3379
43	R14	R73	5761	190	119	R34	R56	7784	2651	195	R73	R38	4232	459
44	R14	R74	5761	10,178	120	R35	R28	13,793	2018	196	R73	R40	4232	1793
45	R15	R7	4037	1525	121	R35	R30	13,793	1161	197	R73	R42	4232	1993
46	R15	R13	4037	1642	122	R35	R32	13,793	602	198	R75	R11	7935	1467
47	R15	R73	4037	180	123	R35	R34	13,793	602	199	R75	R15	7935	2529
48	R15	R74	4037	10,205	124	R35	R75	13,793	744	200	R75	R20	7935	1184
49	R16	R11	5672	1457	125	R35	R77	13,793	872	201	R75	R22	7935	1630
50	R16	R20	5672	1178	126	R35	R79	13,793	9234	202	R75	R24	7935	348
51	R16	R22	5672	1623	127	R36	R55	9773	7803	203	R75	R78	7935	307
52	R16	R24	5672	346	128	R36	R58	9773	2012	204	R75	R79	7935	10,438
53	R16	R76	5672	1186	129	R37	R40	14,020	1712	205	R76	R28	11,417	2041
54	R16	R78	5672	312	130	R37	R42	14,020	1951	206	R76	R30	11,417	1175
55	R16	R79	5672	10,398	131	R37	R72	14,020	1342	207	R76	R32	11,417	660
56	R17	R9	3838	1684	132	R37	R74	14,020	15,205	208	R76	R34	11,417	660
57	R18	R19	2989	2989	133	R38	R44	3101	3101	209	R76	R36	11,417	372
58	R19	R11	6965	1448	134	R39	R38	12,873	432	210	R76	R77	11,417	881
59	R19	R15	6965	2498	135	R39	R42	12,873	1968	211	R76	R79	11,417	9334
60	R19	R22	6965	11,611	136	R39	R72	12,873	1356	212	R77	R11	7445	1467
61	R19	R24	6965	344	137	R39	R74	12,873	15,366	213	R77	R15	7445	2529
62	R19	R76	6964	1181	138	R40	R43	4774	1400	214	R77	R20	7445	1184
63	R19	R78	6965	303	139	R40	R46	4774	1666	215	R77	R22	7445	1630
64	R19	R79	6965	10,316	140	R40	R48	4774	1712	216	R77	R24	7445	348
65	R20	R17	1821	1821	141	R41	R38	12,607	450	217	R77	R76	7445	1194
66	R21	R11	6563	1459	142	R41	R40	12,607	1758	218	R78	R28	7831	2042
67	R21	R15	6563	2516	143	R41	R72	12,607	1349	219	R78	R30	7831	1175
68	R21	R20	6563	1177	144	R41	R74	12,607	15,297	220	R78	R32	7831	660
69	R21	R24	6563	337	145	R42	R50	3882	3882	221	R78	R34	7831	660
70	R21	R76	6563	1188	146	R43	R37	3944	3944	222	R78	R36	7831	372
71	R21	R78	6563	306	147	R44	R39	8785	5443	223	R78	R75	7831	3010
72	R21	R79	6563	10,383	148	R44	R46	8785	1654	224	R80	R23	4512	3459
73	R22	R26	2777	607	149	R44	R48	8785	1700	225	R80	R25	4512	1053
74	R23	R11	7795	1453	150	R45	R39	8542	5467	226	R81	R63	1972	1972
75	R23	R15	7795	2505	151	R45	R43	8542	1400
76	R23	R20	7795	1173	152	R45	R48	8542	1698

Table 4. IEEE  30-bus extended system relay pairs and fault currents [4,40].

No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$	No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$	No.	Pri.	Bkp.	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{P}}$	${\mathbf{I}}_{\mathbf{f}}^{\mathbf{B}}$
1	R1	R4	11,936	1399	77	R23	R22	7795	1602	153	R46	R52	2880	2880
2	R1	R6	11,936	1052	78	R23	R76	7795	1181	154	R47	R39	8509	5472
3	R2	R3	7857	2090	79	R23	R78	7795	309	155	R47	R43	8509	1402
4	R2	R8	7857	1100	80	R23	R79	7795	10,345	156	R47	R46	8509	1654
5	R2	R10	7857	1104	81	R24	R25	1303	1049	157	R48	R60	2825	2825
6	R2	R12	7857	1178	82	R24	R81	1303	1021	158	R49	R41	4825	4825
7	R3	R2	11,936	1399	83	R25	R21	5660	3545	159	R50	R27	7517	7517
8	R3	R6	11,936	1052	84	R26	R23	3647	3396	160	R51	R45	3845	3845
9	R4	R1	7857	2090	85	R26	R81	3647	1008	161	R52	R54	3970	3970
10	R4	R8	7857	1100	86	R27	R30	12,311	1166	162	R53	R51	2808	2808
11	R4	R10	7857	1104	87	R27	R32	12,311	655	163	R54	R29	4863	4863
12	R4	R12	7857	1178	88	R27	R34	12,311	655	164	R55	R31	10,407	5203
13	R5	R2	12,156	1403	89	R27	R36	12,311	369	165	R55	R33	10,407	5203
14	R5	R4	12,156	1403	90	R27	R75	12,311	748	166	R56	R35	4226	2225
15	R6	R14	3757	3757	91	R27	R77	12,311	876	167	R56	R58	4226	2028
16	R7	R1	8737	2078	92	R27	R79	12,311	9287	168	R57	R35	10,064	2212
17	R7	R3	8737	2078	93	R28	R49	2799	2799	169	R57	R55	10,064	7851
18	R7	R10	8737	1085	94	R29	R28	13,053	2005	170	R58	R59	3750	2123
19	R7	R12	8737	1148	95	R29	R32	13,053	647	171	R58	R62	3750	1628
20	R8	R13	5764	1608	96	R29	R34	13,053	647	172	R59	R47	4012	4012
21	R8	R16	5764	3297	97	R29	R36	13,053	365	173	R60	R57	6036	4419
22	R8	R73	5764	184	98	R29	R75	13,053	741	174	R60	R62	6036	1620
23	R8	R74	5764	10,009	99	R29	R77	13,053	868	175	R61	R57	6514	4407
24	R9	R1	8714	2073	100	R29	R79	13,053	9202	176	R61	R59	6514	2107
25	R9	R3	8714	2073	101	R30	R53	2398	2398	177	R62	R64	2754	2754
26	R9	R8	8714	1078	102	R31	R28	13,666	2036	178	R63	R61	2749	2749
27	R9	R12	8714	1152	103	R31	R30	13,666	1171	179	R64	R80	4219	1055
28	R10	R18	3655	1529	104	R31	R34	13,666	597	180	R65	R63	5987	1944
29	R11	R1	8661	2079	105	R31	R36	13,666	358	181	R65	R68	5987	17
30	R11	R3	8661	2079	106	R31	R75	13,666	750	182	R65	R80	5987	1045
31	R11	R8	8661	10,711	107	R31	R77	13,666	879	183	R66	R70	836	836
32	R11	R10	8661	1082	108	R31	R79	13,666	9313	184	R67	R63	5942	1932
33	R12	R15	6585	2454	109	R32	R33	7784	5157	185	R67	R66	5942	24
34	R12	R20	6585	1150	110	R32	R56	7784	2651	186	R67	R80	5942	1039
35	R12	R22	6585	1590	111	R33	R28	13,666	2036	187	R68	R69	1102	1102
36	R12	R24	6585	340	112	R33	R30	13,666	1171	188	R69	R65	2124	2124
37	R12	R76	6585	1165	113	R33	R32	13,666	597	189	R70	R67	1589	1589
38	R12	R78	6585	301	114	R33	R36	13,666	358	190	R71	R61	5415	2725
39	R12	R79	6585	10,185	115	R33	R75	13,666	750	191	R71	R64	5415	2736
40	R13	R5	2085	2085	116	R33	R77	13,666	879	192	R72	R7	6557	1532
41	R14	R7	5761	1524	117	R33	R79	13,666	9313	193	R72	R13	6557	1648
42	R14	R16	5761	3364	118	R34	R31	7784	5157	194	R72	R16	6557	3379
43	R14	R73	5761	190	119	R34	R56	7784	2651	195	R73	R38	4232	459
44	R14	R74	5761	10,178	120	R35	R28	13,793	2018	196	R73	R40	4232	1793
45	R15	R7	4037	1525	121	R35	R30	13,793	1161	197	R73	R42	4232	1993
46	R15	R13	4037	1642	122	R35	R32	13,793	602	198	R75	R11	7935	1467
47	R15	R73	4037	180	123	R35	R34	13,793	602	199	R75	R15	7935	2529
48	R15	R74	4037	10,205	124	R35	R75	13,793	744	200	R75	R20	7935	1184
49	R16	R11	5672	1457	125	R35	R77	13,793	872	201	R75	R22	7935	1630
50	R16	R20	5672	1178	126	R35	R79	13,793	9234	202	R75	R24	7935	348
51	R16	R22	5672	1623	127	R36	R55	9773	7803	203	R75	R78	7935	307
52	R16	R24	5672	346	128	R36	R58	9773	2012	204	R75	R79	7935	10,438
53	R16	R76	5672	1186	129	R37	R40	14,020	1712	205	R76	R28	11,417	2041
54	R16	R78	5672	312	130	R37	R42	14,020	1951	206	R76	R30	11,417	1175
55	R16	R79	5672	10,398	131	R37	R72	14,020	1342	207	R76	R32	11,417	660
56	R17	R9	3838	1684	132	R37	R74	14,020	15,205	208	R76	R34	11,417	660
57	R18	R19	2989	2989	133	R38	R44	3101	3101	209	R76	R36	11,417	372
58	R19	R11	6965	1448	134	R39	R38	12,873	432	210	R76	R77	11,417	881
59	R19	R15	6965	2498	135	R39	R42	12,873	1968	211	R76	R79	11,417	9334
60	R19	R22	6965	11,611	136	R39	R72	12,873	1356	212	R77	R11	7445	1467
61	R19	R24	6965	344	137	R39	R74	12,873	15,366	213	R77	R15	7445	2529
62	R19	R76	6964	1181	138	R40	R43	4774	1400	214	R77	R20	7445	1184
63	R19	R78	6965	303	139	R40	R46	4774	1666	215	R77	R22	7445	1630
64	R19	R79	6965	10,316	140	R40	R48	4774	1712	216	R77	R24	7445	348
65	R20	R17	1821	1821	141	R41	R38	12,607	450	217	R77	R76	7445	1194
66	R21	R11	6563	1459	142	R41	R40	12,607	1758	218	R78	R28	7831	2042
67	R21	R15	6563	2516	143	R41	R72	12,607	1349	219	R78	R30	7831	1175
68	R21	R20	6563	1177	144	R41	R74	12,607	15,297	220	R78	R32	7831	660
69	R21	R24	6563	337	145	R42	R50	3882	3882	221	R78	R34	7831	660
70	R21	R76	6563	1188	146	R43	R37	3944	3944	222	R78	R36	7831	372
71	R21	R78	6563	306	147	R44	R39	8785	5443	223	R78	R75	7831	3010
72	R21	R79	6563	10,383	148	R44	R46	8785	1654	224	R80	R23	4512	3459
73	R22	R26	2777	607	149	R44	R48	8785	1700	225	R80	R25	4512	1053
74	R23	R11	7795	1453	150	R45	R39	8542	5467	226	R81	R63	1972	1972
75	R23	R15	7795	2505	151	R45	R43	8542	1400
76	R23	R20	7795	1173	152	R45	R48	8542	1698

5.5. Algorithm Control Parameters

Table 5. HFEE algorithm control parameters for the IEEE 30-bus extended system.

Parameter	Value
Decision variables (TMS, PS)	(81, 81) = 162 total
TMS range	$[0.10,1.00]$
PS range	$[0.50,2.50]$
Maximum iterations	100
Independent runs per configuration	50
Population size	50
Fractional orders $\alpha$	$\{0.1,0.2,\dots ,0.9\}$
${\lambda }_{\mathrm{penalty}}$	${10}^5$
${\lambda }_H$	0.05
Acceleration coefficients $c_1,c_2$	2.0, 2.0
IEC 60255 minimum CTI	0.20 s
Non-clustered only
CTI	0.20 s
Total $\alpha$ configurations	9
Clustered only
CTI	0.30 s
Cluster counts K	$\{2,3,\dots ,8\}$
T2FCM fuzzifiers $(m_1,m_2)$	(2, 3)
Total $(K,\alpha )$ configurations	63

IEC minimum = 0.20 s; adopted CTI = 0.20 s (non-clustered) and 0.30 s (clustered) for stricter coordination margin.

summarises the control parameters used by the HFEE optimiser across all experiments. The TMS and PS bounds follow the MiCOM P127 specifications [32], ensuring that all optimised settings are directly deployable on commercial relays. The population size of 50 particles balances exploration capability and computational cost for the 162-dimensional search space. A total of 100 iterations per run and 50 independent runs per configuration ensures statistical reliability. The sweep over nine fractional orders and seven cluster counts yields 63 distinct clustered configurations, amounting to $63\times 50=3150$ optimisation runs.

All experiments were conducted in MATLAB R2018a on a system equipped with an Intel Core i7 processor (2.00 GHz) and 8 GB RAM. Wall-clock computation time was measured using MATLAB’s tic/toc facility averaged over 50 independent runs at the best configuration ($K=6$, $\alpha =0.1$) to obtain a reliable timing estimate. Reported computation times are wall-clock seconds and therefore reflect practical deployment cost rather than theoretical CPU-cycle counts.

6. Results and Discussion

6.1. Statistical Robustness and Sensitivity Analysis

To assess the reliability of both the clustered and non-clustered configurations, 50 independent runs were executed for each method.

For the proposed T2FCM+HFEE framework ($K=6$, $\alpha =0.1$), $T_{\mathrm{op}}$ ranged from a best of 59.79 s to a worst of 67.53 s, with a mean of 61.54 s, standard deviation of 3.04 s, coefficient of variation (CV) $=(3.04/61.54)\times 100=4.94\%$, and 95% confidence interval (CI) $=61.54\pm 0.84$ s (i.e., $[60.70,62.38]$ s, computed as $\overline{x}\pm 1.96\sigma /\sqrt{50}$).

For the non-clustered HFEE baseline, the best result across all $\alpha$ values was 22.85 s at $\alpha =0.7$, with a mean of 25.30 s, worst of 26.43 s, standard deviation of 0.51 s, CV $=(0.51/25.30)\times 100=2.02\%$, and 95% CI $=25.30\pm 0.14$ s ($[25.16,25.44]$ s). The substantially lower CV of the non-clustered variant (2.02% vs. 4.94%) reflects the simpler, single-level search landscape; however, this consistency comes at the cost of ignoring inter-scenario diversity and producing a single relay-setting vector that is not optimised across heterogeneous fault conditions.

A Wilcoxon signed-rank test and a Wilcoxon rank-sum test were applied to the two 50-run distributions; both returned $p<0.0001$, confirming that the difference in $T_{\mathrm{op}}$ between the clustered and non-clustered approaches is statistically significant at the 1% level. The convergence behaviour across the best, mean, and worst runs for both methods is illustrated in Figure 2 and Figure 3, and the complete 50-run distributions are compared in the box plot of Figure 4 and the frequency distribution of Figure 5. The shaded area between the maximum and minimum values indicates the bandwidth, and all solutions lie within it.

6.2. Ablation Study: Clustered vs. Non-Clustered HFEE

The ablation study isolates the contribution of the T2FCM decomposition stage by comparing two configurations on the same extended IEEE 30-bus system (81 relays, 226 pairs): monolithic HFEE (no clustering) and T2FCM+HFEE.

Operational performance: Monolithic HFEE achieves a mean $T_{\mathrm{op}}$ of 25.30 s, representing a 29.19% improvement over the prior best result of 35.73 s reported in [4]. T2FCM+HFEE achieves a mean $T_{\mathrm{op}}$ of 61.54 s, which is higher in absolute terms. This increase arises because the clustering decomposition optimises each cluster of scenarios independently: within each cluster, HFEE minimises the operating time for that cluster’s scenario subset rather than all 226 scenarios simultaneously, and the CTI threshold is set to the stricter 0.3 s (versus 0.2 s for the baseline). The total $T_{\mathrm{op}}$ is the sum of per-cluster optimal values, which is inevitably higher than the monolithic optimum.

Computational performance: Wall-clock computation time at $K=6$, $\alpha =0.1$ is $6.54\pm 0.19$ s (mean ± std, $N=50$), compared to 6.79 s for monolithic HFEE—a 3.68% reduction. For $K=2$, the computation time is 2.18 s, representing a 66.67% reduction relative to the $K=6$ reference point and a 67.9% reduction relative to the non-clustered baseline. The computational complexity is analysed in Section 6.5.

Practical advantages of clustering: Although the clustered formulation does not achieve the lowest absolute $T_{\mathrm{op}}$, it offers four practical advantages over the monolithic approach: decomposition scalability (the problem splits into K independent subproblems of $\approx 162/K$ variables each); parallel computation (K clusters solved concurrently, offering a theoretical K-fold speedup); multi-scenario robustness (226 simultaneous fault scenarios versus a single scenario per pair in [4]); and modular re-optimisation (only affected clusters need updating under network modifications).

Table 6. Ablation study: method comparison on the extended IEEE 30-bus system (81 relays, 226 pairs).

Method	CTI $\ge 0.2$	Best (s)	Mean (s)	Std (s)	CTI Viol.	Wall-Clock (s)
LP [5]	√	—	—	—	0	—
H-ACO [23]	√	113	120.85	6.60	0	7.8
ACO [23]	√	141.15	150.39	4.83	0	132.7
HEPSO/TFWO [4]	√	—	35.73	—	0	—
HFEE (no clustering, proposed)	√	22.85	25.30	0.51	0	6.79
T2FCM+HFEE (proposed)	√	59.79	61.54	3.04	0	6.54

“—” = not reported in original study; √ = satisfied.

presents the ablation comparison. The proposed standalone HFEE achieves a mean $T_{\mathrm{op}}$ of 25.30 s, representing a 29.19% improvement over the prior best result of HEPSO/TFWO (35.73 s [4]) at the same CTI level of 0.20 s. This confirms that HFEE, applied monolithically without any decomposition, establishes a new state-of-the-art baseline for the extended IEEE 30-bus benchmark. The T2FCM+HFEE configuration (CTI = 0.30 s, K = 6, $\alpha =0.1$) achieves 61.54 s with zero violations under the stricter coordination margin, with a 3.68% reduction in wall-clock computation time and full multi-scenario robustness across 226 simultaneous fault scenarios. K-means+HFEE and FCM+HFEE were excluded from this comparison as preliminary evaluations showed consistent CTI violations on the 226-constraint system, indicating that crisp and type-1 fuzzy partitioning cannot reliably decompose the coordination problem at this scale; this finding further validates the choice of T2FCM as the clustering stage.

6.3. T2FCM Parameter Sweep and Heatmap Analysis

Figure 6 presents the heatmap of $T_{\mathrm{op}}$ across all 63 $(K,\alpha )$ configurations. Each cell reports the mean $T_{\mathrm{op}}$ from 50 independent runs; configurations producing CTI violations are marked with a prohibited symbol. The global feasible minimum is $T_{\mathrm{op}}=61.54$ s at $K=6$, $\alpha =0.1$.

Two principal trends are evident. First, $\alpha =0.1$ consistently yields the best results across most cluster counts. This indicates that reduced velocity memory promotes broader search-space exploration, which is consistent with the nature of the decomposed subproblems, where each cluster requires the independent discovery of a local optimum in a distinct region of the setting space. Second, $T_{\mathrm{op}}$ is non-monotonic in K: performance improves from $K=2$ towards $K=6$ as finer decomposition reduces per-cluster coordination conflict, then degrades for $K>6$ as clusters become too small to span coherent protection zones; wall-clock computation time rises proportionally throughout (Section 6.5).

For a full visualisation of all 3150 solutions, the 3D surface plot in Appendix (Figure A1) shows the $\alpha$–K–$T_{\mathrm{op}}$ landscape; the majority of results cluster near the lower operational-time boundary, confirming the robustness of the proposed method.

6.4. Cluster Decomposition for the Best Configuration ($K=6$, $\alpha =0.1$)

Table 7. Cluster-wise relay summary for the best T2FCM configuration ($K=6$, $\alpha =0.1$, mean $T_{\mathrm{op}}=61.54$ s).

Cluster	Count	Relay Numbers	${\mathit{T}}_{\mathbf{op}}$ (s)	H
1	12	12,14,16,19,21,25,60,61,65,67,71,72	2.28	1.272
2	10	1,3,5,8,22,36,55,57,76,79	29.68	3.573
3	15	2,4,7,9,11,23,32,34,44,45,47,50,75,77,78	11.62	2.538
4	22	6,10,15,17,18,26,38,40,42,43,46,48,49,51,52,54,56,58,59,64,73,80	15.23	3.353
5	9	27,29,31,33,35,37,39,41,74	1.86	0.811
6	13	13,20,24,28,30,53,62,63,66,68,69,70,81	0.85	2.015
Total	81		61.54

summarises the relay-to-cluster assignment for the best configuration. The 81 relays are partitioned into six groups, each with its own TMS and PS values.

Cluster 4 has the highest relay count (22) and per-cluster $T_{\mathrm{op}}$ (15.23 s), corresponding to distribution-level relays in the 33 kV section where meshed topology creates complex coordination requirements. Cluster 6 has the lowest per-cluster $T_{\mathrm{op}}$ (0.85 s), comprising transformer-adjacent relays operating under simpler coordination constraints. The entropy range (0.811–3.573) indicates some variation in relay load distribution across clusters; this is expected given the heterogeneous nature of the six protection zones. The relay clusters are also shown in Figure 7, where different colours indicate distinct cluster groupings.

Approximately 15–20% of the 226 scenarios lie near protection-zone boundaries. Crisp clustering methods force these scenarios into a single cluster, potentially splitting operationally coupled relay pairs across cluster boundaries and causing CTI violations. T2FCM’s interval-valued memberships allow such scenarios to maintain partial participation in adjacent clusters, preserving coordination chain integrity. The FOU width furthermore serves as a quantitative confidence indicator: wide intervals flag scenarios where accurate coordination is most critical.

PhysicalInterpretation of Relay Settings at Boundary Values

The 81 relays are partitioned into six groups, each with its own TMS and PS values, reveals that several relays converge to boundary values of the decision space, specifically $\mathrm{PS}=0.5$ (minimum) or $\mathrm{PS}=2.5$ (maximum), and $\mathrm{TMS}=0.10$ (minimum) or $\mathrm{TMS}=1.00$ (maximum). These settings are physically meaningful rather than indicative of optimiser failure, for two reasons. First, relays assigned $\mathrm{PS}=0.5$ and $\mathrm{TMS}=0.10$ are configured for maximum sensitivity and minimum time delay; this is appropriate for relays positioned close to the fault source, where rapid isolation is critical and the risk of false tripping is low due to the high PSM ratio. Second, relays assigned $\mathrm{TMS}=1.00$ (maximum delay) serve as remote backup protection for fault paths where primary relays already provide fast clearance; the large time delay ensures selectivity without requiring tight TMS precision. The fact that all 219 active relay pairs satisfy the 0.3 s CTI requirement with zero violations confirms that these boundary-value settings are coordination-compliant and reflect a genuine engineering optimum for their respective cluster subproblems. The assignment of boundary values to certain relays is further consistent with the discrete-variable treatment of $\mathrm{PS}$ [4], where the coarse tap-step resolution naturally drives solutions towards the nearest feasible boundary point when that point satisfies all constraints.

6.5. Computational Complexity and Wall-Clock Time

The total number of fitness evaluations for the monolithic non-clustered approach is

$$P\times \mathrm{maxIt}\times n_{\mathrm{scen}}\times n_{\mathrm{pairs}}=50\times 100\times 226\times 226=255,380,000$$

For the clustered approach ($K=6$),

$$K\times P\times \mathrm{maxIt}\times (n_{\mathrm{scen}}/K)\times n_{\mathrm{pairs}}=6\times 50\times 100\times 38\times 226=257,640,000$$

Although the total evaluation counts are similar, each per-cluster evaluation call processes only $n_{\mathrm{scen}}/K\approx 38$ scenarios rather than all 226, yielding a benchmark cost ratio of approximately $5.95\times$ cheaper per cluster call. Furthermore, the $K=6$ clusters are fully independent and can be solved concurrently on K cores, offering a theoretical $6\times$ parallel speedup.

Table 8. Estimated wall-clock computation time (mean ± std, s) for all $9\times 7=63$ combinations of $\alpha$ and K.

K	$\mathit{\alpha }=0.1$	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.3$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.5$	$\mathit{\alpha }=0.6$	$\mathit{\alpha }=0.7$	$\mathit{\alpha }=0.8$	$\mathit{\alpha }=0.9$
2	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$	$2.18\pm 0.11$
3	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$	$3.27\pm 0.13$
4	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$	$4.36\pm 0.16$
5	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$	$5.45\pm 0.17$
6	†  $\mathbf{6.54}\pm \mathbf{0.19}$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$	$6.54\pm 0.19$
7	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$	$7.63\pm 0.21$
8	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$	$8.72\pm 0.22$

^† = measured point; $K=6$, $\alpha =0.1$ (Bold text) measured at $N=50$ runs.

reports estimated wall-clock computation times for all 63 $(K,\alpha )$ configurations, linearly scaled from the measured point $K=6$, $\alpha =0.1$ ($6.54\pm 0.19$ s, $N=50$). Note that wall-clock time scales linearly with K because each additional cluster adds one independent HFEE run (each of the same 100-iteration budget); $\alpha$ does not affect computation time, so all columns are identical. A comparison of clustered and non-clustered wall-clock times is shown in Figure 8.

Table 8 also reveals an important trade-off: selecting $K=2$ reduces computation time to 2.18 s, a 66.67% reduction relative to $K=6$ and a 67.9% reduction relative to non-clustered HFEE.

However, at $K=2$ and $\alpha =0.2$, $T_{\mathrm{op}}$ rises to 86.39 s—a significant penalty in operational performance. This trade-off is visualised in Figure 9, where each point represents one of the 63 configurations; the non-clustered result is marked with a star. Practitioners can use this figure to select an appropriate $(K,\alpha )$ pair based on their specific balance of operational and computational requirements.

Table 8 and Figure 9 together provide a practical decision guide for selecting $(K,\alpha )$ based on engineering priorities. If total operational time is the primary objective, $K=6$ and $\alpha =0.1$ yield the best $T_{\mathrm{op}}$ of 61.54 s, with a moderate wall-clock time of $6.54\pm 0.19$ s. Conversely, if computational speed is the priority, $K=2$ and $\alpha =0.2$ reduce wall-clock time to $2.18\pm 0.11$ s at the cost of a higher $T_{\mathrm{op}}$ of 86.39 s. For applications where neither objective is dominant, $K=3$ and $\alpha =0.2$ offer an intermediate trade-off, achieving $T_{\mathrm{op}}=77.96$ s with a wall-clock time of $3.27\pm 0.13$ s. This flexibility allows protection engineers to select a configuration that best suits their specific operational and computational constraints, without re-running the full parameter sweep.

6.6. CTI Margin Validation

To verify coordination integrity for the best configuration ($K=6$, $\alpha =0.1$), the CTI margin ${\mathsf{\Delta }}_p=T_{b,p}-T_{p,p}$ was evaluated for all 219 active primary–backup pairs using the optimised relay settings. All 219 pairs satisfied the 0.3 s CTI requirement, with zero violations.

Table 9. Comparison of HFEE with reported non-clustered methods and HFEE CTI margin $\ge 0.20$ s per IEC 60255.

	ACO [23]		H-ACO [23]		HEPSO/TFWO [4]		HFEE	HFEE
Relay	TMS	PS	TMS	PS	TMS (PS N/R)	TMS	PS	CTI (s)
	(68 Relays)		(68 Relays)
R1	0.8421	3	0.2839	3	0.129	0.1767 †	0.9334 †	0.237
R2	1.2619	3.5	0.2528	3.5	0.095	0.1742 †	0.6424 †	0.224
R3	1.2579	3	0.2730	3	0.129	0.1956 †	0.7268 †	0.234
R4	1.2616	2	0.2341	2	0.095	0.1704 †	0.6364 †	0.211
R5	0.9704	2.5	0.2665	2.5	0.184	0.2000	0.6965	0.356
R6	0.8986	1.5	0.2385	1.5	0.058	0.1231	0.8299	0.200
R7	1.2466	3	0.2485	3	0.204	0.1809	0.7707	0.243
R8	1.1603	3.5	0.1908	3.5	0.124	0.1639	0.7612	0.200
R9	1.1382	3	0.2591	3	0.154	0.1982	0.7004	0.200
R10	0.8576	4.5	0.2036	4.5	0.074	0.1014	1.1658	0.200
R11	1.3487	2	0.2802	2	0.156	0.1513	0.8592	0.200
R12	1.3376	3	0.2586	3	0.105	0.1513	0.7426	0.456
R13	1.2707	2	0.2933	2	0.139	0.1709	0.7401	0.200
R14	1.3071	2.5	0.2754	2.5	0.120	0.1728	0.5557	0.481
R15	1.0537	2	0.2485	2	0.150	0.1770	0.8326	0.253
R16	1.2186	2.5	0.2155	2.5	0.142	0.1720	0.9416	0.200
R17	1.3975	2.5	0.2533	2.5	0.310	0.1836	0.9472	0.200
R18	0.9174	5	0.2175	5	0.165	0.1137	0.7041	0.200
R19	0.8843	3.5	0.2040	3.5	0.214	0.1222	0.8094	0.200
R20	1.0738	1.5	0.2683	1.5	0.181	0.1438	0.7955	0.200
R21	0.8568	4.5	0.2355	4.5	0.239	0.1527	0.8515	0.346
R22	1.0633	5	0.1000	5	0.220	0.1900	0.7700	0.200
R23	0.9770	2.5	0.1584	2.5	0.312	0.1770	0.7083	0.219
R24	0.9150	2	0.2184	2	0.084	0.1758	0.2621	0.467
R25	1.2428	5	0.3324	5	0.209	0.1464	0.7565	0.200
R26	1.2771	3.5	0.3320	3.5	0.182	0.1362	0.6555	0.229
R27	1.2334	5	0.3348	5	0.300	0.2000	0.7480	0.220
R28	1.0859	3.5	0.2745	3.5	0.150	0.1816	0.5130	0.200
R29	1.2157	4.5	0.3489	4.5	0.333	0.1990	0.6030	0.200
R30	1.0570	2	0.2491	2	0.112	0.1243	0.8632	0.200
R31	0.8866	2.5	0.3281	2.5	0.378	0.2000 †	0.5266 †	0.210
R32	1.1907	4	0.2470	4	0.074	0.1066 †	0.7586 †	0.231
R33	1.1362	3	0.3204	3	0.378	0.1945 †	0.6632 †	0.221
R34	1.1950	1.5	0.2669	1.5	0.074	0.1039	0.7949	0.200
R35	1.1376	5	0.3119	5	0.334	0.1867	0.8466	0.494
R36	0.9802	2.5	0.2976	2.5	0.116	0.1977	0.1090	0.892
R37	1.1773	4	0.3423	4	0.374	0.1843	0.5078	0.468
R38	1.2921	2	0.3167	2	0.100	0.1998	0.0529	0.200
R39	1.2917	1.5	0.3179	1.5	0.327	0.1864	0.9129	0.200
R40	1.2974	5	0.2566	5	0.156	0.1503	0.6472	0.311
R41	0.9903	4.5	0.2905	4.5	0.347	0.1739	0.7041	0.200
R42	1.0073	1.5	0.3372	1.5	0.222	0.1437	0.8511	0.200
R43	1.2919	4.5	0.2815	4.5	0.336	0.1584	0.7060	0.200
R44	1.0659	4	0.2548	4	0.216	0.1459	0.7983	0.705
R45	0.8946	3	0.3283	3	0.379	0.1666	0.6071	0.214
R46	1.0247	3	0.3180	3	0.212	0.1514	0.8247	0.200
R47	0.9769	3.5	0.2832	3.5	0.271	0.1699	0.8198	0.200
R48	0.9603	2	0.3428	2	0.261	0.1534	0.8065	0.200
R49	1.0435	3.5	0.2750	3.5	0.248	0.1609	0.7106	0.200
R50	0.9856	5	0.3104	5	0.267	0.1731	0.8039	0.200
R51	1.0420	3.5	0.3351	3.5	0.312	0.1283	0.7628	0.200
R52	1.0510	2.5	0.2791	2.5	0.288	0.1927	0.7208	0.200
R53	1.1647	3.5	0.3015	3.5	0.169	0.1212	0.9013	0.200
R54	1.1698	1.5	0.3028	1.5	0.279	0.1999	0.6733	0.200
R55	1.1331	2.5	0.2757	2.5	0.395	0.1845	0.6966	0.200
R56	0.9555	2.5	0.2875	2.5	0.111	0.1677	0.7715	0.200
R57	1.1630	4	0.2558	4	0.285	0.1969	0.7903	0.200
R58	1.0910	4.5	0.2348	4.5	0.101	0.1979	0.5402	0.200
R59	1.1213	1.5	0.1424	1.5	0.176	0.1731	0.9350	0.200
R60	1.1260	1.5	1.0715	1.5	0.307	0.1953	0.7484	0.200
R61	1.1407	3.5	0.2373	3.5	0.167	0.1885	0.8457	0.562
R62	1.1375	2	0.2842	2	0.172	0.1731	0.7313	0.200
R63	0.9398	4	0.2151	4	0.129	0.1898	0.7654	0.200
R64	0.8979	3	0.1001	3	0.259	0.1979	0.9184	0.200
R65	0.9310	3	0.2171	3	0.150	0.1787	0.6779	0.541
R66	0.9310	2	0.1009	2	0.025	0.1314	0.0505	0.200
R67	0.8903	4	0.1748	4	0.134	0.1547	1.0469	0.570
R68	0.8403	4.5	0.1742	4.5	0.025	0.1306	0.0502	0.200
R69					0.098	0.1028	0.7308	0.200
R70					0.091	0.1070	0.6486	0.200
R71					0.025	0.1000	0.8038	1.303
R72					0.145	0.2000	0.5954	0.220
R73					0.025	0.1322	0.0826	1.256
R74					0.374	0.2000	2.2756	TTR
R75					0.139	0.1816 †	0.7434 †	0.215
R76					0.049	0.1185 †	0.6575 †	0.226
R77					0.092	0.1790 †	0.7404 †	0.233
R78					0.025	0.1682 †	0.1017†	0.239
R79					0.241	0.2000	0.9304	T.P
R80					0.204	0.1873	0.8782	0.200
R81					0.025	0.1135	0.6197	0.200
$T_{\mathrm{op}}$:    ACO = 150.39s    H-ACO = 120.8 s    HEPSO/TFWO = 35.73 s     HFEE = 25.3 s

^† Pair resolved by directional element; TTR: R74, R79 are transformer tertiary protection relays that never act as primary relays; N/R = Not reported; Bold = Optimal values.

presents the TMS and PS values alongside the minimum CTI margin for selected relays. The ^† symbol denotes pairs where the directional element resolves the coordination constraint, as described in Section 5.

The minimum observed CTI margin across all 219 active pairs is 0.200 s (R81), which satisfies the IEC 60255 minimum of 0.2 s [30] with no surplus; this relay is transformer-adjacent and its margin is constrained by the physical fault-current ratio. The maximum margin is 3.332 s (R80), reflecting the conservative settings assigned to line relays with low fault-current exposure. The relatively wide margin range (0.200–3.332 s) indicates that T2FCM clustering successfully preserves the coordination hierarchy across all active pairs, including those near protection-zone boundaries.

6.7. Comparison with State-of-the-Art Methods

The proposed T2FCM+HFEE method achieves a mean $T_{\mathrm{op}}$ of 61.54 s with zero CTI violations on the 81-relay extended system with six groups have their corresponding TMS and PS values listed in

Table 10. PS and TMS values for all clusters and CTI margins [≥0.30 s, stricter coordination].

Relay	Cluster 1		Cluster 2		Cluster 3		Cluster 4		Cluster 5		Cluster 6		CTI (s)
	PS	TMS	PS	TMS	PS	TMS	PS	TMS	PS	TMS	PS	TMS
R1 †	2.5	1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	2.249
R2 †	2.5	1	2.5	0.1	2.5	0.1	2.5	0.1	2.5	1	0.5	1	0.348
R3 †	2.5	0.1	2.5	0.1	2.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	1.989
R4 †	2.5	1	2.5	1	2.5	0.1	0.5	1	2.5	0.1	2.5	1	0.354
R5	0.5	1	0.5	0.1	2.5	1	2.5	0.1	0.5	1	0.5	1	2.500
R6	0.5	1	0.5	1	0.5	0.1	0.5	1	2.5	0.1	2.5	1	0.344
R7	2.5	0.1	2.5	1	2.5	0.1	0.5	1	2.5	1	0.5	0.1	0.345
R8	2.5	1	0.5	0.1	2.5	1	2.5	0.1	2.5	1	2.5	0.1	1.662
R9	0.5	0.1	0.5	1	2.5	0.1	0.5	1	0.5	1	0.5	0.1	0.314
R10	2.5	1	0.5	1	2.5	1	0.5	0.1	2.5	1	2.5	1	2.500
R11	2.5	0.1	0.5	0.1	0.5	0.1	0.5	1	2.5	1	0.5	0.1	0.329
R12	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	1	0.5	0.1	1.659
R13	2.5	0.1	2.5	1	2.5	0.1	2.5	1	0.5	0.1	2.5	1	4.349
R14	0.5	0.1	2.5	0.1	2.5	0.1	2.5	0.1	0.5	0.1	2.5	0.609	0.325
R15	0.5	1	0.5	1	2.5	0.1	0.5	0.1	0.5	0.1	2.5	1	1.065
R16	0.5	0.1	2.5	0.1	0.5	0.1	2.5	0.1	2.5	0.1	2.5	1	0.352
R17	0.5	1	2.5	0.1	0.5	0.1	2.5	0.1	0.5	0.1	2.5	0.1	2.225
R18	0.5	1	0.5	1	0.5	1	2.5	1	2.5	1	2.5	1	0.338
R19	0.5	1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.347
R20	0.5	0.1	0.5	1	0.5	0.1	2.5	0.1	0.5	0.1	0.5	1	2.109
R21	2.5	1	2.5	1	2.5	0.1	0.5	1	0.5	1	0.5	1	0.329
R22	0.5	1	0.5	0.1	2.5	0.1	2.5	0.1	0.890	0.1	0.5	1	2.500
R23	2.5	1	2.5	0.1	2.5	1	2.5	1	0.5	0.1	2.5	1	0.319
R24	0.5	0.1	2.5	1	2.5	1	0.5	0.1	2.5	1	0.5	1	2.500
R25	2.5	1	0.5	0.1	2.5	0.1	0.5	1	2.5	1	2.5	0.775	3.338
R26	2.5	0.1	0.5	1	0.5	0.1	0.5	0.1	0.5	0.1	2.5	1	3.239
R27	0.5	1	0.5	0.1	2.5	0.1	0.5	0.1	2.5	1	0.5	1	0.482
R28	0.5	0.1	2.5	1	0.5	0.1	2.5	0.1	0.5	0.1	2.5	1	3.595
R29	0.5	1	0.5	0.1	2.5	0.1	2.5	0.1	2.5	1	2.5	0.1	1.665
R30	0.5	0.1	0.5	1	0.5	0.1	0.5	1	0.5	0.1	2.5	1	4.182
R31 †	2.5	0.29	2.5	0.1	0.5	0.1	2.5	1	0.5	1	2.5	1	0.347
R32 †	2.5	0.1	0.5	0.1	0.5	1	0.5	1	0.5	0.1	0.5	1	0.344
R33 †	2.5	1	0.5	0.1	1.88	1	0.5	1	0.5	0.1	0.5	0.1	1.368
R34	0.5	0.1	2.5	1	2.5	1	0.5	0.1	2.5	1	2.5	0.1	2.682
R35	2.5	1	0.5	1	2.5	0.1	0.5	1	0.5	1	2.5	1	0.354
R36	0.5	1	2.5	0.1	0.5	0.1	2.5	0.1	0.5	1	0.5	0.93	0.350
R37	0.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.345
R38	2.5	0.1	0.5	1	0.5	0.1	0.5	0.1	2.5	1	2.5	0.1	3.390
R39	2.5	0.1	0.5	0.1	0.5	0.1	0.5	1	2.5	1	0.5	1	0.317
R40	2.5	1	0.5	0.1	2.5	1	2.5	0.1	2.5	0.1	0.5	0.1	0.499
R41	0.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	1	0.5	1	0.330
R42	2.5	0.1	0.5	0.1	2.5	0.1	2.5	1	0.5	0.1	2.5	0.1	3.162
R43	2.5	0.1	2.5	0.1	0.5	1	2.5	0.1	2.5	0.1	0.5	1	0.310
R44	2.5	1	2.5	0.1	0.5	0.1	2.5	0.1	0.5	0.1	2.5	1	2.252
R45	2.5	0.1	2.5	0.1	2.5	1	0.5	1	0.5	0.1	0.5	0.1	0.522
R46	0.5	0.1	0.5	1	0.5	1	2.5	0.1	2.5	1	2.5	1	3.535
R47	0.5	0.1	2.5	0.1	0.5	1	2.5	1	2.5	1	2.5	0.1	1.440
R48	0.5	0.1	2.5	1	2.5	0.1	0.5	1	0.5	1	0.5	0.1	3.770
R49	0.5	1	0.5	1	0.5	1	2.5	1	0.5	0.1	2.5	1	1.563
R50	0.5	0.1	2.5	1	0.5	1	2.5	1	2.5	0.1	2.5	1	2.262
R51	0.5	1	0.5	0.1	2.5	1	2.5	1	2.5	1	0.5	0.1	0.340
R52	0.5	0.1	0.5	1	0.5	1	2.5	1	0.5	1	2.5	1	1.477
R53	2.5	1	0.5	1	2.5	0.1	2.5	0.1	2.5	1	2.5	0.1	3.588
R54	2.5	0.1	2.5	1	2.5	0.1	2.5	0.1	0.5	1	0.5	0.1	1.432
R55	0.5	0.1	0.5	0.1	0.5	0.1	0.5	1	0.5	0.1	0.5	0.1	0.703
R56	0.5	0.1	2.5	0.1	0.5	1	0.5	0.1	0.5	1	2.5	1	0.343
R57	0.5	0.1	0.5	0.1	2.5	1	2.5	0.1	0.5	1	0.5	1	1.309
R58	0.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	2.036
R59	0.5	1	2.5	0.1	2.5	0.1	0.5	1	0.5	0.1	2.5	1	3.109
R60	2.5	1	0.5	0.1	0.5	0.1	2.5	0.1	0.5	1	2.5	0.1	2.979
R61	0.5	0.1	2.5	0.1	2.5	0.1	2.5	0.1	2.5	0.1	0.5	0.1	1.619
R62	0.5	1	2.5	0.37	2.5	1	2.5	0.1	2.5	0.1	0.5	0.1	3.630
R63	2.5	0.1	2.5	0.1	2.5	0.1	0.5	1	2.5	1	2.5	0.1	0.353
R64	2.5	0.1	0.5	0.1	0.5	1	2.5	1	2.5	1	0.5	0.1	2.341
R65	0.5	1	0.5	0.1	0.5	1	0.5	0.1	2.5	1	0.5	1	2.335
R66	0.5	1	0.5	0.1	0.5	0.1	2.5	1	0.5	0.1	0.69	1	2.500
R67	2.5	0.1	0.75	1	2.5	0.1	2.5	1	2.5	1	2.5	1	2.350
R68	0.5	0.1	0.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	2.5	0.1	2.500
R69	2.5	0.1	2.5	1	2.5	1	0.5	1	2.5	1	0.5	1	1.744
R70	0.5	0.1	2.5	0.1	2.5	0.1	0.5	0.1	2.5	1	2.5	1	2.500
R71	0.5	1	2.5	0.1	2.5	1	0.5	0.1	0.5	0.1	2.5	1	3.880
R72	2.5	1	0.5	1	2.5	0.1	0.5	0.1	0.5	1	0.5	1	0.325
R73	2.5	1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	1	2.5	1	0.330
R74	2.5	0.1	2.5	1	0.5	0.1	0.5	1	0.5	1	0.5	1	TTR
R75 †	0.5	0.1	2.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	1	0.314
R76 †	0.5	1	0.5	0.1	0.5	1	2.5	1	2.5	1	2.5	0.1	0.319
R77 †	0.5	0.1	0.5	0.1	0.5	1	0.5	0.1	0.5	0.1	0.5	1	0.329
R78 †	0.5	0.1	2.5	0.1	2.5	1	0.5	0.1	0.5	0.1	2.5	0.1	2.179
R79	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5	0.1	2.5	1	TTR
R80	2.5	0.1	2.5	1	0.5	0.1	2.5	0.1	2.5	1	0.5	0.1	3.332
R81	0.5	1	0.5	1	0.5	0.1	0.5	0.1	2.5	1	0.5	1	1.979
$T_{\mathrm{op}}$: T2FCM+HFEE (mean) = 61.54 s

^† Pair resolved by directional element. TTR: R74, R79 are transformer tertiary protection relays that never act as primary relays.

. Compared with methods limited to the 38-relay configuration (fewer than 120 constraints), the proposed framework handles a problem that is $2.1\times$ larger in decision variables and $3.3\times$ more constrained, demonstrating scalability for large-scale protection systems. The SQP-based method of Birla et al. [7] achieved 116.50 s on only 38 relays; the proposed method achieves 61.54 s on 81 relays—a 47.2% reduction despite a significantly larger problem, and a 67% improvement over Al-Bhadely and Inan [25] on the same relay count.

A direct comparison between T2FCM+HFEE and HEPSO/TFWO is not straightforward, as the two methods operate under different problem formulations and objectives: HEPSO/TFWO targets absolute $T_{\mathrm{op}}$ minimisation per pair, whereas the proposed clustering framework prioritises computational scalability and multi-scenario robustness, accepting a trade-off in absolute operational time. A valid comparison is possible between the non-clustered HFEE and HEPSO/TFWO under identical conditions (CTI = 0.2 s, 81 relays, 226 pairs): non-clustered HFEE achieves a mean $T_{\mathrm{op}}$ of 25.30 s, representing a 29.19% improvement over HEPSO/TFWO at the same constraint level. Among clustering-based methods, Zand et al. [18] achieved 52.33 s on the 38-relay system using a Fmincon algorithm with K-means; the proposed approach addresses a significantly larger and more constrained problem (226 pairs vs. fewer than 120).

The entropy term $H_D\left(X\right)$ in the velocity update (Equation (16)) enhances both swarm exploration and coordination balance: the $(1+H_D\left(X\right))$ factor amplifies the exploratory component when diversity is high, while a low $H_D\left(X\right)$ triggers reinitialisation, preventing premature convergence. Simultaneously, the relay-time entropy $H\left(X\right)$ in the objective (Equation (6)) promotes uniform relay utilisation, consistent with [20,21].

Figure 10 presents the 4-quadrant performance analysis. Methods in the top-left quadrant (e.g., SQP) exhibit high $T_{\mathrm{op}}$ at low constraint counts—the weak region. Methods in the top-right quadrant (e.g., HACO) are constrained but operationally slow. Methods in the bottom-left quadrant (e.g., IDA, EEO, EWSO) achieve low $T_{\mathrm{op}}$ but on low-constraint systems. The optimal trade-off region (bottom-right) captures methods that achieve low $T_{\mathrm{op}}$ at high constraint counts; TFWO and both proposed methods (non-clustered HFEE and T2FCM+HFEE) fall in this region. Non-clustered HFEE achieves the lowest $T_{\mathrm{op}}$ (25.30 s) at the highest constraint count (226), with high computational time; the clustered proposal achieves low computational time at the cost of higher $T_{\mathrm{op}}$, with the added benefits described in Section 4.2.4.

7. Conclusions

This study presented and validated both a standalone HFEE framework and a T2FCM clustering-decomposed HFEE framework for the coordination of directional overcurrent relays on the extended IEEE 30-bus system (81 relays, 226 primary—backup coordination pairs).

Standalone HFEE (monolithic, no clustering) achieves a 29.19% improvement over the reported best result of 35.73 s on the same system [4]—establishing a new state-of-the-art baseline for the extended IEEE 30-bus benchmark. T2FCM+HFEE achieves a mean $T_{\mathrm{op}}$ of 61.54 s with zero CTI violations across all active relay pairs. The clustering decomposition reduces wall-clock computation time by 3.68–66.67% relative to the non-clustered baseline depending on K, at the cost of a higher absolute $T_{\mathrm{op}}$. The fractional order $\alpha =0.1$ consistently yields the best results, confirming that reduced velocity memory promotes effective exploration in the decomposed subproblems.

The two-level hierarchical architecture demonstrates scalability for simultaneous sub-transmission and distribution relay coordination, handling 162 decision variables and 226 constraints through six independently optimisable subproblems. The communicationless, entirely offline multi-group deployment strategy—compatible with the MiCOM P127 relay’s eight-group capability—requires no hardware upgrades or real-time communication links.

The primary limitations of this study are as follows: (1) the clustered $T_{\mathrm{op}}$ is higher than monolithic methods, accepted in exchange for scalability and parallel decomposability; (2) the clustered framework requires each relay to store K distinct (PS, TMS) pairs—one per cluster—meaning a $K=6$ configuration demands six independent setting groups per relay; this is compatible with the MiCOM P127’s eight-group capability [32] but may exceed the capacity of legacy relay hardware; (3) highly imbalanced clusters may reduce the theoretical K-fold parallel speedup.

Future work will extend the framework to dynamic fault scenarios, integrate distance relays alongside DOCRs, and explore multi-objective formulations balancing $T_{\mathrm{op}}$, computational cost, and protection reliability.