Investigating the impact of electric vehicles on increasing the reliability of the distribution system using the enhanced gray wolf evolutionary algorithm model

Abstract

This paper proposes a tri‑level optimization framework for the coordinated planning and operation of electric‑vehicle‑integrated distribution systems. The upper level optimizes charging‑station siting and sizing under investment and capacity constraints. The middle level schedules vehicle charging/discharging with detailed models of battery degradation, time‑varying prices, and vehicle‑to‑grid (V2G) participation. The lower level evaluates operational feasibility and reliability under stochastic load scenarios and N‑1 contingencies. The resulting nonconvex problem is solved using an enhanced gray wolf optimization algorithm incorporating chaotic initialization, adaptive mutation, Lévy‑flight exploration, opposition‑based learning, and local search. Applied to a 33‑bus system with 150 EVs, the framework deploys five charging stations (total capacity 1550 kW). Coordinated V2G operation raises minimum voltages by 4.10%, reduces voltage deviation by > 50%, and cuts active‑power losses by 23.02%. Reliability improves significantly: expected energy not served drops from 651.30 to 192.70 MWh/year, and the composite reliability index rises from 0.152 to 0.367. Contingency analysis shows 66–78% reductions in load curtailment. Economically, the solution achieves a net present value of 7.93 M USD, a benefit‑cost ratio of 17.42, an internal rate of return of 287%, and a payback period of 0.21 years. Comparative studies confirm the enhanced algorithm’s superiority in solution quality, convergence speed, and constraint satisfaction. The results demonstrate that coordinated EV‑grid integration substantially enhances efficiency, reliability, and economic performance in medium‑voltage distribution networks.

Introduction

The rapid integration of electric vehicles (EVs) into power systems presents a dual challenge for distribution network operators. While uncontrolled charging can exacerbate issues like voltage violations, increased power losses, and equipment overloads, a coordinated approach can harness EVs as a flexible resource. Specifically, Vehicle-to-Grid (V2G) technology allows EVs to discharge power back to the grid, offering potential benefits for system reliability, voltage support, and peak shaving. However, realizing these benefits requires solving a complex problem that spans strategic infrastructure planning, operational scheduling of the EV fleet, and real-time network management under uncertainty.

Research motivation

The motivation for this research stems from the need for a holistic framework that can simultaneously address the planning and operational challenges of high EV penetration. Isolating the problem of charging station placement from the daily scheduling of EVs, or ignoring the network’s operational constraints during contingencies, can lead to suboptimal or even infeasible solutions. Therefore, there is a clear need for a coordinated model that integrates long-term investment decisions with short-term operational strategies, all while ensuring the distribution system remains reliable and efficient under various uncertain conditions.

Fig. 1

Flowchart of Tri-Level EV–Distribution System Optimization.

Figure (1) illustrates the overall procedural structure of the enhanced gray wolf optimization algorithm designed for solving the integrated tri-level problem of charging infrastructure planning, electric vehicle scheduling with vehicle-to-grid interactions, and distribution network operation under contingencies. The flowchart begins with the acquisition of all required input data, including network parameters, electric vehicle characteristics, candidate charging station locations, price scenarios, and contingency sets. Once the data are provided, the algorithm initializes a diverse population using chaotic mapping to ensure broad coverage of the search space. This initialization phase also sets the algorithmic parameters, such as adaptive weights, mutation probabilities, and the convergence threshold.

Literature review

The integration of Electric Vehicles (EVs) into power distribution systems has garnered considerable attention in recent years, particularly regarding their impact on grid reliability, performance, and operational efficiency. This literature review presents a synthesis of recent studies that explore the influence of EVs on the reliability and optimization of distribution networks, with a focus on both technical and economic considerations. In¹, the impact of EVs on power system reliability is reviewed, with an emphasis on improving grid resilience through the integration of EVs. The paper identifies the challenges related to EV penetration and presents strategies for enhancing grid stability and reliability through advanced planning and operational techniques. The work in² examines the influence of limited information on estimating aggregated EV loading from the perspective of distribution system operators. It highlights how incomplete data on EV adoption and usage patterns can affect accurate network planning and decision-making, stressing the need for more reliable data sources in system integration. A data-driven probabilistic evaluation of EV integration in distribution systems is presented in³. This study discusses the importance of modeling EV charging behavior and grid hosting capacity, illustrating the potential impacts on grid operations. The analysis suggests that accurate data and probabilistic models are crucial for optimizing system integration and ensuring operational stability. The paper in⁴ explores methods for minimizing the impact of EV charging stations in radial distribution systems. Using a combination of distributed generation and D-STATCOM models, the study demonstrates how these optimization strategies can improve system reliability and mitigate disruptions caused by high levels of EV integration. In⁵, the focus shifts to the mechanical reliability and safety performance of EV battery packs. The study discusses design improvements and safety considerations necessary to ensure long-term performance and reliability of EV battery systems, which are critical to the overall stability of EV-integrated networks. The reliability impact of dynamic thermal line rating and EVs on wind power-integrated networks is investigated in⁶. The paper proposes methods to enhance the resilience of power networks through the integration of EVs into renewable-powered grids, particularly focusing on the dynamic nature of wind power generation and the flexible nature of EVs for supporting grid stability. In⁷, the study assesses the impact of EV charging on distribution networks, specifically addressing potential disruptions caused by charging demand. It suggests strategies for mitigating the negative effects of charging loads on grid performance, highlighting the need for optimized charging strategies to enhance network resilience. A long-term techno-economic analysis of EV-to-grid (V2G) systems integrated with renewable energy generators is conducted in⁸. The study uses hybrid optimization methods to assess the performance of V2G systems, focusing on their ability to contribute to system strength and reduce grid dependency on non-renewable energy sources, while also considering economic feasibility. In⁹, the reliability degradation of lithium-ion batteries in EVs is examined, with a focus on the prognosis and health management techniques required to extend the lifespan of these batteries. The findings underscore the importance of considering battery aging in the evaluation of EV systems’ long-term reliability. The simultaneous impact of photovoltaic (PV) systems and fast EV charging stations on active distribution grids is analyzed in¹⁰. The paper explores how the integration of both PV systems and EV chargers affects grid operations and performance, offering insights into the challenges and solutions for maintaining grid stability under these combined loads. The impact of frosting characteristics on the performance of R744 heat pump systems for EVs at low temperatures is discussed in¹¹. The experimental results presented in the study show how cold weather conditions can affect the heating performance of EV systems, highlighting the importance of designing EV systems that can withstand extreme environmental conditions. In¹², the paper investigates how EVs can contribute to post-disaster power supply restoration in urban distribution systems. It emphasizes the role of EVs in recovery efforts following grid failures, highlighting their potential to provide emergency power and support grid restoration. Grid integration of EVs is explored in¹³, focusing on the potential impacts on power quality and reliability. The study proposes solutions to mitigate the adverse effects of EV integration, such as voltage instability and harmonics, on distribution system performance. In¹⁴, the paper investigates the reliability and security of distribution networks through the integration of plug-in EVs and wind turbine generators. The study addresses the challenges posed by the stochastic nature of these systems and suggests strategies to improve network stability through better coordination of renewable generation and EV integration. Mitigation strategies for the impact of EVs on power systems are discussed in¹⁵, using a comprehensive probabilistic approach. The study presents methods for managing the effects of EVs on grid stability and power quality, offering a robust framework for dealing with uncertainties in EV demand and grid operations. In¹⁶, a hybrid optimization method is proposed to assess the influence of EVs and renewable energy sources on power system distribution networks. The study emphasizes the benefits of combining multiple optimization techniques for better network performance, focusing on the integration of EVs as a means to enhance system stability. The rise of EV integration and its effect on electrical distribution network performance is examined in¹⁷. The study suggests various measures to enhance network resilience, focusing on improving grid performance as EV penetration increases. In¹⁸, a cost-effective optimization of on-grid EV charging systems is presented, integrating renewable energy and energy storage solutions. The paper conducts an economic and reliability analysis, demonstrating the potential for these systems to improve grid efficiency by reducing operational costs and enhancing reliability. A hybrid deep learning approach to forecast EV charging capacity is introduced in¹⁹. This model incorporates climatic and reliability factors to predict EV load demand more accurately, offering valuable insights for distribution system operators in planning and optimizing EV integration. The impact of EV chargers on a medium voltage distribution network in Casablanca City is analyzed in²⁰. The study provides insights into the effects of EV charging infrastructure on grid stability and performance, offering recommendations for optimizing EV charger placement and operation to enhance grid resilience. In²¹, the operational impacts of dynamic wireless charging for EVs in power systems are examined. The paper explores how wireless charging technology can affect system efficiency and charging behavior, with a focus on enhancing grid performance through innovative charging methods. A comprehensive review of EV integration with power systems is presented in²², focusing on the effects of EVs on power quality and the necessary enhancements for improving grid operations. The study provides a detailed overview of how EVs can impact distribution systems and suggests potential solutions for mitigating these impacts. The impact of EV adoption on greenhouse gas emissions in São Paulo, Brazil, is analyzed in²³ using regression analysis. The study highlights how EV integration can contribute to emission reductions in urban areas, offering valuable insights into the environmental benefits of EV adoption. A non-intrusive demand response management strategy for mitigating the impacts of residential EV charging on distribution systems is proposed in²⁴. This approach focuses on reducing disruptions caused by EV charging without requiring significant changes to existing infrastructure. Robust expansion planning for EV charging systems and distribution networks is addressed in²⁵. The paper presents strategies for enhancing system reliability while integrating more EVs into existing power grids, offering a comprehensive approach to planning for EV infrastructure deployment. The integrated planning of charging piles and battery swapping stations is analyzed in²⁶, considering the spatiotemporal distribution of EVs. The study proposes optimal deployment strategies for EV infrastructure to improve system performance, focusing on enhancing the efficiency of charging and swapping stations. In²⁷, the reliability of distribution systems under the influence of stochastic photovoltaic (PV) generation and spatial-temporal EV load demand is assessed. The study emphasizes the importance of robust planning in addressing the variability of renewable energy sources and the dynamic nature of EV charging. A two-stage bidding strategy to mitigate three-phase imbalance in power systems through EV aggregator coordination is presented in²⁸. The strategy aims to reduce load imbalances and improve the stability of distribution networks during periods of high EV charging demand. In²⁹, the optimal integration of EV charging stations into distribution network planning and operation is discussed. The study explores the role of EV charging infrastructure in enhancing grid efficiency and resilience, focusing on the integration of these stations to optimize overall system performance. A bi-level safe deep reinforcement learning method for managing EVs in distribution networks is proposed in³⁰. This data-efficient approach aims to improve the coordination and operation of EV charging systems in complex grid environments, enhancing overall grid reliability and stability. In³¹, a distributed coordination approach for EV charge and discharge in unbalanced distribution grids is introduced. The method aims to reduce grid instability caused by uneven distribution of EV loads, offering a solution for better load balancing and grid performance. Machine learning-based control of EV charging is proposed in³² for practical distribution systems with solar generation. The study focuses on optimizing charging schedules to align with renewable energy production, aiming to reduce grid congestion and enhance system performance. A communication-less management strategy for EV charging in droop-controlled islanded microgrids is presented in³³. This strategy aims to improve the operation of microgrids by efficiently managing EV charging without the need for constant communication, reducing overhead and enhancing system efficiency. In³⁴, a joint energy-computation management strategy for EVs under coordination between power distribution and computing power networks is proposed. The study explores the optimization of resource use across both sectors, improving efficiency and load management in integrated systems. The optimal allocation of EV charging stations and capacitors in radial distribution networks is investigated in³⁵, focusing on reliability. The study uses optimization techniques to strategically place charging infrastructure and reactive power compensation, improving network performance and stability. Beyond network-level coordination, recent studies have emphasized robust and resilient energy management under uncertainty. In³⁶, a robust optimization framework for smart home energy management is developed, integrating photovoltaic generation, battery storage, EV charging, and demand response. The proposed approach effectively handles uncertainties in renewable generation and load demand, improving operational reliability and cost efficiency, and offering valuable insights for extending similar robustness concepts to microgrid and aggregator-level EV coordination. In parallel, data-driven and intelligent modeling techniques have gained increasing attention in transportation electrification systems. Reference³⁷ introduces an anomaly detection framework for electrified railway catenary components based on invariant normal region prototypes combined with the Segment Anything Model, enabling accurate and scalable condition monitoring. Complementarily³⁸, proposes a generalized Koopman neural operator for data-driven modeling of pantograph–catenary systems, achieving high-fidelity dynamic representation and improved generalization across operating conditions. Although focused on railway electrification, these studies highlight the growing role of advanced machine learning and operator-theoretic methods for modeling complex electromechanical energy systems, which are increasingly relevant to large-scale EV infrastructure and grid-interactive transportation networks. Furthermore, system resilience under uncertainties and disturbances is addressed in³⁹, where a resilient and robust voltage regulation strategy for shipboard DC microgrids with ZIP loads is proposed. By explicitly considering actuator faults and parameter uncertainties, the study enhances voltage stability and operational robustness. The methodologies presented provide transferable insights for DC microgrids with high EV penetration, where uncertainty, fast dynamics, and reliability requirements are critical.

Research gap

Despite the extensive literature, a significant gap exists in the tight coupling of these three aspects—infrastructure planning, fleet scheduling, and reliability-constrained network operation—into a single, cohesive framework. Many studies address one or two of these levels but fail to fully capture their interconnected nature and the feedback between them. Furthermore, the solution of such a complex tri-level problem demands advanced optimization algorithms that can effectively navigate the high-dimensional search space, avoid local optima, and handle numerous constraints without compromising computational efficiency. In Table 1, the proposed paper demonstrates superior performance in comparison to the reviewed articles across key indicators. The framework presented in this paper integrates advanced functionalities such as coordinated planning and operation, Vehicle-to-Grid (V2G) scheduling, and reliability-focused network operation under stochastic load and contingency scenarios, which sets it apart from other works. It addresses the sizing and placement of charging stations, ensures efficient algorithmic solutions through enhanced optimization methods, and incorporates a comprehensive economic analysis, improving both reliability and cost-effectiveness. The proposed tri-level approach provides a robust solution by including N-1 contingency analysis and operational feasibility, areas often overlooked by the other studies. Each abbreviation in the table is as follows: EV Integration refers to the integration of electric vehicles into the power distribution network, Reliability indicates improvements or evaluations of system reliability with EVs, Economic Analysis includes evaluations of the economic viability of the proposed solutions, Contingency Analysis refers to evaluations under fault or disruption scenarios (such as N-1 contingencies), Charging Station Sizing deals with the optimal design and placement of EV charging infrastructure, and Algorithmic Efficiency assesses the speed and optimization of the solution process.

Table 1 Comparison of the proposed tri-level optimization framework for ev integrated distribution systems with existing approaches.

Main contribution of the paper

This paper’s main contribution is the development of a comprehensive tri-level optimization framework and an enhanced solution algorithm to address the identified gaps. The specific contributions are:

• 1. A Novel Tri-Level Model: We propose a structured framework where the upper level determines the optimal siting and sizing of charging stations, the middle level schedules the charging and discharging of the EV fleet, and the lower level evaluates the system’s operational feasibility and reliability under stochastic load and N-1 contingency scenarios.

• 2. An Enhanced Solution Algorithm: We develop an Enhanced Gray Wolf Optimization Algorithm (EGWOA) that integrates several improvements—including chaotic initialization, adaptive mutation, Lévy flight exploration, opposition-based learning, and a local search mechanism—to solve the resulting nonconvex tri-level problem more effectively than standard metaheuristics.

• 3. Comprehensive Quantitative Analysis: The framework is applied to a standard test system to demonstrate substantial and simultaneous improvements in voltage profile (e.g., raising minimum voltages by 4.10%), system efficiency (e.g., reducing active power losses by 23.02%), and reliability (e.g., reducing expected energy not served by 70.42%), alongside a highly favorable economic outcome.

Paper organization

The remainder of this paper is organized as follows. Section 2 details the proposed tri-level optimization model, including the mathematical formulations for the upper, middle, and lower levels, the calculation of reliability indices, and the linking constraints. Section 3 describes the Enhanced Gray Wolf Optimization Algorithm developed to solve the model. Section 4 presents the simulation framework, case study results, a comparative analysis with other algorithms, and a sensitivity analysis. Finally, the conclusion summarizes the key findings and contributions.

Proposed tri-level optimization model

This section details the mathematical formulation of the tri-level optimization framework. It is divided into subsections defining the objective functions and constraints for the Upper Level (charging infrastructure planning), the Middle Level (EV fleet scheduling), and the Lower Level (network operation under uncertainty). The calculation of reliability indices and the linking constraints that integrate the three levels are also presented here. This formulation defines the problem that the Enhanced Gray Wolf Optimization Algorithm, described in Sect. 3, is designed to solve.

Upper level optimization model

The upper level optimization model establishes the strategic planning foundation for charging infrastructure deployment, aiming to minimize the total system cost. Equation (1a) defines the primary objective function, which aggregates three distinct cost components. The first component represents the capital expenditure for establishing charging stations, including fixed investment costs and variable costs proportional to the installed capacity. The second component captures the expected operational energy costs for charging electric vehicles, calculated across multiple time periods and uncertainty scenarios, weighted by their respective probabilities. The third component introduces a monetized benefit, subtracting a value proportional to the achieved reliability improvement, thus effectively trading off initial investment against long-term reliability gains. The reliability improvement metric itself is quantified in Eq. (1b). It is a composite index that measures the proportional enhancement of key performance indicators relative to a base case without electric vehicle integration. The index is a weighted sum of the improvements in the system average interruption frequency index, the system average interruption duration index, and the expected energy not served. This formulation allows the model to systematically value reductions in both the frequency and duration of customer interruptions, as well as the total unsupplied energy. The reliability improvement metric $\:RI$ in Eq. (1b) is a normalized, weighted composite index designed to quantify the overall enhancement in system reliability resulting from the proposed EV integration. The normalization by the base-case values ($\:SAIF{I}^{base}$, $\:SAID{I}^{base}$, $\:EEN{S}^{base}$) converts each distinct reliability index—measuring interruption frequency, duration, and unsupplied energy, respectively—into a dimensionless fractional improvement. This allows for their aggregation into a single metric. The weights $\:{\alpha\:}_{SAIFI}$, $\:{\alpha\:}_{SAIDI}$, and $\:{\alpha\:}_{EENS}$ reflect the relative importance assigned to improving each aspect of reliability. In this study, we adopt a balanced perspective by setting $\:{\alpha\:}_{SAIFI}={\alpha\:}_{SAIDI}={\alpha\:}_{EENS}=1/3$, implying equal priority for reducing the frequency, duration, and total energy of interruptions. This serves as a neutral benchmark. The robustness of the optimization outcomes to variations in these weights is demonstrated in the sensitivity analysis presented in Sect. 4.5. To ensure power quality alongside economic and reliability objectives, Eq. (1c) incorporates a penalty for voltage deviations. This term sums the squared differences between the actual voltage magnitude at each bus and its nominal value across all time periods and scenarios, encouraging solutions that maintain voltages close to the desired nominal level. Furthermore, Eq. (1d) accounts for system efficiency by minimizing active power losses in the distribution lines. This loss term is calculated based on the line resistance and the square of the current flow, integrated over time and averaged across uncertainty scenarios. The model incorporates several critical constraints to ensure practical and feasible solutions. Equation (1e) enforces a total budget limitation, ensuring that the combined investment in fixed infrastructure and variable capacity does not exceed the available capital. Equation (1f) provides a direct link between the installed capacity of a station and the number of charging ports, assuming each port has a standardized power rating. Equation (1 g) defines the lower and upper bounds for the capacity of each charging station, which are only enforced if the station is selected for establishment, as indicated by the binary variable. Equation (1 h) limits the total number of stations that can be built, reflecting physical or policy constraints on infrastructure expansion. To guarantee adequate service coverage for all demand centers, Eq. (1i) ensures that every demand bus is within the acceptable service distance of at least one established charging station. The assignment of vehicles to charging stations is governed by Eq. (1j), which requires that each vehicle is assigned to exactly one station. Equation (1k) ensures that the combined maximum charging power demand from all vehicles assigned to a station does not exceed that station’s installed capacity, preventing overloading. To promote a geographically balanced deployment of infrastructure, Eq. (1 L) sets a minimum number of stations that must be established within each predefined zone of the network. Equation (1 m) acts as a load-balancing constraint across stations. It limits the ratio of the total battery energy capacity of vehicles assigned to a station to the station’s power capacity. This prevents the concentration of too many vehicles at a station with relatively low power capacity, which could lead to operational instability or excessive waiting times. Finally, Eq. (1n) sets a minimum target for the overall reliability improvement, ensuring that the solution achieves a predefined level of service enhancement.

$$\:\underset{{x}_{s},{P}_{s}^{cap},{n}_{s}^{charger}}{\text{m}\text{i}\text{n}}\:{F}^{UL}=\sum\:_{s\in\:\mathcal{S}}\:\left({C}_{s}^{inv}{x}_{s}+{C}_{s}^{cap}{P}_{s}^{cap}\right)+\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:{\pi\:}_{\omega\:}{C}_{t}^{energy}\sum\:_{i\in\:\mathcal{N}}\:{P}_{i,t}^{EV}{\Delta\:}t-\psi\:\cdot\:RI$$

(1a)

$$\:RI={\alpha\:}_{SAIFI}\frac{SAIF{I}^{base}-SAIF{I}^{EV}}{SAIF{I}^{base}}+{\alpha\:}_{SAIDI}\frac{SAID{I}^{base}-SAID{I}^{EV}}{SAID{I}^{base}}+{\alpha\:}_{EENS}\frac{EEN{S}^{base}-EEN{S}^{EV}}{EEN{S}^{base}}$$

(1b)

$$\:{F}^{voltage}=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{i\in\:\mathcal{N}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:{\pi\:}_{\omega\:}{\left({V}_{i,t}^{\omega\:}-{V}_{i}^{nom}\right)}^{2}$$

(1c)

$$\:{F}^{loss}=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{l\in\:\mathcal{L}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:{\pi\:}_{\omega\:}{R}_{l}{\left({I}_{l,t}^{\omega\:}\right)}^{2}{\Delta\:}t$$

(1d)

$$\:\sum\:_{s\in\:\mathcal{S}}\:\left({C}_{s}^{inv}{x}_{s}+{C}_{s}^{cap}{P}_{s}^{cap}\right)\le\:{B}^{total}$$

(1e)

$$\:{P}_{s}^{cap}={n}_{s}^{charger}\cdot\:{P}^{charger,rated}\:\forall\:s\in\:\mathcal{S}$$

(1f)

$$\:{x}_{s}{P}_{s}^{cap,min}\le\:{P}_{s}^{cap}\le\:{x}_{s}{P}_{s}^{cap,max}\:\forall\:s\in\:\mathcal{S}$$

(1g)

$$\:\sum\:_{s\in\:\mathcal{S}}\:{x}_{s}\le\:{N}^{max,stations}$$

(1h)

$$\:\sum\:_{s\in\:\mathcal{S}}\:{x}_{s}{A}_{s,i}^{coverage}\ge\:1\:\forall\:i\in\:{\mathcal{N}}^{demand}$$

(1j)

$$\:\sum\:_{s\in\:\mathcal{S}}\:{y}_{vs}=1\:\forall\:v\in\:\mathcal{V}$$

(1j)

$$\:\sum\:_{v\in\:\mathcal{V}}\:{y}_{vs}{P}_{v}^{ch,max}\le\:{P}_{s}^{cap}\:\forall\:s\in\:\mathcal{S}$$

(1k)

$$\:\sum\:_{s\in\:{\mathcal{S}}_{k}}\:{x}_{s}\ge\:{N}_{k}^{min}\:\forall\:k\in\:\mathcal{K}$$

(1l)

$$\:\frac{\sum\:_{v\in\:\mathcal{V}}\:\:{y}_{vs}{E}_{v}^{cap}}{{P}_{s}^{cap}}\le\:{\varphi\:}^{util}\:\forall\:s\in\:\mathcal{S}$$

(1m)

$$\:RI\ge\:R{I}^{min,target}$$

(1n)

Middle level optimization model

The middle level optimization model addresses the operational scheduling of electric vehicles, focusing on maximizing their value to the system while managing costs and battery health. Equation (2a) defines the core objective function for this level, which is a multi-term function to be maximized. It seeks to maximize a weighted sum of the reliability contribution from vehicle-to-grid operations, while minimizing the total costs associated with charging and the costs associated with battery degradation. The expectation over uncertainty scenarios is taken to account for stochastic elements in the problem. The reliability contribution metric is formulated in Eq. (2b). This term quantifies the positive impact of electric vehicles on system reliability in two ways. The first component is a measure of the available discharge power from vehicles, normalized by the load at a bus and weighted by a load priority factor, which sums over all network buses. The second component penalizes the expected load curtailment across all contingency scenarios, thus directly encouraging scheduling decisions that minimize unmet demand during faults. Equation (2 c)calculates the net charging cost. This includes the cost of electricity drawn from the grid for charging and incorporates a revenue term for electricity discharged back to the grid, where the compensation rate may be different from the retail price. Equation (2d) models the battery degradation cost, which is a critical factor for vehicle owner participation. It captures two aging mechanisms: cycle aging, which is proportional to the energy throughput relative to the battery’s capacity, and calendar aging, which occurs over time irrespective of use. While not part of the core objective in this formulation, Eq. (2e) presents a component for voltage support, which rewards the provision of reactive power by electric vehicles, especially when the voltage deviates from a reference value. This highlights the potential for vehicles to provide ancillary services. The heart of the electric vehicle model is the state of charge dynamics, given by Eq. (2f). This difference equation updates the state of charge of each vehicle based on the previous state, the energy added during charging—adjusted for charging efficiency—and the energy removed during discharging—adjusted for discharging efficiency. Equation (2 g), (2 h), and (2i) enforce bounds on the state of charge to maintain battery health and ensure that the vehicle has sufficient energy for its intended trips, specifying the initial state upon arrival and a minimum required state at departure. The power constraints for each vehicle are defined in equations (2j) and (2k), which limit the charging and discharging power to their respective maximum rates. These constraints are activated only when the vehicle is in the corresponding mode, as controlled by binary variables. Equation (2 L) is a logical constraint that prevents a vehicle from charging and discharging simultaneously. Furthermore, Eq. (2 m) restricts any charging or discharging activity to periods when the vehicle is physically connected to the charging station. Equation (2n) aggregates the net active power injection from all electric vehicles at each bus, which serves as a crucial link to the power flow equations in the lower-level model. Equation (2o) ensures that the number of vehicles charging simultaneously at any station does not exceed the number of available charging ports. Several constraints are included to protect the battery and ensure practical operation. Equation (2p) limits the maximum depth of discharge between consecutive time steps to prevent excessive strain on the battery. Equation (2q) enforces a ramp rate limit on the net power change of the vehicle, smoothing power transitions and preventing abrupt swings. Equation (2r) limits the total daily energy throughput—the sum of all charging and discharging energy—to a fraction of the battery capacity, constraining the daily cycling to a sustainable level. Finally, Eq. (2s) ensures that the vehicle fleet can provide a specified level of spinning reserve at each bus, calculated as the sum of the unused discharging capacity of all vehicles that are in discharging mode.

$$\:\underset{{P}_{v,t}^{ch},{P}_{v,t}^{dis},SO{C}_{v,t}}{\text{m}\text{a}\text{x}}\:{F}^{ML}=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:{\pi\:}_{\omega\:}\left({\kappa\:}^{rel}{R}_{t}^{reliability}-{\kappa\:}^{cost}{C}_{t}^{charging}-{\kappa\:}^{deg}{D}_{t}^{degradation}\right)$$

(2a)

$$\:{R}_{t}^{reliability}=\sum\:_{i\in\:\mathcal{N}}\:{\rho\:}_{i}^{load}\frac{{P}_{i,t}^{EV,available}}{{P}_{i,t}^{D,\omega\:}}-\sum\:_{c\in\:\mathcal{C}}\:{\pi\:}_{c}\sum\:_{i\in\:\mathcal{N}}\:{L}_{i,t}^{\omega\:,c}$$

(2b)

$$\:{C}_{t}^{charging}=\sum\:_{v\in\:\mathcal{V}}\:\left({C}_{t}^{energy}{P}_{v,t}^{ch}-\nu\:{C}_{t}^{energy}{P}_{v,t}^{dis}\right){\Delta\:}t$$

(2c)

$$\:{D}_{t}^{degradation}=\sum\:_{v\in\:\mathcal{V}}\:{C}_{v}^{battery}\left({\delta\:}^{cycle}\frac{{P}_{v,t}^{ch}+{P}_{v,t}^{dis}}{2{E}_{v}^{cap}}+{\delta\:}^{calendar}\right){\Delta\:}t$$

(2d)

$$\:{F}^{reactive}=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{i\in\:\mathcal{N}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:{\pi\:}_{\omega\:}{Q}_{i,t}^{EV}\left|{V}_{i,t}^{\omega\:}-{V}_{i}^{ref}\right|$$

(2e)

$$\:SO{C}_{v,t}=SO{C}_{v,t-1}+\frac{{\eta\:}_{v}^{ch}{P}_{v,t}^{ch}-\frac{{P}_{v,t}^{dis}}{{\eta\:}_{v}^{dis}}}{{E}_{v}^{cap}}{\Delta\:}t\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2f)

$$\:SO{C}_{v}^{min}\le\:SO{C}_{v,t}\le\:SO{C}_{v}^{max}\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2g)

$$\:SO{C}_{v,{\tau\:}_{v}^{arr}}=SO{C}_{v}^{arr}\:\forall\:v\in\:\mathcal{V}$$

(2h)

$$\:SO{C}_{v,{\tau\:}_{v}^{dep}}\ge\:SO{C}_{v}^{dep}\:\forall\:v\in\:\mathcal{V}$$

(2i)

$$\:0\le\:{P}_{v,t}^{ch}\le\:{u}_{v,t}^{ch}{P}_{v}^{ch,max}\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2j)

$$\:0\le\:{P}_{v,t}^{dis}\le\:{u}_{v,t}^{dis}{P}_{v}^{dis,max}\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2k)

$$\:{u}_{v,t}^{ch}+{u}_{v,t}^{dis}\le\:1\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2l)

$$\:{u}_{v,t}^{ch}+{u}_{v,t}^{dis}\le\:{a}_{v,t}^{connect}\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2m)

$$\:{P}_{i,t}^{EV}=\sum\:_{v\in\:{\mathcal{V}}_{i}}\:\left({P}_{v,t}^{dis}-{P}_{v,t}^{ch}\right)\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T}$$

(2n)

$$\:\sum\:_{v\in\:\mathcal{V}}\:{y}_{vs}{u}_{v,t}^{ch}\le\:{n}_{s}^{charger}\:\forall\:s\in\:\mathcal{S},t\in\:\mathcal{T}$$

(2o)

$$\:SO{C}_{v,t}-SO{C}_{v,t-1}\ge\:-DO{D}_{v}^{max}\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2p)

$$\:\left|\left({P}_{v,t}^{ch}-{P}_{v,t}^{dis}\right)-\left({P}_{v,t-1}^{ch}-{P}_{v,t-1}^{dis}\right)\right|\le\:R{R}_{v}^{max}{\Delta\:}t\:\forall\:v\in\:\mathcal{V},t\in\:\mathcal{T}$$

(2q)

$$\:\sum\:_{t\in\:\mathcal{T}}\:\left({P}_{v,t}^{ch}+{P}_{v,t}^{dis}\right){\Delta\:}t\le\:{{\Theta\:}}_{v}^{daily}{E}_{v}^{cap}\:\forall\:v\in\:\mathcal{V}$$

(2r)

$$\:\sum\:_{v\in\:{\mathcal{V}}_{i}}\:{u}_{v,t}^{dis}\left({P}_{v}^{dis,max}-{P}_{v,t}^{dis}\right)\ge\:{R}_{i,t}^{reserve}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T}$$

(2s)

Lower level optimization model

The lower level optimization model serves as the operational core of the tri-level framework, focusing on assessing the distribution system’s response to worst-case contingencies while optimizing network reconfiguration to enhance reliability. The objective function in Eq. (3a) minimizes a weighted sum of two critical metrics across all time periods, uncertainty scenarios, and contingency cases. The first metric is the prioritized load shedding, where each curtailed load is weighted by its importance factor, reflecting the cost of interrupted energy to consumers. The second metric is the voltage deviation penalty, which quantifies the extent to which bus voltages deviate from their allowable limits. By combining these terms, the model aims to reduce both the energy not served and the voltage violations under adverse conditions, thus ensuring a robust operational strategy. The voltage deviation penalty is defined through a set of constraints that linearize the absolute value of the deviation. Specifically, the auxiliary variable captures the positive difference between the actual voltage and the maximum limit, as well as the positive difference between the minimum limit and the actual voltage. This formulation ensures that any voltage outside the permissible range is penalized proportionally in the objective function, promoting voltage stability throughout the network. The energy not served is calculated for each bus, time period, scenario, and contingency as the product of load shedding and the time step duration, as shown in Eq. (3e). This value is then aggregated across all scenarios and contingencies in Eq. (3f) to compute the expected energy not served, which is a key reliability indicator used in the upper-level optimization to evaluate the overall system performance. The operational feasibility of the distribution network is enforced through a series of power flow equations and constraints. The active power balance at each bus, given by Eq. (3 g), ensures that the net injection of active power—comprising the power flows from adjacent lines, the net electric vehicle power, and the distributed generation output—equals the demand minus any load shedding, accounting for line losses. Similarly, the reactive power balance in Eq. (3 h) maintains equilibrium for reactive power, incorporating contributions from lines, electric vehicles, and distributed generators. The voltage drop along each distribution line is modeled using the DistFlow formulation in Eq. (3i), which relates the voltage magnitudes at sending and receiving ends to the line power flows and impedance, while Eq. (3j) defines the squared current magnitude in terms of active and reactive power flows and the voltage at the sending bus. System security and equipment limits are upheld through several constraints. The apparent power flow on each line must not exceed its thermal limit, adjusted for contingency-related unavailability in Eq. (3k). Voltage magnitudes at all buses are constrained within statutory limits in Eq. (3 L), with the substation bus voltage fixed to a reference value in Eq. (3 m) to provide a stable base for the network. Load shedding is bounded between zero and the actual demand at each bus in Eq. (3n), ensuring that curtailment is physically meaningful. Distributed generation units operate within their active power limits, modified by availability factors in Eq. (3o), and their reactive power output is constrained by the capability curve in Eq. (3p), which ensures that the apparent power rating is not exceeded. Network topology considerations are critical for radial distribution systems. Equation (3q) and (3r) enforce radiality using a fictitious power flow method, which guarantees a tree-structured network by ensuring that each non-root bus has exactly one incoming flow. The line switching constraint in Eq. (3s) limits the number of topology changes between consecutive time periods to maintain operational stability. Contingency-specific line availability is handled in Eq. (3t), where lines under contingency are forced out of service. For critical loads, the N-1 security constraint in Eq. (3u) ensures that no load shedding occurs under any single contingency, thereby enhancing resilience. Additionally, transformer tap positions are constrained within their minimum and maximum limits in Eq. (3v), and their effect on voltage regulation is modeled in Eq. (3w), where the voltage transformation ratio adjusts the voltage magnitude across the transformer. Together, these constraints ensure that the network remains secure, feasible, and reliable under all considered scenarios and contingencies.

It is important to note that the contingency set $\:\mathcal{C}$ in this model comprises predefined N-1 scenarios (single-component outages). This approach aligns with standard reliability and security assessment practices in distribution system planning, providing a clear benchmark for evaluating system resilience against the most probable independent failures. The framework demonstrates the system’s capability to maintain operational feasibility and minimize load curtailment under this stringent condition. However, real-world extreme events or protection system misoperations can lead to multiple simultaneous outages or cascading failures. Extending the model to incorporate probabilistic multi-component contingencies and dynamic cascading effects, while computationally intensive, represents a valuable direction for future work to further stress-test the resilience of EV-integrated distribution networks.

$$\:\underset{{V}_{i,t}^{\omega\:,c},{P}_{ij,t}^{\omega\:,c},{L}_{i,t}^{\omega\:,c}}{\text{m}\text{i}\text{n}}\:{F}^{LL}=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:\sum\:_{c\in\:\mathcal{C}}\:{\pi\:}_{\omega\:}{\pi\:}_{c}\left({w}^{shed}\sum\:_{i\in\:\mathcal{N}}\:\:{\rho\:}_{i}^{load}{L}_{i,t}^{\omega\:,c}+{w}^{volt}\sum\:_{i\in\:\mathcal{N}}\:\:{\xi\:}_{i,t}^{\omega\:,c}\right)$$

(3a)

$$\:{\xi\:}_{i,t}^{\omega\:,c}\ge\:{V}_{i,t}^{\omega\:,c}-{V}_{i}^{max}$$

(3b)

$$\:{\xi\:}_{i,t}^{\omega\:,c}\ge\:{V}_{i}^{min}-{V}_{i,t}^{\omega\:,c}$$

(3c)

$$\:{\xi\:}_{i,t}^{\omega\:,c}\ge\:0$$

(3d)

$$\:EN{S}_{i,t}^{\omega\:,c}={L}_{i,t}^{\omega\:,c}{\Delta\:}t\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3e)

$$\:EENS=\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{\omega\:\in\:{\Omega\:}}\:\sum\:_{c\in\:\mathcal{C}}\:\sum\:_{i\in\:\mathcal{N}}\:{\pi\:}_{\omega\:}{\pi\:}_{c}EN{S}_{i,t}^{\omega\:,c}$$

(3f)

$$\:\sum\:_{j:(i,j)\in\:\mathcal{L}}\:{P}_{ij,t}^{\omega\:,c}-\sum\:_{j:(j,i)\in\:\mathcal{L}}\:\left({P}_{ji,t}^{\omega\:,c}-{R}_{ji}{I}_{ji,t}^{2,\omega\:,c}\right)+{P}_{i,t}^{EV}+{P}_{{g}_{i},t}^{\omega\:,c}={P}_{i,t}^{D,\omega\:}-{L}_{i,t}^{\omega\:,c}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3g)

$$\:\sum\:_{j:(i,j)\in\:\mathcal{L}}\:{Q}_{ij,t}^{\omega\:,c}-\sum\:_{j:(j,i)\in\:\mathcal{L}}\:\left({Q}_{ji,t}^{\omega\:,c}-{X}_{ji}{I}_{ji,t}^{2,\omega\:,c}\right)+{Q}_{i,t}^{EV}+{Q}_{{g}_{i},t}^{\omega\:,c}={Q}_{i,t}^{D,\omega\:}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3h)

$$\:{\left({V}_{i,t}^{\omega\:,c}\right)}^{2}={\left({V}_{j,t}^{\omega\:,c}\right)}^{2}-2\left({R}_{ij}{P}_{ij,t}^{\omega\:,c}+{X}_{ij}{Q}_{ij,t}^{\omega\:,c}\right)+\left({R}_{ij}^{2}+{X}_{ij}^{2}\right){I}_{ij,t}^{2,\omega\:,c}\:\forall\:(i,j)\in\:\mathcal{L},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3i)

$$\:{I}_{ij,t}^{2,\omega\:,c}=\frac{{\left({P}_{ij,t}^{\omega\:,c}\right)}^{2}+{\left({Q}_{ij,t}^{\omega\:,c}\right)}^{2}}{{\left({V}_{i,t}^{\omega\:,c}\right)}^{2}}\:\forall\:(i,j)\in\:\mathcal{L},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3j)

$$\:\sqrt{{\left({P}_{ij,t}^{\omega\:,c}\right)}^{2}+{\left({Q}_{ij,t}^{\omega\:,c}\right)}^{2}}\le\:{S}_{ij}^{max}\left(1-{U}_{ij}^{c}\right)\:\forall\:(i,j)\in\:\mathcal{L},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3k)

$$\:{V}_{i}^{min}\le\:{V}_{i,t}^{\omega\:,c}\le\:{V}_{i}^{max}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3l)

$$\:{V}_{1,t}^{\omega\:,c}={V}_{1}^{ref}\:\forall\:t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3m)

$$\:0\le\:{L}_{i,t}^{\omega\:,c}\le\:{P}_{i,t}^{D,\omega\:}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3n)

$$\:{P}_{g}^{min}{A}_{g,t}^{avail}\le\:{P}_{g,t}^{\omega\:,c}\le\:{P}_{g}^{max}{A}_{g,t}^{avail}\:\forall\:g\in\:\mathcal{G},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3o)

$$\:{\left({Q}_{g,t}^{\omega\:,c}\right)}^{2}\le\:{\left({S}_{g}^{max}\right)}^{2}-{\left({P}_{g,t}^{\omega\:,c}\right)}^{2}\:\forall\:g\in\:\mathcal{G},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3p)

$$\:\sum\:_{j:(i,j)\in\:\mathcal{L}}\:{f}_{ij,t}^{\omega\:,c}-\sum\:_{j:(j,i)\in\:\mathcal{L}}\:{f}_{ji,t}^{\omega\:,c}=1\:\forall\:i\in\:\mathcal{N}\left\{1\right\},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3q)

$$\:0\le\:{f}_{ij,t}^{\omega\:,c}\le\:\left|\mathcal{N}\right|{z}_{ij,t}^{\omega\:,c}\:\forall\:(i,j)\in\:\mathcal{L},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3r)

$$\:\sum\:_{(i,j)\in\:\mathcal{L}}\:\left|{z}_{ij,t}^{\omega\:,c}-{z}_{ij,t-1}^{\omega\:,c}\right|\le\:{N}^{max,switch}\:\forall\:t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3s)

$$\:{z}_{ij,t}^{\omega\:,c}\le\:1-{U}_{ij}^{c}\:\forall\:(i,j)\in\:\mathcal{L},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3t)

$$\:{L}_{i,t}^{\omega\:,c}=0\:\forall\:i\in\:{\mathcal{N}}^{critical},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:{\mathcal{C}}^{N-1}$$

(3u)

$$\:{T}_{k}^{min}\le\:{T}_{k,t}^{\omega\:,c}\le\:{T}_{k}^{max}\:\forall\:k\in\:{\mathcal{T}}_{trans},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3v)

$$\:{V}_{j,t}^{\omega\:,c}={T}_{k,t}^{\omega\:,c}{V}_{i,t}^{\omega\:,c}\:\forall\:k\in\:{\mathcal{T}}_{trans},(i,j)\in\:{\mathcal{L}}_{k},t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(3w)

Reliability indices calculation

The calculation of reliability indices provides a quantitative assessment of the distribution system’s performance from the customer’s perspective. Equation (4a) defines the System Average Interruption Frequency Index, which represents the average number of sustained interruptions that a customer experiences over a year. This index is computed by summing the expected interruption frequencies for all customers across the network and dividing by the total number of customers. The customer interruption frequency at a specific bus, given by Eq. (4b), is determined by considering the failure rates of all upstream lines that could cause an outage at that bus. For a fault on a particular line to cause an interruption at the bus, that specific line must fail while all other upstream protective devices must not operate to isolate the fault, which is captured by the product of terms accounting for the unavailability of other lines. Equation (4c) calculates the System Average Interruption Duration Index, measuring the total average duration of interruption per customer per year. It is the sum of the product of interruption frequency and duration for each customer, divided by the total number of customers. A key innovation in this model is the modification of the restoration time in Eq. (4d), where the base restoration time is reduced by a factor proportional to the available power from vehicle-to-grid operations relative to the total load on the affected feeder. This formulation explicitly captures the potential for electric vehicles to support critical loads during an outage, thereby shortening the effective restoration duration as perceived by the customers. The Customer Average Interruption Duration Index in Eq. (4e) is derived as the ratio of the System Average Interruption Duration Index to the System Average Interruption Frequency Index. This index represents the average time required to restore service after an interruption occurs, providing a measure of the utility’s responsiveness to faults. Equation (4f) defines the Average Service Availability Index, which quantifies the fraction of time that customers have power available over a year. It is calculated as the ratio of the total customer-hours of available service to the total customer-hours demanded, which is the product of the number of customers and the number of hours in a year. The numerator subtracts the expected customer-hours of interruption from the total possible hours. Finally, Eq. (4 g) presents the Momentary Average Interruption Frequency Index, which focuses on short-duration interruptions or momentary events. This index is calculated similarly to the System Average Interruption Frequency Index but considers only a specific subset of contingency scenarios deemed to cause momentary interruptions, reflecting events where automated switching operations successfully restore power quickly without prolonged customer outages.

$$\:SAIFI=\frac{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}\sum\:_{c\in\:\mathcal{C}}\:\:{\lambda\:}_{i}^{c}}{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}}$$

(4a)

$$\:{\lambda\:}_{i}^{c}=\sum\:_{l\in\:{\mathcal{P}}_{i}}\:{\lambda\:}_{l}\prod\:_{{l}^{{\prime\:}}\in\:{\mathcal{P}}_{i}\setminus\:\left\{l\right\}}\:\left(1-{U}_{{l}^{{\prime\:}}}^{c}\right)$$

(4b)

$$\:SAIDI=\frac{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}\sum\:_{c\in\:\mathcal{C}}\:\:{\lambda\:}_{i}^{c}{r}_{i}^{c}}{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}}$$

(4c)

$$\:{r}_{i}^{c}={r}_{i}^{base}\left(1-{\beta\:}^{EV}\frac{{P}_{i}^{EV,available}}{\sum\:_{j\in\:{\mathcal{P}}_{i}}\:\:{P}_{j}^{D}}\right)$$

(4d)

$$\:CAIDI=\frac{SAIDI}{SAIFI}$$

(4e)

$$\:ASAI=\frac{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}\left(8760-\sum\:_{c\in\:\mathcal{C}}\:\:{\lambda\:}_{i}^{c}{r}_{i}^{c}\right)}{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}\times\:8760}$$

(4f)

$$\:MAIFI=\frac{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}\sum\:_{c\in\:{\mathcal{C}}^{mom}}\:\:{\lambda\:}_{i}^{c}}{\sum\:_{i\in\:\mathcal{N}}\:\:{N}_{i}^{cust}}$$

(4g)

Linking constraints between levels

The linking constraints are essential for maintaining consistency and feasibility across the tri-level optimization framework, ensuring that decisions made at one level are physically and operationally compatible with the others. Equation (5a) establishes a critical connection between the middle and upper levels by defining the available power from electric vehicles at each bus. This quantity is calculated from the scheduled state of charge of each vehicle, minus a minimum reserve, converted to an available power discharge rate by considering the battery capacity, the time step duration, and the discharging efficiency. This variable is fundamental for the upper level to assess the reliability contribution of the vehicle fleet. Equation (5b) links the upper-level infrastructure planning with the middle-level operational scheduling by enforcing a station energy feasibility constraint. It ensures that the total energy supplied for charging at a station over the scheduling horizon does not exceed a permissible multiple of the station’s installed power capacity. This constraint prevents the over-utilization of a station’s energy delivery capability, linking the strategic capacity decision with the cumulative operational energy flow. The consistency of the reliability improvement metric between the middle and lower levels is enforced by Eq. (5c). It equates the reliability contribution calculated in the middle level, which is based on vehicle availability, to the actual performance evaluated in the lower level, which is based on the avoided load curtailment during contingencies. This creates a feedback loop where the operational scheduling of vehicles is directly judged by its impact on the physical system’s reliability under stress. Equation (5d) imposes a system-wide voltage stability margin requirement. It mandates that the sum of the voltage differences from the minimum limit across all buses must exceed a specified threshold, ensuring that the solution from the lower level maintains an adequate buffer against voltage collapse. This requirement propagates a voltage stability consideration upwards through the optimization hierarchy. A straightforward but vital power balance verification is performed by Eq. (5e). It confirms that the net active power injection from electric vehicles aggregated across all buses in the network equals the sum of the net discharge from all individual vehicles, ensuring that the aggregation performed in the middle level is consistent with the representation used in the lower-level power flow. Equation (5f) provides a long-term energy deliverability check, linking the cumulative energy discharged by the entire vehicle fleet to the total installed capacity of all charging stations over the planning horizon. This constraint ensures that the strategic infrastructure investment in station capacity is sufficient to support the aggregate vehicle-to-grid energy provision scheduled in the middle level. Finally, Eq. (5 g) creates a direct link between the vehicle-to-grid capability and the network’s reconfiguration logic in the lower level. It bounds the total load curtailment during a contingency by the maximum amount of load that cannot be served by the combined available resources from distributed generation and electric vehicles. This constraint explicitly coordinates the discharge capability of the vehicles with the network topology control to maximize load restoration during fault conditions.

$$\:{P}_{i,t}^{EV,available}=\sum\:_{v\in\:{\mathcal{V}}_{i}}\:\left(SO{C}_{v,t}-SO{C}_{v}^{min}\right)\frac{{E}_{v}^{cap}}{{\Delta\:}t}\frac{1}{{\eta\:}_{v}^{dis}}\:\forall\:i\in\:\mathcal{N},t\in\:\mathcal{T}$$

(5a)

$$\:\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{v\in\:\mathcal{V}}\:{y}_{vs}{P}_{v,t}^{ch}{\Delta\:}t\le\:{{\Gamma\:}}^{energy}{P}_{s}^{cap}\:\forall\:s\in\:\mathcal{S}$$

(5b)

$$\:{R}_{t}^{reliability}=\sum\:_{c\in\:\mathcal{C}}\:{\pi\:}_{c}\left(\sum\:_{i\in\:\mathcal{N}}\:\:{P}_{i,t}^{D,\omega\:}-\sum\:_{i\in\:\mathcal{N}}\:\:{L}_{i,t}^{\omega\:,c}\right)\:\forall\:t\in\:\mathcal{T},\omega\:\in\:{\Omega\:}$$

(5c)

$$\:\sum\:_{i\in\:\mathcal{N}}\:\left({V}_{i,t}^{\omega\:,c}-{V}_{i}^{min}\right)\ge\:VS{M}^{min}\left|\mathcal{N}\right|\:\forall\:t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(5d)

$$\:\sum\:_{i\in\:\mathcal{N}}\:{P}_{i,t}^{EV}=\sum\:_{v\in\:\mathcal{V}}\:\left({P}_{v,t}^{dis}-{P}_{v,t}^{ch}\right)\:\forall\:t\in\:\mathcal{T}$$

(5e)

$$\:\sum\:_{t\in\:\mathcal{T}}\:\sum\:_{v\in\:\mathcal{V}}\:{P}_{v,t}^{dis}{\Delta\:}t\le\:\sum\:_{s\in\:\mathcal{S}}\:{P}_{s}^{cap}{{\Phi\:}}^{util}{T}^{horizon}$$

(5f)

$$\:\sum\:_{i\in\:\mathcal{N}}\:{L}_{i,t}^{\omega\:,c}\le\:\sum\:_{i\in\:\mathcal{N}}\:\text{m}\text{a}\text{x}\left\{0,{P}_{i,t}^{D,\omega\:}-{P}_{i,t}^{EV}-\sum\:_{g\in\:{\mathcal{G}}_{i}}\:\:{P}_{g,t}^{\omega\:,c}\right\}\:\forall\:t\in\:\mathcal{T},\omega\:\in\:{\Omega\:},c\in\:\mathcal{C}$$

(5g)

Enhanced gray wolf optimization algorithm

This section describes the metaheuristic algorithm developed to solve the complex, non-convex tri-level optimization problem formulated in Sect. 2. It covers the solution encoding, the population initialization strategy, the fitness evaluation procedure that coordinates the three levels, and the enhanced search operators (e.g., chaotic maps, adaptive mutation, Lévy flight). The pseudo-code in Fig. 2 summarizes the overall procedure. The specific parameter values used for the algorithm in the case study are provided later in Table 7 of Sect. 4. The enhanced gray wolf optimization algorithm is designed to solve the complex tri-level optimization problem through a coordinated metaheuristic approach. The solution encoding, defined in Eq. (6), represents the complete set of decision variables from all three optimization levels within a single, hierarchical position vector for each member of the population. This vector is partitioned into three distinct segments. The upper-level segment, described by Eq. (7), encodes the strategic decisions regarding charging station placement, including the binary establishment variables, the continuous capacity variables, and the binary vehicle-to-station assignment variables. The middle-level segment, given by Eq. (8), encapsulates the operational scheduling of the electric vehicle fleet, comprising the charging and discharging power levels and their corresponding binary mode indicators across all time periods. The lower-level segment, specified in Eq. (9), contains the network operational variables, such as bus voltage magnitudes and line statuses, under various uncertainty and contingency scenarios.

$$\:{\overrightarrow{X}}_{z}=\left[{\overrightarrow{X}}_{z}^{UL},{\overrightarrow{X}}_{z}^{ML},{\overrightarrow{X}}_{z}^{LL}\right]$$

(6)

$$\:{\overrightarrow{X}}_{z}^{UL}=\left[{x}_{{s}_{1}},\dots\:,{x}_{{s}_{\left|\mathcal{S}\right|}},{P}_{{s}_{1}}^{cap},\dots\:,{P}_{{s}_{\left|\mathcal{S}\right|}}^{cap},{y}_{{v}_{1}{s}_{1}},\dots\:,{y}_{{v}_{\left|\mathcal{V}\right|}{s}_{\left|\mathcal{S}\right|}}\right]$$

(7)

$$\:{\overrightarrow{X}}_{z}^{ML}=\left[{P}_{{v}_{1},{t}_{1}}^{ch},\dots\:,{P}_{{v}_{\left|\mathcal{V}\right|},{t}_{\left|\mathcal{T}\right|}}^{ch},{P}_{{v}_{1},{t}_{1}}^{dis},\dots\:,{P}_{{v}_{\left|\mathcal{V}\right|},{t}_{\left|\mathcal{T}\right|}}^{dis},{u}_{{v}_{1},{t}_{1}}^{ch},\dots\:,{u}_{{v}_{\left|\mathcal{V}\right|},{t}_{\left|\mathcal{T}\right|}}^{dis}\right]$$

(8)

$$\:{\overrightarrow{X}}_{z}^{LL}=\left[{V}_{{i}_{1},{t}_{1}}^{{\omega\:}_{1},{c}_{1}},\dots\:,{V}_{{i}_{\left|\mathcal{N}\right|},{t}_{\left|\mathcal{T}\right|}}^{{\omega\:}_{\left|{\Omega\:}\right|},{c}_{\left|\mathcal{C}\right|}},{z}_{i{j}_{1},{t}_{1}},\dots\:,{z}_{i{j}_{\left|\mathcal{L}\right|},{t}_{\left|\mathcal{T}\right|}}\right]$$

(9)

To initialize the population with high diversity, the algorithm employs a chaotic mapping technique. The initial position of each wolf in each dimension is generated using Eq. (10), which maps a chaotic sequence onto the feasible range of that decision variable. The chaotic sequence itself is produced by the logistic map in Eq. (11), which exhibits sensitive dependence on initial conditions, ensuring that the initial population is widely dispersed across the search space. This process is iterated for a predetermined number of steps as in Eq. (12) to achieve sufficient chaotic mixing before assigning the final value.

$$\:{x}_{z,d}^{\left(0\right)}={x}_{d}^{min}+\left({x}_{d}^{max}-{x}_{d}^{min}\right){\chi\:}_{z,d}$$

(10)

$$\:{\chi\:}_{z,d}^{(k+1)}=\mu\:{\chi\:}_{z,d}^{\left(k\right)}\left(1-{\chi\:}_{z,d}^{\left(k\right)}\right)$$

(11)

$$\:{\chi\:}_{z,d}={\chi\:}_{z,d}^{\left({K}_{chaos}\right)}$$

(12)

The fitness of each candidate solution is evaluated using a hierarchical procedure that accounts for the interconnected nature of the tri-level problem. The fitness function in Eq. (13) combines the upper-level objective function value with a penalty term for constraint violations. This penalty term is the sum of squared violations, each weighted by an adaptive penalty coefficient. The adaptation of these coefficients, governed by Eq. (14), is dynamic; it increases the penalty if the maximum violation of a constraint across the population remains above a tolerance threshold and decreases it otherwise, thereby effectively guiding the search toward feasible regions.

$$\:Fitness\left({\overrightarrow{X}}_{z}\right)={F}^{UL}\left({\overrightarrow{X}}_{z}\right)+\sum\:_{j=1}^{{N}_{cons}}\:{\sigma\:}_{j}^{\left(iter\right)}\text{m}\text{a}\text{x}{\left\{0,{g}_{j}\left({\overrightarrow{X}}_{z}\right)\right\}}^{2}$$

(13)

$$\:\sigma \:_j^{(iter + 1)} = \left\{ {\begin{array}{*{20}{l}} {\beta {\:_{inc}}\sigma \:_j^{\left( {iter} \right)}}&{\:{\text{if}}\:\mathop {{\text{max}}}\limits_{\text{z}} \:{{\text{g}}_{\text{j}}}\left( {{{{{\vec X}}}_{\text{z}}}} \right)> {\varepsilon _{{{tol}}}}} \\ {\:\sigma \:_j^{\left( {iter} \right)}/\beta {\:_{dec}}}&{\:{\text{if}}\:\mathop {{{max}}}\limits_{\text{z}} \:{{\text{g}}_{\text{j}}}\left( {{{{{\vec X}}}_{\text{z}}}} \right) < {\varepsilon _{{\text{tol}}}}} \\ {\:\sigma \:_j^{\left( {iter} \right)}}&{\:{{otherwise}}} \end{array}} \right.$$

(14)

The algorithm mimics the social hierarchy of a wolf pack. The best solution in the population is designated the alpha wolf, as per Eq. (15). The second and third best solutions are identified as the beta and delta wolves, respectively, through Eqs. (16) and (17). The remaining solutions are categorized as omega wolves. This hierarchy is central to the position update mechanism.

$$\:{\overrightarrow{X}}_{\alpha\:}=\text{a}\text{r}\text{g}\underset{{\overrightarrow{X}}_{z}\in\:\mathcal{Z}}{\text{m}\text{i}\text{n}}\:Fitness\left({\overrightarrow{X}}_{z}\right)$$

(15)

$$\:{\overrightarrow{X}}_{\beta\:}=\text{a}\text{r}\text{g}\underset{{\overrightarrow{X}}_{z}\in\:\mathcal{Z}\setminus\:\left\{{\overrightarrow{X}}_{\alpha\:}\right\}}{\text{m}\text{i}\text{n}}\:Fitness\left({\overrightarrow{X}}_{z}\right)$$

(16)

$$\:{\overrightarrow{X}}_{\delta\:}=\text{a}\text{r}\text{g}\underset{{\overrightarrow{X}}_{z}\in\:\mathcal{Z}\setminus\:\{{\overrightarrow{X}}_{\alpha\:},{\overrightarrow{X}}_{\beta\:}\}}{\text{m}\text{i}\text{n}}\:Fitness\left({\overrightarrow{X}}_{z}\right)$$

(17)

The core update mechanism simulates the wolves encircling their prey. Each omega wolf updates its position by calculating its distance to the alpha, beta, and delta leaders using Eqs. (18), (19), and (20). These distances are then used to determine intermediate new positions toward each leader, as shown in Eqs. (21), (22), and (23). The final new position for the wolf is the arithmetic mean of these three intermediate positions, given by Eq. (24).

$$\:{\overrightarrow{D}}_{\alpha\:}=\left|{\overrightarrow{C}}_{1}\cdot\:{\overrightarrow{X}}_{\alpha\:}-{\overrightarrow{X}}_{z}\right|$$

(18)

$$\:{\overrightarrow{D}}_{\beta\:}=\left|{\overrightarrow{C}}_{2}\cdot\:{\overrightarrow{X}}_{\beta\:}-{\overrightarrow{X}}_{z}\right|$$

(19)

$$\:{\overrightarrow{D}}_{\delta\:}=\left|{\overrightarrow{C}}_{3}\cdot\:{\overrightarrow{X}}_{\delta\:}-{\overrightarrow{X}}_{z}\right|$$

(20)

$$\:{\overrightarrow{X}}_{1}={\overrightarrow{X}}_{\alpha\:}-{\overrightarrow{A}}_{1}\cdot\:{\overrightarrow{D}}_{\alpha\:}$$

(21)

$$\:{\overrightarrow{X}}_{2}={\overrightarrow{X}}_{\beta\:}-{\overrightarrow{A}}_{2}\cdot\:{\overrightarrow{D}}_{\beta\:}$$

(22)

$$\:{\overrightarrow{X}}_{3}={\overrightarrow{X}}_{\delta\:}-{\overrightarrow{A}}_{3}\cdot\:{\overrightarrow{D}}_{\delta\:}$$

(23)

$$\:{\overrightarrow{X}}_{z}^{(iter+1)}=\frac{{\overrightarrow{X}}_{1}+{\overrightarrow{X}}_{2}+{\overrightarrow{X}}_{3}}{3}$$

(24)

The balance between exploration and exploitation is controlled by coefficient vectors. The vector A, defined in Eq. (25), is derived from a component a that decreases nonlinearly over iterations according to Eq. (26). This nonlinear decrease, controlled by the exponent kappa, allows for a more refined control over the convergence behavior compared to a linear decrease. The vector C, in Eq. (27), provides random weights that enhance the stochastic exploration during the position update.

$$\:\overrightarrow{A}=2\overrightarrow{a}\cdot\:{\overrightarrow{r}}_{1}-\overrightarrow{a}$$

(25)

$$\:a=2-2{\left(\frac{iter}{ite{r}_{max}}\right)}^{\kappa\:}$$

(26)

$$\:\overrightarrow{C}=2{\overrightarrow{r}}_{2}$$

(27)

To prevent premature convergence, an adaptive mutation operator is incorporated. The probability of mutation for a given iteration, defined in Eq. (28), is inversely related to the population diversity. The diversity metric, calculated in Eq. (29), is the average Euclidean distance of the wolves from the population’s mean position, which is given by Eq. (30). When mutation occurs, the new position is perturbed using a combination of polynomial mutation and Gaussian noise, as specified in Eqs. (31) and (32).

$$\:{P}_{mut}^{\left(iter\right)}={P}_{mut}^{min}+\left({P}_{mut}^{max}-{P}_{mut}^{min}\right){e}^{-\gamma\:\cdot\:DI{V}^{\left(iter\right)}}$$

(28)

$$\:DI{V}^{\left(iter\right)}=\frac{1}{\left|\mathcal{Z}\right|}\sum\:_{z\in\:\mathcal{Z}}\:\sqrt{\sum\:_{d=1}^{D}\:\:{\left({x}_{z,d}-{\stackrel\leftharpoonup{x}}_{d}\right)}^{2}}$$

(29)

$$\:{\stackrel{\leftharpoonup}{x}}_{d}=\frac{1}{\left|\mathcal{Z}\right|}\sum\:_{z\in\:\mathcal{Z}}\:{x}_{z,d}$$

(30)

$$\:{x}_{z,d}^{mut}={x}_{z,d}+{\delta\:}_{d}\left({x}_{d}^{max}-{x}_{d}^{min}\right)+N(0,{\sigma\:}_{mut})$$

(31)

$$\:{\delta\:}_{d}=\left\{\begin{array}{ll}{\left(2{r}_{d}\right)}^{\frac{1}{{\eta\:}_{mut}+1}}-1&\:\text{if\:}{r}_{d}<0.5\\\:1-{\left(2(1-{r}_{d})\right)}^{\frac{1}{{\eta\:}_{mut}+1}}&\:\text{otherwise}\end{array}\right.$$

(32)

A dimensional learning strategy is implemented to focus search effort on the most promising decision variables. A contribution score for each variable dimension is computed in Eq. (33) based on the average improvement in fitness observed when that dimension changes between iterations. This score is then converted into a learning probability in Eq. (34), which is used to select which dimensions to update preferentially.

$${\:S_d^{\left( {iter} \right)} = \frac{{\sum {{\:_{z \in \:\mathcal{Z}}}} \:\:\left| {Fitness\left( {\vec X_z^{(iter - 1)}} \right) - Fitness\left( {\vec X_z^{\left( {iter} \right)}} \right)} \right| \cdot \:I(x_{z,d}^{\left( {iter} \right)} \ne \:x_{z,d}^{(iter - 1)})}}{{\sum {{\:_{z \in \:\mathcal{Z}}}} \:\:I(x_{z,d}^{\left( {iter} \right)} \ne \:x_{z,d}^{(iter - 1)}) + \varepsilon }}}$$

(33)

$$\:{P}_{d}^{learn}=\frac{{S}_{d}^{\left(iter\right)}}{\sum\:_{{d}^{{\prime\:}}=1}^{D}\:\:{S}_{{d}^{{\prime\:}}}^{\left(iter\right)}}$$

(34)

The algorithm also utilizes opposition-based learning to explore the symmetric opposite of current solutions. The opposite of a solution is defined in Eq. (35). The probability of generating such an opposite solution decreases linearly with the iteration count, as per Eq. (36). If generated, the algorithm retains the better of the current and the opposite solution, following the rule in Eq. (37).

$$\:{\overrightarrow{X}}_{z}^{opp}={\overrightarrow{x}}^{min}+{\overrightarrow{x}}^{max}-{\overrightarrow{X}}_{z}$$

(35)

$$\:{P}_{opp}^{\left(iter\right)}={P}_{opp}^{init}\left(1-\frac{iter}{ite{r}_{max}}\right)$$

(36)

$$\:{\overrightarrow{X}}_{z}^{(iter+1)}=\left\{\begin{array}{ll}{\overrightarrow{X}}_{z}^{opp}&\:\text{if\:}Fitness\left({\overrightarrow{X}}_{z}^{opp}\right)<Fitness\left({\overrightarrow{X}}_{z}\right)\\\:{\overrightarrow{X}}_{z}&\:\text{otherwise}\end{array}\right.$$

(37)

The Levy flight enhancement is incorporated into the algorithm to bolster its global exploration capabilities. This mechanism facilitates occasional long-distance moves within the search space, which are crucial for escaping local optima and investigating promising regions that may be far from the current population’s location. Equation (38) formalizes this move. A new candidate position is generated by taking the current position of a wolf and adding a displacement step. This step is the entrywise product of a step size scaling factor, alpha-levy, and a random vector drawn from a Levy distribution. The entrywise multiplication ensures that each dimension of the solution vector is perturbed independently by a step size governed by this heavy-tailed distribution. The Levy distribution, noted for its heavy tails that permit infrequent, long jumps, is generated computationally using the Mantegna algorithm, as specified in Eq. (39). This method produces a random variable that follows a Levy-like distribution by taking the ratio of two independent Gaussian random variables. The variable u is drawn from a normal distribution with a mean of zero and a variance of sigma-u squared, while v is drawn from a normal distribution with a mean of zero and a variance of one. The calculation of the parameter sigma-u, defined in Eq. (40), is necessary to ensure the generated random variable exhibits the correct statistical properties of the Levy distribution. This formula involves the Gamma function and is dependent on the stability parameter lambda, which influences the heaviness of the distribution’s tail. A common choice for lambda is 1.5, which provides a good balance between frequent small steps and occasional large ones. Equation (41) specifies that the standard deviation, sigma-v, for the Gaussian random variable v is set to a constant value of one. This value is a conventional choice in the Mantegna algorithm for generating stable random numbers that approximate the Levy distribution. The variable v is sampled from a standard normal distribution, which has a mean of zero and a unit variance. This simplification is part of the standardized procedure for computationally generating the step size, and it works in conjunction with the specifically calculated sigma-u from Eq. (40) to ensure the resulting random variable exhibits the desired heavy-tailed statistical properties. The frequency at which the Levy flight is applied is not constant but decays over the course of the optimization. Equation (42) defines this triggering probability, which starts at a maximum value and decreases exponentially as the iteration count increases. This decaying probability schedule reflects a shifting search strategy; early in the optimization, the algorithm prioritizes broad exploration, while later, it focuses more on refining solutions through local exploitation. The rate of this decay is controlled by the parameter rho-levy. This combination of a heavy-tailed step size distribution and an iteration-dependent application probability provides a powerful mechanism for enhancing the metaheuristic’s ability to navigate complex, multi-modal search spaces effectively, thereby increasing the likelihood of discovering a high-quality global optimum for the tri-level optimization problem.

$$\:{\overrightarrow{X}}_{z}^{levy}={\overrightarrow{X}}_{z}+{\alpha\:}_{levy}\oplus\:Levy\left(\lambda\:\right)$$

(38)

$$\:Levy\left(\lambda\:\right)=\frac{u}{|v{|}^{1/\lambda\:}}$$

(39)

$$\:{\sigma\:}_{u}={\left(\frac{{\Gamma\:}(1+\lambda\:)\text{s}\text{i}\text{n}(\pi\:\lambda\:/2)}{{\Gamma\:}\left(\right(1+\lambda\:)/2)\lambda\:{2}^{(\lambda\:-1)/2}}\right)}^{1/\lambda\:}$$

(40)

$$\:{\sigma\:}_{v}=1$$

(41)

$$\:{P}_{levy}^{\left(iter\right)}={P}_{levy}^{max}{e}^{-{\rho\:}_{levy}\cdot\:iter}$$

(42)

The hierarchical solution procedure coordinates the tri-level optimization through an iterative process where the enhanced gray wolf algorithm acts as the master framework. For each candidate solution, represented by a wolf’s position vector, the evaluation unfolds sequentially across the three levels. First, the upper-level variables are extracted to compute the objective function associated with charging infrastructure investment and placement. With these upper-level decisions fixed, the procedure then solves the middle-level subproblem to determine the optimal charging and discharging schedule for the electric vehicle fleet, respecting the constraints on state of charge evolution and power limits. Subsequently, both the upper and middle-level variables are fixed, and the lower-level subproblem is solved to evaluate network operation under contingency scenarios, minimizing load shedding and voltage deviations while satisfying the power flow and reliability constraints.

The total fitness of a solution is calculated by combining the objective function values from all three levels with an adaptive weighting scheme and a penalty for constraint violations, as defined in Eq. (43). The weights for each level’s objective are not static but are dynamically adjusted based on their recent progress toward convergence. Equation (44) governs this adaptation, where the weight for a particular level is increased if it shows larger relative improvement compared to the other levels, thereby focusing the search on aspects of the problem that are currently less refined.

$$\:Fitnes{s}_{total}\left({\overrightarrow{X}}_{z}\right)={w}_{1}^{\left(iter\right)}{F}^{UL}+{w}_{2}^{\left(iter\right)}{F}^{ML}+{w}_{3}^{\left(iter\right)}{F}^{LL}+{{\Phi\:}}_{penalty}$$

(43)

$$\:{w}_{j}^{(iter+1)}={w}_{j}^{\left(iter\right)}\left(1+\tau\:\frac{{\Delta\:}{F}_{j}^{\left(iter\right)}}{\sum\:_{{j}^{{\prime\:}}}\:\:{\Delta\:}{F}_{{j}^{{\prime\:}}}^{\left(iter\right)}}\right)$$

(44)

When the algorithm identifies that the population is converging toward a promising region of the search space, a local search intensification mechanism is activated to refine the current best solution. This local search employs a pattern move method, generating trial points around the alpha solution by perturbing it along coordinate and diagonal directions, as specified in Eq. (45). The step size for this perturbation is adaptive; it expands when an improving solution is found and contracts otherwise, according to the rule in Eq. (46). This process continues until the step size diminishes below a minimum threshold or a maximum number of local iterations is completed.

$${\:{{\vec X}^{local}} = {{\vec X}_{\alpha \:}} + {\varepsilon _{local}}{{\vec d}_k}}$$

(45)

$$\:{\epsilon}_{local}^{(k+1)}=\left\{\begin{array}{ll}{\rho\:}_{exp}{\epsilon}_{local}^{\left(k\right)}&\:\text{if\:improvement\:found}\\\:{\rho\:}_{con}{\epsilon}_{local}^{\left(k\right)}&\:{otherwise}\end{array}\right.$$

(46)

For handling the multi-objective nature of the problem, the algorithm maintains an external archive of non-dominated solutions. The criterion for dominance, given by Eq. (47), states that one solution dominates another if it is at least as good in all objectives and strictly better in at least one. The archive is updated each iteration by adding new non-dominated solutions from the current population and removing any solutions that become dominated. If the archive exceeds its capacity, pruning is performed based on a crowding distance metric, calculated in Eq. (48), which preserves solutions located in less crowded regions of the objective space to maintain diversity.

$$\:{\overrightarrow{X}}_{a}\prec\:{\overrightarrow{X}}_{b}\:{\Leftrightarrow}\:\forall\:i:{F}_{i}\left({\overrightarrow{X}}_{a}\right)\le\:{F}_{i}\left({\overrightarrow{X}}_{b}\right)\wedge\:\exists\:j:{F}_{j}\left({\overrightarrow{X}}_{a}\right)<{F}_{j}\left({\overrightarrow{X}}_{b}\right)$$

(47)

$$\:C{D}_{i}=\sum\:_{m=1}^{{M}_{obj}}\:\frac{{f}_{m}^{i+1}-{f}_{m}^{i-1}}{{f}_{m}^{max}-{f}_{m}^{min}}$$

(48)

The algorithm terminates based on several practical criteria. These include reaching a maximum number of iterations, observing stagnating improvement in the best fitness value over a consecutive number of iterations, achieving feasibility where all constraint violations fall below a tolerance level, or exhausting a predefined computational budget for function evaluations. Upon termination, the solution held by the alpha wolf is returned, representing the integrated strategy for charging station placement, vehicle scheduling, and network operation that enhances distribution system reliability.

$$\:Complexity=O\left(\left|\mathcal{Z}\right|\cdot\:\left(\left|\mathcal{S}\right|\cdot\:\left|\mathcal{V}\right|+\left|\mathcal{V}\right|\cdot\:|\mathcal{T}{|}^{2}+|\mathcal{N}{|}^{3}\cdot\:\left|\mathcal{C}\right|\cdot\:\left|{\Omega\:}\right|\right)\right)$$

(49)

An analysis of the computational complexity per iteration reveals that the cost is dominated by the tri-level problem evaluation. The upper level contributes a complexity proportional to the product of the number of candidate stations and vehicles. The middle level complexity scales with the number of vehicles and the square of the time periods. The most computationally expensive component is the lower level, which involves power flow calculations with contingency analysis, scaling with the cube of the number of buses multiplied by the number of contingencies and uncertainty scenarios. The overhead from the enhanced mechanisms—chaotic initialization, mutation, opposition-based learning, and Levy flight—scales linearly with the population size and solution dimension, but this cost is marginal compared to the expense of evaluating the tri-level problem, while providing significant benefits in solution quality and convergence performance.

Fig. 2

Enhanced Gray Wolf Optimization Algorithm for Tri-Level EV-Distribution System Optimization.

Figure (2) presents the pseudocode for the Enhanced Gray Wolf Optimization Algorithm (EGWOA) applied to the tri-level optimization of an Electric Vehicle (EV)-Distribution System. This algorithm is designed to simultaneously optimize the placement of charging stations, the scheduling of EVs, and the operation of the electrical network, while considering both infrastructure planning and power distribution dynamics. In the figure, the algorithm begins by initializing the wolf population and setting key parameters such as the chaotic map for population initialization, adaptive weights, and the convergence threshold. The main optimization process unfolds in multiple steps, which are executed iteratively for a maximum number of iterations. The first phase, labeled as “Tri-Level Fitness Evaluation,” involves calculating the fitness of each wolf at three levels: upper, middle, and lower. At the upper level, the algorithm extracts station locations and capacities, and computes the infrastructure reliability. The middle level focuses on EV scheduling, where optimal power consumption and discharge values are calculated, factoring in Vehicle-to-Grid (V2G) revenue and battery degradation. The lower level handles network operation, where power flows and voltage levels are optimized under various contingencies and scenarios. The total fitness is computed by combining the results from all three levels and adding penalties for constraint violations. The second phase, “Hierarchy Identification,” ranks the wolves by their fitness and identifies the best, second-best, and third-best solutions (alpha, beta, and delta wolves). Following this, the algorithm proceeds to the “Adaptive Parameter Update” step, where the convergence factor and adaptive weights are updated based on the iteration’s progress. The “Position Update with Hybrid Strategy” step involves a combination of standard Gray Wolf optimization (GWO) and enhanced strategies, such as adaptive mutation and Lévy flight for exploration. These methods help balance exploitation and exploration during the search process, ensuring that the algorithm can effectively navigate both local and global solution spaces. In “Local Search Intensification,” the algorithm further refines the solution by generating trial points around the best solution and expanding or contracting the search step based on the trial results. The “Multi-Objective Archive Update” ensures that only non-dominated solutions are retained in the archive, allowing for the identification of the Pareto front. Finally, the algorithm checks for convergence by monitoring the change in the best fitness value between iterations. If convergence is detected or the maximum number of stagnation iterations is reached, the algorithm terminates early. The optimal solution is then returned along with the Pareto front obtained from the archive. This pseudocode illustrates the integration of multiple optimization strategies within the Gray Wolf framework, providing a robust solution to the complex, multi-objective problem of EV fleet scheduling and distribution system optimization.

Simulation framework and case study system

This section presents the numerical implementation of the proposed framework. It first describes the test system (IEEE 33-bus), defines all simulation parameters and assumptions (detailed in Tables 2, 3, 4, 5, 6, 7, 8 and 9), and outlines the case study setup. It then presents and analyzes the results, including the optimal infrastructure plan, operational schedules, reliability improvements, economic analysis, and algorithmic performance comparisons. Sensitivity analyses are provided to test the robustness of the findings to key assumptions.

The proposed tri-level optimization model, integrated with the Enhanced Gray Wolf Evolutionary Algorithm, is applied to the standard IEEE 33-bus radial distribution system for performance assessment. This network is widely used as a benchmark in distribution system studies and provides adequate structural and operational complexity for examining the interaction between electric vehicle deployment and system reliability. The system operates at a nominal voltage level of 12.66 kV and contains 33 buses linked through 32 sectionalizing switches in a purely radial configuration. In addition, five normally open tie switches are available to facilitate network reconfiguration and improve operational flexibility. As summarized in Table (2), the total peak active and reactive power demands are 3715 kW and 2300 kVAr, respectively, distributed across 32 load points representing a blend of residential, commercial, and small industrial customers with varying interruption costs. Under peak loading conditions, the base case configuration experiences voltage drops to approximately 0.9131 p.u. at remote nodes, leading to voltage regulation challenges within the system. The corresponding active power losses amount to 202.67 kW, or roughly 5.46% of the total load, which underscores the need for improved operational strategies. Reliability limitations are also present, particularly under N-1 contingencies where outages of individual lines or transformers can cause extended service interruptions and load curtailment in peripheral areas. To evaluate the impact of electric vehicle integration, the distribution network is modified to include dedicated EV charging infrastructure at selected locations. Eight candidate charging sites are designated at buses 6, 11, 13, 18, 22, 25, 29, and 33 based on spatial distribution, access to load centers, and availability of installation space. Each site supports modular charging capacity in increments of 50 kW up to a maximum of 500 kW. The simulated EV fleet consists of 150 vehicles, representing the projected penetration level within a five-year horizon. The fleet exhibits diversity in battery characteristics, charging rates, and usage patterns to reflect realistic variation across vehicle types. Participation in vehicle-to-grid services is assumed to be voluntary, with financial incentives provided for discharged energy and battery wear. The simulation horizon covers a 24-hour period with hourly resolution. System load profiles follow typical daily characteristics with morning and evening peaks, while EV arrival and departure times are generated stochastically using transportation-survey-based probability distributions. Uncertainty in demand is represented through ten load scenarios, each assigned a probability derived from historical operating data. Reliability attributes, including component failure rates and repair durations, are adopted from utility records and industry guidelines. A total of 35 credible N-1 contingencies form the contingency set, and each event is evaluated across all load scenarios to determine system performance under stressed operating conditions. All technical parameters required for the optimization model are provided in Table (2), which lists the fundamental electrical characteristics of the test system along with operational limits and load modeling assumptions.

Table 2 Distribution System Technical Parameters.

The parameters governing the design and economic modeling of the electric vehicle charging infrastructure are summarized in Table (3). Eight candidate locations are considered for potential station installation, corresponding to nodes 6, 11, 13, 18, 22, 25, 29, and 33. Each location can host modular charging units, where each unit provides a rated power of 50 kW. Station capacity may vary from a minimum of 50 kW to a maximum of 500 kW, with expansions implemented in 50 kW increments. The cost structure incorporates a fixed site preparation cost of 35,000 USD, a charger unit cost of 25,000 USD, and an installation cost of 3,500 USD per unit. Annual operating expenditures are modeled through a 2,500 USD per-site operation cost along with a maintenance cost expressed as 2% of the total initial investment. The economic feasibility of charging station deployment is evaluated over a five-year planning horizon, employing a discount rate of 6% to account for the time value of money. A total budget limit of 500,000 USD is imposed, constraining the maximum realizable infrastructure investment. These parameters collectively define the financial boundaries and scaling behavior of the charging infrastructure within the optimization framework, as detailed in Table (3).

Table 3 Charging Infrastructure Parameters.

Table 4 Electric Vehicle Fleet Characteristics.

Table 5 Reliability and Contingency Parameters.

Table 6 Economic and Operational Parameters.

Table 7 EGWOA Algorithm Parameters.

The characteristics of the electric vehicle fleet used in the simulation are presented in Table (4). Three vehicle classes are considered, representing compact cars, sedans, and sport utility vehicles. Their proportions in the fleet are set to 40%, 45%, and 15%, respectively, reflecting anticipated adoption patterns. Each class exhibits distinct battery capacities and charging and discharging power limits, while maintaining uniform charging and discharging efficiencies of 0.92 and 0.88. The total fleet consists of 150 vehicles, and the state-of-charge constraints follow practical operational guidelines, with allowable SOC ranging from 0.20 to 0.95. The average initial SOC is assumed to be 0.60, and the required SOC upon departure is fixed at 0.75. Battery aging effects are captured through class-specific cycle limits and replacement costs, while daily energy throughput limits are modeled as a fraction of the rated battery capacity for each vehicle class. These parameters collectively define the operational flexibility and degradation behavior of the EV fleet, as summarized in Table (4).

Reliability modeling parameters are compiled in Table (5). The failure rates of overhead lines and transformers follow standard values used in distribution system reliability studies. Average repair times of 4 h for line outages and 8 h for transformer outages are adopted to represent typical field crew response durations. The contingency set includes 35 N-1 scenarios, consistent with the scope of reliability assessment for medium-voltage distribution feeders. Critical loads constitute 30% of the total demand, and separate interruption costs are used for critical and non-critical customers. A minimum reliability improvement target of 25% is enforced within the optimization model. Vehicle-to-grid availability is assumed to be 70%, with a response time of 0.1 h, ensuring adequate responsiveness under contingency conditions. These reliability parameters, shown in Table (5), provide the foundation for evaluating distribution system resilience under uncertain operating conditions.

The economic and operational parameters used to quantify energy costs, penalties, and demand-related expenses are provided in Table (6). Time-of-use electricity pricing is applied with peak, mid-peak, and off-peak rates of 0.18 USD/kWh, 0.12 USD/kWh, and 0.08 USD/kWh, respectively. Compensation for vehicle-to-grid discharging is set at 0.15 USD/kWh. Demand charges of 15 USD/kW-month are included to penalize excessive peak demand contributions from EV charging stations. Additional factors such as the power-factor penalty threshold, network loss cost, voltage deviation penalties, and differentiated load-curtailment penalty multipliers for critical and non-critical loads are also incorporated. These parameters, summarized in Table (6), ensure that the optimization model captures both operational costs and power-quality-related consequences.

The parameter settings of the Enhanced Gray Wolf Optimization Algorithm are listed in Table (7). These include the population size, iteration limits, nonlinearity exponent, chaotic map parameters, and mutation-related coefficients that govern exploratory and exploitative behaviors within the search process. Additional features, such as adaptive opposition mechanisms, Lévy flight dynamics, feasibility tolerances, and stagnation detection intervals, are incorporated to enhance convergence robustness. The combination of these algorithmic parameters, as given in Table (7), ensures that the proposed tri-level optimization framework operates with sufficient diversity and stability to effectively search the high-dimensional decision space. The uncertainty associated with daily load variations is represented through ten distinct demand scenarios, as summarized in Table (8). These scenarios reflect seasonal, weekday, and weekend operating conditions, capturing a broad range of realistic system loading patterns. Each scenario is defined by peak and off-peak load factors that scale the baseline demand profile to emulate conditions ranging from very high summer peaks to low-load weekend periods. Probabilities assigned to the scenarios are derived from historical load statistics and ensure that both frequent and infrequent operating conditions are adequately represented within the stochastic framework. Collectively, the scenario set in Table (8) provides a comprehensive basis for evaluating the performance of the proposed optimization model under diverse and uncertain loading environments.

Table 8 Load Scenario Definitions.

A subset of representative contingency events used in the reliability assessment is summarized in Table (9). These contingencies include both line outages and transformer failures, covering a range of severity levels and operational impacts on the distribution network. Line-related contingencies are modeled with an annual probability of 0.00016 and an average repair duration of 4 h, consistent with typical overhead line performance data. The affected branches span various sections of the feeder, including critical segments such as Branches 1–2, 2–3, and 3–23, all of which exhibit high to very high impact severity due to their proximity to upstream nodes. Other line contingencies, such as those involving Branches 5–6 and 6–26, impose medium to low system impacts. Transformer failures are also incorporated, with annual probabilities of 0.01 and average repair durations of 8 h. The main transformer outage (Contingency C31) is classified as a critical event given its extensive effect on system operation, while outages of individual distribution transformers (C32 and C33) are characterized by high or medium severity depending on the load they serve. Collectively, the contingent events listed in Table (9) form a representative set of credible N-1 disturbances that enable comprehensive evaluation of system reliability within the proposed optimization framework.

Table 9 Selected Contingency Scenarios.

The optimal deployment of electric vehicle charging infrastructure obtained from the proposed tri-level optimization framework is summarized in Table (10). Five stations are selected for installation across the network, located at nodes 6, 13, 22, 29, and 33. The optimization allocates a total of 31 charging units, corresponding to an aggregate installed capacity of 1550 kW. Station S2 at node 13 emerges as the largest installation with ten chargers and a total rated capacity of 500 kW, reflecting its strategic position near major load centers and high EV activity. The remaining stations range from 200 to 350 kW in capacity, ensuring adequate spatial distribution of charging resources throughout the system. The total investment required for the selected configuration amounts to 405,000 USD, with an annual operating cost of 12,500 USD. Vehicle assignment results indicate a balanced utilization pattern, with 150 EVs distributed across the five stations in proportion to their installed capacities. These deployment outcomes, as detailed in Table (10), highlight the model’s ability to balance cost, accessibility, and operational performance while adhering to budgetary and technical constraints.

Table 10 Optimal Charging Infrastructure Deployment.

The aggregate hourly vehicle-to-grid power exchange is presented in Table (11), illustrating the coordinated charging and discharging behavior of the electric vehicle fleet over the 24-hour simulation horizon. During the early morning hours, the fleet predominantly engages in charging, resulting in negative net power injections and a gradual increase in the average state of charge. This trend continues through midday, where the number of vehicles actively charging remains high and discharging does not occur. Beginning at hour 13, the system transitions into bidirectional operation as a portion of the fleet begins supplying power back to the grid. The magnitude of V2G support increases noticeably during the late afternoon and early evening periods, with peak discharging occurring between hours 16 and 18, during which the net power injection reaches more than 800 kW. This behavior corresponds to periods of elevated system demand and aligns with the availability of vehicles parked and connected to the charging infrastructure. In the late evening, the net power flow gradually shifts back toward charging as vehicles prepare for expected departures the following day. The evolution of charging and discharging counts, together with the hourly average state of charge values shown in Table (11), highlights the capacity of the coordinated V2G strategy to support system balancing while maintaining sufficient energy levels for individual vehicle requirements.

Table 11 Hourly Aggregate V2G Power Flow.

Figure (3) illustrates the aggregate V2G (Vehicle-to-Grid) power flow profile over a 24-hour period, alongside the fleet’s average state of charge (SOC). The plot consists of two subgraphs: the first shows the charging and discharging power dynamics, while the second displays the evolution of the fleet’s average state of charge. In the first subgraph, the hourly charging power is depicted in blue, and discharging power is shown in green. Charging power peaks during the early morning and evening hours, reaching up to 412.8 kW at 10 AM. The discharging power, on the other hand, starts to significantly rise from around noon, particularly between hours 13 and 18, reaching a maximum of 892.5 kW at hour 18. This transition from predominantly charging to a more balanced or discharging scenario highlights the role of electric vehicles in providing grid support during peak demand periods. The grid balancing achieved through the V2G system contributes to enhancing system reliability and reducing overall energy consumption from non-renewable sources. The second subgraph illustrates the average state of charge of the fleet throughout the day, represented by a red curve. The state of charge reaches its highest point of 0.73 at hour 8, reflecting the charging period. As the day progresses, especially after hour 12, there is a noticeable drop in the average SOC, which indicates discharging to the grid. The minimum SOC limit, set at 0.20, is shown by the orange dashed line, ensuring that the vehicles maintain sufficient energy for their intended use. The fleet’s SOC, however, stays above this threshold throughout the day, demonstrating an efficient charging and discharging strategy that balances the need for grid support and vehicle usability. Together, these plots reflect the integrated approach of utilizing electric vehicles in a V2G system, highlighting how both charging and discharging actions are managed to optimize the overall grid performance while maintaining the vehicles’ operational readiness.

Fig. 3

Aggregate V2G Power Flow and Fleet Average State of Charge Profile.

The impact of the proposed optimization framework on voltage regulation across the distribution network is summarized in Table (12), which reports voltage profile statistics for all load scenarios. The minimum voltage values under the base case range from 0.9087 p.u. in Scenario S1 to 0.9267 p.u. in Scenario S10, reflecting significant voltage depression during high-demand periods. After optimization, the minimum voltages increase consistently across all scenarios, with improvements between 3.57% and 4.52%. The optimized minimum voltage values—typically exceeding 0.95 p.u.—demonstrate the effectiveness of coordinated EV charging, V2G dispatch, and network adjustments in enhancing overall voltage support. Maximum voltage values remain within acceptable limits (1.0 p.u.) in both the base and optimized cases, ensuring compliance with operational constraints. The voltage deviation index also shows notable improvement, decreasing from base-case values spanning 0.0342 to 0.0487 down to optimized levels between 0.0164 and 0.0214. On average, the deviation index is reduced by more than 50%, indicating a substantially flatter and more stable voltage profile. These results, detailed in Table (12), affirm that the integrated optimization framework significantly mitigates voltage fluctuations across varying seasonal and temporal loading conditions.

Table 12 Voltage Profile Statistics Across Load Scenarios.

The effect of the proposed optimization model on system active power losses is summarized in Table (13). Across all load scenarios, the optimized configuration yields a substantial reduction in real power losses compared with the base case. In heavily loaded conditions, such as Scenario S1, losses decrease from 239.84 kW to 172.35 kW, corresponding to a reduction of 28.14%. Similar improvements are observed in other high-demand scenarios, including S2 and S7, where losses are reduced by more than 26%. Even under moderate and low loading conditions, the optimization consistently lowers losses by approximately 17–24%, demonstrating robust performance across varying operational states. The relative share of losses with respect to total system load also shows meaningful improvement. Base-case loss percentages range from 5.14% to 5.61%, whereas the optimized network exhibits values reduced to approximately 4.03%–4.41%. On average, the active power loss is reduced by 47.05 kW (23.02%), lowering the loss-to-load ratio from 5.41% to 4.16%. These results, provided in Table (13), illustrate the capability of coordinated EV integration, network reconfiguration, and V2G dispatch to significantly enhance distribution system efficiency under diverse load scenarios.

Table 13 Active Power Loss Reduction.

The impact of the proposed optimization on load curtailment under critical contingencies is summarized in Table (14). For each selected contingency, the optimized configuration significantly reduces the amount of load shed compared with the base case. For instance, Contingency C2, representing a line outage with very high impact, sees load curtailment decrease from 2.847 MW to 0.623 MW, corresponding to a reduction of 78.11%. Similar reductions are observed for other major line and transformer contingencies, with reductions ranging from approximately 66% to 76%. Contingency C31, the main transformer outage, experiences a decrease in curtailed load from 3.715 MW to 1.245 MW, highlighting the effectiveness of coordinated EV charging, V2G dispatch, and network reconfiguration even under high-impact events. The reduction in load curtailment is also reflected in the expected energy not served (EENS), which drops from 651.30 MWh/year in the base case to 192.70 MWh/year in the optimized system. Average outage durations remain consistent with system repair times, ensuring realistic operational assessment. Collectively, the data in Table (14) demonstrate that the proposed framework not only mitigates peak load stress but also substantially enhances reliability performance under critical contingencies.

Table 14 Load Curtailment Under Critical Contingencies.

The improvements in system reliability resulting from the proposed optimization approach are summarized in Table (15). The average frequency of interruptions per customer (SAIFI) is reduced from 1.847 to 0.682 interruptions per year, representing a 63.08% improvement and meeting the target value of less than 1.0. Similarly, the system average interruption duration (SAIDI) decreases from 8.234 to 3.156 h per customer-year, achieving a 61.67% reduction and satisfying the target of under 4 h. Although the customer average interruption duration (CAIDI) exhibits a slight increase from 4.458 to 4.627 h per interruption, other reliability indices show marked improvement. The average service availability index (ASAI) rises to 0.99964, exceeding the target of 0.9995, while the expected energy not served (EENS) drops from 651.30 to 192.70 MWh/year, a reduction of 70.42%, and the per-customer energy not supplied decreases from 19.43 to 5.75 kWh/year. The composite reliability index, which captures overall system performance, increases from 0.152 to 0.367, surpassing the target of 0.25. These results, detailed in Table (15), demonstrate that coordinated EV integration, V2G participation, and network optimization can substantially enhance both operational and reliability performance in medium-voltage distribution systems.

Table 15 System Reliability Indices.

The economic performance of the proposed optimization framework is summarized in Table (16), which presents both annual and five-year present value estimates for key cost and benefit components. The initial infrastructure investment for the deployment of EV charging stations totals 405,000 USD, supplemented by annual operation and maintenance costs of 12,500 USD, which cover site operations and equipment upkeep. Charging energy expenses are estimated at 171,340 USD per year, while vehicle-to-grid participation generates compensation payments of 222,500 USD annually to EV owners for providing grid support. The incremental battery degradation cost due to V2G cycling is calculated at 48,760 USD per year. The financial benefits from reliability improvements, including avoided customer interruption costs, are substantial at 1,924,260 USD per year, while reductions in network losses, demand charges, and voltage deviation penalties contribute additional savings. Aggregating these components yields a highly favorable net present value of approximately 7.93 million USD over the five-year horizon. The benefit-cost ratio of 17.42, internal rate of return of 287%, and a simple payback period of just 0.21 years (about 2.5 months) underscore the exceptional economic viability of coordinated EV integration and V2G-enabled distribution system optimization. These results, detailed in Table (16), confirm that the proposed approach delivers both significant operational benefits and a highly attractive business case.

The battery degradation cost included in the model (Eq. 2 d) and the economic analysis (Table 16) represents the direct financial impact of additional cycling due to V2G participation. This cost is internalized in the optimization, and the V2G compensation rate is set to cover this cost and provide an incentive to the EV owner. However, it is important to note that real-world EV owner participation in V2G programs is influenced by factors beyond direct financial compensation, such as perceived battery health risks, convenience, and trust in the system. The current model assumes rational economic agents and full participation willingness at the calculated compensation rate. In practice, long-term adoption of V2G services may require higher compensation to overcome non-monetary barriers and risk aversion, or the use of alternative business models (e.g., leasing batteries separately). These behavioral aspects are important for the actual implementation of the proposed coordinated planning and operation framework.

Table 16 Economic Analysis Summary.

The economic results presented in Table 16, including the notably short payback period of 0.21 years, are derived from a baseline scenario with static economic parameters over the 5-year horizon. This scenario assumes: (1) stable V2G compensation rates fixed at 0.15 USD/kWh, (2) constant electricity price profiles, and (3) no escalation in infrastructure or maintenance costs. In practice, these factors are subject to significant market, regulatory, and technological uncertainties. To evaluate the robustness of the business case against these risks, we extend the sensitivity analysis in Table 22 to examine the combined impact of varying two key parameters: the V2G compensation rate and the infrastructure cost multiplier. The results, presented in Table 17, show that while the exceptional baseline payback period lengthens under less favorable conditions, the investment remains highly attractive across a wide range of plausible scenarios.

The presented economic results, particularly the high internal rate of return and short payback period, are highly favorable. These metrics are driven by the substantial monetized value assigned to reliability improvement, which is the largest benefit component in our analysis. This reflects a planning perspective that fully internalizes the cost of interruptions. In practice, the realization of these benefits depends on market structures and regulatory frameworks that allow the distribution system operator or third-party aggregators to capture this value. Factors such as project financing costs, taxation, regulatory approval timelines, and evolving tariff structures would moderate the reported returns. The sensitivity analyses (Tables 22 and 23, and 17) demonstrate that the core business case remains strong under a range of assumptions. Therefore, the results should be interpreted as demonstrating the significant potential economic value of coordinated EV-V2G-grid integration, providing a robust justification for developing the market mechanisms and regulatory models needed to unlock this value.

Table 17 Sensitivity of Key Economic Metrics to V2G Compensation and Cost Uncertainty.

This table demonstrates the joint impact of a lower V2G revenue stream and higher capital costs. Even under a “stress scenario” combining a 33% reduction in V2G compensation (0.10 USD/kWh) with a 30% capital cost overrun, the project maintains a strong positive NPV of 4.41 M USD, a benefit-cost ratio > 9, and a payback period under 6 months. This confirms the fundamental economic viability of the coordinated planning approach, albeit with more conservative returns than the optimistic baseline.

The convergence behavior and computational performance of the Enhanced Gray Wolf Optimization Algorithm (EGWOA) are summarized in Table (18). The algorithm converged after 247 iterations, requiring a total computational time of 18.3 h. The initial best fitness value of 8.947 was reduced to 2.847 at the final iteration, corresponding to a fitness improvement of 68.18%. The average fitness of the final population reached 3.156, while the population diversity at convergence was maintained at 0.087, indicating sufficient exploration throughout the optimization process. No constraint violations were observed for the final best solution, and the algorithm archive contained 97 solutions, providing a rich set of near-optimal alternatives. Overall, the EGWOA performed 12,350 function evaluations, with an average time per iteration of 266 s. These convergence statistics, detailed in Table (17), demonstrate the algorithm’s efficiency and robustness in solving the high-dimensional, constrained tri-level optimization problem associated with coordinated EV integration and V2G-enabled distribution system management.

Table 18 EGWOA Convergence Statistics.

Computational stability and robustness analysis

To assess the computational stability and consistency of the proposed Enhanced Gray Wolf Optimization Algorithm (EGWOA), 30 independent runs were executed from different initial populations generated by varying the seed of the chaotic mapping process. Each run used the same algorithmic parameters listed in Table 7. The results are summarized in Table 19, which reports statistics for the final best fitness, the number of iterations to convergence, the total computational time, and the feasibility rate.

Table 19 Stability Analysis over 30 Independent Runs.

The results demonstrate a high degree of stability. The best fitness values exhibit very low variability (standard deviation 0.009), with all runs converging to a near-optimal solution (within 0.8% of the best-found fitness). The iteration count and computational time show moderate variation, which is expected due to the stochastic nature of the metaheuristic, but all runs converged within the predefined maximum iteration limit. Most importantly, all 30 runs produced solutions that fully satisfied the problem constraints (100% feasibility rate). The consistent performance can be attributed to the EGWOA’s design: the chaotic initialization ensures a diverse starting population, while the adaptive mutation and Lévy-flight mechanisms help escape local optima, and the penalty-based constraint handling guides the search toward feasible regions. This multi-run analysis confirms that the proposed algorithm is not only effective but also reliable and robust to different initialization points for the given tri-level optimization problem.

To ensure a fair and statistically rigorous comparison, the proposed EGWOA and all benchmark algorithms (Standard GWO, Particle Swarm Optimization, Genetic Algorithm, Simulated Annealing, and Differential Evolution) were each executed for 30 independent runs from different random seeds. Each run was allowed a maximum of 300 iterations, using algorithm parameters from the literature that yielded the best performance in preliminary tuning. The best fitness value achieved at convergence was recorded for each run.

Table (20) presents a comparative analysis of the proposed EGWOA algorithm against several alternative optimization methods, including standard Gray Wolf Optimization, Particle Swarm Optimization, Genetic Algorithm, Simulated Annealing, and Differential Evolution. The results demonstrate that EGWOA achieves a statistically significant superior performance. The mean best fitness of EGWOA is significantly lower (better) than that of all benchmark algorithms, with p-values far below the 0.05 significance threshold. Furthermore, EGWOA exhibits the lowest standard deviation, indicating the highest robustness and consistency across different runs. This confirms that the enhanced mechanisms in EGWOA are not merely beneficial on a single trial but provide a reliable and significant advantage in navigating the complex tri-level optimization landscape.

Table 20 Statistical Comparison of Optimization Algorithms (Based on 30 Independent Runs).

Fig. 4

Comparison of optimization algorithms based on average performance metrics from 30 independent runs: (a) mean best fitness, (b) average convergence rate, (c) mean computational time, (d) feasibility rate.

Figure (4) presents a comprehensive comparison of the performance of six optimization algorithms, namely the Enhanced Gray Wolf Optimization Algorithm (EGWOA) and five benchmark algorithms: Standard Gray Wolf Optimization (GWO), Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Simulated Annealing (SA), and Differential Evolution (DE). The comparison is made across four key performance metrics: solution quality (best fitness), convergence rate, computational efficiency (time), and solution feasibility. In subplot (a), the best fitness values achieved by each algorithm are compared. The EGWOA (represented in green) demonstrates the best solution quality, achieving the lowest fitness value, which indicates superior performance in finding the optimal solution. The other algorithms show higher fitness values, indicating that they were less effective in solving the problem. Subplot (b) compares the convergence rates of the algorithms. The EGWOA again leads the comparison, achieving the highest convergence rate at 68.18%, suggesting that it reaches a near-optimal solution more quickly compared to the other algorithms. In contrast, the Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) exhibit lower convergence rates, indicating slower progress toward an optimal solution. In subplot (c), the computational time required by each algorithm is presented. The EGWOA stands out by achieving the optimal solution in the shortest amount of time (18.3 h), outperforming all other algorithms. The other algorithms, such as GA and SA, require significantly more time, indicating a trade-off between solution quality and computational efficiency. Finally, subplot (d) presents the feasibility rate of the solutions obtained by each algorithm. EGWOA, along with Standard GWO and SA, achieves a 100% feasibility rate, meaning that the solutions provided by these algorithms satisfy all constraints. However, algorithms such as PSO, GA, and DE fall short of this target, reflecting a lower percentage of feasible solutions. This figure effectively highlights the EGWOA as the most balanced and efficient optimization algorithm in terms of solution quality, convergence rate, computational efficiency, and feasibility, making it a superior choice for the problem at hand.

Sensitivity analysis

Sensitivity analysis is conducted to assess the robustness of the optimal charging infrastructure deployment and associated reliability and economic performance with respect to variations in the size of the electric vehicle fleet. Table (21) summarizes the results for fleet sizes ranging from 100 to 200 vehicles. As the fleet size increases, the number of charging stations and total installed capacity also rises, reflecting the need to accommodate higher demand. Investment costs grow proportionally from 289,500 USD for 100 vehicles to 537,000 USD for 200 vehicles. Correspondingly, the composite reliability index improves from 0.286 to 0.421, indicating enhanced system resilience with larger EV participation. The net present value (NPV) also increases in magnitude with fleet size, reaching approximately 9.76 million USD for the 200-vehicle scenario, while the marginal benefit per additional vehicle gradually decreases from 54,700 USD to 48,800 USD. These findings, detailed in Table (21), illustrate the trade-offs between infrastructure investment, system reliability, and economic benefits, providing decision-makers with insight into the sensitivity of outcomes to EV penetration levels.

Table 21 Sensitivity to EV Fleet Size.

The sensitivity of the system’s economic performance to variations in electricity pricing is examined in Table (22). Four price scenarios, ranging from low to very high, are analyzed to evaluate their impact on annual energy costs, V2G compensation, and net energy expenditures. As peak and off-peak electricity prices increase from 0.14/0.06 USD/kWh to 0.26/0.12 USD/kWh, the annual energy cost rises from 135,920 USD to 242,180 USD. V2G compensation remains constant at 222,500 USD, resulting in net energy costs that shift from a negative value of − 86,580 USD under the low-price scenario to a positive 19,680 USD under the very high-price scenario. These variations translate into corresponding impacts on the net present value (NPV) of the project, with NPV increasing by 3.8% under low prices and decreasing by 6.9% under very high prices. The results in Table (22) highlight the sensitivity of the business case to electricity price fluctuations and underscore the importance of accurate price forecasting for planning economically efficient EV integration and V2G operations.

Fig. 5

Sensitivity Analysis: Impact of EV Fleet Size on System Performance.

Figure 5 illustrates the results of a sensitivity analysis, examining the impact of electric vehicle (EV) fleet size on various system performance metrics. The figure consists of four subplots, each focusing on a different aspect of the system: infrastructure deployment, economic performance, reliability enhancement, and marginal economic benefit. In subplot (a), the relationship between EV fleet size and infrastructure deployment is shown. The number of charging stations and the total installed capacity (in kilowatts) both increase with fleet size, reflecting the need for more infrastructure as the number of vehicles grows. The left axis represents the number of stations (shown in blue), while the right axis shows the total installed capacity (shown in red). The dual-axis approach allows for a clear comparison between these two parameters, both of which grow as the fleet size increases. Subplot (b) presents the economic performance of the system in terms of investment costs and net present value (NPV) as a function of fleet size. The investment cost (shown in green) increases with fleet size, which is expected due to the need for more charging infrastructure. In contrast, the NPV (shown in magenta) also rises with fleet size, reflecting the growing economic benefits of deploying a larger EV fleet. This subplot highlights the trade-off between investment and economic returns as the system scales. Subplot (c) focuses on the system’s reliability enhancement, quantified by the composite reliability index. As the fleet size increases, the reliability index improves, indicating that a larger EV fleet contributes to better overall system reliability. The plot shows a clear upward trend, demonstrating that integrating more EVs into the system leads to a more resilient distribution network. Finally, subplot (d) illustrates the marginal economic benefit per vehicle in terms of fleet size. The marginal benefit (shown in blue bars) initially remains constant and then gradually decreases as the fleet size increases. This trend suggests that while adding more vehicles continues to provide economic benefits, the additional benefit per vehicle diminishes with larger fleets.

Together, these subplots provide valuable insights into how the size of the EV fleet influences various aspects of the system, including infrastructure needs, economic performance, reliability, and marginal returns.

Table 22 Sensitivity to Electricity Price Scenarios.

The sensitivity of the system’s economic and reliability performance to variations in customer interruption cost (CIC) valuation is presented in Table (23). The analysis considers CIC multipliers ranging from 0.5 to 1.5 to represent different assumptions about the monetary value of avoided interruptions. As the CIC multiplier increases, the estimated economic value of expected energy not served (EENS) rises proportionally, from 0.962 million USD/year at 0.5× to 2.886 million USD/year at 1.5×. Correspondingly, the total annual benefit and benefit-cost ratio increase, reflecting enhanced financial justification for investment in charging infrastructure and V2G-enabled reliability improvements. The internal rate of return (IRR) also rises sharply from 142% to 432%, indicating that higher reliability valuation significantly strengthens the economic attractiveness of the project. Across all CIC scenarios, the investment decision remains favorable, demonstrating that the proposed optimization framework maintains robust economic and reliability performance under varying assumptions about the value of service continuity.

Table 23 Sensitivity to Reliability Valuation.

The proposed tri-level optimization framework, integrating coordinated electric vehicle (EV) charging, vehicle-to-grid (V2G) participation, and network reconfiguration via the Enhanced Gray Wolf Optimization Algorithm (EGWOA), demonstrates significant improvements in distribution system performance. Voltage profiles are flattened, with minimum voltages increasing by an average of 4.10% and voltage deviation indices reduced by over 50%, while active power losses decrease by 23.02% on average, enhancing overall efficiency. Reliability under critical contingencies is markedly improved, with total expected energy not served dropping from 651.30 MWh/year to 192.70 MWh/year, the composite reliability index rising from 0.152 to 0.367, and standard indices such as SAIFI, SAIDI, and ASAI meeting target values. Optimally sited EV charging stations balance fleet utilization and system support, while V2G discharging during peak-demand periods mitigates load curtailment. Economic analysis confirms a highly attractive business case, with a net present value of 7.93 million USD, a benefit-cost ratio of 17.42, an internal rate of return of 287%, and a payback period of just 0.21 years. Sensitivity analyses show robust performance under varying fleet sizes, electricity prices, and reliability valuations. Compared with conventional metaheuristics, EGWOA achieves faster convergence, superior solution quality, and full constraint satisfaction. These results collectively reveal that coordinated EV and V2G integration, combined with strategic network reconfiguration, provides an effective, reliable, and economically viable approach for enhancing medium-voltage distribution system performance.

Figure (6) presents a detailed analysis of the impact of vehicle-to-grid (V2G) optimization across multiple operational aspects of a distribution network. The figure consists of six subplots that visualize key performance indicators such as V2G power flow, fleet average state of charge (SOC), voltage profiles, system losses, reliability contributions, and load curtailment under major contingencies. In the first subplot, the V2G power flow is depicted alongside electricity prices. The V2G power, shown in bars, fluctuates throughout the day with periods of charging (negative power values) and discharging (positive power values). This dynamic pattern is shown in conjunction with hourly electricity prices, which affect the overall economic performance of the V2G strategy. The negative bars represent charging, while the positive bars correspond to discharging, and the price curve reveals the varying electricity costs during the day. The second subplot illustrates the fleet’s average state of charge (SOC) across the 24-hour period. The SOC is maintained within predefined limits, ensuring operational flexibility and avoiding extreme values that could lead to battery degradation. The plot includes minimum and maximum SOC limits, highlighting the fleet’s ability to manage energy efficiently without falling below or exceeding these bounds. The third subplot compares the voltage profiles at a critical node (Node 18) under the base case and V2G-optimized conditions. The optimized voltage levels are consistently higher than the base case, with improvements ensuring that the system maintains voltage levels above the critical threshold of 0.95 per unit, which is crucial for maintaining system stability and avoiding voltage-related failures. The fourth subplot presents the comparison of system losses between the base case and the V2G-optimized scenario. The optimized configuration consistently shows a reduction in active power losses, reflecting the improved efficiency achieved by integrating V2G technology into the network. The fifth subplot shows the cumulative reliability contribution from each of the five selected charging stations. The plot reveals the varying contributions of each station to the overall system reliability, with a cumulative trend line demonstrating how the total contribution increases as more stations are integrated. The final subplot examines the load curtailment under major contingency scenarios. By comparing the base case and the V2G-optimized scenario, it is clear that V2G optimization significantly reduces load curtailment during contingency events, which enhances the system’s reliability and reduces the amount of unsupplied energy. Together, these subplots provide a comprehensive overview of how V2G optimization impacts multiple dimensions of a distribution system, including energy efficiency, voltage stability, reliability, and operational costs.

Fig. 6

Comprehensive Analysis of V2G Optimization Impact on Distribution System Performance.

Figure (7) presents a 3D visualization of the electric vehicle (EV) power flow dynamics throughout a 24-hour period. In this figure, the horizontal axis represents the hour of the day, ranging from 1 to 24, while the vertical axis shows the number of vehicles in either charging or discharging modes. The third axis displays the corresponding net power (in kW), which fluctuates based on the state of the EVs and their energy flow. The data plotted in the figure consists of two distinct phases: the charging and discharging periods of the electric vehicles. The charging mode is depicted using a scatter plot with blue color intensity representing the net power, with higher values indicating greater power consumption. The discharging mode, although not explicitly shown here, is represented by the absence of charging activity and the increase in net power during the latter hours of the day, as indicated by the rise in the Z-axis values. The plot also includes lines connecting successive data points, providing a clearer view of the changes in the charging dynamics and net power consumption over time. The color bar on the right side of the figure further emphasizes the relationship between net power and color intensity, reinforcing the visual interpretation of how net power evolves throughout the 24-hour cycle. This figure effectively illustrates the correlation between the number of EVs being charged and the resulting net power fluctuations, allowing for a comprehensive understanding of how EV power flows contribute to the overall energy system at different times of the day. The insights derived from such visualizations are crucial for managing and optimizing power distribution in smart grids, particularly in the context of integrating renewable energy sources and EV charging infrastructure.

Fig. 7

EV Power Flow Dynamics.

Figure (8) presents a heatmap that visualizes the 24-hour activity and power flow pattern of an electric vehicle (EV) fleet. The activity matrix includes four distinct variables: the number of vehicles charging, the number of vehicles discharging, the net power (scaled by a factor of 10 kW), and the average state of charge (SOC) of the vehicles (scaled by a factor of 100). These variables are shown for each of the 24 h in a day, with the data values represented as color-coded cells in the heatmap. On the horizontal axis, the hours of the day are labeled from 1 to 24, providing a clear temporal distribution of the activity levels. The vertical axis categorizes the activities into four distinct groups: Charging Vehicles, Discharging Vehicles, Net Power, and Average SOC. The color intensity in each cell corresponds to the magnitude of the activity level, with the colors ranging from red (high activity) to green (low activity). For the “Charging Vehicles” and “Discharging Vehicles” rows, the heatmap reveals the fluctuating number of vehicles involved in each mode, showing periods of intense charging in the morning and evening hours, while discharging activity increases during late afternoon and early evening. The “Net Power” row captures the dynamic power requirements of the fleet, with negative values indicating periods of charging (power consumption) and positive values indicating discharging (power generation). Finally, the “Avg SOC” row illustrates the average state of charge across the vehicles, showing a trend of charging early in the day and discharging during the evening, reflecting typical usage patterns. This heatmap provides a comprehensive and intuitive understanding of the EV fleet’s power dynamics throughout the day. It effectively highlights the interplay between vehicle charging and discharging behavior, net power requirements, and state of charge, which is crucial for optimizing energy management in smart grid systems.

Fig. 8

24-Hour EV Fleet Activity and Power Flow Pattern.

The integration of 150 EVs with bidirectional V2G capability into the 33-bus system introduces distributed generation-like behavior, which can affect the existing protection coordination. Reverse power flow during vehicle discharging may lead to: (1) altered fault current magnitudes and directions, potentially desensitizing overcurrent relays or causing false tripping; (2) challenges for fault detection and isolation, especially in radial networks where protection schemes often assume a single source substation; and (3) miscoordination between upstream and downstream protective devices. To ensure reliable operation with the proposed optimal V2G schedules, the protection architecture would likely require adjustments. These could include the adoption of directional overcurrent relays, adaptive protection schemes that adjust settings based on real-time network topology and generation patterns, or the implementation of communication-assisted protection strategies. Future work will explicitly model these protection interactions to co-optimize V2G dispatch with adaptive relay settings, ensuring both operational and protection reliability.

The reported computational time of 18.3 h reflects the complexity of the full tri-level planning problem, which simultaneously optimizes infrastructure investment, 24-hour fleet scheduling, and reliability under numerous stochastic scenarios and N-1 contingencies. This is suitable for offline planning studies but not for real-time control. However, the framework possesses inherent features that enable its adaptation for faster, operational timescales. For quasi-real-time dispatch (e.g., hourly or 15-minute ahead scheduling), the problem structure can be significantly simplified: the upper-level (station sizing) becomes fixed, and the lower-level reliability evaluation can be replaced with a simplified set of real-time security constraints. The core optimization then reduces to the middle-level scheduling problem, with a much shorter horizon (e.g., a few hours), using forecasted prices and vehicle availabilities. Furthermore, the EGWOA’s efficiency could be enhanced for this purpose by: (1) Warm-starting the population with the previous solution, drastically reducing iterations; (2) Employing a reduced-order model (e.g., linearized power flow) for the network constraints; and (3) Implementing a parallelized evaluation of scenarios. A promising direction is to use the offline planning model (as presented) to derive optimal policy rules or to train a surrogate machine-learning model that can perform near-instantaneous inference for real-time dispatch, a key avenue for our future work.

Computational scalability and practical implementation pathways

The proposed tri-level framework integrates long-term planning with detailed operational and reliability evaluation, which incurs a non-trivial computational cost, as evidenced in the 33-bus case study. This cost is primarily driven by the repeated AC power flow and contingency analysis in the lower level. For offline planning studies—the primary target of this work—such runtimes are acceptable. However, applying the framework to larger, real-world distribution networks requires scalability strategies. The complexity is managed through several inherent and implementable features:

• 1. Sparsity-Aware Solvers: The radial nature of distribution networks allows for the use of highly efficient backward-forward sweep power flow solvers, reducing per-evaluation complexity.

• 2. Parallelizable Structure: The evaluation of stochastic scenarios and contingencies for a given candidate solution is independent and can be distributed across multiple processors, offering near-linear speedup.

• 3. Model Reduction for Larger Systems: For extensive networks, a hierarchical or zonal approach can be adopted, where the system is decomposed into manageable subsystems optimized independently or iteratively. Furthermore, the non-linear DistFlow model can be replaced with its linearized version (LinDistFlow) for planning studies with minimal loss of accuracy in identifying optimal investment locations and capacities.

• 4. Decomposition for Real-Time Use: For real-time operational scheduling, the problem simplifies dramatically. With fixed infrastructure, the model reduces to a middle-level scheduling problem, which can be solved efficiently using the proposed EGWOA warm-started from previous solutions or via a pre-trained surrogate model, making it suitable for hourly or intra-hourly dispatch.

Therefore, while the full tri-level model is computationally intensive, its structure is amenable to standard large-scale optimization techniques. The framework provides a valuable benchmark and a systematic methodology, and its components can be adapted in complexity to match the computational budget and accuracy requirements of different planning stages, from feasibility studies to detailed design.

Ablation study on EGWOA enhancements

To rigorously evaluate the contribution of each heuristic component within the Enhanced Gray Wolf Optimization Algorithm (EGWOA), a detailed ablation study was conducted. The objective was to isolate the impact of chaotic initialization, adaptive mutation, Lévy-flight exploration, opposition-based learning, and the local search mechanism on the algorithm’s performance in solving the tri-level optimization problem. Six algorithm variants were defined by systematically removing one enhancement at a time from the full EGWOA, alongside the standard Gray Wolf Optimization (GWO) algorithm as a baseline. Each variant was executed for 30 independent runs on the IEEE 33-bus case study, using the parameters in Table 7 and identical initial random seeds where applicable.

Table 24 Ablation Study Results (Average of 30 Independent Runs).

Table 24 analyzes the individual contribution of each enhancement embedded in the EGWOA by comparing its full version with reduced variants and the standard GWO. The full EGWOA clearly achieves the best overall performance, yielding the lowest mean best fitness value with a very small standard deviation, which indicates high solution quality and strong robustness across independent runs. Its convergence occurs earlier than all other variants while maintaining a 100% feasibility rate, confirming its effectiveness in handling the tight coupling and constraint complexity of the tri-level optimization problem. The relatively high final diversity shows that the algorithm preserves sufficient population spread at convergence, avoiding premature stagnation while still enabling effective exploitation. Removing individual components leads to consistent performance degradation, though with different characteristics. Excluding chaotic initialization and opposition-based learning mainly affects early exploration, resulting in slightly worse fitness values and slower convergence, while feasibility remains unaffected. In contrast, removing adaptive mutation has a more significant impact, as reflected by higher fitness values, slower convergence, and a reduced feasibility rate. This indicates that adaptive mutation plays a critical role in maintaining feasibility and escaping local optima during later search stages. Similarly, the absence of Lévy-flight exploration degrades solution quality and convergence speed, highlighting its importance in enhancing global exploration in the highly nonconvex search space. The removal of the local search mechanism leads to slower convergence and inferior final solutions, despite maintaining feasibility, which suggests that local refinement is essential for fine-tuning candidate solutions once promising regions are identified. Compared with all EGWOA variants, the standard GWO exhibits substantially inferior performance, characterized by much higher fitness values, large variability, and failure to converge within the maximum iteration limit. Its very low final diversity further indicates premature convergence and limited search capability. Overall, the results in Table 24 demonstrate that the superior performance of EGWOA arises from the synergistic integration of all enhancement mechanisms, each addressing different aspects of exploration, exploitation, and constraint handling in the tri-level optimization framework.

Sensitivity analysis of weighting factors and robustness

The multi-objective formulation utilizes weighting factors to scalarize the problem. To assess the robustness of the results to these parameters, a comprehensive sensitivity analysis was conducted on the most critical weights.

Table 25 Sensitivity Analysis of Key Weighting Factors.

Table 25 evaluates the sensitivity of the proposed multi-objective framework to variations in the most influential weighting factors used in the scalarized objective function. The results show that changing the reliability improvement weight ψ by ± 30% leads to only moderate variations in the economic and technical outcomes. Increasing ψ slightly improves the composite reliability index and loss reduction, at the expense of a marginally higher investment cost reflected in the NPV, while also yielding a better overall fitness value. Conversely, decreasing ψ reduces the emphasis on reliability, resulting in slightly weaker reliability and loss reduction performance. However, the magnitude of these changes remains limited, indicating that the optimization results are not overly dependent on the exact tuning of this weight. A similar robustness trend is observed for the voltage deviation and load shedding penalty coefficients. Variations of ± 30% in the voltage penalty coefficient produce only small changes in NPV, composite RI, and loss reduction, suggesting that voltage quality is adequately enforced even when the penalty strength is relaxed or intensified. Adjusting the load shedding penalty coefficient shows a slightly stronger influence on both economic and reliability-related metrics, as expected, since load shedding directly impacts service continuity. Nevertheless, the resulting variations in best fitness and performance indicators remain within a narrow range, confirming that the solution is stable and not sensitive to moderate mis-specification of penalty parameters. The analysis of different RI component weight sets further supports the robustness of the framework. Shifting emphasis among the RI subcomponents produces only minor changes in NPV, reliability index, and loss reduction, while preserving comparable fitness values. More importantly, across all tested scenarios, the optimal planning and operational decisions, such as charging station placement and V2G scheduling patterns, remain virtually unchanged. The dynamic constraint penalty mechanism consistently maintains a 100% feasibility rate in all sensitivity cases, demonstrating that the proposed formulation and solution method are inherently robust and reliable, even under significant variations in weighting factors.

Sensitivity analysis of V2G operational assumptions

The practical performance of vehicle-to-grid (V2G) services depends on parameters such as the aggregate availability of the EV fleet for grid support and its response time to dispatch signals. These parameters are subject to uncertainty stemming from driver behavior, battery management strategies, and communication infrastructure. The base-case values (availability factor = 0.70, response time = 0.1 h) used in the study align with typical assumptions in the V2G literature for ancillary services and peak shaving. To assess the robustness of the planning outcomes to these assumptions, we conduct a sensitivity analysis by varying each parameter over a plausible range while keeping the other at its base value. Additionally, a combined pessimistic scenario is evaluated.

Table 26 Sensitivity of Key Outcomes to V2G Operational Parameters.

Table 26 examines the sensitivity of system performance to key V2G operational assumptions, namely the EV fleet availability factor and the response time to dispatch signals. The base-case results confirm a balanced trade-off between reliability improvement and economic performance, with a composite reliability index of 0.367, an expected energy not supplied (EENS) of 192.70 MWh/year, and an attractive NPV of − 7.93 million USD. These values serve as a reference for evaluating how deviations in V2G behavior influence the overall planning and operational outcomes. Variations in the availability factor have a clear and intuitive impact on reliability and economics. Reducing availability to 0.50 leads to a noticeable deterioration in reliability, reflected by a lower composite RI and higher EENS, as fewer EVs are available to provide grid support during critical periods. This reduction also slightly weakens the economic performance, as the diminished V2G contribution lowers the value of reliability-driven benefits. Conversely, increasing availability to 0.90 significantly enhances system reliability, yielding the highest RI and the lowest EENS among all scenarios, while also improving NPV due to the greater utilization of V2G services for peak shaving and ancillary support. The effect of response time is comparatively milder but still meaningful. Increasing the response time from 0.1 to 0.5 h results in only a small decline in RI and a modest increase in EENS, indicating that the system can tolerate moderate communication and control delays without major performance loss. However, a very slow response time of 2.0 h causes a more pronounced reduction in reliability and economic benefits, as delayed V2G activation limits its effectiveness during contingency events. In the pessimistic combined scenario, where both availability is low and response time is long, the reliability metrics reach their weakest values and NPV is least favorable. Nevertheless, the composite RI remains well above the minimum target level, and the NPV remains strongly positive in absolute terms. Importantly, the optimal infrastructure deployment remains unchanged across all scenarios, confirming that the core planning decisions and the study’s main conclusions are robust to realistic uncertainty in V2G operational assumptions.

Conclusion

This paper has presented a comprehensive tri-level optimization framework for the coordinated planning and operation of electric vehicle (EV)-integrated distribution systems. The model effectively addresses the interconnected challenges of charging infrastructure investment, EV fleet scheduling with Vehicle-to-Grid (V2G) capabilities, and distribution network operation under uncertainty and contingencies. The proposed Enhanced Gray Wolf Optimization Algorithm (EGWOA) successfully solved this complex, non-convex problem by incorporating chaotic initialization, adaptive mutation, Lévy flight, and other mechanisms to enhance exploration and exploitation. Application to the IEEE 33-bus test system demonstrated the framework’s significant benefits. The results confirm that coordinated planning and operation can substantially improve system performance. Key outcomes include a 4.10% improvement in minimum voltages, a 23.02% reduction in active power losses, and a 70.42% reduction in Expected Energy Not Served (EENS). Economically, the strategy yields a highly attractive business case with a net present value of 7.93 million USD and a payback period of only 0.21 years. Furthermore, the EGWOA outperformed benchmark algorithms in terms of solution quality, convergence speed, and constraint satisfaction. These findings collectively validate that a tightly coupled approach to EV integration can transform EVs from a potential burden into a valuable asset for enhancing the efficiency, reliability, and economic performance of medium-voltage distribution networks. While this study provides a robust foundation, several avenues remain for future research to enhance the model’s applicability and realism:

• 1. Real-World Validation and Dynamic Operation: The framework should be tested on larger, real-world distribution networks with more complex topologies and real load data. Furthermore, extending the model from a daily scheduling horizon to a rolling-horizon or multi-period planning model would better capture the dynamic evolution of the system and EV fleet over years.

• 2. Advanced Uncertainty Modeling: The current model uses a scenario-based approach for load uncertainty. Future work could incorporate more sophisticated methods for other stochastic variables, such as renewable energy generation (e.g., solar PV), electricity prices, and EV driver behavior, using techniques like robust optimization or stochastic programming.

• 3. Market Integration and Prosumer Behavior: Integrating a detailed model of electricity markets, including real-time pricing and ancillary service markets, would provide a more accurate representation of V2G economics. Additionally, modeling the strategic behavior and price sensitivity of EV owners as prosumers would make the scheduling model more realistic.

• 4. Multi-Energy Systems and Cyber-Security: Expanding the framework to include multi-energy systems, such as coupled electricity and heat networks, could unlock further synergies. As the system becomes more automated and data-dependent, investigating the cyber-security vulnerabilities of the proposed control architecture is also a critical area for future work.

• 5. Algorithm Enhancement for Real-Time Application: Although the EGWOA is effective for planning studies, its computational time may be prohibitive for real-time operation. Developing a simplified or distributed version of the algorithm, or hybridizing it with machine learning techniques for fast decision-making, would be a valuable step toward practical implementation.

• 6. EV Owner Behavior and Degradation Perception: The current model internalizes battery degradation costs but does not capture the full spectrum of prosumer behavior. Future work should incorporate more detailed models of EV owner participation, including risk aversion, non-monetary barriers, and long-term adoption dynamics under different incentive structures and market designs. This would facilitate the development of more effective V2G programs that ensure high participation rates and sustainable growth.

• 7. Extended Contingency Analysis: The current N-1 contingency analysis could be extended to include probabilistic models of correlated multi-component failures and dynamic cascading outages. This would provide a more comprehensive assessment of system resilience against extreme events and interdependent failures, further validating the robustness of the coordinated planning framework.