An α-Cut Optimization Framework for Modular EV Charging Station Design Under Fuzzy Uncertainty

Abstract

This paper develops a unified α-cut optimization framework for modular electric vehicle (EV) fast-charging station design under fuzzy uncertainty. Uncertain peak demand, annual delivered energy, electricity price, ambient temperature, arrival rate, and energy per session are represented by triangular or trapezoidal fuzzy numbers and reformulated through α-cut bounds. The resulting design problem is expressed as a hybrid discrete–continuous model in which the number of modules, the selected catalog module rating, installed power, cooling provision, and a station-volume proxy are jointly optimized. An aggregated representation of interchangeable modules is adopted to remove permutation-equivalent descriptions and preserve a compact search space. Three planning views are examined: minimum CAPEX at a prescribed α-cut level, minimum loss-driven OPEX under a CAPEX budget, and a service-oriented admissibility/coverage analysis that avoids interpreting larger α values as greater robustness. The strengthened numerical study includes a deterministic nominal benchmark, peak demand sensitivity regimes, feasibility threshold and budget sweep results, explicit service stress scenarios, and a queueing sensitivity check against Erlang-C and discrete-event simulation indicators. The results show that baseline CAPEX designs may be dominated by catalog thresholds, whereas OPEX and service-oriented conclusions become informative once budget and traffic regimes are varied. The proposed framework is therefore positioned as a tractable α-cut-based design screening and comparative optimization tool for representative modular EV charging station scenarios, rather than as a universally validated operational design rule.

1. Introduction

The rapid expansion of electric mobility is intensifying the deployment of public DC fast-charging infrastructure. However, the design of EV fast-charging stations remains challenging due to the simultaneous presence of highly variable user flows and peak loads, distribution-grid and contracted power limits, and economic trade-offs involving both capital and operating costs. In practical planning, engineers must often commit to a station design with incomplete information, including selecting the module rating, the number of installed modules, and the associated thermal and spatial resources required to ensure safe and reliable operation.

A modular station architecture offers a natural response to these requirements. It supports scalability, redundancy, flexible power sharing, and improved serviceability [1,2,3,4]. At the same time, modularity introduces discrete design decisions, making the resulting planning problem inherently hybrid and sensitive to uncertainty in demand and operating conditions. Recent literature on charging infrastructure optimization has developed rapidly, ranging from broad reviews on station allocation and sizing to mixed-integer formulations for logistics, corridor planning, and infrastructure deployment [4,5,6,7,8].

Uncertainty in charging station design is often addressed through probabilistic models and simulation-based approaches. While such methods can be effective when well-calibrated data are available, they require sufficiently reliable probability distributions and may not always yield results that are easily interpretable as explicit engineering guarantees. In many practical situations, uncertainty is better characterized as imprecision rather than randomness. An alternative, engineer-friendly approach is therefore to represent uncertain quantities by fuzzy numbers, which capture optimistic, nominal, and pessimistic operating conditions within a unified mathematical framework. Fuzzy sets, introduced by Zadeh, provide the basis for this representation [9].

In fuzzy optimization, α-cuts play a central role because they transform fuzzy quantities into deterministic interval bounds, thereby reducing the original problem to a family of crisp optimization models parameterized by the satisfaction level α. This is especially convenient for engineering design problems, where constraints must often be satisfied under prescribed uncertainty treatment [10,11,12]. Closely related to this, the extension principle provides the mathematical basis for propagating fuzzy uncertainty through functions and constraints, thereby supporting fuzzy arithmetic and the construction of α-cut images for derived quantities [11,13].

In the present paper, the parameter α is interpreted as an α-cut level of the fuzzy input description rather than as a direct measure of robustness or conservatism. Under the adopted α-cut convention, larger α yields narrower interval bounds, and therefore less conservative deterministic upper-bound constraints. Consequently, any service-oriented optimization involving α must be interpreted as an admissibility or coverage analysis, not as a statement that maximizing α automatically produces a more robust design.

Besides capacity and cost, service quality is another important consideration in fast-charging infrastructure planning. Queueing-theoretic models, including Erlang-C-based formulations and related extensions, are widely used to evaluate waiting times and service adequacy [14,15,16]. This makes it natural to include service-oriented criteria alongside CAPEX- and OPEX-related objectives and to study the trade-offs between investment, operational efficiency, and service adequacy under fuzzy uncertainty.

The aim of this article is to develop an α-cut optimization framework for the joint sizing of a modular EV fast-charging station under fuzzy uncertainty and a prescribed grid limit of P_max. Uncertain inputs, including peak demand, annual delivered energy, electricity price, ambient temperature, arrival rate, and energy per session, are represented by fuzzy numbers and propagated through α-cuts. The design variables include the number of installed charging modules, a discrete choice of module rating p from a standard catalog, and continuous proxy quantities related to station dimensioning, such as cooling capacity and volume. The resulting problem is formulated as a hybrid discrete–continuous optimization model with an aggregated representation of modular design choices.

Beyond its immediate engineering application, the present study contributes a unified computational optimization framework for modular charging station design under fuzzy uncertainty. The framework combines α-cut-based deterministic reformulations of uncertain requirements and cost terms with an aggregated, symmetry-reduced representation of interchangeable modules in a single hybrid discrete–continuous model. This structure enables the consistent comparative treatment of investment-oriented design, OPEX-oriented design under budget restrictions, and service-oriented admissibility/uncertainty-coverage analysis within the same mathematical setting. Accordingly, the main contribution of the paper is methodological as well as applied, providing a tractable optimization architecture for uncertainty-aware design screening of modular engineering systems.

The contributions of this paper are fourfold. First, the paper develops a unified hybrid discrete–continuous optimization framework for modular EV fast-charging station design under fuzzy uncertainty, in which discrete architecture decisions and continuous sizing proxies are treated within a common mathematical setting. Second, a systematic α-cut-based reformulation is constructed for requirement-type constraints and cost-driving uncertain terms, with explicit verification of the monotonicity assumptions used by the upper-endpoint treatment. Third, the formulation adopts an aggregated representation of interchangeable charging modules to eliminate redundant permutation-equivalent design descriptions and preserve a compact optimization structure. Fourth, the framework is used to compare CAPEX, OPEX, and service-oriented admissibility regimes through deterministic benchmarking, budget sweeps, stress scenarios, and queueing sensitivity checks, thereby clarifying when the fuzzy treatment changes the selected architecture and when catalog granularity dominates the outcome.

2. Related Work

2.1. Fuzzy Sets, α-Cuts, and Fuzzy Programming

Fuzzy set theory, introduced by Zadeh, provides a mathematically grounded way to represent imprecision through membership functions and remains a standard tool for modeling uncertain engineering quantities when purely probabilistic descriptions are not sufficiently reliable [9]. In decision-making contexts, Bellman and Zadeh interpreted fuzzy goals and fuzzy constraints as interacting fuzzy sets, which naturally motivated satisfaction-based formulations and max–min decision structures [17].

A key mechanism in fuzzy optimization is the use of α-cuts, which transform fuzzy quantities into deterministic interval bounds. This provides a practical bridge between fuzzy uncertainty representation and optimization modeling. In this context, the extension principle is a foundational concept for propagating uncertainty through functions, while α-cut arithmetic in positive domains supports simple and conservative endpoint propagation for products and ratios [11,13].

In optimization, fuzzy linear programming and fuzzy multi-objective programming have developed into established research directions, including α-cut-based reformulations of fuzzy problems into families of crisp deterministic programs [10,11,12]. Zimmermann’s early work remains one of the most influential formulations of satisfaction-based fuzzy programming [10], while Tanaka and Asai further developed fuzzy linear programming with fuzzy-number coefficients in ways that continue to shape contemporary optimization practice [12].

In the present study, this line of research is particularly relevant because it provides the mathematical basis for representing uncertain charging-station inputs as fuzzy numbers and for reformulating the resulting design problem through α-cuts into deterministic optimization settings.

2.2. Aggregated Formulations for Modular Optimization Problems

Interchangeable components are a well-known source of redundancy in integer and mixed-integer optimization. When variables or subsystems are physically identical, explicitly indexed formulations may generate multiple equivalent solutions that differ only by permutation of labels, thereby increasing combinatorial complexity without adding physical meaning [18,19]. Margot’s survey remains a standard reference on symmetry in integer linear programming and on algorithmic and modeling strategies for reducing equivalent search [18]. Liberti provides a broader mathematical programming perspective on structural redundancy and symmetry-related reformulations [19], while Liberti and Ostrowski discuss symmetry-breaking constraints that preserve at least one optimal solution while eliminating redundant equivalent configurations [20].

This line of research motivates the aggregated formulation adopted in the present study. Instead of indexing identical charging modules individually, the station design is represented through the integer module count and a discrete choice of module rating. This preserves the effective design space while avoiding redundant permutation-equivalent representations. In the present context, the purpose of aggregation is not to study symmetry as an end in itself, but to obtain a more compact and transparent optimization model for modular EV charging station design under fuzzy uncertainty [18,19,20,21].

2.3. EV Charging Infrastructure Planning and Sizing Under Constraints

Optimization models for EV charging infrastructure have been developed at multiple planning levels, including location-allocation, corridor deployment, grid-constrained station sizing, and coupled transportation–power-network formulations. Early refueling-location models, such as the flow-refueling location problem, established important modeling principles for alternative-fuel infrastructure under route and range constraints and later influenced EV charging siting and sizing studies [22].

For fast-charging infrastructure in particular, a number of studies have addressed the joint interaction between discrete design choices, network limitations, and demand-related uncertainty. Sadeghi-Barzani et al. proposed a widely cited placing-and-sizing formulation based on mixed-integer nonlinear optimization that shows how infrastructure decisions and grid constraints can be treated within a unified model [7]. Sathaye and Kelley examined corridor planning under uncertain demand, offering an early-stage deployment perspective relevant to charging-network expansion [6]. More recently, integrated planning approaches have emphasized the interaction between transportation systems and electricity distribution networks. In this context, Unterluggauer et al. reviewed EV charging infrastructure planning in coupled transportation and power-distribution settings and identified recurring challenges related to uncertainty treatment and integrated objective formulation [5]. Zhang et al. further studied fast-charging station siting and sizing in coupled transportation and power networks [8].

In contrast to network-level planning studies, the present work focuses on the design of a single modular fast-charging station under a prescribed grid limit. The emphasis is on joint station sizing under fuzzy uncertainty, where uncertain inputs are represented explicitly through α-cuts, and both modular architecture and auxiliary dimensioning variables are incorporated into the optimization model. This perspective complements broader infrastructure-planning studies by addressing uncertainty-aware device- and station-level design within a mathematically structured optimization framework.

2.4. Queueing and Service-Quality Modeling for Charging Stations

Service quality in EV charging stations is commonly assessed using queueing-related indicators such as waiting probability, waiting time, and utilization level. For this reason, queueing models, including Erlang-C-type approximations and related extensions, are increasingly used in charging station analysis and planning [15,16,23]. These models are especially relevant when station adequacy must be evaluated not only in terms of installed capacity but also in terms of user-facing service performance.

At the same time, detailed queueing formulations may be difficult to integrate directly into compact optimization models, particularly when uncertainty is represented through fuzzy inputs rather than stochastic processes. In the present study, this motivates the use of a utilization-based surrogate for queueing risk. The surrogate is not intended to replace a full queueing-theoretic analysis, but to provide a computationally tractable service-oriented constraint that remains compatible with the proposed α-cut optimization framework [16,23].

2.5. Positioning of the Present Contribution

The reviewed literature highlights three main gaps relevant to the present study. First, although fuzzy and α-cut-based methods provide a suitable mathematical basis for handling imprecise engineering inputs, they have only been partially explored in modular EV charging station design [9,10,11,12,13,17]. Second, charging infrastructure planning studies often focus on network-level siting and sizing while giving less attention to uncertainty-aware station-level design with explicit modular architecture [5,6,7,8,22]. Third, although interchangeable modular components naturally motivate aggregated formulations, this modeling perspective is not consistently integrated into charging station optimization under fuzzy uncertainty [18,19,20].

The present article addresses these gaps by developing an α-cut optimization framework for modular EV fast-charging station design under fuzzy uncertainty. The framework combines fuzzy-number-based input representation, hybrid discrete–continuous design optimization, and an aggregated representation of interchangeable charging modules within a unified mathematical formulation. In addition, it examines three complementary planning settings—CAPEX minimization, OPEX minimization under budget, and service-oriented admissibility/uncertainty-coverage analysis—thereby providing a structured basis for uncertainty-aware station design under grid, thermal, and service-related constraints.

3. Fuzzy Mathematical Formulation

3.1. Uncertain Inputs Represented by Fuzzy Numbers

The design of a modular EV fast-charging station is affected by several uncertain operating and environmental factors. In the present study, the main uncertain inputs include peak charging demand, daily energy throughput, electricity price, ambient temperature, arrival rate, and energy per session. Since these quantities are often difficult to describe through sufficiently reliable probabilistic models, they are represented here by fuzzy numbers.

A fuzzy number $\tilde{x}$ on $ℝ$ is characterized by a membership function ${\mu }_{\tilde{x}}(t)$, which assigns to each $t\in ℝ$ a degree of compatibility with the linguistic and uncertain quantity $\tilde{x}$. In the proposed model, uncertain inputs like peak demand, energy throughput, electricity price, ambient temperature, arrival rate, and energy per session are modeled using standard triangular and trapezoidal fuzzy numbers [9,11,24].

A triangular fuzzy number is defined as

$$\tilde{x}=(a,b,c),\hspace{1em}\hspace{1em}\hspace{1em}a\le b\le c.$$

(1)

Its membership function is piecewise linear, increasing on [a, b] and decreasing on [b, c] with a peak value equal to one at t = b.

A trapezoidal fuzzy number is defined as

$$\tilde{x}=(a, b, c, d),\hspace{1em}\hspace{1em}\hspace{1em}a\le b\le c\le d,$$

(2)

Its membership function has a unit membership core on [b, c] and piecewise linear shoulders on [a, b] and [c, d].

The selected fuzzy inputs enter the optimization model through requirement-type constraints, cost-related terms, and service-oriented surrogate expressions. In this way, uncertainty affects the design problem at the level of station sizing, power selection, thermal provision, and operational adequacy.

3.2. α-Cut Representation of Fuzzy Uncertainty

For $\alpha \in [0,1]$, the α-cut of a fuzzy number $\tilde{x}$ is the closed interval

$${\left[\tilde{x}\right]}_{\alpha }=\left[x^L(\alpha ),x^U(\alpha )\right],$$

(3)

where x^L(α) and x^U(α) denote the lower and upper bounds of the interval associated with the satisfaction level α.

The α-cut representation transforms fuzzy quantities into deterministic interval bounds and therefore provides the basis for the optimization framework used in this study [11,12,13,24].

For the fuzzy numbers considered here, α-cuts admit closed-form expressions.

For a triangular fuzzy number $\tilde{x}=(a, b, c)$:

$${\left[\tilde{x}\right]}_{\alpha }=\left[a+\alpha (b-a),c-\alpha (c-b)\right].$$

(4)

For a trapezoidal fuzzy number $\tilde{x}=(a, b, c, d)$:

$${\left[\tilde{x}\right]}_{\alpha }=\left[a+\alpha (b-a),d-\alpha (d-c)\right].$$

(5)

3.3. α-Cut Arithmetic in Positive Domains

Many quantities arising in EV charging station design are nonnegative, including power demand, energy throughput, electricity price, arrival rate, and energy per session. For interval quantities

$$x\in [x^L,x^U],\hspace{1em}\hspace{1em}\hspace{1em}y\in [y^L,y^U],$$

(6)

with x^L ≥ 0 and y^L ≥ 0, conservative interval bounds for products and ratios follow directly from monotonicity over the positive domain [11,13,24].

3.3.1. Product

Let z = xy, with x ≥ 0 and y ≥ 0. Then

$$z\in \left[x^L{y}^L,x^U{y}^U\right].$$

(7)

This rule is used in the present model for quantities such as:

• offered load:

$$P_{off}={\lambda }_e,$$

(8)

• and loss-cost scaling:

$$C_{loss}=c_e{E}_{loss}.$$

(9)

3.3.2. Ratio

Let z = x/y with x ≥ 0 and y > 0. If y^L > 0, then

$$z\in \left[{\displaystyle \frac{x^L}{y^U}},{\displaystyle \frac{x}{y^L}}\right].$$

(10)

This relation becomes relevant when service-oriented or thermal submodels involve ratio-type expressions.

3.4. Conservative Deterministic Counterpart for Requirement-Type Constraints

Given a design decision vector x, constraints involving fuzzy parameters can be enforced at a prescribed satisfaction level α by requiring feasibility with respect to the corresponding α-cut bounds. In the present study, requirement-type constraints are treated conservatively by evaluating the relevant uncertain expression at the upper endpoint of the α-cut interval whenever the underlying function is nondecreasing with respect to the uncertain input. Thus, a constraint of the form $g(x,\tilde{Z})\le b$ is replaced at level α by the deterministic counterpart

$$g(x,Z^U(\alpha ))\le b.$$

(11)

This construction ensures feasibility for all admissible values of the uncertain parameter within the selected α-cut, provided that g is monotone nondecreasing in Z. To formalize the conservative reformulation principle used throughout the paper, the following proposition is stated.

Proposition 1.

Let g(x,Z) be a constraint function, where x is the design vector and Z is a fuzzy parameter with α-cut interval ${[Z]}_{\alpha }=[Z^L(\alpha ),Z^U(\alpha )]$. If g(x,Z) is monotone nondecreasing in Z over ${[Z]}_{\alpha }$, then the deterministic constraint:

$g\left(x,Z^U(\alpha )\right)\le 0$

Implies

$g(x,Z)\le 0$ for all $Z\in {[Z]}_{\alpha }$.

Proof sketch.

Since g(x,Z) is monotone nondecreasing in Z, its largest value over the interval ${[Z]}_{\alpha }$ is attained at the upper endpoint $Z^U(\alpha )$. Therefore, if the constraint is satisfied at $Z^U(\alpha )$, it is satisfied for every admissible value $Z\in {[Z]}_{\alpha }$. □

A similar conservative treatment is adopted for cost-related objective terms driven by nonnegative uncertain multipliers, where the corresponding upper bound at level α is minimized [10,12,17,25].

In the first two optimization settings considered in this paper, α is treated as a prescribed design parameter controlling the degree of uncertainty treatment. Alternatively, the α-cut level may be introduced as a decision variable in a coverage-oriented formulation. Since smaller α corresponds to wider α-cut intervals under the adopted convention, the resulting problem is interpreted as an admissibility or uncertainty-coverage analysis rather than as robustness maximization. In this setting, the smallest feasible α-cut level defines the extent of fuzzy uncertainty coverage, and the corresponding coverage index is interpreted according to the adopted α-cut convention [10,12,17,25].

Remark 1.

Proposition 1 applies only to expressions that are monotone in the uncertain parameter over the corresponding α-cut interval. For non-monotonic mappings, endpoint evaluation does not necessarily provide the exact worst-case value, since the maximum may occur at an interior point of the interval. In such cases, a more general interval-analysis treatment or direct maximization over the α-cut box would be required. In the present study, the proposition is applied only to requirement-type expressions that are monotone in the uncertain quantity over the relevant positive domain. For clarity, the monotonicity of the derived requirement expressions used in Section 4 and Section 5 is summarized in

Table 1. Monotonicity of the uncertain expressions used in the deterministic α-cut reformulation.

Expression Component	Uncertain Input	Role in the Formulation	Monotonicity over the Relevant Positive Domain	Deterministic α-Cut Treatment
Pinst ≥ DU(α)	D	Requirement-type constraint	Nondecreasing in D	Upper endpoint DU(α)
$T_{amb}^U(\alpha )+{\displaystyle \frac{k}{A_{hs}}}{P}_{loss}(\rho )\le T_{j,\max }$	Tamb	Requirement-type constraint	Nondecreasing in Tamb	$\mathrm{Upper} \mathrm{endpoint} T_{amb}^U(\alpha )$
LU(α) = λU(α)sU(α)	λ, s	Derived service-load quantity	Nondecreasing in λ and s for λ ≥ 0, s ≥ 0	Product upper bound via endpoint multiplication
$N\rho \ge {\displaystyle \frac{L^U(\alpha )}{{\rho }_{\max }}}$	λ, s	Service-oriented constraint	Nondecreasing in L, hence Nondecreasing in λ and s	Upper bound LU(α)
$C_{opex}^U(\alpha )=c_e^U(\alpha )E^U(\alpha ){\displaystyle \frac{P_{loss}(\rho )}{\rho }}$	ce, E	Cost objective	Nondecreasing in ce and E for ce ≥ 0, E ≥ 0	Product upper bound via endpoint multiplication
Closs = ceEloss	ce	Cost scaling	Nondecreasing in ce for Eloss ≥ 0	$\mathrm{Upper} \mathrm{endpoint} c_e^U(\alpha )$
Poff = λs	λ, s	Offered-load definition	Nondecreasing in λ and s for λ ≥ 0, s ≥ 0	Product upper bound via endpoint multiplication
Ratio-type service subexpressions z = x/y	x, y	Auxiliary derived quantity	Nondecreasing in x, nonincreasing in y for x ≥ 0, y > 0	Interval ratio bound $\left[{\displaystyle \frac{x^L}{y^U}},{\displaystyle \frac{x^U}{y^L}}\right]$

.

3.5. Queue-Risk Surrogate for Service-Oriented Constraints

To incorporate service-related performance in a computationally tractable manner, a utilization-based surrogate for queueing risk is adopted [14,15,16,23,24,25,26]. The purpose of this surrogate is not to reproduce a full queueing-theoretic model, but to provide a tractable constraint that reflects congestion sensitivity within the proposed α-cut optimization framework. Let:

• $\tilde{\lambda }$ denote the fuzzy arrival rate, [sessions/hour];
• $\tilde{e}$ denote the fuzzy energy per session, [kWh/session].

The fuzzy offered average power is then defined as:

$${\tilde{P}}_{off}=\tilde{\lambda }\tilde{e}.$$

(12)

At satisfaction level α, a conservative upper bound is obtained from the corresponding α-cut arithmetic as

$$P_{off}^U(\alpha ).$$

(13)

Given the installed station power $P_{inst}$ and a prescribed utilization limit p_max, service adequacy is enforced through the constraint

$$P_{off}^U(\alpha )\le {\rho }_{\max }{P}_{inst}.$$

(14)

The adopted utilization-based surrogate should be interpreted as a first-order congestion indicator rather than as a full queueing-theoretic model. It is most informative in moderate-to-high utilization regimes, where increasing offered load leads to a rapid deterioration of service conditions. At the same time, it does not reproduce waiting-time distributions, finite-queue effects, or transient congestion behavior. Relative to standard Erlang-C-type formulations, the surrogate may be conservative or optimistic depending on the arrival regime, the effective number of parallel service channels, and the admissible utilization range [14,16,23,26]. For this reason, the numerical section includes a comparative sensitivity check under representative stress scenarios.

3.6. Summary of Model Quantities and Design Variables

The upper-bound quantities derived from the α-cut representation and used repeatedly in the subsequent optimization models include:

$$D_U^{\alpha } ,\hspace{1em}\hspace{1em}{T}_{amb,U}^{\alpha } ,\hspace{1em}\hspace{1em}{L}_U^{\alpha }={\lambda }_U^{\alpha }{S}_U^{\alpha } ,\hspace{1em}\hspace{1em}{(c_{e}E)}_U^{\alpha }=c_{e,U}^{\alpha }{E}_U^{\alpha } .$$

(15)

Together with the modular station design variables and auxiliary sizing proxies, these quantities define the three optimization settings considered in the paper: CAPEX minimization at a prescribed α, OPEX minimization under a budget constraint, and service-oriented admissibility/uncertainty-coverage analysis [9,10,11,12,13,14,15,16,17,23,24,26].

4. System Model and Aggregated Design Variables

4.1. Modular Station Architecture and Installed Power

A modular DC fast-charging station supplied through a grid connection with a contractual power limit P_max is considered. The station is composed of N identical power modules operating in parallel, where p denotes the rated power of one module [1,2,3,4,27,28,29,30,31]. The total installed station power is therefore given by

$$P_{inst}=N\rho .$$

(16)

The installed power must satisfy the grid-limit constraint

$$P_{inst}\le P_{\max } .$$

(17)

In addition, the station must provide sufficient charging capacity under uncertain peak demand. Let D denote the corresponding fuzzy demand quantity and let D^U(α) be its upper α-cut bound at satisfaction level α. Then, peak-coverage feasibility is enforced through

$$P_{inst}\ge D^U(\alpha ) .$$

(18)

This formulation links the modular station architecture directly to the α-cut-based deterministic treatment of uncertainty and defines the primary sizing requirement used in the subsequent optimization models.

4.2. Discrete Module Rating Selection

In practical charging station design, admissible module ratings are selected from a standardized catalog rather than treated as continuous variables. Accordingly, the module power rating p is chosen from a discrete set P = {p₁, p₂, …, p_K}. To model this choice, binary variables are introduced so that exactly one catalog rating is selected:

$$\rho ={\displaystyle \sum _{k=1}^K{\rho }_k{y}_k},\hspace{1em}\hspace{1em}{\displaystyle \sum _{k=1}^K{y}_k=1},\hspace{1em}\hspace{1em}{y}_k\in \left\{0,1\right\},k=1,\dots ,K .$$

(19)

This representation preserves compatibility with mixed-integer optimization while reflecting the practical procurement and standardization constraints of modular charging hardware [2,3,27,29].

4.3. Auxiliary Sizing Proxies

To incorporate module dimensioning directly into the optimization problem, continuous proxy variables are introduced for two key engineering drivers, namely volumetric power density and thermal feasibility [1,2,3,4,27,30,31,32]. These proxies allow the model to link the selected module rating to physically meaningful sizing requirements without introducing excessive implementation detail.

4.3.1. Power-Density Constraint

Let V denote the module volume proxy and let p_V denote an achievable volumetric power density. The following constraint is imposed:

$$\rho \le {\rho }_{V}V .$$

(20)

This relation links the selected module rating p to a minimum required module volume and provides a compact representation of power-density-based dimensioning [2,27,29,30].

4.3.2. Loss Model at Rated Power

A compact surrogate is adopted for the losses at rated power:

$$P_{loss}=a\rho +b{\rho }^2,\hspace{1em}\hspace{1em}\hspace{1em}a,b\ge 0.$$

(21)

The linear term represents approximately constant conduction and auxiliary losses per unit of rated power, while the quadratic term captures load-dependent loss components. Although simplified, this form provides a tractable approximation suitable for inclusion in the optimization framework [2,3,27,30,32].

4.3.3. Thermal Feasibility via Cooling Proxy

Let Q_cool denote an effective cooling-capacity proxy and let R_θ(Q_cool) denote the corresponding effective thermal resistance. Thermal feasibility is enforced through a monotone surrogate of the form

$$\mathsf{\Delta }T\le R_{\theta }(Q_{cool})P_{loss}.$$

(22)

This relation makes the required cooling provision an endogenous design outcome and links thermal feasibility to both the selected rating and the loss model [1,2,3,4,27,30,31].

4.4. Aggregated Formulation

A key structural property of modular charging stations is the interchangeability of identical modules: any permutation of module labels corresponds to the same physical design and yields the same objective value. If such systems are modeled through explicitly indexed per-module variables, the formulation may contain many redundant permutation-equivalent solutions, which increases combinatorial complexity without adding physical meaning [18,19,20,21].

To avoid this redundancy, the present study adopts an aggregated formulation based on station-level design variables, such as the number of installed modules and the selected module rating, rather than module-indexed decision variables. This aggregated representation preserves the effective design space while yielding a more compact and transparent optimization model [18,19,20,21].

Remark 2.

The aggregated formulation is equivalent to an explicitly indexed formulation up to permutation symmetry of identical modules. More precisely, any feasible indexed design in which the modules are interchangeable induces the same physically distinct station configuration through the pair (N,p), together with the associated continuous sizing proxies, and any relabeling of module indices leaves both feasibility and objective value unchanged. Therefore, aggregation removes redundant design descriptions without excluding physically distinct feasible solutions.

4.5. Cost Modeling Components

The proposed optimization framework distinguishes between capital and operating cost contributions. For the investment-oriented setting, a separable CAPEX model is adopted:

$${\mathrm{C}}_{capex}=C^{fix}+N(c_0+c_V{V}_m+c_A{A}_{hs}) ,$$

(23)

where C^fix captures station-level infrastructure and installation costs, while c₀, c_V and c_A are module-related coefficients associated with the selected architecture and auxiliary sizing proxies [4,7,8,27,29].

For the operating-cost setting, the objective focuses on the annual cost of conversion losses. This term is driven by the fuzzy electricity price ${\tilde{c}}_e$ and the fuzzy annual delivered energy $\tilde{E}$. Using the positive-domain α-cut arithmetic introduced in Section 3, a conservative upper bound at a prescribed satisfaction level α₀ is written as

$${\mathrm{C}}_{opex}^U({\alpha }_0)=c_e^U\left({\alpha }_0\right)E^U({\alpha }_0)\left({\displaystyle \frac{P_{loss}(\rho )}{\rho }}\right).$$

(24)

The OPEX expression remains sensitive to the selected station design through the loss-related term and therefore supports comparison of feasible configurations not only in terms of investment cost, but also in terms of annual operating efficiency [2,3,10,11,12,27,30].

5. Optimization Problems Under α-Cut Uncertainty

This section formulates the three optimization settings derived from the proposed α-cut framework for modular EV fast-charging station design under fuzzy uncertainty. The objective is to translate the deterministic α-level bounds introduced in the previous sections into explicit design problems corresponding to investment-oriented, operating-cost-oriented, and service-oriented planning perspectives. In this way, the same mathematical framework is used to examine how different objective functions interact with grid, thermal, and service-related constraints under uncertain operating conditions.

5.1. Case A: Minimum CAPEX at Satisfaction Level α₀

$$\underset{N,V_m, A_{hs},y}{\min }{C}^{fix}+N(c_0+c_V{V}_m+c_A{A}_{hs})$$

(25)

subject to

$$D^U({\alpha }_0)\le N \rho \le P_{\max } ,$$

(26)

$$\rho ={\displaystyle \sum _{k=1}^K{\rho }_k{y}_k},\hspace{1em}{\displaystyle \sum _{k=1}^K{y}_k}=1,\hspace{1em}{y}_k\in \{0,1\},$$

(27)

$$\rho \le {\rho }_V{V}_m,$$

(28)

$$T_{amb}^U({\alpha }_0)+{\displaystyle \frac{k}{A_{hs}}}(\alpha \rho +b{\rho }^2)\le T_{j,\max } ,$$

(29)

$$N\in {\mathbb{Z}}_{\ge 0} ,\hspace{1em}{V}_m\ge 0 ,\hspace{1em}{A}_{hs}\ge 0 .$$

(30)

5.2. Case B: Minimum Loss-Driven OPEX at α₀ Under a CAPEX Budget

In the second optimization setting, the satisfaction level α₀ is prescribed, while the objective is to minimize the annual cost associated with conversion losses under a capital-investment limit. The problem is formulated as

$$\underset{N,V_m, A_{hs},y}{\min }{c}_e^U({\alpha }_0)E^U({\alpha }_0)\left({\displaystyle \frac{P_{loss}(\rho )}{\rho }}\right)$$

(31)

subject to all constraints of Case A and the additional constraint C_capex ≤ B.

This formulation highlights the trade-off between capital expenditure and operating efficiency, since lower-loss designs may require larger module ratings or increased auxiliary sizing, which in turn affect the admissible solution set under the CAPEX budget.

5.3. Case C: Service-Oriented Admissibility and Uncertainty Coverage

The third optimization setting is revised to avoid the misleading interpretation that larger α automatically implies a more robust design. Under the α-cut convention used in this paper, larger α produces narrower intervals and less conservative deterministic upper bounds. Therefore, direct maximization of α would primarily identify feasibility at the fuzzy core and would not measure uncertainty coverage.

The service-oriented component is instead expressed through the utilization-based condition:

$${\displaystyle \frac{L^U(\alpha )}{P_{\mathrm{inst}}}}\le {\rho }_{\max } \mathrm{or} \mathrm{equivalently} P_{\mathrm{inst}}\ge {\displaystyle \frac{L^U(\alpha )}{{\rho }_{\max }}},$$

(32)

where

$L^U(\alpha )={\lambda }^U(\alpha )s^U(\alpha )$ is the α-cut upper bound of the offered average charging power. This condition is used both as a fixed-α service-feasibility test and as part of a coverage-oriented formulation.

For the coverage-oriented variant, the model maximizes the admissible fuzzy-uncertainty coverage:

$$\mathsf{\Gamma }=1-\alpha$$

(33)

or equivalently minimizes the smallest α-cut level that remains feasible:

$$\max \mathsf{\Gamma }=1-\alpha$$

subject to the constraints of Case A, the service requirement:

$$L^U(\alpha )\le {\rho }_{\max }{P}_{\mathrm{inst}} \mathrm{and} 0\le \alpha \le 1$$

(34)

Because smaller α corresponds to wider α-cuts in the present convention, a smaller feasible ${\alpha }_{\min }$ indicates that the selected station architecture can cover a wider range of fuzzy input values. This formulation should be interpreted as a coverage or admissibility measure, not as a probabilistic reliability guarantee. Among designs attaining the same ${\alpha }_{\min }$, the minimum-CAPEX architecture is selected lexicographically.

5.4. Computational Solution Approach

The proposed optimization settings combine a discrete catalog-based choice of module rating with integer module-count decisions and continuous sizing proxies. Because the admissible module ratings are selected from a finite standard set, the resulting problems can be solved by exhaustive screening of physically distinct aggregated configurations (N,p), followed by closed-form evaluation of the continuous proxies.

For a fixed pair (N,p), installed power is fixed as:

$$P_{\mathrm{inst}}=Np .$$

(35)

The volume proxy enters through the lower-bound relation:

$$V_m\ge {\displaystyle \frac{p}{p_V}} ,$$

(36)

and has a positive coefficient in the CAPEX objective. Therefore, the optimal value is:

$$V_m^{*}={\displaystyle \frac{p}{p_V}}.$$

(37)

Similarly, the thermal feasibility relation can be written as:

$$A_{hs}\ge {\displaystyle \frac{k{P}_{\mathrm{loss}}(p)}{T_j^{\max }-T_{\mathrm{amb}}^U(\alpha )}}$$

(38)

under the adopted monotone thermal-resistance surrogate. Because A_hs also has a positive CAPEX coefficient, its optimal value for fixed (N,p,α) is the smallest feasible value.

Proposition 2.

Exactness of finite screening under the adopted surrogate model. For the model used in this paper, enumeration over all admissible catalog ratings p and all feasible integer module counts N is globally exact over the considered hybrid design space. For each fixed (N,p), the continuous sizing proxies V_m and A_hs are uniquely determined by their active lower-bound feasibility relations, and the objective functions are monotone nondecreasing in these proxies.

Proof sketch.

For fixed (N,p), the remaining continuous variables appear in separable lower-bound constraints and with nonnegative coefficients in CAPEX. Increasing V_m or A_hs beyond their lower feasible values cannot improve CAPEX and does not reduce the loss-driven OPEX term, which depends only on p under the adopted rated-loss surrogate. Hence, the fixed-(N,p) subproblem is solved by the smallest feasible proxy values. Since all physically distinct (N,p) configurations are explicitly screened, the best screened solution is globally optimal for the surrogate model. □

In the numerical implementation, all admissible catalog ratings p ∈ P and all integer module counts N satisfying N_p ≤ P_max are explicitly enumerated. For each pair (N,p), the installed power, volume proxy, cooling proxy, CAPEX, OPEX surrogate, and service-feasibility indicators are computed directly. Designs violating peak demand coverage, grid limits, thermal feasibility, budget feasibility, or service utilization constraints are discarded. Among the remaining feasible configurations, the best solution is selected according to the objective of the corresponding case. In Case A, the minimum CAPEX feasible design is selected; in Case B, the minimum OPEX design satisfying the CAPEX budget is selected; and in Case C, feasibility is checked over the admissible α-cut levels to determine the smallest feasible α_min, from which the coverage index Γ = 1 − α_min is obtained.

This exactness statement is model-specific. If non-convex part-load efficiency maps, detailed thermal-field models, or nonlinear cooling-performance curves were introduced, the fixed-configuration subproblem could cease to be monotone or convex, and a more rigorous global optimization procedure would be required.

6. Numerical Study

6.1. Study Design and Compared Cases

This section evaluates the proposed α-cut optimization framework on a representative modular DC fast-charging station with contractual grid limit P_max. The uncertain input quantities are represented by triangular or trapezoidal fuzzy numbers and are summarized in

Table 2. Numerical parameters and fuzzy input data used in the study.

Symbol	Description	Category	Value or Definition	Units
$P_{\max }$	Grid connection limit	Fixed	500	kW
P	Module rating catalog	Discrete set	{30, 40, 50, 60, 75, 100}	kW
$\tilde{D}$	Peak demand	Fuzzy (Tri)	Tri(320, 420, 520)	kW
$\tilde{E}$	Annual delivered energy	Fuzzy (Trap)	Trap(300,000, 450,000, 850,000, 1,200,000)	kWh/yr
${\tilde{c}}_e$	Electricity price	Fuzzy (Trap)	Trap(0.12, 0.16, 0.22, 0.30)	€/kWh
${\tilde{T}}_{amb}$	Ambient temperature	Fuzzy (Trap)	Trap(−10, 0, 30, 40)	°C
$\tilde{\lambda }$	Arrival rate	Fuzzy (Tri)	Tri(2, 5, 9)	session/h
$\tilde{S}$	Energy per session	Fuzzy (Tri)	Tri(18, 32, 55)	kWh/session
${\rho }_{\max }$	Utilization cap (queue surrogate)	Fixed	0.85	-
$P_{loss}(\rho )$	Rated-power loss model	Model	$\alpha P_m+b{P}_m^2$	kW
a	Linear loss coefficient	Fixed	0.01	-
b	Quadratic loss coefficient	Fixed	1.5 × 10−4	1/kW
$R_{\theta }(A_{hs})$	Thermal resistance surrogate	Model	$k/A_{hs}$	°C/kW
k	Thermal coefficient	Fixed	20	°C.cm2/kW
$T_{j,\max }$	Junction temperature limit	Fixed	110	°C
${\rho }_V$	Power density (volume proxy)	Fixed	2.5	kW/L
Ccapex	CAPEX model	Model	$C^{fix}+N(c_0+c_V{V}_m+c_A{A}_{hs})$	€
$C^{fix}$	Station fixed cost	Fixed	120,000	€
$c_0$	Base cost per module	Fixed	6000	€
$c_V$	Volume cost coefficient	Fixed	900	€/L
$c_A$	Cooling cost coefficient	Fixed	0.35	€/cm2
B	CAPEX budget (Case B)	Fixed	220,000	€

. For each prescribed satisfaction level α, conservative deterministic bounds for requirement-related and cost-related terms are obtained from the corresponding α-cut upper bounds, as summarized in

Table 3. α-level upper bounds used in the optimization models.

${\mathit{\alpha }}_0$	${\mathit{D}}^{\mathit{U}}({\mathit{\alpha }}_0)$	${\mathit{T}}_{\mathit{a}\mathit{m}\mathit{b}}^{\mathit{U}}({\mathit{\alpha }}_0)$	${\mathit{\lambda }}^{\mathit{U}}({\mathit{\alpha }}_0)$	${\mathit{s}}^{\mathit{U}}({\mathit{\alpha }}_0)$	${\mathit{L}}^{\mathit{U}}{({\mathit{\alpha }}_0)}^{\ast }$	${\mathit{c}}_{\mathit{e}}^{\mathit{U}}({\mathit{\alpha }}_0)$	${\mathit{E}}^{\mathit{U}}({\mathit{\alpha }}_0)$
0.70	450 kW	33.0 °C	6.2 1/h	38.90 kWh/session	241.18 kW	0.244 €/kWh	955,000 kWh/yr
0.85	435 kW	31.5 °C	5.6 1/h	35.45 kWh/session	198.52 kW	0.232 €/kWh	902,500 kWh/yr
0.95	425 kW	30.5 °C	5.2 1/h	33.15 kWh/session	172.38 kW	0.224 €/kWh	867,500 kWh/yr

* Note: $L^U({\alpha }_0)={\lambda }^U({\alpha }_0)s^U({\alpha }_0)$.

.

The fuzzy input descriptions used in the numerical study are intended to represent optimistic, nominal, and conservative engineering operating conditions for a representative modular DC fast-charging station. The selected triangular and trapezoidal fuzzy numbers were chosen to preserve analytical transparency of the α-cut bounds while remaining consistent with literature-informed engineering ranges for demand, energy throughput, electricity price, ambient temperature, and charging session characteristics. To show that the reported behavior is not an artifact of a single hand-selected parameter set, the numerical study is further extended with a deterministic nominal benchmark, a budget-sweep analysis, and stress scenarios involving increased traffic load and tighter utilization limits.

Three optimization settings are compared. In Case A, the objective is to minimize CAPEX subject to α-level peak demand coverage and thermal feasibility. In Case B, the objective is to minimize an α-level upper bound on annual loss-driven OPEX under a prescribed CAPEX budget B. In Case C, the service-related constraint is used in two complementary ways: first, as a service-stress design check at selected α-cut levels, and second, as a coverage-oriented formulation that maximizes the width of the admissible fuzzy uncertainty representation, Γ = 1 − α. This revised interpretation avoids equating larger α with stronger robustness.

In all cases, the module rating is selected from a discrete catalog P. The module volume and cooling-related variable are treated as continuous sizing proxies and are optimized jointly with the number of installed modules and the selected module rating. This hybrid discrete–continuous setting reflects practical equipment-selection constraints while allowing the station dimensioning variables to emerge directly from the optimization model.

6.2. α-Levels and Conservative Bounds

The numerical results are reported for the prescribed satisfaction levels ${\alpha }_0\in \{0.70, 0.85, 0.95\}$. For each selected level, Table 3 summarizes the corresponding α-level upper bounds used in the optimization models. These include: (i) the peak demand ambient-temperature bounds, $D^U({\alpha }_0)$ and $T_{amb}^U({\alpha }_0)$, used in the peak-coverage and thermal-feasibility constraints; (ii) the offered-load bound, $L^U({\alpha }_0)$, used in the service-oriented surrogate; and (iii) the electricity-price and annual-energy bounds, $c_e^U({\alpha }_0)$ and $E^U({\alpha }_0)$, used in the OPEX objective.

6.3. Reported Outputs and Evaluation Metrics

For each optimization run, the reported outputs include the resulting station architecture, auxiliary sizing variables, and the corresponding economic or service-oriented performance indicators. In particular, the numerical results include: (i) the number of installed modules N and the resulting installed power P_inst; (ii) the auxiliary sizing proxies, namely the module volume V_m and the cooling-related variable A_hs; (iii) the relevant economic metrics, including CAPEX for Case A and the loss-driven OPEX upper bound for Case B, with CAPEX also acting as a feasibility constraint in the budget-limited setting; (iv) deterministic benchmark values; and (v) for the service-oriented setting, the active service or peak demand driver and the uncertainty-coverage index Γ = 1 − α_min.

6.4. Parameter Set and Reproducibility

All numerical experiments are conducted using the parameter values reported in Table 2. The uncertain inputs are represented by standard triangular or trapezoidal fuzzy numbers, and the corresponding α-cuts are obtained analytically from their closed-form expressions. This ensures full reproducibility of the deterministic bounds used in the optimization models.

Because the main uncertain quantities considered in the study, such as demand, energy, electricity price, and arrival rate, are nonnegative, conservative upper bounds for product-type expressions are obtained by endpoint multiplication over the corresponding α-cuts. As a result, the mapping from fuzzy inputs to deterministic optimization instances remains transparent and directly traceable.

7. Results

7.1. α-Level Bounds and Uncertainty Treatment

The α-level upper bounds reported in Table 3 show how the conservative deterministic quantities vary with the prescribed satisfaction level α₀. As α₀ increases from 0.70 to 0.95, the corresponding upper bounds on peak demand and ambient temperature, D^U(α₀), and $T_{amb}^U$(α₀), decrease, reflecting the narrower α-cut intervals associated with higher satisfaction levels. The same tendency is observed for the offered-load bound L^U(α₀), which is used in the service-oriented surrogate.

In the representative parameter setting considered here, the peak demand coverage constraint remains the dominant sizing requirement across the tested α₀ values, whereas the service-oriented bound is less restrictive. This can be seen directly from the relative magnitudes reported in Table 3. As a result, the service-related constraint plays a secondary role in most of the reported designs, while peak demand coverage largely determines the selected station size.

7.2. Case A: CAPEX Design, Catalog Thresholds, and Peak Demand Sensitivity

Table 4. CAPEX-optimal designs for Case A, including the core α-cut level.

α	DU(α)	N*	p*	Pinst	Vm	Ahs	CAPEX
0.70	450 kW	6	75 kW	450 kW	30 L	0.414 cm2	318,000.87 €
0.85	435 kW	6	75 kW	450 kW	30 L	0.406 cm2	318,000.85 €
0.95	425 kW	6	75 kW	450 kW	30 L	0.401 cm2	318,000.84 €
1.00	420 kW	7	60 kW	420 kW	24 L	0.285 cm2	313,200.70 €

summarizes the CAPEX-optimal modular configurations obtained for the prescribed α-cut levels and for the core case α = 1.00. The inclusion of the core case makes the catalog-threshold effect explicit. For α = 0.70, α = 0.85, and α = 0.95, the peak demand upper bound remains above 420 kW, so the same 6 × 75 kW architecture is selected. At α = 1.00, the peak demand requirement decreases to 420 kW, making the 7 × 60 kW configuration feasible and less expensive. Thus, the baseline CAPEX results do not indicate that uncertainty is irrelevant; rather, they show that the selected architecture changes only when an α-dependent requirement crosses a discrete catalog threshold.

To demonstrate that the observed baseline stability is not merely an artifact of a single hand-selected parameter set,

Table 5. Peak demand sensitivity at α = 0.85.

Regime	Peak Demand Fuzzy Number	DU(0.85)	Selected Design	Pinst	CAPEX	Interpretation
R1 relaxed peak	Tri(300, 400, 480)	412 kW	7 × 60 kW	420 kW	313,200.70 €	lower requirement admits 7 × 60 kW
R0 baseline	Tri(320, 420, 520)	435 kW	6 × 75 kW	450 kW	318,000.85 €	threshold crossed; 6 × 75 kW selected
R2 stressed peak	Tri(360, 460, 560)	475 kW	5 × 100 kW	500 kW	336,000.93 €	grid-near requirement favors 5 × 100 kW

reports a peak demand sensitivity check at α = 0.85. A relaxed demand description leads to a seven-module 60 kW architecture, the baseline description selects six 75 kW modules, and a stressed demand description pushes the design to a five-module 100 kW architecture. The α-cut framework, therefore, changes the selected architecture when the fuzzy demand regime shifts relative to the discrete catalog thresholds.

7.3. Case B: Feasibility Thresholds and Budget-Sweep Behavior

7.3.1. Minimum Feasible Investment Level

Before minimizing loss-driven OPEX under a budget, the feasible investment threshold must be identified.

Table 6. Minimum feasible CAPEX threshold as a function of α.

α	Minimum-Feasible Design	Pinst	Minimum CAPEX	Dominant Feasibility Driver
0.70	6 × 75 kW	450 kW	318,000.87 €	peak demand/catalog threshold
0.85	6 × 75 kW	450 kW	318,000.85 €	peak demand/catalog threshold
0.95	6 × 75 kW	450 kW	318,000.84 €	peak demand/catalog threshold
1.00	7 × 60 kW	420 kW	313,200.70 €	peak demand/catalog threshold

shows that the originally specified budget B = 220,000 € is infeasible for all tested α-levels. This is not a numerical inconvenience but a design-relevant result: below the minimum feasible CAPEX, no modular architecture can simultaneously satisfy the peak demand, grid-limit, and thermal constraints.

7.3.2. Budget Sweep at α = 0.85

Table 7. Budget sweep for Case B at α = 0.85.

Budget B	Status	OPEX-Optimal Design	Pinst	CAPEX	OPEX Bound	Design Regime
220,000.00 €	Infeasible	–	–	–	–	below the feasibility threshold
300,000.00 €	Infeasible	–	–	–	–	below the feasibility threshold
318,500.00 €	Feasible	6 × 75 kW	450 kW	318,000.85 €	4449.72 €	CAPEX-dominated
330,000.00 €	Feasible	6 × 75 kW	450 kW	318,000.85 €	4449.72 €	CAPEX-dominated
340,000.00 €	Feasible	9 × 50 kW	450 kW	336,000.57 €	3664.15 €	transition
350,000.00 €	Feasible	11 × 40 kW	440 kW	344,400.44 €	3350.08 €	OPEX-oriented
375,000.00 €	Feasible	15 × 30 kW	450 kW	372,000.28 €	3035.99 €	low-loss flexible
400,000.00 €	Feasible	15 × 30 kW	450 kW	372,000.28 €	3035.99 €	low-loss flexible

gives an explicit budget sweep at α = 0.85. As the budget increases, the feasible OPEX-minimizing architecture moves from the CAPEX-minimum six-module 75 kW design to smaller module ratings with a lower loss ratio. This makes the CAPEX–OPEX trade-off visible: the objective can only select lower-loss 50 kW, 40 kW, and 30 kW module designs after the budget becomes sufficiently large.

7.4. Benchmark Against a Nominal Deterministic Design

A deterministic benchmark was added to separate the effect of the discrete catalog from the effect of fuzzy uncertainty treatment. The benchmark uses the modal triangular values D₀ = 420 kW, λ₀ = 5 sessions/h, and s₀ = 32 kWh/session. For trapezoidal quantities, the midpoint of the unit-membership core is used, namely E₀ = 650,000 kWh/year and c_e,0 = 0.19 €/kWh. The ambient-temperature benchmark uses the upper edge of the core, T_amb,0 = 30 °C, for consistency with thermal sizing.

The comparison between the deterministic nominal benchmark and the α-cut-based CAPEX-optimal designs is summarized in

Table 8. Deterministic nominal benchmark versus α-cut CAPEX designs.

Design Case	Input Treatment	Selected Design	Pinst	CAPEX	Loss-Cost Metric	Main Insight
Nominal deterministic	modal/core midpoint inputs	7 × 60 kW	420 kW	313,200.70 €	2347.50 €	catalog threshold at 420 kW
α-cut Case A, α = 0.95	upper α-cut bounds; DU = 425 kW	6 × 75 kW	450 kW	318,000.84 €	4156.91 €	uncertainty bound crosses catalog threshold
α-cut Case A, α = 0.85	upper α-cut bounds; DU = 435 kW	6 × 75 kW	450 kW	318,000.85 €	4449.72 €	uncertainty bound crosses catalog threshold
α-cut Case A, α = 0.70	upper α-cut bounds; DU = 450 kW	6 × 75 kW	450 kW	318,000.87 €	5372.44 €	uncertainty bound crosses catalog threshold

.

The benchmark shows a concrete design effect: the nominal deterministic solution is 7 × 60 kW, whereas the α-cut designs at α = 0.95, α = 0.85, and α = 0.70 require 6 × 75 kW because their peak demand upper bounds exceed 420 kW. The fuzzy framework therefore changes the selected architecture when the deterministic nominal solution lies close to a catalog feasibility threshold.

7.5. Case C: Service-Oriented Stress Scenarios and Uncertainty Coverage

Because α = 1 corresponds to the core of the fuzzy inputs and not to maximum robustness, the service-oriented analysis is reported explicitly as an admissibility and stress design check.

Table 9. Service stress designs at α = 1.

Scenario	Traffic and Service Setting	Service Req. at α = 1	Active Driver	Selected Design	Pinst	CAPEX
Baseline	λ = Tri(2, 5, 9); s = Tri(18, 32, 55); ρmax = 0.85	188.24 kW	peak demand	7 × 60 kW	420 kW	313,200.70 €
S1 heavier traffic	λ = Tri(3, 6, 10); s = Tri(18, 32, 55); ρmax = 0.85	225.88 kW	peak demand	7 × 60 kW	420 kW	313,200.70 €
S2 higher session energy	λ = Tri(2, 5, 9); s = Tri(22, 36, 60); ρmax = 0.85	211.76 kW	peak demand	7 × 60 kW	420 kW	313,200.70 €
S3 tighter utilization	λ = Tri(2, 5, 9); s = Tri(18, 32, 55); ρmax = 0.75	213.33 kW	peak demand	7 × 60 kW	420 kW	313,200.70 €
S4 combined stress	λ = Tri(3, 6, 10); s = Tri(22, 36, 60); ρmax = 0.75	288.00 kW	peak demand	7 × 60 kW	420 kW	313,200.70 €
S5 severe service stress	λ = Tri(4, 8, 12); s = Tri(25, 40, 60); ρmax = 0.75	426.67 kW	service	6 × 75 kW	450 kW	318,000.83 €

evaluates the minimum-CAPEX service-feasible design at α = 1 under progressively stronger traffic assumptions. In the baseline and moderate stress scenarios, peak demand coverage remains the active driver. In the severe service stress scenario S5, however, the service requirement exceeds the 420 kW peak demand core and changes the design from 7 × 60 kW to 6 × 75 kW.

The same scenarios were also evaluated through the coverage-oriented index Γ = 1 − α_min. Here, α_min is the smallest α-cut level for which at least one design can satisfy the grid limit, peak demand coverage, thermal feasibility, and service condition. A smaller α_min, or a larger Γ, means that the design can cover a wider fuzzy interval.

Table 10. Coverage-oriented service analysis under stress scenarios.

Scenario	αmin	Γ = 1 − αmin	DU(αmin)	Service Req.	Active Limit	Lexicographic Design
Baseline	0.200	0.800	500 kW	410.16 kW	peak/grid	5 × 100 kW (500 kW)
S1 heavier traffic	0.200	0.800	500 kW	485.65 kW	peak/grid	5 × 100 kW (500 kW)
S2 higher session energy	0.251	0.749	494.88 kW	494.88 kW	service	5 × 100 kW (500 kW)
S3 tighter utilization	0.384	0.616	481.63 kW	481.63 kW	service	5 × 100 kW (500 kW)
S4 combined stress	0.638	0.362	456.18 kW	456.18 kW	service	10 × 50 kW (500 kW)
S5 severe service stress	0.795	0.205	440.52 kW	440.52 kW	service	9 × 50 kW (450 kW)

shows that service stress reduces uncertainty coverage even before it changes the minimum-CAPEX architecture.

7.6. Queueing Sensitivity Check for the Utilization Surrogate

The utilization surrogate is computationally convenient, but it does not guarantee a specific waiting time target. To clarify its interpretation,

Table 11. Queueing sensitivity check for a six-channel 75 kW station.

Scenario	ρ = λs/Pinst	Erlang-C P(Wait)	Erlang-C Mean Wq	DES Mean Wq	DES P(Wq > 10 min)	Implication
Baseline core	0.356	0.028	0.3 min	0.2 min	0.006	low waiting risk
Heavy but below cap	0.711	0.425	7.1 min	7.0 min	0.213	surrogate feasible but visible waits
Near utilization cap	0.840	0.719	22.7 min	22.7 min	0.469	surrogate optimistic for waiting time
At capacity limit	1.000	1.000	∞	not simulated	not simulated	unstable/infeasible

compares the surrogate utilization with Erlang-C and a discrete-event M/M/c simulation for representative six-channel cases with 75 kW per channel. The comparison is not used to recalibrate the optimization model; it is included to show when the surrogate may be optimistic or conservative relative to waiting time indicators.

The table shows that the utilization cap is best interpreted as a first-order screening rule. At low utilization, it is consistent with negligible waiting times. Near the cap, however, the design may still satisfy ρ ≤ 0.85 while producing substantial waiting probabilities. Therefore, the surrogate is conservative only with respect to avoiding utilization overload; it can be optimistic if the design target is a strict waiting time percentile or finite-queue blocking probability.

8. Discussion and Limitations

The revised numerical evidence clarifies the role and the limits of the proposed α-cut optimization framework. The baseline CAPEX results are strongly affected by catalog granularity: for α = 0.70, 0.85, and 0.95, the upper bound on peak demand remains above 420 kW, and the same 6 × 75 kW configuration is selected. However, the deterministic benchmark and the α = 1 core case show that this behavior reflects a catalog-threshold effect rather than a lack of model sensitivity. When the requirement falls to the 420 kW threshold, the preferred design changes from 6 × 75 kW to 7 × 60 kW. The peak demand sensitivity study further confirms that different fuzzy demand regimes can produce different modular architectures.

The Case B budget sweep shows that the CAPEX budget is a structural part of the design problem. Budgets below approximately 318,000 € are infeasible in the baseline α range. Slightly above this threshold, the feasible set is narrow, and the result is dominated by minimum investment. As the budget increases, the OPEX objective gains the freedom to select smaller module ratings with a lower loss ratio. This makes the CAPEX–OPEX trade-off explicit rather than relying on a single representative budget.

The interpretation of α is central. Under the adopted α-cut convention, a larger α yields narrower bounds and less conservative deterministic constraints. Therefore, the revised Case C does not interpret a larger α as stronger robustness. Instead, service-oriented performance is analyzed through active stress scenarios and through the coverage index Γ = 1 − α_min. This formulation shows that stronger traffic assumptions and tighter utilization caps reduce uncertainty coverage and may make the service constraint active.

The physical surrogates influence the reported designs in specific ways. The rated-power loss model ranks module ratings by the loss ratio P_loss(p)/p; because this ratio increases with p under the selected coefficients, OPEX-oriented designs favor smaller module ratings when the CAPEX budget allows them. A more detailed part-load efficiency map could change this ranking if converters have different efficiency behavior at partial utilization. The thermal proxy affects CAPEX through A_hs but does not dominate the architecture in the reported parameter range; a detailed thermal field model or a nonlinear cooling curve could produce different auxiliary sizing. The service surrogate controls utilization but not waiting time percentiles, as shown by the queueing sensitivity check. Thus, the optimization results should be interpreted as design screening outcomes under transparent surrogates, not as full operational performance predictions.

The fuzzy input parameters are best understood as design study descriptions of optimistic, nominal/core, and conservative operating regimes. The added deterministic benchmark, demand regime sensitivity, budget sweep, and service stress scenarios were included to reduce dependence on a single baseline setting. Nevertheless, practical deployment would require calibration of the membership functions using station logs, local demand forecasts, electricity price data, and climate records.

Overall, the strength of the framework is not that it proves a universal optimal architecture for EV charging stations. Its value is that it provides a compact and reproducible way to determine when a modular design is controlled by fuzzy uncertainty bounds, by catalog thresholds, when a budget makes the problem infeasible, and when service adequacy becomes an active sizing driver.

9. Conclusions

This paper presents an α-cut-based optimization framework for modular EV fast-charging station design under fuzzy uncertainty and a prescribed grid limit. Uncertain peak demand, annual delivered energy, electricity price, ambient temperature, arrival rate, and energy per charging session are represented by triangular or trapezoidal fuzzy numbers and converted into deterministic α-cut bounds.

The design model combines discrete catalog-based module selection, integer module counts, and continuous sizing proxies for volume and cooling. The aggregated representation of interchangeable modules removes permutation-equivalent descriptions and makes exhaustive screening over physically distinct (N,p) configurations exact for the adopted surrogate model. A formal argument was added to show that, for fixed (N,p), the continuous proxies are uniquely determined by monotone active lower-bound constraints.

The strengthened numerical study provides a more balanced interpretation of the framework. In the baseline Case A results, CAPEX-optimal designs remain stable across α = 0.70, α = 0.85, and α = 0.95 because the same catalog threshold is active. However, the deterministic benchmark, the α = 1 core case, and the demand regime sensitivity study show that the α-cut treatment can change the selected modular architecture when requirement bounds cross catalog thresholds.

For Case B, the budget sweep shows that the originally specified B = 220,000 € is infeasible and that the OPEX objective becomes meaningful only after the budget exceeds the minimum feasible CAPEX threshold. Above that threshold, increasing the budget allows the optimizer to select smaller, lower-loss module ratings, thereby exposing a clear CAPEX–OPEX trade-off.

For the service-oriented setting, the interpretation of α was clarified. Because larger α corresponds to narrower and less conservative α-cuts, larger α is not interpreted as stronger robustness. The paper therefore reports service stress designs and uses the coverage-oriented index Γ = 1 − α_min to quantify the width of the admissible fuzzy uncertainty range. The added stress scenarios and queueing sensitivity check show when the utilization surrogate remains secondary, when it becomes the active sizing driver, and why it should not be interpreted as a full waiting time model.

The resulting contribution is a tractable modeling and optimization approach for uncertainty-aware modular charging station design screening. Future work should calibrate fuzzy inputs using measured charging data, incorporate part-load efficiency maps, replace or augment the utilization surrogate with validated queueing or simulation-based constraints, and test the method on multiple station classes and grid-connection regimes. The proposed framework should therefore be interpreted as a transparent design screening and comparative optimization tool based on simplified yet traceable surrogates. It is not intended to replace calibrated operational simulation, detailed thermal modeling, or full queueing-based service validation for real charging stations.