Enhancing electric vehicle charging station design using multifidelity simulations

Abstract

We study simulation-assisted service system design, where stochastic simulation is used to select the best design from a finite set of structural or parametric alternatives. Since high-fidelity simulation can be prohibitively time-consuming, we adopt a multifidelity approach that combines expensive high-fidelity runs with cheaper, coarser low-fidelity runs to estimate system performance and compare designs. This research is motivated by the design of an integrated electric vehicle (EV) fast-charging station. We formulate the design problem under the fixed-budget ranking and selection framework, in which the simulation budget is allocated across fidelity levels and design alternatives to maximize the probability of correct selection of the best design. We derive an asymptotic solution, develop a selection algorithm that satisfies the resulting optimality conditions, and establish its consistency and asymptotic optimality. We further demonstrate the algorithm’s empirical performance through an EV fast-charging station case study and a set of synthetic examples. These theoretical and empirical results provide actionable guidance on when and how multifidelity simulation can improve best-design selection in complex service system design problems.

1. Introduction

Simulation is a powerful technique for evaluating and optimizing real-world complex service systems that lack mathematically tractable models. It allows operators to replicate and study system dynamics over time, particularly in environments with complexity, uncertainty, and variability, and provides valuable insights into performance under different operating conditions. Accordingly, simulation-assisted service system design has become an important research area and has been widely applied across service systems, including supply chain management (Yao and Chen, 2018), portfolio optimization (Köksalan and Şakar, 2016), traffic system design (Shone et al., 2021), and healthcare resource allocation (Ordu et al., 2021).

The use of simulation for service system design comes with a notable issue: it can be excessively time-consuming, particularly for simulations that strive for high accuracy and details to replicate real-world systems. For example, a simulation run of 24-h dynamics of a transportation network with 20 intersections can take up to 2 h (Ho et al., 2007). Considering that multiple design alternatives need to be evaluated and multiple simulation runs need to be conducted for each design to obtain reasonable performance estimates, the total duration of the design process can be quite long. It limits the application of simulation when decisions are expected in a short period of time.

On the other hand, there often exist certain input parameters to the simulation model of the service system that govern the model’s accuracy. Operators can adjust these parameters to improve the model’s accuracy, which typically incurs higher time or monetary costs when running the model; alternatively, they can also adjust the parameters to produce a coarser model, which reduces run times or costs of the model. A common example of such parameters is the simulation time. A longer simulation run of the service system leads to more accurate estimates of the system performance, while a shorter simulation run results in coarser estimates.

Observing the trade-off between the output accuracy and computing efficiency when controlling these input parameters of the simulation models, the concept of multifidelity simulation has emerged as a promising approach for simulation-based performance evaluation and system design (Xu et al., 2016). Here, a fidelity level corresponds to certain values of these parameters. It shows the extent to which the simulation can mimic the system’s behavior. The basic idea of multifidelity simulation is that, for time-consuming simulation models, the operator does not have to conduct each simulation run at the highest-fidelity level (the output of which can be treated as samples of the system’s true performance). Instead, the operator might adaptively conduct the experiments at different fidelity levels (the highest or one of the lower levels), to obtain informative samples for system design while significantly reducing the computing time.

A case in point is the design of an electric vehicle (EV) fast-charging station. As shown in Figure 1, such a station integrates direct current (DC) fast chargers with on-site photovoltaic (PV) generation and battery storage to supply charging demand. Customers queue for service and may abandon when waiting times exceed their tolerance, while station operation must coordinate congestion and energy supply under equipment constraints. Due to the complexity of such stations, the interaction between queuing dynamics and power flows is difficult to model and optimize analytically, making simulation the preferred tool. The simulation can be conducted at different fidelity levels by varying the simulation horizon $T_{\text{days}}$ , the time resolution $\mathrm{\Delta }t$ , and the arrival segmentation $K_{\text{arr}}$ , which affect the accuracy and computational cost of the simulation samples.

Figure 1.

System architecture of an integrated EV charging station. EV = electric vehicle.

In this research, we study the problem of EV charging station design using multifidelity simulations, and assume that the operator can choose to run the simulation at a finite number of fidelity levels. The goal is to correctly and efficiently identify the best system design via simulation samples obtained from various fidelity levels.

We formulate this problem using the fixed-budget ranking and selection (R&S) model. This is a well-established branch of research in simulation, statistics, and machine learning (Gao et al., 2017b; Wu et al., 2024), which seeks to determine an efficient allocation of the simulation budget to the system designs so as to maximize the probability of correct selection (PCS) for the best design. The solution approach will be based on the optimal computing budget allocation (OCBA) method (Chen and Lee, 2011), a widely used framework for PCS optimization due to its intuitive allocation principle and practical effectiveness (Chen et al., 2023; Hu et al., 2025; Wang et al., 2025; Zhou et al., 2024). Traditional OCBA was developed to determine the allocation of the simulation budget at a single (the highest) fidelity level. In this research, we will extend it to incorporate simulation samples from multiple fidelity levels.

To estimate the true (highest-fidelity) system performance of each design, we employ the fusion estimator (Giles, 2015; Peherstorfer et al., 2018). Compared with standard estimators that rely on samples from a single fidelity level, the fusion estimator combines inexpensive but biased low-fidelity samples with a small number of unbiased but computationally expensive high-fidelity samples. It provides an unbiased estimate of the highest-fidelity performance without imposing strong assumptions, and can substantially reduce variance for a limited simulation budget. The fusion estimator has been successfully applied in various fields, including aerospace (Geraci et al., 2017), financial engineering (Giles and Szpruch, 2018), and so on.

Although motivated by integrated EV fast-charging stations, our setting arises in many simulation-assisted service system design problems where performance is evaluated at the highest fidelity, and the simulation model admits controllable fidelity parameters. Examples include adjusting fidelity via the simulation horizon, time-step resolution, demand or arrival aggregation, or the level of agent aggregation in agent-based simulations. In such problems, decision makers often compare a finite set of candidate designs under a limited simulation budget, making it important to combine information across fidelity levels in a principled way.

We highlight two points regarding the setting and methodology of this research.

First, we assume that there are input parameters to the simulation model that directly control its fidelity levels. Therefore, samples from different fidelity levels can be obtained conveniently by changing the values of these parameters when conducting simulation experiments. This is distinguished from some existing multifidelity simulation research, which is based on a number of simulation models with different structures and construction methodologies, each representing a level of fidelity (Peng et al., 2018). We argue that simulation models with different structures and construction methodologies are not always available for real service systems, and the costs for constructing them cannot be easily ignored.

Second, existing studies on multifidelity simulations have observed that incorporating low-fidelity levels into sampling may improve the effectiveness of optimizing service system performance (Kang et al., 2020; Peng et al., 2018; Xu et al., 2016). However, these studies generally lack a thorough analysis of this phenomenon, particularly with regard to the specific roles that different fidelity levels play in the performance optimization. As a result, several critical practical questions about multifidelity simulations remain unanswered, such as whether multifidelity simulations should be used to support service system design, and which fidelity levels should be prioritized when sampling them. The mathematical model and analytical results presented in this research can offer insights into these questions. This contributes to a deeper understanding among scholars and practitioners of the role and potential of multifidelity simulations in service system design.

We make four main contributions in this research. First, we provide a simulation-assisted design framework for integrated EV charging stations. This framework incorporates capacity planning, service quality, and adaptive operating decisions within station design, and relates equipment sizing decisions, such as the charger capacity and battery limits, to congestion effects and uncertainty arising from renewable generation. It further models the simulation fidelity, including simulation horizon, time-step resolution, and arrival segmentation, as a set of controllable inputs, and formulates the design problem as a fixed-budget R&S problem.

Second, we construct an unbiased fusion estimator for estimating the station’s highest-fidelity performance using samples from various fidelity levels, and derive allocation rules that determine the sample sizes to collect for each design–fidelity pair in order to maximize the PCS of selecting the best design. The resulting optimal sample sizes are proportional to a simple index that integrates key factors such as mean performance differences, sampling variances, and computational costs. This index quantifies the relative importance of sampling each fidelity level when comparing station designs under multifidelity simulations and provides good insights into how to balance sampling efforts across fidelity levels and designs.

Third, we develop a selection algorithm to implement the derived budget allocation rules, called the asymptotic optimal multifidelity sampling (AOMS) algorithm. We show that AOMS is consistent and asymptotically optimal, and provides a stopping rule that guarantees a target PCS upon termination. The superior performance of AOMS is also demonstrated by numerical experiments on an EV charging station case study and a set of synthetic examples.

Last, our empirical results provide practical guidance for EV charging station design under multifidelity simulation. In particular, they show that the sampling process adaptively allocates the simulation budget across fidelity levels according to the estimated optimal budget proportions. In the early stage of sampling, lower-fidelity levels are often more undersampled relative to their optimal allocation targets and should therefore receive more simulation effort. As the sampling proceeds, relatively more budget should be allocated to higher-fidelity levels in order to distinguish small performance differences at the target fidelity. Our results also suggest that service-capacity decisions are the main drivers of system performance because they directly shape congestion and service-level agreement (SLA) compliance under peak demand, while PV, battery, and inverter sizing play a secondary and regime-dependent role by affecting operating cost and feasibility, and become critical mainly when charging intensity and concurrent power demand are high.

The remainder of the paper is organized as follows. Section 2 reviews the related literature. Section 3 provides preliminaries of this research and then formulates the EV charging station design problem. Section 4 derives the optimality conditions of the design problem built in Section 3 and extracts some useful insights from the conditions. In Section 5, we propose an efficient sequential selection algorithm and analyze its theoretical properties. Numerical examples and computational results are reported in Section 6, followed by conclusions and discussion in Section 7. In the E-Companion, Sections EC.1–EC.8 provide technical proofs of all lemmas, theorems, and propositions. Section EC.9 further discusses the stopping rule that guarantees the PCS. Section EC.10 provides a detailed version of AOMS, and Section EC.11 provides additional numerical tests and results to the main paper.

2. Literature review

This research relates to two streams of literature: simulation-based service system design and R&S.

2.1. Simulation-based service system design

Simulation-based service system design formulates the design task as an optimization problem, where simulation is used to evaluate the system’s mean performance and decision variables govern the structural and parametric aspects of the design (Mefteh, 2018). These problems typically involve large solution spaces and therefore require search-based optimization methods to guide exploration (Fu, 2015). In this research, we follow this modeling paradigm but focus on a finite set of system designs.

In the context of EV charging station design, existing studies often use empirical data and optimization models to characterize interactions among charging access, grid constraints, and user behavior. They address decisions such as station location, the number and types of chargers, and operating rules for pricing, queue management, and energy use (Guillet and Schiffer, 2023; Valogianni et al., 2020), subject to constraints imposed by the grid and travel demand. For example, Powell et al. (2022) studied how infrastructure build-out and operating strategies affect peak loads and renewable integration. Related work has examined fast-charging network expansion and installation (Hanig et al., 2025; Yu et al., 2022), transportation-power planning for facility siting and sizing (Mao et al., 2020), joint infrastructure and local energy planning (Jiao et al., 2020), congestion and service analysis (Pourvaziri et al., 2024), and charging operations under uncertainty (Chen et al., 2024; Punt et al., 2024).

These studies provide an important foundation for EV charging system design, but to maintain tractability, many of them rely on aggregate or stylized representations of station capacity and service. Such abstractions are useful for planning, but less suitable for comparing station layouts when performance depends on complex operational dynamics. In this research, we develop a multifidelity simulation framework for fast-charging stations that allows detailed modeling of operational dynamics and system performance. The resulting solution method is simulation-based and therefore fundamentally different from the approaches described above.

2.2. Ranking and selection

The other related stream is R&S, which aims to identify the best design from a finite set of alternatives. Its performance is commonly measured by the PCS (Chen et al., 2008) or the expected opportunity cost (Gao et al., 2017a). Existing R&S procedures mainly include fixed-budget methods (Chen et al., 2000; Frazier and Powell, 2008; Gao and Chen, 2015; Ryzhov, 2016), which optimize selection quality under a fixed simulation budget, and fixed-confidence methods (Fan et al., 2016; Kim and Nelson, 2006; Zhong and Hong, 2022), which seek a target guarantee with minimal sampling effort.

In multifidelity simulation, R&S approaches can be grouped into adaptation, filtering, and fusion (Peherstorfer et al., 2018). Adaptation methods build low-fidelity surrogates (often based on Gaussian processes (GPs)) to guide most of the search. A representative example here is the multifidelity knowledge gradient (KG) procedure of Wu and Frazier (2017). Filtering methods use low-fidelity models for preliminary screening and invoke the highest-fidelity model only for promising designs, as in Peng et al. (2018) and Xu et al. (2016).

Fusion methods instead combine low- and high-fidelity samples to obtain more accurate or lower-variance estimators. Despite their potential, fusion-based R&S procedures remain relatively limited. Our method falls into this category. We treat fidelity as a controllable input and use a fusion estimator that exploits inexpensive low-fidelity simulations to improve the efficiency of selecting the best design.

3. Problem formulation

In this section, we first introduce the basic setup of the EV charging station design problem, and then provide a formulation of this problem under multifidelity simulations.

3.1. EV charging station design

3.1.1. System and operations

We consider an integrated EV charging station as introduced in Section 1. Compared with a traditional charging station, the integrated charging station incorporates five main physical assets and interfaces: (i) a bank of identical DC fast chargers; (ii) a PV array; (iii) a stationary battery energy storage system (BESS); (iv) a bidirectional grid-tied inverter that aggregates site alternating current (AC) power; and (v) the utility interconnection. Customers arrive according to a stochastic process, join a finite-capacity queue at the chargers, and may abandon if their waiting time exceeds a tolerance threshold. Energy is dispatched adaptively based on the station state, subject to equipment ratings and interconnection constraints.

Customer arrivals are modeled as a nonhomogeneous Poisson process with a time-varying rate that captures systematic intraday demand variation. Each customer is characterized by an energy demand, a vehicle-side charging-power cap, and a patience time. Vehicles queue for the chargers, with admission decisions based on accrued waiting time, remaining charging requirement, and short-horizon forecasts of congestion and energy conditions. A charging session is completed once its required energy has been delivered. Vehicles whose waiting time exceeds their patience depart without service, which then results in lost demand and service penalties.

Energy supply to the station is provided by on-site PV generation, the utility grid, and a stationary BESS. PV generation and battery power are represented on the DC side, while a bidirectional inverter with rating $P_{\text{inv}}$ constrains the total AC power available for EV charging, battery operations, and grid exports. Grid imports and exports are subject to interconnection limits $P_{\text{grid}}^{max}$ and $P_{\text{export}}^{max}$ . Available supply is allocated among active sessions using state-dependent dispatch weights that balance charging needs against near-term price and supply conditions.

Such a station generates cash flows from electricity transactions. Electricity sold to vehicles generates revenue at a fixed retail tariff, while electricity purchased from the grid is settled at time-of-use prices and electricity sold back to the grid is credited at a discounted rate. Capital expenditures (CAPEX) for chargers, PV, the inverter, and the battery are required and converted into equivalent daily costs, with operations and maintenance (O&M) costs accounted for on a per-day basis. SLA violations incur monetary penalties, as determined by the 95th-percentile waiting time and the abandonment rate.

3.1.2. Station designs

We consider a finite set of potential designs, denoted by $\{{\mathit{z}}_i,i\in \mathcal{K}\}$ , each of which is defined by a vector of structural and parametric configurations that specify the station’s integrated architecture (i)–(v). Specifically, we adopt the following design variables: the number of DC fast chargers $n_{\text{c}}$ , the PV array area $A_{\text{pv}}$ (m ${}^2$ ), the battery capacity $E_{\text{b}}$ (kWh), the battery charge/discharge power rating $P_{\text{b}}$ (kW), and the site inverter capacity $P_{\text{inv}}$ (kW). Each design is restricted to a predefined set of capacity levels based on standard engineering specifications and common design choices.

3.1.3. Fidelity controls

Let $\mathit{s}=(T_{\text{days}},\mathrm{\Delta }t,K_{\text{arr}})$ denote the fidelity parameters of the simulation model for the EV charging station, where $T_{\text{days}}$ denotes the simulation horizon in days, $\mathrm{\Delta }t$ denotes the discrete-time step (in min), and $K_{\text{arr}}$ denotes the number of piecewise-constant segments used to approximate the intraday arrival-rate curve. Each parameter in $\mathit{s}$ takes a finite number of values. Denote $m$ as the total number of values of $\mathit{s}$ , and let $\mathcal{S}=\{{\mathit{s}}_q\mid q=1,\dots ,m\}$ be the set of these values. Note that these parameters control the resolution of the simulation samples for a given station design, and varying their values does not change the candidate design set, the objective function, the operating policy, or the stochastic primitives of the underlying station model.

Treating these input parameters as adaptive controls allows the operator to conduct experiments at different fidelity levels and to manage the trade-off between estimation accuracy and computational cost. Specifically, extending the simulation horizon improves the stability of performance estimates by averaging out short-term fluctuations, but increases computational cost from longer simulated operations. Reducing the time step enhances waiting time and congestion estimates through more precise state representation, at the cost of more frequent state updates. In addition, increasing arrival segmentation improves the estimation of service quality and revenue by capturing within-day demand variation, but causes higher computational cost due to more granular evaluation of demand and operational decisions.

3.1.4. Objective

The operator’s objective is to identify a design that minimizes the expected total cost over the simulated horizon. The cost is measured on a per-day basis and combines amortized CAPEX and O&M costs, adjusted for SLA penalties (denoted by ${\text{Pen}}_{\text{SLA}}$ ) and operating profit (denoted by ${\pi }_{\text{oper}}$ ) from charging services and electricity trades with the grid, which is given by:

$$\begin{array}{rl}\underset{{\mathit{z}}_i,i\in \mathcal{K}}{min}\mu ({\mathit{z}}_i,{\mathit{s}}^{*},P)& \triangleq ({\text{CAPEX}}_{\text{day}}({\mathit{z}}_i)+{\text{O\&M}}_{\text{day}}({\mathit{z}}_i))T_{\text{days}}\\ & -{\mathbb{E}}_{\xi \sim P}[{\pi }_{\text{oper}}({\mathit{z}}_i,{\mathit{s}}^{*},\xi )-{\text{Pen}}_{\text{SLA}}({\mathit{z}}_i,{\mathit{s}}^{*},\xi )].\end{array}$$

(1)

Here, each design is evaluated by its mean performance $\mu ({\mathit{z}}_i,{\mathit{s}}^{*},P)$ over the simulated horizon $T_{\text{days}}$ , where ${\mathit{s}}^{*}$ is the highest-fidelity level. The amortized CAPEX and O&M costs admit explicit closed-form expressions and can be computed directly from the design-specific cost parameters multiplied by $T_{\text{days}}$ . In contrast, the station’s operating profit and SLA penalties depend on stochastic demand and operational conditions, which are not available in analytical form. These components are estimated by simulation. Given the input distribution $P$ of the simulation model, operators can then evaluate $\mu ({\mathit{z}}_i,{\mathit{s}}^{*},P)$ for each design ${\mathit{z}}_i$ using simulation samples of ${\pi }_{\text{oper}}({\mathit{z}}_i,{\mathit{s}}^{*},\xi )$ and ${\text{Pen}}_{\text{SLA}}({\mathit{z}}_i,{\mathit{s}}^{*},\xi )$ under realizations $\xi \sim P$ and the operating decisions implied by them. Based on these estimates, the best design (with the lowest expected total cost) can be selected. For notational simplicity, we let ${\mu }_i(\mathit{s})\triangleq \mu ({\mathit{z}}_i,\mathit{s},P)$ and refer to ${\mathit{z}}_i$ as design $i$ thereafter.

3.2. Multifidelity estimation and fixed-budget R&S

3.2.1. Problem setup

For the system designs under comparison, let $X({\mathit{z}}_i,{\mathit{s}}_q,{\xi }_l)$ be the simulation sample collected in the $l$ -th replication of design $i$ at fidelity level ${\mathit{s}}_q$ , with $i=1,2,\dots ,k$ and $q=1,2,\dots ,m$ , and let $c_{i,q}$ denote the cost for obtaining one simulation sample for design $i$ at that fidelity level. Without loss of generality, suppose that for $q=1,\dots ,m$ , the fidelity level decreases from high to low, and assume that the simulation costs of the $m$ fidelity levels are in a decreasing order, that is, $c_{i,1}\ge c_{i,2}\ge \cdots \ge c_{i,m}$ for each station design $i$ . In the EV problem, these fidelity levels follow the nested hierarchy defined in Section 3.1.3. The goal is to find the best design $i^{*}\triangleq {\arg \min }_i{\mu }_i({\mathit{s}}_1)$ with a given simulation budget $n$ . The cost for simulations at ${\mathit{s}}_1$ is often expensive, and a fixed simulation budget permits only a limited number of highest-fidelity replications. Therefore, we must carefully allocate the budget across fidelity levels to identify the best design efficiently.

Define the mean performance difference for design $i$ between fidelity levels ${\mathit{s}}_q$ and ${\mathit{s}}_{q+1}$ as ${\delta }_{i,q}\triangleq {\mu }_i({\mathit{s}}_q)-{\mu }_i({\mathit{s}}_{q+1})$ , $q=1,\dots ,m-1$ , and define ${\delta }_{i,m}\triangleq {\mu }_i({\mathit{s}}_m)$ for the lowest-fidelity level ${\mathit{s}}_m$ . Then, ${\mu }_i({\mathit{s}}_1)$ admits the telescoping decomposition $\sum _{q=1}^m{\delta }_{i,q}$ , which we estimate by sampling each term at its corresponding fidelity levels.

Let $Y_{iql}\triangleq X({\mathit{z}}_i,{\mathit{s}}_q,{\xi }_{iql})-X({\mathit{z}}_i,{\mathit{s}}_{q+1},{\xi }_{iql})$ denote the $l$ -th simulation sample for ${\delta }_{i,q}$ , $q=1,\dots ,m-1$ , and $Y_{iml}\triangleq X({\mathit{z}}_i,{\mathit{s}}_m,{\xi }_{iml})$ denote the $l$ -th simulation sample for ${\delta }_{i,m}$ . Let ${\kappa }_{i,q}$ be the cost of obtaining one sample of ${\delta }_{i,q}$ . The per-sample costs $c_{i,q}$ and ${\kappa }_{i,q}$ satisfy the following relationship:

$${\kappa }_{i,m}=c_{i,m},c_{i,q}\le {\kappa }_{i,q}\le 2c_{i,q},q=1,\dots ,m-1.$$

(2)

Equation ${\kappa }_{i,m}=c_{i,m}$ holds because each sample $Y_{iml}$ for ${\delta }_{i,m}$ is obtained from a single simulation run at the lowest-fidelity level ${\mathit{s}}_m$ . For $q=1,\dots ,m-1$ , each sample $Y_{iql}$ for ${\delta }_{i,q}$ is computed by evaluating the same design and performance measure at two adjacent fidelity levels (i.e., ${\mathit{s}}_q$ and ${\mathit{s}}_{q+1}$ ) under the same environmental realization ${\xi }_{iql}$ . Therefore, when $c_{i,q}\ge c_{i,q+1}$ , the sampling cost ${\kappa }_{i,q}$ satisfies $c_{i,q}\le {\kappa }_{i,q}\le 2c_{i,q}$ .

In the EV charging simulation, adjacent fidelity levels can be coupled within this hierarchy, so paired sampling often requires only a small incremental cost. When two fidelity levels differ only in horizon length, the lower-fidelity output is an intermediate result of the higher-fidelity run. When they differ in time step or arrival segmentation, the two fidelity levels can share the same environmental realization, which limits redundant computation.

To facilitate the analysis, we make the following technical assumptions.

Assumption 1

The best system design $i^{*}$ is unique.

Assumption 2

Samples $Y_{iql}$ are independent across different simulation replications $l$ , designs $i$ , and fidelity levels ${\mathit{s}}_q$ . Moreover, $Y_{iql}$ ’s are normally distributed with mean ${\delta }_{iq}$ and variance ${\eta }_{i,q}^2$ .

Assumption 1 requires that the best design is unique. This assumption is reasonable because if two designs have the same mean performance, they cannot be distinguished in terms of optimality.

Assumption 2 is also standard in the simulation optimization literature (Law and Kelton, 2000). Independence across simulation runs can be realized by using independent random number streams. Note that this independence does not apply to the paired samples used to construct $Y_{iql}$ . The two samples $X({\mathit{z}}_i,{\mathit{s}}_q,{\xi }_{iql})$ and $X({\mathit{z}}_i,{\mathit{s}}_{q+1},{\xi }_{iql})$ share the same realizations of arrivals, weather, and prices and are therefore generally correlated. Normality is also reasonable because simulation outputs are often averages (e.g., average performance measures or batch means) and therefore are approximately normally distributed by the Central Limit Theorem. Our analysis also extends to sub-Gaussian samples with only minor technical modifications, but in this research we focus on the normal case for simplicity.

3.2.2. Fusion estimator

In this research, we estimate ${\mu }_i({\mathit{s}}_1)$ for each station design $i$ by fusing less expensive but potentially biased samples from lower-fidelity levels with a small number of unbiased but expensive samples at the highest-fidelity level. Let $n_{i,q}\in \mathbb{N}$ denote the number of samples used to estimate ${\delta }_{i,q}$ . The budget spent on estimating ${\delta }_{i,q}$ is ${\kappa }_{i,q}{n}_{i,q}$ , and the total simulation budget is $n\triangleq \sum _{q=1}^m\sum _{i=1}^k{\kappa }_{i,q}{n}_{i,q}$ . For notational simplicity, define $p_{i,q}\triangleq {\kappa }_{i,q}{n}_{i,q}/n$ as the proportion of the total budget allocated to estimating ${\delta }_{i,q}$ , and let $\mathit{p}$ denote the vector of all $p_{i,q}$ ’s. We estimate ${\mu }_i({\mathit{s}}_1)$ using the fusion estimator:

$${\hat{\mu }}_i^n({\mathit{s}}_1)=\sum _{q=1}^m{\overline{Y}}_{i,q}^n,$$

(3)

where ${\overline{Y}}_{i,q}^n=(1/n_{i,q})\sum _{l=1}^{n_{i,q}}{Y}_{iql}$ is the sample mean estimator of ${\delta }_{i,q}$ based on $n_{i,q}$ samples. The estimator ${\hat{\mu }}_i^n({\mathit{s}}_1)$ is adapted from the multilevel Monte Carlo framework (Giles, 2008, 2015). By construction, this estimator is unbiased.

Given the simulation budget proportion $p_i\triangleq \sum _{q=1}^m{p}_{i,q}$ for each design $i$ , we can further allocate this budget across fidelity levels to minimize the variance of the fused estimator. The following lemma characterizes the optimal allocation within each design that achieves the minimal variance.

Lemma 1

The budget proportion to minimize the variance of ${\hat{\mu }}_i^n({\mathit{s}}_1)$ satisfies:

$$\frac{p_{i,q}}{p_{i,q^{\prime }}}=\frac{{\mathcal{V}}_{i,q}}{{\mathcal{V}}_{i,q^{\prime }}},$$

(4)

and the minimal variance is $(\sum _{q=1}^m{\mathcal{V}}_{i,q}{)}^2/(p_{i}n)$ where ${\mathcal{V}}_{i,q}\triangleq ({\kappa }_{i,q}{\eta }_{i,q}^2{)}^{1/2}$ .

According to Lemma 1, given a design $i$ , the optimal budget proportion assigned to fidelity level ${\mathit{s}}_q$ is proportional to ${\mathcal{V}}_{i,q}$ . Similar fusion estimators have been used in real-world applications (Geraci et al., 2017; Giles and Szpruch, 2018) to reduce variance. In our setting, variance reduction is not only important for accurate performance estimation, but also directly improves the ability of the algorithm to rank competing designs, and thus enhances the quality of the selected design. This benefit is strengthened in the EV application where the nested fidelity hierarchy allows the telescoping decomposition to successively correct the bias from shorter horizons, coarser time grids, and rougher arrival-rate approximations. To illustrate why the estimator can be effective, consider a simple case with two fidelity levels $m=2$ : a low-fidelity level ${\mathit{s}}_2$ and a high-fidelity level ${\mathit{s}}_1$ . The telescoping decomposition ${\mu }_i({\mathit{s}}_1)={\delta }_{i,1}+{\delta }_{i,2}$ , where ${\delta }_{i,1}={\mu }_i({\mathit{s}}_1)-{\mu }_i({\mathit{s}}_2)$ and ${\delta }_{i,2}={\mu }_i({\mathit{s}}_2)$ , suggests estimating a coarse level at ${\mathit{s}}_2$ and correcting it using paired differences between ${\mathit{s}}_1$ and ${\mathit{s}}_2$ :

• (a)

Low-fidelity samples are inexpensive. With a small simulation budget, one can collect many samples at ${\mathit{s}}_2$ to obtain a stable estimate of ${\mu }_i({\mathit{s}}_2)$ , which serves as a baseline even though ${\mu }_i({\mathit{s}}_2)$ is biased relative to ${\mu }_i({\mathit{s}}_1)$ .

(b)

The variance of estimating ${\delta }_{i,1}\triangleq {\mu }_i({\mathit{s}}_1)-{\mu }_i({\mathit{s}}_2)$ can be small when the two fidelities are under the same environmental realization. In particular, using common random numbers makes the paired outputs $X({\mathit{z}}_i,{\mathit{s}}_1,\xi )$ and $X({\mathit{z}}_i,{\mathit{s}}_2,\xi )$ highly correlated, so the increment $Y_{i,1}=X({\mathit{z}}_i,{\mathit{s}}_1,\xi )-X({\mathit{z}}_i,{\mathit{s}}_2,\xi )$ may exhibit substantially lower variance than either marginal output.

These two effects allow the fusion estimator to remain unbiased for ${\mu }_i({\mathit{s}}_1)$ while achieving a lower variance per unit of computation than using high-fidelity simulations alone. The same intuition extends to multiple fidelity levels, where adjacent-level coupling can produce a sequence of low-variance increments. Section EC.2 of the E-Companion provides a detailed variance comparison and verifies that the fusion estimator indeed leads to variance reduction in the EV simulation.

Another advantage of the fusion estimator is that its variance-minimizing allocation (4) across fidelity levels also serves the objective in multifidelity simulation-based service system design. For each design $i$ , given its total budget proportion $p_i=\sum _{q=1}^m{p}_{i,q}$ , the allocation of $\{p_{i,q}{\}}_{q=1}^m$ that minimizes $\mathrm{Var}({\hat{\mu }}_i^n({\mathit{s}}_1))$ is consistent with the rate-optimal allocation that maximizes the PCS. This will be discussed in more detail in Section 4.

3.2.3. Selection quality measure

Based on the fusion estimators ${\hat{\mu }}_i^n({\mathit{s}}_1)$ under a total simulation budget $n$ , we can compare the $k$ system designs and select the estimated best one ${\hat{i}}_n^{*}={\arg \min }_i{\hat{\mu }}_i^n({\mathit{s}}_1)$ . A correct selection happens when the estimated best design ${\hat{i}}_n^{*}$ is the real best one $i^{*}$ . However, with a finite simulation budget, the correct selection can never be guaranteed. In this research, we seek to maximize the PCS for the best design:

$${\mathrm{PCS}}^n=\mathbb{P}({\hat{i}}_n^{*}=i^{*})=\mathbb{P}\left(\bigcap _{i=1,i\ne i^{*}}^k\{{\hat{\mu }}_{i^{*}}^n({\mathit{s}}_1)<{\hat{\mu }}_i^n({\mathit{s}}_1)\}\right),$$

by finding the optimal allocation of the total simulation budget to the design–fidelity pairs. Essentially, we will solve the following simulation budget allocation problem:

$$\begin{array}{rl}\underset{\mathit{p}}{max}& {\mathrm{PCS}}^n\\ \text{s.t.}& \sum _{i=1}^k\sum _{q=1}^m{p}_{i,q}=1,p_{i,q}\ge 0,i=1,\dots ,k,q=1,\dots ,m.\end{array}$$

(5)

In this optimization problem, the decision variables are the budget proportions $p_{i,q}$ , $i=1,\dots ,k$ , $q=1,\dots ,m$ , which control the number of simulation samples $n_{i,q}$ allocated to estimate each ${\delta }_{i,q}$ . Because of the lack of analytical expressions for the objective function $\mathrm{PCS}$ , it is challenging to find the exact optimizer of (5). Instead, a common practice in the R&S literature (Chen et al., 2000; Lee et al., 2012) is to pursue the asymptotic optimizer (as $n\to \mathrm{\infty }$ ). That is, we can look for the solution that maximizes the rate at which $\mathrm{PCS}$ converges to one. Accordingly, we define the probability of false selection ( $\mathrm{PFS}$ ) for the best design as

$${\mathrm{PFS}}^n=\mathbb{P}(\bigcup _{i=1,i\ne i^{*}}^k\{{\hat{\mu }}_{i^{*}}^n({\mathit{s}}_1)\ge {\hat{\mu }}_i^n({\mathit{s}}_1)\}).$$

(6)

It is obvious that ${\mathrm{PCS}}^n=1-{\mathrm{PFS}}^n$ and the rate at which ${\mathrm{PCS}}^n$ converges to one is identical to the rate of ${\mathrm{PFS}}^n$ converging to zero. Below, we characterize the convergence rate of $\mathrm{PFS}$ .

Lemma 2

Under Assumptions 1 and 2, the rate function for the $\mathrm{PFS}$ can be expressed as:

$$\begin{array}{rl}-\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\log {\mathrm{PFS}}^n=& \underset{i\in \{1,2,\dots ,k\},i\ne i^{*}}{min}{\mathcal{R}}_{i^{*},i}\left(\mathit{p}\right),\end{array}$$

(7)

where ${\mathcal{R}}_{i^{*},i}(\mathit{p})$ is the rate function for the probability $\mathbb{P}({\hat{\mu }}_i^n({\mathit{s}}_1)\le {\hat{\mu }}_{i^{*}}^n({\mathit{s}}_1))$ of incorrectly ranking the best design $i^{*}$ and a nonbest one $i\ne i^{*}$ .

This lemma shows that the PFS converges exponentially fast with the exponent:

$$\begin{array}{rl}& \underset{i\in \{1,2,\dots ,k\},i\ne i^{*}}{min}{\mathcal{R}}_{i^{*},i}\left(\mathit{p}\right)\\ & =\underset{i\in \{1,2,\dots ,k\},i\ne i^{*}}{min}-\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\log \mathbb{P}({\hat{\mu }}_i^n({\mathit{s}}_1)\le {\hat{\mu }}_{i^{*}}^n({\mathit{s}}_1)).\end{array}$$

In other words, the PFS and $\exp (-n\cdot \underset{i\in \{1,2,\dots ,k\},i\ne i^{*}}{min}{\mathcal{R}}_{i^{*},i}(\mathit{p}))$ have the same convergence rate. The exponent $\underset{i\in \{1,2,\dots ,k\},i\ne i^{*}}{min}{\mathcal{R}}_{i^{*},i}(\mathit{p})$ is determined by the most difficult comparison of the mean estimates between the best design and nonbest designs at the highest-fidelity level ${\mathit{s}}_1$ . The reason for this effect is that the most difficult comparison has the slowest convergence rate, which dominates the convergence rates of the other comparisons.

4. Solution methodology

In this section, we solve the simulation budget allocation problem.

4.1. Rate-optimal budget allocation rule

Based on (7), the problem that optimizes the rate function of PCS is given by

$$\begin{array}{rl}& \underset{\mathit{p}}{max}\underset{i\in \{1,\dots ,k\},i\ne i^{*}}{min}{\mathcal{R}}_{i^{*},i}\left(\mathit{p}\right)\\ & \text{s.t.}\sum _{i=1}^k\sum _{q=1}^m{p}_{i,q}=1,p_{i,q}\ge 0,i=1,\dots ,k,q=1,\dots ,m.\end{array}$$

(8)

Comparing problem (8) with problem (5), the main difference lies in the substitution of $\mathrm{PCS}$ with its rate function.

Problem (8) is a convex optimization model. We can solve it below by investigating its dual problem. To further simplify the notation, we let $\mathit{\mu }=({\mu }_1({\mathit{s}}_1),\dots ,{\mu }_k({\mathit{s}}_1){)}^{\mathrm{\top }}$ , ${\mathit{\eta }}_i^2=({\eta }_{i,1}^2,\dots ,{\eta }_{i,m}^2{)}^{\mathrm{\top }}$ , and ${\mathit{\eta }}^2=(({\mathit{\eta }}_1^2{)}^{\mathrm{\top }},\dots ,({\mathit{\eta }}_k^2{)}^{\mathrm{\top }}{)}^{\mathrm{\top }}$ .

Theorem 1

The optimal solution to (8) is

$$\begin{array}{rl}& \frac{p_{i,q}^{*}}{p_{i^{\prime },q^{\prime }}^{*}}=\frac{{\mathcal{W}}_{i,q}({\mathit{\gamma }}^{*},\mathit{\mu },{\mathit{\eta }}^2)}{{\mathcal{W}}_{i^{\prime },q^{\prime }}({\mathit{\gamma }}^{*},\mathit{\mu },{\mathit{\eta }}^2)},i,i^{\prime }=1,\dots ,k,q,q^{\prime }=1,\dots ,m;\\ & \sum _{i=1}^k\sum _{q=1}^m{p}_{i,q}^{*}=1,\end{array}$$

(9)

where ${\mathit{\gamma }}^{*}$ is the optimal solution to

$$\begin{array}{rl}& \underset{\mathit{\gamma }}{min}b(\mathit{\gamma };\mathit{\mu },{\mathit{\eta }}^2)\triangleq -\sum _{i=1}^k\sum _{q=1}^m{\mathcal{W}}_{i,q}(\mathit{\gamma },\mathit{\mu },{\mathit{\eta }}^2)\\ & \text{s.t.}\sum _{i\ne i^{*}}{\gamma }_i=1,{\gamma }_i\ge 0,i\ne i^{*},\end{array}$$

(10)

and ${\mathcal{W}}_{i,q}(\mathit{\gamma },\mathit{\mu },{\mathit{\eta }}^2)$ is

$$\begin{array}{rl}& {\mathcal{W}}_{i,q}(\mathit{\gamma },\mathit{\mu },{\mathit{\eta }}^2)\\ & \triangleq \{\begin{array}{ll}{\mathcal{V}}_{i,q}(\frac{{\gamma }_i}{({\mu }_{i^{*}}({\mathit{s}}_1)-{\mu }_i({\mathit{s}}_1){)}^2}{)}^{1/2},& i\ne i^{*},q=1,\dots ,m\\ {\mathcal{V}}_{i,q}(\sum _{i^{\prime }=1,i^{\prime }\ne i^{*}}^k\frac{{\gamma }_{i^{\prime }}}{({\mu }_{i^{*}}({\mathit{s}}_1)-{\mu }_{i^{\prime }}({\mathit{s}}_1){)}^2}{)}^{1/2},& i=i^{*},q=1,\dots ,m,\end{array}\end{array}$$

(11)

where ${\mathcal{V}}_{i,q}=({\kappa }_{i,q}{\eta }_{i,q}^2{)}^{1/2}$ . In addition, when the budget allocation is ${\mathit{p}}^{*}$ , ${\mathcal{R}}_{i^{*},i}({\mathit{p}}^{*})={\mathcal{R}}_{i^{*},i^{\prime }}({\mathit{p}}^{*})$ for any two nonbest designs $i,i^{\prime }\ne i^{*}$ .

Theorem 1 shows that, under the budget allocation rule (9), the selected station design is most likely to be the best among all the alternatives, in the sense that it leads to the fastest rate at which the PFS converges to zero. Intuitively, allocating more samples to estimate ${\delta }_{i,q}$ improves the accuracy of ${\hat{\mu }}_i^n({\mathit{s}}_1)$ and hence facilitates the comparisons at the highest-fidelity level ${\mathit{s}}_1$ . However, because the total simulation budget is fixed, allocating more samples to one design–fidelity pair necessarily reduces the samples available for others, and the allocation rule (9) captures the optimal balance across all design–fidelity pairs.

This rule implies that the optimal budget proportion for the pair of design $i$ and fidelity level ${\mathit{s}}_q$ is proportional to a simple index ${\mathcal{W}}_{i,q}({\mathit{\gamma }}^{*},\mathit{\mu },{\mathit{\eta }}^2)$ , which integrates the sampling variances, computation costs, and the differences in mean performance. In the EV application, this index indicates how important the corresponding correction term is for estimating the highest-fidelity performance of a given station design. To interpret this index more concretely, note from (11) that ${\mathcal{W}}_{i,q}$ increases with ${\mathcal{V}}_{i,q}=({\kappa }_{i,q}{\eta }_{i,q}^2{)}^{1/2}$ . Thus, an increment ${\delta }_{i,q}$ that is noisy (large ${\eta }_{i,q}^2$ ) and expensive to sample (large ${\kappa }_{i,q}$ ) should receive a larger proportion of the budget. For nonbest designs $i\ne i^{*}$ , ${\mathcal{W}}_{i,q}$ scales inversely with the mean gap $|{\mu }_{i^{*}}({\mathit{s}}_1)-{\mu }_i({\mathit{s}}_1)|$ . This is because designs whose highest-fidelity performances are closer to the best are harder to distinguish and therefore receive more budget. In addition, ${\mathcal{W}}_{i,q}$ depends on the dual weights ${\mathit{\gamma }}^{*}$ , which endogenously capture which nonbest designs constitute the most difficult comparisons in the rate function optimization and the allocation should focus effort on these comparisons to accelerate the PFS decay. It is also worth noting that, for each design–fidelity pair, the computational cost and sampling variance always affect the allocation multiplicatively through the product ${\kappa }_{i,q}{\eta }_{i,q}^2$ . It means that if ${\kappa }_{i,q}$ increases by a factor of $r$ ( $r>1$ ) while ${\eta }_{i,q}^2$ is reduced to $1/r$ of its original value, the optimal budget allocation remains unchanged.

In addition, Theorem 1 implies that, for any fixed design $i$ , the optimal allocation across fidelity levels (see Proposition 1 in the E-Companion for more details) satisfies:

$$\frac{p_{i,q}^{*}}{p_{i,q^{\prime }}^{*}}=\frac{{\mathcal{V}}_{i,q}}{{\mathcal{V}}_{i,q^{\prime }}},q,q^{\prime }=1,\dots ,m.$$

Note that this rule is identical to (4) in Lemma 1. It suggests that, when the goal is to optimize the rate function of PCS, the rate-optimal budget allocation within each design simultaneously minimizes the variance of the fusion estimator. This is intuitive because given the total budget proportions $\{p_i{\}}_{i=1}^k$ of each design, the rate function ${\mathcal{R}}_{i^{*},i}(\mathit{p})$ is inversely proportional to the variances of the two fusion estimators ${\hat{\mu }}_{i^{*}}^n({\mathit{s}}_1)$ and ${\hat{\mu }}_i^n({\mathit{s}}_1)$ . Then, minimizing the variance of each fusion estimator according to the allocation (4) also maximizes the rate function of PCS. With this consistency, the remaining optimization is reduced to determining how to allocate $\{p_i^{*}{\}}_{i=1}^k$ across designs. Moreover, variance reduction induced by the fusion estimator directly translates into an improved convergence rate of PCS, as discussed in detail in Section EC.6 of the E-Companion.

In this study, the fusion estimator exploits the nested fidelity hierarchy of the EV charging simulation, where simulation inputs jointly determine precision and computational cost. If, instead, different fidelity levels are generated by different simulation models, the fusion estimator (3) may still be applicable, but other widely used multifidelity estimators (such as multifidelity Monte Carlo (Peherstorfer et al., 2016) and approximate control variates (Gorodetsky et al., 2020)) may be more appropriate.

In the literature, there have been studies (Wu and Frazier, 2017) that consider fidelities as continuous variables and fit a GP (Williams and Rasmussen, 2006) to the joint space of design and fidelity to predict the service system performance at other fidelity levels. However, as more samples are collected at the highest-fidelity level ${\mathit{s}}_1$ , the GP prediction at ${\mathit{s}}_1$ becomes dominated by the ${\mathit{s}}_1$ samples, and low-fidelity samples do not asymptotically reduce the predictive variance at ${\mathit{s}}_1$ . For this reason, we do not pursue the GP approach. Additional discussions about this issue are provided in Remark 2 of Section EC.5.

5. Sequential selection algorithm

To implement the optimal budget allocation rule indicated by Theorem 1, we propose an algorithm called the asymptotic optimal multifidelity sampling (AOMS). This algorithm iteratively updates the estimated mean and variance for each design–fidelity pair, uses the current estimates to compute the target budget proportions prescribed by the allocation rule, and then allocates the next sample to the design–fidelity pair that is most undersampled relative to that target. Repeating this loop makes the sampling decisions increasingly aligned with those in Theorem 1 as more samples are collected. Note that the presentation in this section focuses on a high-level overview to highlight the main logic of the algorithm. The implementation details (e.g., the complete step-by-step pseudocode, the dual-ascent routine for solving (10), and other practical mechanisms used in the algorithm) are discussed in the E-Companion for completeness.

Next, we analyze the theoretical properties of the AOMS algorithm. We will establish its consistency and asymptotic optimality through the following two theorems.

Theorem 2

As the total budget $n$ of the AOMS algorithm goes to infinity, $\underset{t\to \mathrm{\infty }}{lim}{N}_{i,q}^t\to \mathrm{\infty }$ almost surely for all $i\in \{1,\dots ,k\}$ and $q\in \{1,\dots ,m\}$ .

Theorem 2 indicates that the number of simulation samples $N_{i,q}^t$ allocated to estimate ${\delta }_{i,q}$ of station design $i$ at fidelity level ${\mathit{s}}_q$ up to iteration $t$ will go to infinity as $t\to \mathrm{\infty }$ , $i=1,\dots ,k$ and $q=1,\dots ,m$ . It guarantees sufficient exploration for every design–fidelity pair. According to the Law of Large Numbers, estimates will converge to their true values ${\delta }_{i,q}$ , ${\eta }_{i,q}^2$ and ${\mu }_i({\mathit{s}}_1)$ . In particular, the estimated best design ${\hat{i}}_t^{*}$ will become the real best one $i^{*}$ eventually, that is, AOMS will correctly select the best design in the long run.

Theorem 3

In the AOMS algorithm, $\underset{t\to \mathrm{\infty }}{lim}({\kappa }_{i,q}{N}_{i,q}^t/N^t)=p_{i,q}^{*}$ almost surely for all $i\in \{1,\dots ,k\}$ and $q\in \{1,\dots ,m\}$ , where $p_{i,q}^{*}$ corresponds to the allocation rule in (9) and $N^t=\sum _{i=1}^k\sum _{q=1}^m{\kappa }_{i,q}{N}_{i,q}^t$ .

Theorem 3 indicates that the budget allocation generated by the AOMS algorithm can recover the allocation rule that maximizes the rate function of PFS, that is, the algorithm is asymptotically optimal. In other words, from a large-sample perspective, the AOMS algorithm can identify the best service system design with the highest possible probability.

Remark 1

Operators may sometimes want a guarantee on the PCS (Fan et al., 2016; Wang et al., 2024; Zhong and Hong, 2022) to justify the selected design under multifidelity simulation. In general, as with fixed-budget methods in the single-fidelity setting (Hong et al., 2021), the AOMS algorithm that simply stops when the total budget $n$ is exhausted does not support statistical inference for the PCS. In Section EC.9 of the E-Companion, we therefore propose a stopping rule in the spirit of the fixed-confidence framework (to replace the “stop at budget $n$ ” criterion), which guarantees the PCS of AOMS is at least the prespecified level $1-\alpha$ upon termination.

6. Numerical experiments

We conduct a series of numerical experiments on the EV charging station design problem in Section 3.1 to evaluate the effectiveness and robustness of the proposed AOMS algorithm. Due to space limitations, results for the synthetic examples and additional (less important) EV charging station experiments are deferred to the E-Companion.

A station design is characterized by $(n_{\text{c}},A_{\text{pv}},E_{\text{b}},P_{\text{b}},P_{\text{inv}})$ and is selected from a finite candidate set. Specifically, we enumerate $n_{\text{c}}\in \{4,5,6,7\}$ , $A_{\text{pv}}\in \{200,250,300,350\}$ m ${}^2$ , $E_{\text{b}}\in \{200,250,300,350\}$ kWh, $P_{\text{b}}\in \{20,40\}$ kW, and $P_{\text{inv}}\in \{150,200\}$ kW. This produces $256$ candidate designs in total. Note that these candidate designs are constructed subject to feasibility constraints reflecting budget, land, and interconnection considerations, with detailed formulations and parameter definitions provided in the E-Companion (Table EC.1). The fidelity variables are parameterized as $\mathit{s}=(T_{\text{days}},\mathrm{\Delta }t,K_{\text{arr}})$ , where $T_{\text{days}}$ controls the simulation horizon, $\mathrm{\Delta }t$ the time resolution, and $K_{\text{arr}}$ the number of segments in the piecewise-constant approximation of the intraday arrival-rate curve. We consider $m=5$ fidelity levels summarized in Table 1, where the highest fidelity is ${\mathit{s}}^{*}={\mathit{s}}_1=(100,5,24)$ . Unless stated otherwise, we use a total simulation budget of $n=6.0\times {10}^4$ , with each design–fidelity pair initially allocated $n_0=5$ samples. Computing costs are defined as $c({\mathit{s}}_q)=T_{\text{days}}\cdot (60/\mathrm{\Delta }t)\cdot K_{\text{arr}}$ , reflecting that longer horizons, finer time steps, and more arrival segments increase per-sample runtime. Within each fidelity level, costs are set to be identical for different designs. For AOMS, the cost of generating one sample of ${\delta }_{i,q}$ for design $i$ and fidelity level ${\mathit{s}}_q$ is ${\kappa }_{i,q}=c({\mathit{s}}_q)+c({\mathit{s}}_{q+1})$ for $q=1,\dots ,m-1$ and ${\kappa }_{i,m}=c({\mathit{s}}_m)$ .

Table 1.

Parameter settings of the fidelity levels.

Fidelity $\mathit{s}$	${\mathit{s}}_5$	${\mathit{s}}_4$	${\mathit{s}}_3$	${\mathit{s}}_2$	${\mathit{s}}_1({\mathit{s}}^{*})$
$(T_{\text{days}},\mathrm{\Delta }t,K_{\text{arr}})$	$(50,60,6)$	$(50,15,6)$	$(75,15,6)$	$(75,5,24)$	$(100,5,24)$

For each test scenario, the benchmark best design used for PCS evaluation is identified by evaluating all 256 candidate designs at the highest-fidelity level ${\mathit{s}}^{*}$ with 2,000 independent replications per design. Because this benchmark is simulation estimated, the detailed identification results and reliability checks are reported in Section EC.11.2.1 of the E-Companion.

6.1. Effectiveness evaluation

This subsection evaluates the effectiveness of AOMS under a fixed simulation budget by comparing it with representative single-fidelity and multifidelity benchmarks:

Single-fidelity OCBA (SFOCBA) (Chen and Lee, 2011). The original OCBA method was designed for fixed-budget R&S in the single-fidelity setting. It employs the following budget allocation rule:

$$\begin{array}{rl}\frac{n_i}{n_j}& =\frac{{\sigma }_i^2({\mathit{s}}^{*}){({\mu }_{i^{*}}({\mathit{s}}^{*})-{\mu }_j({\mathit{s}}^{*}))}^2}{{\sigma }_j^2({\mathit{s}}^{*}){({\mu }_{i^{*}}({\mathit{s}}^{*})-{\mu }_i({\mathit{s}}^{*}))}^2},\\ & i,j\in \{1,2,\dots ,k\}\setminus \{i^{*}\}\\ n_{i^{*}}& ={\sigma }_{i^{*}}({\mathit{s}}^{*})\sqrt{\sum _{i=1,i\ne i^{*}}^k{\left(\frac{n_i}{{\sigma }_i({\mathit{s}}^{*})}\right)}^2}.\end{array}$$

Single-fidelity expected improvement (SFEI) (Jones et al., 1998; Shu et al., 2021). The original expected improvement (EI) was developed for the single-fidelity setting. It iteratively samples the design with the largest expected improvement:

$$\begin{array}{rl}{i}^{\text{EI},t+1}& =\underset{i\in \mathcal{K}}{\arg max}{\mathbb{E}}_B\\ & \times \{max[{\mu }_i({\mathit{s}}^{*})-\underset{i^{\mathrm{\prime }}\in \mathcal{K}}{max}{y}_{i^{\mathrm{\prime }}}^t({\mathit{s}}^{*}),0]|i^{t+1}=i\},\end{array}$$

where $y_{i^{\prime }}^t({\mathit{s}}^{*})$ represents the posterior mean of ${\mu }_{i^{\prime }}({\mathit{s}}^{*})$ in iteration $t$ and ${\mathbb{E}}_B$ denotes the expectation with respect to the Bayesian belief.

Multifidelity knowledge gradient (MFKG) (Wu and Frazier, 2017). This procedure extends the classic KG algorithm to incorporate continuous fidelity parameters and samples the design–fidelity pair with the largest normalized improvement:

$$\begin{array}{rl}& \text{MFKG}(i,q)\\ & =\frac{\begin{array}{l}{\mathbb{E}}_B[\underset{i^{\mathrm{\prime }}\in \mathcal{K}}{max}{y}_{i^{\prime }}^{t+1}({\mathit{s}}^{*})|i^{t+1}=i,q^{t+1}=q]\\ -\underset{i^{\mathrm{\prime }}\in \mathcal{K}}{max}{y}_{i^{\prime }}^t({\mathit{s}}^{*})\end{array}}{c_{i,q}}.\end{array}$$

For implementation, we evaluate MFKG under 12 specifications that cover key modeling and tuning choices at manageable computational cost. These specifications combine three GP covariance kernels, Matérn- $5/2$ , Matérn- $3/2$ , and radial basis function (RBF), two KG fantasy sample sizes, 8 and 16, and two GP hyperparameter re-optimization intervals, 25 and 50 iterations. Here, fantasy samples refer to simulated possible observations used to approximate the expectation ${\mathbb{E}}_B$ over the updated posterior in the KG calculation. For the EV charging station design problem, the reported MFKG benchmark uses the RBF kernel, 16 KG fantasy samples, and re-optimization every 50 iterations, which achieves the highest PCS among the 12 specifications.

It is noted that the lower-fidelity settings are used only within the multifidelity procedures (i.e., AOMS and MFKG) to obtain cheaper approximate evaluations of the same objective.

Figure 2 reports the PCS values of the compared algorithms. AOMS consistently achieves the best performance. In particular, to reach a PCS of 90%, AOMS uses less than half the simulation budget required by SFOCBA and SFEI. MFKG exhibits strong early performance but improves only marginally as the budget increases. Its adoption of GP makes it tend to allocate more-than-enough samples to the highest-fidelity level ${\mathit{s}}_1$ when predicting ${\mu }_i({\mathit{s}}_1)$ , which compromises its efficacy in selecting the best design. Details of MFKG’s budget allocation are reported in Section EC.11.2.2 of the E-Companion. As a result, MFKG can lose efficiency in late-stage ranking among close competitors. By contrast, the two single-fidelity methods (SFOCBA and SFEI) gradually improve with more high-fidelity sampling, but remain inferior to AOMS over the tested budget range. These results highlight the effectiveness of AOMS. Its adaptive multifidelity sampling allows more reliable identification of the best design under tight budgets rather than relying on a particular budget region or specific setting.

Figure 2.

Performance comparison on the EV charging station problem. EV = electric vehicle.

In Section EC.11.1 of the E-Companion, we further test all algorithms on three standard synthetic benchmark functions (Griewank, Bohachevsky, and Sphere). The results show that AOMS consistently achieves higher PCS across all three benchmarks over the full budget range. It suggests that the advantage of AOMS is not specific to the EV charging station case, but extends to a broader class of problems with various underlying structures.

6.2. Robustness under different demand regimes

This subsection evaluates the performance of AOMS under three representative demand regimes: (a) an early-adoption regime with low penetration and mild intraday variation; (b) a rush-hour regime with sharp morning and evening peaks; and (c) a highly volatile regime driven by seasonal and weather-related variability. These regimes are implemented within a fixed simulation model by varying only the exogenous inputs, including the daily arrival volumes, intraday peak profiles, and weather-driven PV generation and energy-demand conditions. We compare AOMS with SFOCBA for the three regimes by evaluating their PCS values, discussing the operational implications of the recommended design, and quantifying how the PCS improvements of AOMS translate into expected profit gains. The PCS for each algorithm is computed based on 100 replications.

As shown in the first row of Figure 3, the proposed AOMS outperforms SFOCBA for all demand regimes by a significant margin. Notably, in the volatile regime (Regime 3), AOMS achieves a PCS of nearly 85% when the total budget is exhausted, while SFOCBA reaches only about 65%. The second row of Figure 3 displays the accumulated budget allocated to each fidelity level (of all designs) throughout the simulations for the three regimes. Initially, the lowest-fidelity level ${\mathit{s}}_5=(50,60,6)$ receives the most simulation. However, as the sampling process continues, the two highest-fidelity levels ${\mathit{s}}_1=(100,5,24)$ and ${\mathit{s}}_2=(75,5,24)$ become the primary recipients of the simulation budget. Therefore, AOMS is robust to different demand regimes and consistently demonstrates better performance than SFOCBA. Its advantage is more obvious as the demand process becomes more volatile.

Beyond performance comparison, we also derive actionable guidance for the EV charging station design. Table 2 summarizes the performance improvements achieved by AOMS in the three demand regimes and translates these gains into regime-specific design recommendations for the number of chargers, the battery capacity, the battery power rating, and the inverter capacity. For all regimes, AOMS achieves substantial PCS improvements ranging from 3% to 18%, with the largest gains observed as demand uncertainty increases. In Regime 1, where demand is relatively stable, the gains primarily come from avoiding systematic overcapacity. Accordingly, it is recommended that station designs use fewer chargers, smaller battery capacity, and lower inverter power. In Regimes 2 and 3, demand is more volatile. This not only amplifies the PCS benefits, but also calls for higher charging capacity, higher inverter power, and a higher battery power rating, while keeping battery capacity relatively conservative.

We next demonstrate how the advantage of AOMS in PCS translates into improvements in operational performance of the charging station. Specifically, we compare the expected mean performance (profit) of the estimated recommended design (i.e., the best design ${\hat{i}}^{*}$ ) delivered by AOMS and SFOCBA under the same simulation budget. That is, we will compare $\mathbb{E}[{\mu }_{{\hat{i}}^{*}}({\mathit{s}}^{*})]$ between the two algorithms, where

$$\mathbb{E}[{\mu }_{{\hat{i}}^{*}}({\mathit{s}}^{*})]=\sum _{i=1}^k\mathbb{P}({\hat{i}}^{*}=i){\mu }_i({\mathit{s}}^{*}),$$

and $\mathbb{P}({\hat{i}}^{*}=i)$ denotes the probability of design $i$ being selected as the best design. Table 3 provides the comparison results. It can be observed that AOMS consistently produces higher expected profits than SFOCBA across the three demand regimes. These results indicate the effectiveness of AOMS in improving the charging station’s profit-generating capability by identifying more profitable designs under the same simulation budget.

Figure 3.

Performance comparison under different demand regimes on the EV charging station problem. EV = electric vehicle.

Table 2.

Comparison results and design recommendations under different demand regimes.

Scenario	PCS improvements (%)	Design recommendations $(n_{\text{c}},E_{\text{b}},P_{\text{b}},P_{\text{inv}})$
Regime 1	3	$(n_{\text{c}}\downarrow ,E_{\text{b}}\downarrow ,P_{\text{b}}\uparrow ,P_{\text{inv}}\downarrow )$
Regime 2	14	$(n_{\text{c}}\uparrow ,E_{\text{b}}\downarrow ,P_{\text{b}}\uparrow ,P_{\text{inv}}\uparrow )$
Regime 3	18	$(n_{\text{c}}\uparrow ,E_{\text{b}}\downarrow ,P_{\text{b}}\uparrow ,P_{\text{inv}}\uparrow )$

Note. PCS = probability of correct selection.

Arrows indicate directional changes in recommended design dimensions relative to the average design structure across all candidate designs. $\uparrow$ denotes an increase and $\downarrow$ denotes a decrease.

Table 3.

Expected profit improvements of AOMS relative to SFOCBA under different demand regimes.

Scenario	Profit of SFOCBA	Profit of AOMS	Profit increase over SFOCBA
Regime 1	82,703.80	86,574.44	3,870.64
Regime 2	39,198.88	48,996.71	9,797.83
Regime 3	214,556.71	227,926.66	13,369.95

Note. AOMS = asymptotic optimal multifidelity sampling; SFOCBA = Single-fidelity optimal computing budget allocation.

To complement these observations, we conduct additional numerical experiments in Sections EC.11.2.3 and EC.11.2.4 of the E-Companion to assess the robustness of AOMS under different operator-preference scenarios and understand how the best design varies under different operator preferences. The results reveal that AOMS remains robust across operator preferences, achieving PCS gains of up to 23% over SFOCBA under service- and profit-oriented objectives. These results also lead to consistent sizing conclusions, showing that with PV capacity fixed, lower charging demand calls for fewer chargers, and top designs tend to use smaller battery capacity but higher battery power rating and inverter capacity.

7. Conclusions and discussion

This research studies simulation-assisted service system design with a focus on integrated EV fast-charging stations. These systems integrate capacity planning, service quality, and energy operations, connecting equipment sizing with issues like congestion and the uncertainties introduced by renewable energy sources. Since high-resolution simulation can be computationally expensive, we formulate the design problem under the fixed-budget R&S framework. We develop a fusion estimator that uses multifidelity samples to estimate the highest-fidelity system performance, and a rate-optimal budget allocation rule that maximizes the PCS for identifying the best design. In addition, we introduce a sequential sampling algorithm (AOMS) to effectively implement the proposed budget allocation rule.

Although motivated by integrated EV charging stations, the proposed framework applies broadly to service systems in which a finite set of alternatives is evaluated through expensive stochastic simulation and simulation accuracy can be controlled via input fidelity parameters. The formulation in Section 3 and the rate-optimal R&S development in Section 4 indicate that, when such controllable fidelity exists, fusion-based multifidelity estimation combined with principled budget allocation can systematically improve selection quality under limited simulation budgets.

The multifidelity problem considered in this research shares some similarities with the contextual R&S in the literature. In contextual R&S, simulation samples are allocated to design–context pairs to identify the best design for each context. In multifidelity R&S, samples are allocated to design–fidelity pairs, with the objective of correctly selecting the design that performs the best at the highest-fidelity level. This distinction is important, because contextual R&S requires correct selection uniformly across contexts, while our goal concerns only the highest-fidelity comparison. As a result, although both problems may use similar performance prediction models (e.g., GPs or linear models), the multifidelity fusion estimator developed here does not directly extend to contextual problems. It will be an interesting research direction to develop a unified framework that bridges the two problems.

We close this paper by summarizing some observations that emerge from our theoretical development and numerical experiments, which can guide practitioners in deciding when and how to use multifidelity simulation in the design of EV charging stations and other service systems.

First, under a fixed simulation budget, the quality of the selected service system design is primarily determined by how efficiently one converts computation into effective comparison at the highest fidelity, rather than by running the highest fidelity as often as possible. In our setting, this efficiency comes from combining a multifidelity fusion estimator that targets ${\mu }_i({\mathit{s}}_1)$ while making use of information from inexpensive lower-fidelity samples, and a rate-optimal budget allocation that prioritizes the design–fidelity pairs that are most critical for improving PCS. From an implementation perspective, the key is to (i) pair adjacent fidelity levels using common random numbers so that the increment estimators have small variance, (ii) retain a modest number of replications at the highest fidelity ${\mathit{s}}_1$ to keep the fusion estimator grounded in the target performance, and (iii) allocate the remaining budget based on quantities that can be estimated online, including the mean gaps at fidelity level ${\mathit{s}}_1$ , the sampling variances of the increments, and the associated per-sample costs. Under these conditions, multifidelity simulation effectively converts a limited simulation budget into stronger evidence for ranking at ${\mathit{s}}_1$ , and facilitates efficient selection of the best design.

Second, AOMS often exhibits an adaptive pattern during the sampling process. This pattern arises because the algorithm starts from a uniform initial allocation and then gradually moves toward the nonuniform target allocation characterized by the theoretical results. An implication of it for practice is that lower-fidelity levels can be valuable when the goal is to compare designs at low cost, especially when they provide cost-effective variance reduction. Higher-fidelity levels receive more budget when the estimated optimal allocation indicates that their contribution to reducing uncertainty at the target fidelity is sufficiently large relative to their sampling cost.

Third, the benefits of fusion and adaptive allocation become most significant in intrinsically difficult regimes, which include volatile demand, high output noise, or small performance gaps between the best and near-best designs. These are precisely the cases in which allocating most of the budget to ${\mathit{s}}_1$ can be especially inefficient (see the robustness experiments in Section 6.2). This implies that multifidelity simulation should be prioritized when operators face tight decision timelines, computational constraints, or regime uncertainty (e.g., when adoption rates are changing quickly, arrival patterns are less predictable, or the weather causes more variation). In these environments, the value of a multifidelity R&S procedure lies not only in achieving a higher PCS under the same simulation budget, but also in supporting more robust decision making when system dynamics are harder to predict in advance.

Last, numerical experiments on the EV charging station suggest a practical order for sizing decisions. Within the tested design ranges and cost-service trade-offs, service capacity (especially the number of fast chargers) often plays the most critical role, because queuing congestion directly affects waiting times, abandonment, and SLA penalties during peak demand. In contrast, the sizing of PV, battery, and inverter assets usually has a secondary and more situation-dependent impact, which mainly influences operating costs and feasibility. For example, inverter limits typically become important only when charging intensity and overall power demand are high, while battery sizing tends to favor relatively low energy capacity and high power rating for short-run flexibility. These patterns are evident in the robustness experiments under different demand regimes and operator preferences in Sections 6.2 and EC.11.2.3 of the E-Companion. This leads to a practical guideline for designing integrated EV charging stations, which is to first provide sufficient service capacity to keep congestion risk under control, and then adjust energy assets to reduce operating costs and meet equipment and interconnection constraints. Given these candidate designs, multifidelity simulation can be used to efficiently select the best one.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261468125 - Supplemental material for Enhancing electric vehicle charging station design using multifidelity simulations

Supplemental material, sj-pdf-1-pao-10.1177_10591478261468125 for Enhancing electric vehicle charging station design using multifidelity simulations by Jiang Li, Jianzhong Du, Siyang Gao, Qiang Ye and Guangxin Jiang in Production and Operations Management