Robust Scheduling of Multi-Service-Area PV-ESS-Charging Systems Along a Highway Under Uncertainty

Abstract

Against the backdrop of China’s dual-carbon goals, traditional road transportation has relatively high carbon emissions and is in urgent need of a low-carbon transition. The intermittency of photovoltaic (PV) power generation and the stochastic nature of electric vehicle (EV) charging demand introduce significant uncertainty for PV-energy storage-charging systems in highway service areas. Existing approaches often struggle to balance economic efficiency and reliability. This study develops a min-max-min robust optimization model for a full-route PV-energy storage-charging system. A box uncertainty set is used to characterize uncertainties in PV output and EV load, and a tunable uncertainty parameter is introduced to regulate risk. The model is solved using a column-and-constraint generation (C&CG) algorithm that decomposes the problem into a master problem and a subproblem. Strong duality, combined with a big-M formulation, enables an alternating iterative solution between the master problem and the subproblem. Simulation results demonstrate that the proposed algorithm attains the optimal solution and, relative to deterministic optimization, achieves a desirable trade-off between economic performance and robustness.

1. Introduction

Against the backdrop of global climate change, China has put forward the “dual carbon” goals: peaking carbon emissions by 2030 and achieving carbon neutrality by 2060. In China, the transportation sector is the second-largest source of carbon emissions [1], second only to the industrial sector, and urgently needs a green and low-carbon transformation. Within this sector, road transportation accounts for 79.15% of total emissions [2], rendering its green transition a pivotal link in realizing the “dual carbon” goals.

In recent years, new energy vehicle (NEV) technology has advanced rapidly, leading to a substantial increase in electric vehicle (EV) ownership. The growing penetration of EVs has sharply increased charging demand, which can impose considerable pressure on the power grid, especially during peak charging periods [3]. On highways, large-scale charging loads further reshape the load profile of service areas, while the existing transportation energy system—still highly dependent on the utility grid—often lacks sufficient flexibility and resilience to accommodate such changes. As a practical response, integrating photovoltaic-energy storage-charging (PESC) systems into highway service areas has emerged as a promising solution to enhance local energy security and enable low-carbon operation. Related techno-economic studies have demonstrated the feasibility of hybrid renewable energy–based EV charging infrastructure. Abodwair et al. developed and compared multiple scenarios using HOMER Grid, in which solar and wind generation were jointly integrated with battery energy storage under utility-grid support to ensure reliable power supply for EV charging. Their results indicate that, although storage-coupled configurations generally require higher upfront investment, they may offer superior long-term economic performance when evaluated from a life-cycle cost perspective [4]. These findings highlight the potential of “source-grid-load-storage-charging” integration, which is also encouraged by China’s transportation authorities through policy guidance.

However, photovoltaic power generation is significantly affected by weather conditions, which makes it difficult for the power supply of photovoltaic-energy storage-charging (PESC) systems to stably match the demand. Furthermore, the large-scale grid connection of EVs creates significant stochastic loads, posing substantial challenges to the optimization of PESC systems. Therefore, to improve the operational efficiency and security of integrated PESC systems, research on robust scheduling optimization in the operational phase of PESC systems that accounts for uncertainties is essential. This research will not only contribute to achieving the “dual carbon” goals but also promote the green upgrade of transportation infrastructure.

Numerous scholars have conducted research on the optimal scheduling of PESC systems, which can be broadly categorized into deterministic optimization and optimization considering uncertainty. Traditional deterministic optimization methods have limitations in addressing renewable energy uncertainties, while stochastic optimization approaches pose significant challenges for practical engineering implementation. Consequently, the research focus has gradually shifted towards robust optimization for microgrids that considers uncertainties in both renewable generation and load capacity [5]. Reference [6] established a generalized energy optimization model by constructing a renewable energy output uncertainty set and a “source-grid-load-storage” coordination framework. They employed a boundary scenario generation technique to transform the bi-level programming into a single-level problem for solution, ultimately minimizing operational costs while ensuring system robustness. Reference [7] developed a capacity allocation model considering the dual uncertainties of wind/solar power output and load, utilizing a bi-layer architecture where the outer layer optimizes investment cost and the inner layer handles operational constraints. The dual decomposition and Benders decomposition algorithms were used to achieve efficient cooperative optimization of the master–slave problems, significantly enhancing the microgrid planning’s ability to balance economy and robustness. Reference [8] addresses the source–load uncertainty from high renewable energy penetration and large-scale EV integration. It proposes an EV charging demand model based on trip chain theory, establishes a multi-RIES optimization model with SESS and IDR, develops an improved time-varying robust optimization (ROOT) algorithm via scenario method to minimize daily operation cost, verifies the method through case studies, and finds the proposed EV charging model reduces charging cost by 3.5% and system operation cost by 11.7% compared with traditional methods, suggesting future research on complex trip chain-based charging demand modeling.

Although substantial progress has been made in related studies, clear limitations remain when these methods are applied to the large-scale grid integration of electric vehicles (EVs) in highway scenarios. Conventional uncertainty modeling approaches often fail to accurately capture the uncertainties of photovoltaic (PV) generation and EV charging demand. In addition, existing robust optimization models usually lack a dynamic risk-adjustment mechanism, making it difficult to achieve a flexible trade-off between economic performance and robustness. Moreover, prior work has paid limited attention to coordinated scheduling and optimization across multiple highway service areas.

To address these issues, this study develops a robust optimization framework for source–load uncertainties based on a box uncertainty set, and optimizes the operating strategy of photovoltaic-storage-charging (PESC) microgrids to maintain both economic efficiency and reliability under uncertainty. By applying strong duality theory and the Big-M method, the proposed model is reformulated as a mixed-integer linear programming (MILP) problem for direct solution.

The main contributions of this study are as follows:

(1)

Focusing on the highway-specific scenario, this study develops a three-stage robust optimization framework for the operation of a corridor-wide, multi-service-area photovoltaic-storage-charging (PESC) system under uncertainties in PV generation and charging demand, and employs a box uncertainty set to characterize the source–load uncertainties in a unified manner.

(2)

This study introduces an adjustable uncertainty parameter to control the risk level by limiting the extent to which uncertain variables can reach extreme values, allowing the conservativeness of the operating scheme to be flexibly tuned as needed. As the uncertainty parameter increases, the resulting solution becomes more robust and can better mitigate the high compensation cost caused by real-time electricity price fluctuations. In addition, the proposed model is solved using the C&CG algorithm combined with duality-based reformulation and Big-M linearization, and the simulations indicate that the solution procedure converges quickly and yields an operating schedule for the worst-case scenario.

2. Highway Service Area Photovoltaic-Energy Storage-Charging System

The overall architecture of the long-route Highway service area microgrid is shown in Figure 1. Vehicles enter or exit the Highway through toll station nodes and stop at service area nodes for charging. In this study, toll stations are used only as the origin and destination (O–D) nodes for EV-flow modeling and Monte Carlo-based forecasting of EV charging load; they do not participate in the energy exchange of the microgrids. This architecture consists of three parts: the power grid, the information flow, and the transportation network [9], with an upper-level dispatch center responsible for information collection and issuing dispatch commands. The synergistic interaction of these three components enables the realization of energy management and operational optimization of the service areas.

The power grid primarily includes the microgrid systems at service area nodes, the load from toll stations, and the distribution network. The integrated photovoltaic-energy storage-charging (PESC) system within the service area microgrid forms the core, combining photovoltaic generation, energy storage systems, and electric vehicle charging stations, the system architecture is shown in Figure 2. On the supply side, the power grid includes the distribution network, photovoltaic units, and the energy storage system. On the demand side, it comprises the dynamic load from EVs and the daily fixed load of the service areas. All units are connected to the tie-line via converters. The microgrid interacts with the distribution grid through this line, operating on a principle of self-consumption with surplus electricity fed into the grid. The microgrid dispatch center collects real-time operational data from various energy equipment. It uses optimization algorithms to generate dispatch commands, dynamically adjusting energy storage charging/discharging, photovoltaic output allocation, and power purchase/sale from/to the grid to meet the load demand [10].

The transportation network comprises the Highway and the vehicles operating on it. The dispatch center collects traffic flow data and the power load status at various nodes, dynamically transmitting the optimization strategies to the power grid.

3. Establishment of System Uncertainty Model

In practical energy systems, photovoltaic (PV) output and electric vehicle (EV) charging load are often affected by various factors such as weather conditions and load fluctuations, which involve considerable uncertainty. As a result, conventional deterministic optimization methods may not adequately handle such uncertainty, potentially leading to solutions that fail to satisfy economic efficiency and reliability requirements in real-world operation. To better address these uncertainties, this study adopts a box uncertainty set model. By bounding uncertain factors within predetermined ranges, the model can characterize the possible variation intervals of PV output and EV charging load while maintaining computational tractability, thereby providing more robust decision support for system optimization. However, because a box uncertainty set can be overly conservative and uncertain parameters do not necessarily reach their extreme values simultaneously in every time period, an uncertainty budget parameter $\Gamma$ is introduced to regulate the conservatism of the solution. For example, when the EV-load uncertainty budget parameter is set to $\Gamma =5$, the EV load attains its extreme value (the upper bound for load, and the lower bound for renewable generation) in five time periods. This modeling approach not only accounts for uncertainties but also guarantees that the system maintains satisfactory operational performance even under extreme or unexpected conditions.

In this chapter, we first establish deterministic calculation models for PV power generation and EV charging load. On this basis, we further develop uncertainty models for PV output and EV charging load, respectively.

3.1. Photovoltaic Power Output Calculation

Photovoltaic (PV) output is affected by module performance, solar irradiance, and ambient temperature. Considering these factors, the actual PV power at time t is calculated as [11]:

$$P_{pv}=\eta P_s{\displaystyle \frac{A}{A_{st}}}[1+{\theta }_T(T-T_{ref})]$$

(1)

Here, $P_{pv}$ denotes the actual PV power output and $P_s$ is the rated power of the PV module. $A$ represents the measured solar irradiance, while $A_t$ is the irradiance under standard test conditions. T is the PV module (surface) temperature, and $T_{ref}$ is the reference temperature at standard conditions (typically 25 °C). $\eta$ is the PV array efficiency. ${\theta }_T$ is the temperature coefficient of power, which is set to −0.004 in this study.

All symbols and their meanings involved in this paper are summarized in Appendix A.

3.2. Dynamic Estimation of EV Charging Load

Based on highway traffic flow and the historical OD (Origin–Destination) matrix, this study employs a Monte Carlo simulation approach to model EV charging behaviors at service areas and to derive the hourly EV charging-load profiles of all service areas. The simulation uses an hourly time step $t\in \mathcal{T}$ (e.g., $t = 1, \dots , 24$) and considers all service areas $i\in \mathcal{N}$ (in this paper, $\mid \mathcal{N}\mid = 12$) [12].

First, the initial traffic flow is allocated based on the total vehicle volume and the historical OD (Origin–Destination) matrix. Using an hourly time step, the traffic-flow distribution across different road sections is calculated for each hour, thereby generating a time-varying arrival-flow input for the expressway service-area nodes. After the inflow is established, an initial SOC is randomly assigned to each vehicle following a normal distribution to represent the battery state when entering the expressway. As a vehicle passes each service-area node, its SOC is updated according to the traveled distance and the energy consumption per kilometer, and the corresponding charging probability and charging demand at that node are evaluated. When the SOC falls below a predefined charging threshold, the vehicle is considered to have a charging tendency and the charging behavior is recorded. If charging is required, charging resources are assigned following a first-come-first-served (FCFS) rule, and the charging load of the service area is accumulated and updated dynamically.

The resulting hourly EV charging-load profile is used as the pre-dispatch load in the subsequent optimization and is denoted by $P_{i,t}^{before}$.

(i)

OD-based traffic flow and hourly vehicle arrivals

Daily traffic volume is first allocated to toll-station nodes using the OD matrix, and then converted to hourly traffic flow using the 24-h distribution. This yields the number of EVs that may pass each service area in each hour, which serves as the input for charging-load simulation.

(ii)

Initial SOC assignment

For initial SOC, 30% of EVs are assumed to enter the highway with 90% SOC, while the remaining 70% follow a normal distribution $\mathcal{N}(0.7,0.1)$ [13].

(iii)

Calculation of EV’s SOC upon Arrival at the Service Area

As an EV travels along the expressway, its SOC is updated according to the traveled distance and the energy consumption per kilometer. To reflect random fluctuations caused by traffic conditions and weather, a stochastic factor is introduced. The arrival SOC of an EV at service area $i$ can be expressed as:

$$SO{C}_i^{arr}=SO{C}^{ini}-{\displaystyle \frac{d_i\cdot e}{C_b}}\cdot \xi$$

(2)

where $d_i$ is the travel distance to service area $i$, $e$ is the reference energy consumption per kilometer, $C$ is the battery capacity, and $\xi$ is a random variable used to capture stochastic fluctuation ($\xi \sim \mathcal{N}\left(\mathrm{1,0.1}\right)$, corresponding to approximately ±10% variation around the reference consumption).

(iv)

Charging probability as a function of arrival SOC

The charging probability is determined by the EV’s SOC upon arrival. The relationship between $SO{C}^{arr}$ and the charging probability is given by Equation (3) [14]. Based on this probability, each EV makes a charging decision at the current service-area node, and its SOC is updated in subsequent simulation steps.

$$P=\left\{\begin{array}{c}1,SOC\le 0.3\\ -SO{C}^2+1.09,0.3<SOC\le 0.5\\ 3.36{(SOC-1)}^2,0.5<SOC\le 1\end{array}\right.$$

(3)

(v)

Charging time calculation

For EVs that decide to charge, the charging time is calculated by:

$$\tau ={\displaystyle \frac{(SO{C}^{full}-SO{C}^{arr})\cdot C}{{\eta }_{EV}{P}_{EV}^{ch}}}$$

(4)

where $SO{C}^{full}$ is the target (fully charged) SOC, ${\eta }_{EV}$ is charging efficiency, and $P_{EV}^{ch}$ is charger power.

(vi)

Charging-load allocation across hours

Because $\tau$ may exceed one hour, the charging energy is allocated to consecutive hourly intervals to obtain an hourly charging-load curve for each service area. In addition, charging resources are assigned following a first-come-first-served (FCFS) rule, and the charging load of each service area is accumulated and updated dynamically.

(vii)

Monte Carlo averaging

After repeating the above simulation for S trials, the expected charging power is obtained by averaging:

$${\hat{P}}_{EV,i,t}^{MC}={\displaystyle \frac{1}{S}}{\displaystyle \sum _{s=1}^S{P}_{EV,i,t}^{s,MC}},\hspace{1em}\forall i\in \mathcal{N},\forall t\in \mathcal{T}$$

(5)

where $S$ is the number of Monte Carlo runs, and $P_{EV,i,t}^{s,MC}$ denotes the total EV charging power of service area $i$ during hour $t$ in the $s$-th simulation.

In this study, the Monte Carlo averaged profile is used as the pre-dispatch EV load for subsequent optimization, i.e.,

$$P_{EV,i,t}^{before}={\hat{P}}_{EV,i,t}^{MC}$$

(6)

3.3. Photovoltaic Power Output Uncertainty Model

Photovoltaic (PV) output is uncertain due to weather influences. This paper employs a box uncertainty set to describe the fluctuation range of the PV output at the i-th service area during time period t [15].

$$u_{\mathrm{PV}}=\left\{P_{\mathrm{PV},i,t}\right|(1-{\gamma }_{\mathrm{PV}})P_{\mathrm{PV},i,t}^{fore}\le P_{\mathrm{PV},i,t}\le (1+{\gamma }_{\mathrm{PV}})P_{\mathrm{PV},i,t}^{fore},\forall i\in \mathcal{N},\forall t\in \mathcal{T}\}$$

(7)

where $P_{\mathrm{PV},i,t}^{fore}$ is the forecasted PV output value, and ${\gamma }_{\mathrm{PV}}$ is the PV output uncertainty coefficient. Considering the characteristics of PV generation, this coefficient is set to 0.15 [16], indicating a potential variation of ±15% from the forecast value.

Equation (1) represents the feasible range of all possible PV power outputs, which can also be expressed as:

$$u_{\mathrm{PV}}=\left\{P_{\mathrm{PV},i,t}\right|{\hat{u}}_{\mathrm{PV},i,t}-B_{\mathrm{PV},i,t}△u_{\mathrm{PV},i,t}\le P_{\mathrm{PV},i,t}\le {\hat{u}}_{\mathrm{PV},i,t}+B_{\mathrm{PV},i,t}△u_{\mathrm{PV},i,t}\}$$

(8)

where ${\hat{u}}_{\mathrm{PV},i,t}$ is the forecasted PV output value; $△u_{\mathrm{PV},i,t}={\gamma }_{\mathrm{PV}}{P}_{\mathrm{PV},i,t}^{fore}$ is the maximum allowed fluctuation deviation of the PV output; and $B_{\mathrm{PV},i,t}$ is a binary (0/1) variable. A value of 1 indicates that the PV output at the service area reaches the boundary value in that time period. The constraint for the $B_{\mathrm{PV},i,t}$ is given below, where ${\mathsf{\Gamma }}_{\mathrm{PV}}$ is the uncertainty budget parameter, representing the maximum number of time periods in which the PV output at the service area can attain its extreme value.

$$\sum _{t\in \mathcal{T}}{B}_{\mathrm{PV},t}\le {\mathsf{\Gamma }}_{\mathrm{PV}}$$

(9)

3.4. Uncertainty Model for EV Loads

To exploit the spatio-temporal flexibility of EV charging among multiple service areas, we first generate the pre-dispatch hourly EV charging-load profile, denoted by $P_{EV,i,t}^{before}$, via Monte Carlo simulation based on traffic-flow data and the OD matrix. A coordinated charging dispatch strategy is then applied to adjust this baseline profile, yielding the post-dispatch load profile $P_{EV,i,t}^{after}$. In the robust optimization, uncertainties are modeled as deviations around the post-dispatch profile.

The uncertainty of EV charging load is influenced by various factors, such as the number of vehicles, battery state, and charging behavior. In this study, a box uncertainty set is adopted to characterize this uncertainty. Specifically, uncertainties are modeled as deviations around the post-dispatch EV load $P_{EV,i,t}^{after}$. The uncertainty set of the EV load can be expressed as [15]:

$$u_{\mathrm{EV}}=\left\{P_{\mathrm{EV},i,t}\right|(1-{\gamma }_{\mathrm{EV}})P_{\mathrm{EV},i,t}^{after}\le P_{\mathrm{EV},i,t}\le (1+{\gamma }_{\mathrm{EV}})P_{\mathrm{EV},i,t}^{after},\forall i\in \mathcal{N},\forall t\in \mathcal{T}\}$$

(10)

where $P_{EV,i,t}^{after}$ denotes the coordinated post-dispatch EV load; ${\gamma }_{EV}$ is the EV-load uncertainty coefficient, set to 0.1 [17], representing a possible fluctuation of ±10%; $\mathcal{T}$ denotes the dispatch horizon with an hourly time step; and $\mathcal{N}$ is the set of service areas (12 service areas in this paper).

Equation (10) represents the entire possible value range of the EV load, and it can also be expressed as:

$$u_{\mathrm{EV}}=\left\{P_{\mathrm{EV},i,t}\right|{\hat{u}}_{\mathrm{EV},i,t}-B_{\mathrm{EV},i,t}△u_{\mathrm{EV},i,t}\le P_{\mathrm{EV},i,t}\le {\hat{u}}_{\mathrm{EV},i,t}+B_{\mathrm{EV},i,t}△u_{\mathrm{EV},i,t}\}$$

(11)

Among them, ${\hat{u}}_{EV,i,t}$ denotes the center value of the EV load in the uncertainty set, which is taken as the post-dispatch EV load, ${\hat{u}}_{EV,i,t}=P_{EV,i,t}^{after}$. $\mathsf{\Delta }{u}_{EV,i,t}={\gamma }_{EV}{P}_{EV,i,t}^{after}$ represents the maximum allowable deviation of the EV load, where ${\gamma }_{EV}$ is the uncertainty coefficient. The binary variable $B_{EV,i,t}\in \left\{\mathrm{0,1}\right\}$ is used to indicate whether the realized EV load $P_{EV,i,t}$ at service area $i$ and time period $t$ reaches the boundary of the uncertainty set. The constraints for it are as follows:

$$\sum _{t\in \mathcal{T}}{B}_{\mathrm{EV},t}\le {\mathsf{\Gamma }}_{\mathrm{EV}}$$

(12)

4. Robust Optimization Model and Solution for the Operation Stage of PV-Energy Storage-Charging Systems in Service Areas

In this section, a robust optimization model for the operation stage of the PV-ESS-charging system in service areas is established. With an hourly time interval, the charging and discharging states of energy storage, power purchase and sale states with the grid, energy storage charging power, energy storage discharging power, purchased electricity from the grid, sold electricity to the grid, EV load, and PV output are optimized for each hour of the whole day, so as to obtain a day-ahead operation scheme capable of coping with uncertainties. This scheme can effectively reduce the total operation cost of the system during intraday operation when there are errors in the prediction of PV output and EV load.

4.1. Robust Optimization Model for the Operation Stage of PV-Energy Storage-Charging Systems

4.1.1. Objective Function

The objective function comprehensively considers the grid transaction cost, energy storage loss cost, and EV dispatch cost.

$$f=\min (C_{\mathrm{grid}}+C_{\mathrm{ESS}}+C_{\mathrm{EV}})$$

(13)

Among them, $C_{\mathrm{grid}}$ is the cost of transactions with the distribution grid, including the revenue from selling surplus electricity and the cost of purchasing electricity when PV power supply is insufficient. $C_{\mathrm{ESS}}$ is the loss expenditure from the charging and discharging of the energy storage system [18].

Based on the dispatch results of the ordered charging of electric vehicles, the EV load can serve as an adjustable load connecting various service areas. $C_{\mathrm{EV}}$ is the dispatch cost for electric vehicles. The calculations of various costs are as follows:

$$C_{\mathrm{grid}}=\sum _{t\in \mathcal{T}}\sum _{i\in \mathcal{N}}{\lambda }_t(P_{\mathrm{grid},i,t}^{\mathrm{buy}}-P_{\mathrm{grid},i,t}^{\mathrm{sell}})\mathsf{\Delta }t$$

(14)

$$C_{\mathrm{ESS}}=c_{\mathrm{ESS}}\sum _{t\in \mathcal{T}}\sum _{i\in \mathcal{N}}(P_{\mathrm{ESS},i,t}^{\mathrm{ch}}{\eta }_{\mathrm{ESS}}^{\mathrm{ch}}+P_{\mathrm{ESS},i,t}^{\mathrm{dis}}/{\eta }_{\mathrm{ESS}}^{\mathrm{dis}})\mathsf{\Delta }t$$

(15)

$$C_{\mathrm{EV}}=\sum _{t\in \mathcal{T}}\sum _{i\in \mathcal{N}}{c}_{adj}\cdot |P_{EV,i,t}^{adj}|\mathsf{\Delta }t$$

(16)

Among them, $P_{\mathrm{PV},i,t}$, $P_{\mathrm{ESS},i,t}^{\mathrm{ch}}$, $P_{\mathrm{ESS},i,t}^{\mathrm{dis}}$ and $P_{\mathrm{EV},i,t}$ are the PV output power, energy storage system (ESS) charging power, ESS discharging power, and charging pile charging power in the i-th service area during time period t. $P_{\mathrm{grid},i,t}^{\mathrm{buy}}$, $P_{\mathrm{grid},i,t}^{\mathrm{sell}}$ are the purchased power when PV output is insufficient and the sold power for surplus power feeding into the grid in the i-th service area during time period t, respectively. ${\lambda }_t$ is the transaction price with the distribution grid in each time period; $c_{\mathrm{ESS}}$ is the cost coefficient for calculating the charging and discharging losses of ESS equipment. $c_{adj}$ is the EV dispatch cost coefficient, which is 0.1 CNY/kWh, and $\left|P_{EV,i,t}^{adj}\right|$ is the load difference before and after dispatch. Among them, there is an absolute value term in Equation (16). It is necessary to introduce auxiliary variables $P_{\mathrm{EV},i,t}^1$ and $P_{\mathrm{EV},i,t}^2$ to linearize $\left|P_{\mathrm{EV},i,t}^{adj}\right|=|P_{\mathrm{EV},i,t}-P_{\mathrm{EV},i,t}^{\mathrm{before}}|$ Equation (16):

$$C_{\mathrm{EV}}=\sum _{t\in \mathcal{T}}\sum _{i\in \mathcal{N}}{c}_{adj}\cdot (P_{\mathrm{EV},i,t}^1+P_{\mathrm{EV},i,t}^2)\mathsf{\Delta }t$$

(17)

The constraints for $P_{\mathrm{EV},i,t}^1$ and $P_{\mathrm{EV},i,t}^2$ are:

$$P_{\mathrm{EV},i,t}-P_{\mathrm{EV},i,t}^{\mathrm{before}}+P_{\mathrm{EV},i,t}^1-P_{\mathrm{EV},i,t}^2=0$$

(18)

$$P_{\mathrm{EV},i,t}^1(t)\ge 0,P_{\mathrm{EV},i,t}^2(t)\ge 0$$

(19)

4.1.2. Constraint Conditions

(1)

System Power Balance Constraints

$$P_{\mathrm{PV},i,t}+P_{\mathrm{grid},i,t}^{\mathrm{buy}}-P_{\mathrm{grid},i,t}^{\mathrm{sell}}+P_{\mathrm{ESS},i,t}^{\mathrm{dis}}-P_{\mathrm{ESS},i,t}^{\mathrm{ch}}=P_{\mathrm{EV},i,t}$$

(20)

(2)

ESS Charging and Discharging Power Constraints.

$$0\le P_{\mathrm{ESS},i,t}^{\mathrm{ch}}$$

(21)

$$P_{\mathrm{ESS},i,t}^{\mathrm{ch}}\le P_{\mathrm{ESS},i,t}^{\mathrm{ch},\max }\cdot U_{i,t}^{\mathrm{ch}}$$

(22)

$$0\le P_{\mathrm{ESS},i,t}^{\mathrm{dis}}$$

(23)

$$P_{\mathrm{ESS},i,t}^{\mathrm{dis}}\le P_{\mathrm{ESS},i}^{\mathrm{dis},\max }\cdot (1-U_{i,t}^{\mathrm{ch}})$$

(24)

$$U_{i,t}^{\mathrm{ch}}\in \{0,1\},\forall i\in \mathcal{N},\forall t\in \mathcal{T}$$

(25)

Among them, $U_{i,t}^{\mathrm{ch}}$ is a 0/1 variable representing the ESS charging-discharging state, where 1 denotes charging and 0 denotes discharging. $P_{\mathrm{ESS},i,t}^{\mathrm{ch},\max }$ represents the maximum charging power of the i-th service area in time period t, and $P_{\mathrm{ESS},i}^{\mathrm{dis},\max }$ represents the maximum discharging power.

(3)

ESS Charging-Discharging Balance Constraint. It ensures the ESS capacity is equal at the start and end of dispatch, reducing battery damage from cyclic ESS dispatch.

$$\eta \sum _{N_{\mathrm{T}}}^{t=1}\left(P_{\mathrm{ESS},i,t}^{\mathrm{ch}}\mathsf{\Delta }t\right)-{\displaystyle \frac{1}{\eta }}\sum _{N_{\mathrm{T}}}^{t=1}\left(P_{\mathrm{ESS},i,t}^{\mathrm{dis}}\mathsf{\Delta }t\right)=0$$

(26)

(4)

ESS SOC Constraint. It ensures the ESS SOC stays within a reasonable range, avoiding over-charging or over-discharging to extend its service life.

$$SO{C}_{i,t}=SO{C}_{i,t-1}+{\eta }_{\mathrm{ch}}\cdot P_{\mathrm{ESS},i,t}^{\mathrm{ch}}\cdot \mathsf{\Delta }t/E_{\mathrm{ESS},i}-P_{\mathrm{ESS},i,t}^{\mathrm{dis}}\cdot \mathsf{\Delta }t/(E_{\mathrm{ESS},i}\cdot {\eta }_{dis})$$

(27)

$$SO{C}_i^{\min }\le SO{C}_{i,t}\le SO{C}_i^{\max }$$

(28)

Among them, ${\eta }_{\mathrm{ch}}$ and ${\eta }_{dis}$ are the ESS charging and discharging efficiencies, respectively, taking a value of 0.95; $E_{\mathrm{ESS},i}$ is the capacity of the ESS in the i-th service area; $SO{C}_i^{\min }$ and $SO{C}_i^{\max }$ are the lower and upper limits of SOC, respectively, taking values of 0.1 and 0.9.

(5)

Grid Interaction Constraints

$$0\le P_{\mathrm{grid},i,t}^{\mathrm{buy}}$$

(29)

$$P_{\mathrm{grid},i,t}^{\mathrm{buy}}\le P_{\mathrm{grid},i}^{\max }\cdot U_{i,t}^{\mathrm{buy}}$$

(30)

$$0\le P_{\mathrm{grid},i,t}^{\mathrm{sell}}$$

(31)

$$P_{\mathrm{grid},i,t}^{\mathrm{sell}}\le P_{\mathrm{grid},i}^{\max }\cdot (1-U_{i,t}^{\mathrm{buy}})$$

(32)

$$U_{i,t}^{\mathrm{buy}}\in \{0,1\}$$

(33)

Among them, $P_{\mathrm{grid},i}^{\max }$ is the upper limit of grid interaction power, set at 500 kW. $U_{i,t}^{\mathrm{buy}}$ is a 0/1 variable representing power purchase and sale with the distribution grid, where 1 denotes power purchase and 0 denotes power sale.

(6)

EV Load Constraints

With the EV dispatch cost coefficient set to 0.1 CNY/kWh, the total EV load is constrained to be no greater than 1.05 times the total pre-dispatch EV load.

$$\sum _{i\in \mathcal{N}}{P}_{\mathrm{EV},i,t}\le 1.05\times \sum _{i\in \mathcal{N}}{P}_{\mathrm{EV},i,t}^{before}$$

(34)

(7)

Uncertain Parameter Constraints

The constraints regarding the uncertainties of PV output and EV load are shown in Equations (2) and (5).

4.1.3. Robust Optimization Formulation

Based on the aforementioned objective function and constraints, if uncertainty factors are not considered, the problem to be solved is a minimization problem aiming for the lowest comprehensive cost:

$$\begin{array}{ll}& \underset{x}{\min }{c}^{\mathrm{T}}y\\ \hfill \mathrm{s}.\mathrm{t}.& Dy\ge d\\ & Ky=k\\ & Fx+Gy\ge h\\ & I_{u}y=\hat{u}\end{array}$$

(35)

Among them, $\mathit{x}$ represents the decision variables to be made in the operation stage, including decisions related to grid interaction and ESS charging-discharging, and these are 0/1 variables; $\mathit{y}$ represents the decision variables to be observed, including ESS charging power, ESS discharging power, purchased electricity from the grid, sold electricity to the grid, EV load, auxiliary variables of EV load, and PV output, which are the specific powers adjusted after the realization of uncertain factors; For PV generation, $\hat{u}$ denotes the PV output calculated using the model in Section 3.1 and is treated as a deterministic value. For EV charging load, $\hat{u}$ represents the center value of the EV-load uncertainty set, which is taken as the post-dispatch EV load and is also deterministic.

$$x=[U_{i,t}^{\mathrm{ch}};U_{i,t}^{\mathrm{buy}}]\in {\{0,1\}}^{2\mathrm{TN}}$$

(36)

$$y=[P_{\mathrm{ESS},i,t}^{\mathrm{ch}};P_{\mathrm{ESS},i,t}^{\mathrm{dis}};P_{\mathrm{grid},i,t}^{\mathrm{buy}};P_{\mathrm{grid},i,t}^{\mathrm{sell}};P_{\mathrm{EV},i,t};P_{\mathrm{EV},i,t}^1;P_{\mathrm{EV},i,t}^2;P_{\mathrm{PV},i,t}]\in {ℝ}^{8\mathrm{TN}}$$

(37)

When considering the uncertainties of power sources and loads, it is necessary to solve for the minimum cost under the worst-case scenario, which is a max-min problem.

$$\begin{array}{ll}& \underset{u\in U}{\max }\underset{y\in \mathsf{\Omega }(x,u)}{\min }{c}^{\mathrm{T}}y\\ \hfill \mathrm{s}.\mathrm{t}. & Dy\ge d\\ & Ky=k\\ & Fx+Gy\ge h\\ & I_{u}y=u\end{array}$$

(38)

U represents the uncertainty set; $\Omega (x,u)$ is the feasible region of y given the decision variable x and the uncertain quantity u. $I_u$ is the coefficient matrix of uncertain variables, indicating that under a certain uncertain scenario u, $P_{EV}=u_{EV}$, $P_{pv}=u_{pv}$ in y.

$$u=[u_{\mathrm{PV}};u_{\mathrm{EV}}]\in {ℝ}^{2\mathrm{TN}}$$

(39)

By combining Equations (35) and (38), the robust optimization problem for the PV–energy storage–charging operation stage considering uncertainties can be formulated as a min-max-min optimization problem:

$$\begin{array}{c}\underset{x}{\min }\left\{\underset{u\in U}{\max }\underset{y\in \mathsf{\Omega }(x,u)}{\min }{c}^{\mathrm{T}}y\right\}\\ \mathrm{s}.\mathrm{t}.\hspace{1em}\left\{\begin{array}{l}Dy\ge d,\hfill \\ Ky=k,\hfill \\ Fx+Gy\ge h,\hfill \\ I_{u}y=u\hfill \end{array}\right.\end{array}$$

(40)

In the equations, $T=24$, representing the total dispatch time periods in a day; $N = 12$, representing the number of service areas in the solved problem. In the calculation, x, $\mathit{y}$, and u need to be converted into column vectors, and sparse matrices are constructed by blocking according to service areas to facilitate subsequent calculations.

$c^{\mathrm{T}}y$ represents the total system operation cost function, including Equations (13)–(17). Both c and $y$ are column vectors with a dimension of 8TN.

The inequality constraint $Dy\ge d$ includes the lower-bound constraints of various variables, the SOC constraints of the energy storage system, and the total-load constraints of EVs, corresponding to Equations (19), (21), (23), (28), (29), (31) and (34). The dimension of matrix D is $(2\mathrm{TN}+\mathrm{N})\times 8\mathrm{TN}$, and d is a column vector with a dimension of $(2\mathrm{TN}+\mathrm{N})$.

The equality constraint $Ky=k$ includes the auxiliary-variable constraints introduced by EV loads, the charging-discharging balance constraints of energy storage, and the power-balance constraints of the PV–energy storage–charging system, corresponding to Equations (18), (20) and (26). The dimension of matrix K is $(2\mathrm{TN}+\mathrm{N})\times 8\mathrm{TN}$, and k is a column vector with a dimension of $(2\mathrm{TN}+\mathrm{N})$.

The constraint $Fx+Gy\ge h$ involving decision variables corresponds to Equations (22), (24), (30) and (32). The dimension of matrix F is $4\mathrm{TN}\times 2\mathrm{TN}$, the dimension of matrix G is $4\mathrm{TN}\times 8\mathrm{TN}$, and h is a column vector with a dimension of $8\mathrm{TN}$.

In summary, this robust optimization problem is formally a three-level optimization problem: the outermost level minimizes the cost of the first-stage decision x, the middle level maximizes the cost caused by the uncertain parameter u, and the innermost level minimizes the cost of the second-stage decision $\mathit{y}$. Constructing this structure can ensure that the system can still achieve optimal performance under the worst-case scenario.

4.2. Robust Optimization Solution Based on the C&CG Algorithm

4.2.1. Description of the Column-and-Constraint Generation (C&CG) Algorithm

For the aforementioned min-max-min robust optimization problem, this paper adopts the Column-and-Constraint Generation (C&CG) algorithm for solution [19]. The C&CG algorithm originates from the cutting-plane method and has been widely applied in solving large-scale linear and nonlinear optimization problems.

In this study, the master problem is responsible for optimizing the charging and discharging of energy storage, EV dispatch, and grid-interactive power, with the objective of minimizing the system cost. In each iteration, the C&CG algorithm generates new columns and gradually optimizes the objective function by incorporating new decision variables, thus improving economic efficiency.

The subproblem evaluates the extreme operating states of the system by calculating the worst-case scenarios, i.e., when the PV output is low and the EV load is high. To address such high-dimensional uncertainties, this paper introduces the strong duality theory and the Big-M method. By introducing dual variables, the max-min subproblem is transformed into a standard linear programming problem, enabling a more precise robust optimization solution. Additionally, by performing overall optimization for the multi-service-area system, this method can effectively handle the complex coupling relationships among multiple service areas.

Compared with traditional robust optimization methods, the C&CG algorithm can significantly improve the efficiency of solving large-scale problems when dealing with the coordinated optimization problem of multi-service areas involving multiple uncertainty sources and complex constraints. Moreover, it can ensure that the global optimal solution is still found under the worst-case scenarios of low PV output and high load demand.

4.2.2. Solution of the Master Problem

The master problem is the outermost cost-minimization problem. By decomposing Equation (34), the master problem can be derived as:

$$\begin{array}{c}\underset{x}{\min }\alpha \\ \mathrm{s}.\mathrm{t}.\alpha \ge c^{\mathrm{T}}{y}_l\\ D{y}_l\ge d\\ K{y}_l=k\\ Fx+G{y}_l\ge h\\ I_u{y}_l=u_l^{*}\\ \forall l\le m\end{array}$$

(41)

Among them, $\alpha$ is an auxiliary variable of the objective function; $m$ is the current iteration number, and the solution $y_l$ after l iterations can be obtained in advance; $u_l^{*}$ is the value of the uncertain variable passed from the subproblem after the l-th iteration. The master problem is a Mixed Integer Linear Programming (MILP) problem and can be solved directly.

Introducing the variables involved in this study, the master problem can be further specified as follows:

First, we define the following:

$$\begin{array}{l}{c}_{i,t}^{ch}=c_{ESS}{\eta }_{ESS}^{ch}\mathsf{\Delta }t,c_{i,t}^{dis}=c_{ESS}{\displaystyle \frac{\mathsf{\Delta }t}{{\eta }_{ESS}^{dis}}},c_{i,t}^{buy}={\lambda }_t\mathsf{\Delta }t,c_{i,t}^{sell}=-{\lambda }_t\mathsf{\Delta }t\\ c_{i,t}^{EV}=0,c_{i,t}^{EV1}=c_{adj}\mathsf{\Delta }t,c_{i,t}^{EV2}=c_{adj}\mathsf{\Delta }t,c_{i,t}^{PV}=0\end{array}$$

(42)

$$c=[\mathrm{vec}(c_{i,t}^{ch});\mathrm{vec}(c_{i,t}^{dis});\mathrm{vec}(c_{i,t}^{buy});\mathrm{vec}(c_{i,t}^{sell});\mathrm{vec}(0);\mathrm{vec}(c_{i,t}^{EV1});\mathrm{vec}(c_{i,t}^{EV2});\mathrm{vec}(0)]\in {ℝ}^{8TN}$$

(43)

Accordingly, the problem can be further written as follows:

$$\underset{x}{\min }\alpha$$

(44)

$$\alpha \ge c^T\left[\begin{array}{c}\mathrm{vec}\left(P_{ESS,i,t}^{ch,(l)}\right)\\ \mathrm{vec}\left(P_{ESS,i,t}^{dis,(l)}\right)\\ \mathrm{vec}\left(P_{grid,i,t}^{buy,(l)}\right)\\ \mathrm{vec}\left(P_{grid,i,t}^{sell,(l)}\right)\\ \mathrm{vec}\left(P_{EV,i,t}^{(l)}\right)\\ \mathrm{vec}\left(P_{EV,i,t}^{1,(l)}\right)\\ \mathrm{vec}\left(P_{EV,i,t}^{2,(l)}\right)\\ \mathrm{vec}\left(P_{PV,i,t}^{(l)}\right)\end{array}\right]$$

(45)

$$P_{PV,i,t}^{(l)}+P_{grid,i,t}^{buy,(l)}-P_{grid,i,t}^{sell,(l)}+P_{ESS,i,t}^{dis,(l)}-P_{ESS,i,t}^{ch,(l)}-P_{EV,i,t}^{(l)}=0$$

(46)

$$P_{ESS,i,t}^{ch,(l)}\ge 0$$

(47)

$$P_{ESS,i,t}^{dis,(l)}\ge 0$$

(48)

$${\eta }_{ESS}^{ch}{\displaystyle \sum _{t=1}^{N_T}{P}_{ESS,i,t}^{ch,(l)}}\mathsf{\Delta }t-{\displaystyle \sum _{t=1}^{N_T}{P}_{ESS,i,t}^{dis,(l)}}\mathsf{\Delta }t=0$$

(49)

$$P_{grid,i,t}^{buy,(l)}\ge 0$$

(50)

$$P_{grid,i,t}^{sell,(l)}\ge 0$$

(51)

$$1.05{\displaystyle \sum _{i\in N}{P}_{EV,i,t}^{bfore}}-{\displaystyle \sum _{i\in N}{P}_{EV,i,t}^{(l)}}\ge 0$$

(52)

$$\left(P_{EV,i,t}^{(l)}-{\hat{P}}_{EV,i,t}\right)+P_{EV,i,t}^{1,(l)}-P_{EV,i,t}^{2,(l)}=0$$

(53)

$$P_{EV,i,t}^{1,(l)}\ge 0$$

(54)

$$P_{EV,i,t}^{2,(l)}\ge 0$$

(55)

$$SO{C}_{i,t-1}^{(l)}+{\eta }_{ESS}^{ch}{\displaystyle \frac{P_{ESS,i,t}^{ch,(l)}\mathsf{\Delta }t}{E_{ESS,i}}}-{\displaystyle \frac{1}{{\eta }_{ESS}^{dis}}}{\displaystyle \frac{P_{ESS,i,t}^{dis,(l)}\mathsf{\Delta }t}{E_{ESS,i}}}=SO{C}_{i,t}^{(l)}$$

(56)

$$SO{C}^{min}\le SO{C}_{i,t}^{(l)}\le SO{C}^{max}$$

(57)

$$U_{i,t}^{ch},U_{i,t}^{buy}\in \{0,1\}$$

(58)

$$P_{ESS,i,t}^{ch,max}{U}_{i,t}^{ch}-P_{ESS,i,t}^{ch,(l)}\ge 0$$

(59)

$$P_{ESS,i,t}^{dis,max}\left(1-U_{i,t}^{ch}\right)-P_{ESS,i,t}^{dis,(l)}\ge 0$$

(60)

$$P_{grid}^{max}{U}_{i,t}^{buy}-P_{grid,i,t}^{buy,(l)}\ge 0$$

(61)

$${\hat{u}}_{PV,i,t}-B_{PV,i,t}\mathsf{\Delta }{u}_{PV,i,t}^{\max }=u_{PV,i,t}$$

(62)

$${\hat{u}}_{EV,i,t}+B_{EV,i,t}\mathsf{\Delta }{u}_{EV,i,t}^{\max }=u_{EV,i,t}$$

(63)

$$\forall l\le m$$

(64)

4.2.3. Solution of the Subproblem

The subproblem is a max-min problem that seeks the worst-case scenario given the first-stage decisions:

$$\underset{u\in U}{\max }\underset{y\in \mathsf{\Omega }(x,u)}{\min }{c}^{\mathrm{T}}y$$

(65)

This is a bi-level optimization problem, where the inner layer seeks the optimal response to the realization of uncertainties, and the outer layer seeks the realization of uncertainties that maximizes the cost. To solve it effectively, the strong duality theory can be used to convert the inner min problem into its dual form, thus transforming the original problem into a single-level max problem.

According to the strong duality theory, for the given $x_i$ and u, the dual problem of Equation (36) is:

$$\begin{array}{ll}& \underset{u\in U,\gamma ,\lambda ,\nu ,\pi }{\max }{d}^{\mathrm{T}}\gamma +k^{\mathrm{T}}\lambda +{(h-Fx)}^{\mathrm{T}}v+u^{\mathrm{T}}\pi \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathit{D}}^{\mathrm{T}}\mathit{\gamma }+{\mathit{K}}^{\mathrm{T}}\mathit{\lambda }+{\mathit{G}}^{\mathrm{T}}\mathit{\nu }+{\mathit{I}}_{\mathrm{u}}^{\mathrm{T}}\mathit{\pi }\le \mathit{c}\\ & \mathit{\gamma }\ge 0,\mathit{\nu }\ge 0\end{array}$$

(66)

In the dual reformulation, the variables $\mathit{\gamma },\mathit{\lambda },\mathit{\nu },\mathit{\pi }$ are introduced. Specifically, $\mathit{\gamma }$ is associated with the inequality constraints of the recourse problem (e.g., lower-bound and operational inequality limits), $\mathit{\lambda }$ corresponds to the equality constraints (e.g., power balance, storage energy balance, and auxiliary-variable equalities), $\mathit{\nu }$ is associated with the constraints that couple the recourse variables with the first-stage binary decisions, and $\mathit{\pi }$ is associated with the uncertainty-coupling constraints.

In the dual objective, the terms ${\mathit{d}}^{\mathrm{\top }}\mathit{\gamma }$, ${\mathit{k}}^{\mathrm{\top }}\mathit{\lambda }$, and $\left(\mathit{h}−\mathit{F}\mathit{x}{)}^{\mathrm{\top }}\mathit{\nu }\right.$ represent the contributions of the recourse constraints’ right-hand sides, with $\left(\mathit{h}−\mathit{F}\mathit{x}{)}^{\mathrm{\top }}\mathit{\nu }\right.$ explicitly reflecting how the fixed first-stage decision $\mathit{x}$ affects the feasible operating region and the resulting worst-case cost. The term ${\mathit{u}}^{\mathbf{\top }}\mathit{\pi }$ captures the impact of the uncertainty realization on the objective.

Here, $\mathit{\pi }$ is the dual variable associated with the uncertainty-coupling constraints. Since the dual objective contains ${\mathit{u}}^{\mathrm{\top }}\mathit{\pi }$, and after merging the outer maximization $u$ and $\pi$ appear as decision variables in the same level, ${\mathit{u}}^{\mathrm{\top }}\mathit{\pi }$ becomes a typical bilinear term. According to [20], for a polyhedral uncertainty set $U$, the optimal worst-case solution is attained at an extreme point of $U$. That is, when Equation (66) reaches its maximum, the uncertainty realization u must lie on the boundaries of the fluctuation intervals described in Equation (8) and (11). In this paper, the operating cost becomes higher when PV generation takes the lower bound of its interval and EV load takes the upper bound of its interval, which better matches the definition of an adverse scenario. The optimal solution to this problem is the boundary values within U, i.e., the scenarios of minimum PV output and maximum EV load described in Equations (67) and (68). At this time, the operation cost of the PV–energy storage–charging system is the highest, which is the worst-case scenario.

$$u_{\mathrm{PV},i,t}={\hat{u}}_{\mathrm{PV},i,t}-B_{\mathrm{PV},i,t}△u_{\mathrm{PV},i,t}^{\max }$$

(67)

$$u_{\mathrm{EV},i,t}={\hat{u}}_{\mathrm{EV},i,t}+B_{\mathrm{EV},i,t}△u_{\mathrm{EV},i,t}^{\max }$$

(68)

After substituting the boundary-selection representation in Equations (67) and (68) into Equation (66), products between the 0–1 variables $\left(B_{PV,t},B_{EV,t}\right)$ and the continuous dual variables arise. Following the standard Big-M linearization method in [21], we introduce auxiliary variables ($B_{PV,t}^{\prime },B_{EV,t}^{\prime }$) to replace these binary–continuous products and add the corresponding linear constraints, thereby converting the subproblem into a mixed-integer linear program (MILP). Detailed step-by-step constraints and the equivalence proof can be found in [21]. At this time, the subproblem can be described as:

$$\begin{array}{ll}\hfill \underset{B,B,\gamma ,A,\nu ,\pi }{\max }& {\mathit{d}}^{\mathrm{T}}\mathit{\gamma }++k^{\mathrm{T}}\lambda +{(\mathit{h}-\mathit{F}\mathit{x})}^{\mathrm{T}}\mathit{\nu }+{\hat{\mathit{u}}}^{\mathrm{T}}\mathit{\pi }+\mathsf{\Delta }{\mathit{u}}^{\mathrm{T}}{\mathit{B}}^{\mathbf{\prime }}\\ \hfill \mathrm{s}.\mathrm{t}.& {\mathit{D}}^{\mathrm{T}}\mathit{\gamma }+{\mathit{K}}^{\mathrm{T}}\mathit{\lambda }+{\mathit{G}}^{\mathrm{T}}\mathit{\nu }+{\mathit{I}}_{\mathrm{u}}^{\mathrm{T}}\mathit{\pi }\le \mathit{c}\\ & 0\le {\mathit{B}}^{\mathbf{\prime }}\le \overline{\mathit{\pi }}\mathit{B}\\ & \mathit{\pi }-\overline{\mathit{\pi }}(1-\mathit{B})\le {\mathit{B}}^{\mathbf{\prime }}\le \mathit{\pi }\\ & {\displaystyle \sum _{t\in \mathcal{T}}}{B}_{\mathrm{PV},t}\le {\mathsf{\Gamma }}_{\mathrm{PV}}\\ & {\displaystyle \sum _{t\in \mathcal{T}}}{B}_{\mathrm{EV},t}\le {\mathsf{\Gamma }}_{\mathrm{EV}}\end{array}$$

(69)

Among them, $\mathsf{\Delta }u={\left[\mathsf{\Delta }{u}_{\mathrm{PV}}^{\max }(t),\mathsf{\Delta }{u}_{\mathrm{EV}}^{\max }(t)\right]}^{\mathrm{T}}$, $B^{\prime }={[B_{\mathrm{PV}}^{\prime }(t),B_{\mathrm{EV}}^{\prime }(t)]}^{\mathrm{T}}$, and $\overline{\mathit{\pi }}$ is the maximum value of variable $\mathit{\pi }$, taking a sufficiently large positive real number. After the above transformation, the subproblem is converted into a directly solvable MILP problem.

4.2.4. Solution Procedure

After the above transformations, the PV–energy storage–charging robust model represented by Equation (40) can be decomposed into the master problem shown in Equation (41) and the subproblem shown in Equation (69), both of which are Mixed Integer Linear Programming (MILP) problems. Figure 3 illustrates the procedural framework of the C&CG algorithm. The basic steps of the C&CG algorithm are as follows:

Step 1: Initialization: Set the iteration counter m = 0, the upper bound UB = +∞, the lower bound LB = −∞, and set the initial worst-case scenario according to the boundary values of uncertain variables.

Step 2: Solve the Relaxed Master Problem: Based on the given worst-case scenario, solve the relaxed master problem in Equation (40) to obtain the optimal solution $(x_m,{\alpha }_m,y_1,y_2,\dots y_m)$, and update the lower bound as $LB={\alpha }_m$.

Step 3: Solve the Subproblem: Fix $x=x_m$, solve the subproblem shown in Equation (69) to obtain the objective value $f_m\left(x_m\right)$ of the worst-case scenario under this decision and the value $u_m$ of the uncertain variable, and update the upper bound as $UB=\min \{UB,f_m(x_m)\}$.

Step 4: Convergence Check: Iteratively solve the master problem and the subproblem, and perform a convergence check. If $UB-LB\le \epsilon$, the algorithm terminates; otherwise, update the worst-case scenario set as $\mathcal{S}=\mathcal{S}\cup \left\{m\right\},m=m+1$, and return to Step 2.

5. Case Study Analysis

In this chapter, considering the dual uncertainties of PV output and EV load, a two-stage robust optimization model based on the C&CG algorithm is established and solved for the PV-energy storage-charging system of the Highway service areas.

Simulations are conducted based on the MATLAB 2018b platform and GUROBI solver. Through detailed simulation analysis, the operating characteristics and cooperative effects of each service area are systematically explored, the performance differences between robust optimization and deterministic optimization are compared, and the adaptability and system robustness of the proposed scheme under different harsh conditions are verified.

5.1. Case Parameters

The specific highway section selected in this study is shown in Figure 4. It has a total of 17 nodes, including 12 service area nodes and 5 toll station nodes, with an overall length of approximately 640 km. The distance between each pair of adjacent nodes is about 40 km.

The configurations of PV, energy storage, and charging piles in each service area are shown in

Table 1. PV–energy storage–charging Capacity Configuration in Different Service Areas.

Service Area Number	Number of PV Modules	ESS Capacity (kWh)	Number of Charging Piles
1	16	1824	5
2	29	2946	7
3	17	2956	13
4	25	3352	14
5	23	2712	11
6	30	3348	20
7	25	1154	16
8	32	2516	14
9	31	2732	17
10	26	3496	12
11	25	3038	10
12	29	3442	10

. In the budgeted uncertainty set, ${\mathsf{\Gamma }}_{PV}$ and ${\mathsf{\Gamma }}_{EV}$ denote the maximum numbers of hours (out of 24) during which PV generation and EV load, respectively, are allowed to take their interval boundaries (worst-case deviations); in this study, we set ${\mathsf{\Gamma }}_{PV}={\mathsf{\Gamma }}_{EV}= 6$, meaning that at most 6 h (about one quarter of the day) can reach extreme deviations, representing a moderate risk level that balances economy and robustness.

The relevant calculation parameters of the PV-energy storage-charging system are shown in

Table 2. Simulation Parameter Values.

Parameter	Value	Parameter	Value
$C_{ESS}$	0.15 CNY/(kWh)	${\mathsf{\Gamma }}_{PV}$	6
$C_{adj}$	0.1 CNY/(kWh)	${\mathsf{\Gamma }}_{EV}$	6
$P_{grid, i}^{max}$	1000 kW	$\eta$	0.95
$P_{ESS,i,t}^{ch,max}/P_{ESS,i,t}^{dis,max}$	$0.5 E_{ESS,i}$	$\epsilon$	1%

. The transaction electricity prices between the system and the distribution grid are shown in Figure 5 [15].

The relevant parameter information of photovoltaic (PV) modules, electric vehicle (EV) charging piles, and EV charging is shown in the

Table 3. Parameters of PV modules and EV charging settings.

Parameter	Value
Operational lifespan of photovoltaic (PV) Module	25
Capacity of a Single PV Module	30 kW
Photovoltaic (PV) Module’s Standard Test Conditions (STC) Irradiance	1000 W/m2
Photovoltaic (PV) Module’s Standard Test Conditions (STC) Temperature	25 °C
Photovoltaic (PV) Module’s Power Temperature Coefficient	−0.004
Electric Vehicle’s Battery Capacity	55 kW·h
Electric Vehicle’s Energy Consumption per Kilometer	0.15 kW·h/km
Charging Pile Power	80 kW
Electric Vehicle’s Charging Efficiency	90%

.

In the case study, on the PV side, hourly meteorological data provided by the NASA Langley Research Center’s Prediction Of Worldwide Energy Resources (POWER) project (including solar irradiance and ambient temperature) are used. The meteorological time series corresponding to the location of the case expressway are extracted and, together with the installed PV capacity of each service area, substituted into the PV power calculation model presented in Section 3 to obtain the forecast PV generation profiles for each service area.

On the EV-load side, based on expressway traffic-flow data and the OD matrix, the EV load dynamic calculation model in Section 3 is used to simulate vehicle charging behaviors at service areas, thereby deriving the hourly forecast EV charging-load profiles for the service areas. The EV OD distribution is given in

Table 4. EV Origin–Destination (OD) Distribution Information.

Origin	Destination
	Node 1	Node 5	Node 9	Node 13	Node 17
Node 1	/	13%	7%	5%	2%
Node 5	/	/	22%	8%	4%
Node 9	/	/	/	18%	9%
Node 13	/	/	/	/	12%
Node 17	/	/	/	/	/

, and the time-of-day traffic proportion is shown in Figure 6 (reflecting the share of EVs in each time period within a day) [9].

Assuming a PV output deviation of 15%, the corresponding PV uncertainty profile (with the red curve representing the optimized realized PV output) is shown in Figure 7. Assuming an EV-load deviation of 10% for each service area, the corresponding EV-load uncertainty profiles (with the red curve representing the optimized realized EV load) are shown in Figure 8. The uncertainty budget parameter $\Gamma$ for both PV output and EV load is set to 6, meaning that within a day the realized PV output reaches the lower bound of the uncertainty set in six time periods, while the EV load reaches the upper bound of the uncertainty set in six time periods.

5.2. Simulation Result Analysis

(1)

Optimization Result Analysis of All Service Areas Along the Highway

Figure 9 presents the optimal power dispatch results of the PV–energy storage–charging systems in 12 highway service areas under uncertainty considerations. It can be observed that the PV power generation in all service areas mainly concentrates on daytime hours (6:00–18:00), consistent with the variation law of solar irradiance intensity.

Meanwhile, all service areas generally adopt economically driven dispatch strategies. During low-electricity-price periods such as nighttime and early morning (0:00–6:00), most service areas choose to purchase electricity from the grid to charge their energy storage systems, reducing the system operation costs. During daytime periods with sufficient PV output and high electricity prices (10:00–16:00), the systems prioritize using PV electricity to meet load demands and sell surplus electricity to the grid, improving economic benefits. During evening and nighttime peak electricity price periods (18:00–22:00), PV output decreases while load demands increase. At this time, the systems prioritize using energy storage discharge (dark green downward bars) to meet loads, avoiding purchasing electricity from the grid during high-price periods and thus reducing operation costs.

The ESS charging and discharging curves of each service area are shown in Figure 10. Different service areas exhibit differences in ESS dispatch strategies due to variations in their PV installed capacities, ESS scales, and load characteristics. For service areas with sufficient PV output (e.g., Service Area 2), the ESS charges extensively during low electricity price periods, fully leveraging the price advantage. During nighttime when electricity prices are high, the ESS discharges to sustain the load, reducing the need for high-cost electricity purchase. Moreover, there are obvious intervals in the ESS charging and discharging processes, avoiding frequent charge–discharge switching and contributing to extending the ESS lifespan. For service areas with insufficient PV output (e.g., Service Area 7), the resulting dispatch strategy follows a more conventional operating pattern. The ESS conducts preventive charging during low electricity price periods to cope with harsh scenarios such as insufficient PV output or increased load. During peak load periods with high electricity prices, the ESS discharges instead of relying on the grid, saving electricity purchase costs. The dispatch strategy balances economic efficiency and the ability to handle uncertainties.

The robust-optimized dispatch strategy can maximize economic value while ensuring self-supply needs through “charging at off-peak times and discharging at peak times.” After considering uncertainties, the strategy is preventive: the ESS charging and discharging are more conservative, and the interaction with the grid is more flexible; service areas of different scales implement differentiated dispatch strategies. Overall, this strategy achieves a good balance between economic efficiency and reliability, significantly enhancing the system’s ability to handle uncertainties.

(2)

Comparison with Deterministic Optimization

To verify the effectiveness and superiority of the robust optimization method proposed in this chapter, a comparative analysis is conducted between it and the deterministic optimization method. Deterministic optimization uses the predicted values of PV output and EV load as input parameters, without considering their uncertainties, and the objective function can be directly solved using the Mixed Integer Linear Programming (MILP) method. In contrast, robust optimization considers the fluctuations of these parameters within the box uncertainty set.

To compare the operational impacts of robust scheduling and deterministic scheduling,

Table 5. Comparison of operating characteristics under deterministic and robust scheduling.

Method	Average Equivalent Full Cycles (All Service Areas) 1	Total Grid Exchange Energy (All Service Areas) [kWh]
Robust Optimization	0.629	67,377.949
Deterministic Optimization	1.226	71,383.809

¹ Average equivalent full cycles of the ESS across all service areas is computed as: $\overline{N_{EFC}}= {\displaystyle \frac{1}{12\sum _{i=1}^{12}{E}_{n,i}}}\left(\sum _{i=1}^{12}{E}_{ch,i}+ \sum _{i=1}^{12}{E}_{dis,i}\right), E_{Ch,i}= \sum _{t=0}^{23}{p}_{ESS,i,t}^{ch}∆t$, $E_{dis,i }= \sum _{t=0}^{23}{p}_{ESS,i,t}^{dis}∆t$, $E_{n,i}$ represents the energy storage capacity configured for the i-th service area.

summarizes the average equivalent full charge–discharge cycles of ESSs across all service areas and the total energy exchanged with the distribution grid.

Compared with the deterministic solution, the robust solution yields lower average equivalent full cycles and lower total grid-exchanged energy, indicating that robust scheduling can suppress excessive ESS cycling under uncertainty, extend ESS lifetime, and reduce reliance on the grid, thereby improving overall operational performance.

By selecting different ${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}$ and ${\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}$, the operating costs of the day-ahead PV–energy storage–charging system obtained from simulations are as follows.

It can be seen that when the uncertain parameters of power sources and loads are all 0, robust optimization is equivalent to deterministic optimization. Moreover, the larger the values of the uncertain parameters, the higher the operating costs of the PV-energy storage-charging system. That is, when formulating dispatch plans, the more uncertainties are considered, the more conservative the developed plans will be.

According to the comparative analysis in

Table 6. Day-ahead Operation Costs of the PV-ESS-Charging System Under Different Uncertainty Parameters.

Optimization Model	System Day-Ahead Operation Cost (CNY)
Deterministic Optimization Model	53,886
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=0,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=0$	53,886
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=6,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=3$	57,666
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=6,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=6$	59,722
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=12,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=6$	60,185

, although the deterministic optimization strategy shows better economic efficiency in day-ahead dispatch plans, this does not directly equate to the overall superiority of this strategy. Specifically, the deterministic scheme corresponds to the power generation and consumption plans declared by the microgrid to the day-ahead market; however, prediction deviations in actual operation will cause a discrepancy between the actual power generation/consumption and the planned values on the next day, and such deviations must be balanced by electricity in the real-time market [22]. Since real-time transaction electricity prices usually have a premium margin—i.e., the electricity purchase price is higher than that in the day-ahead market, while the electricity sales price is relatively lower [23]—such deviation adjustment will increase the system’s comprehensive transaction costs.

Assuming a 10% prediction error for PV output and EV load, during the real-time dispatch stage, it is necessary to compensate for this part of the power difference through electricity purchase and sale with the grid. The real-time electricity purchase and sales prices are set to 1.5 times and 0.5 times the day-ahead electricity price [15], respectively, and real-time compensation cost calculations are performed for the different optimization models in

Table 7. Comprehensive Operation Costs of the PV-ESS-Charging System Under Different Uncertainty Parameters.

Optimization Model	System Day-Ahead Operation Cost (CNY)	Intra-Day Compensation Cost (CNY)	Comprehensive Cost (CNY)
Deterministic Optimization Model	53,886	28,637	82,523
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=0,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=0$	53,886	28,637	82,523
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=6,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=3$	57,666	22,444	80,110
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=6,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=6$	59,722	19,933	79,655
${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=12,{\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=6$	60,185	18,856	79,041

.

Through comparative analysis, it can be seen that although the day-ahead economic indicators of the robust optimization model are relatively high, when prediction deviations occur on both the power source and load sides, the total expenditure required for intra-day electricity compensation is significantly reduced, resulting in a lower full-cycle operating cost. This indicates that when formulating dispatch plans, robust optimization can effectively cope with electricity price fluctuations in the real-time market by constructing a more abundant safety adjustment space (considering multiple uncertainty scenarios).

The above involves calculating a 10% prediction error for the predicted values of all time periods. To further analyze the impact of prediction errors on the comprehensive cost, the Monte Carlo method is used to randomly calculate the prediction errors for different time periods under the same certainty parameters.

Taking the optimization model with ${\mathsf{\Gamma }}_{\mathrm{P}\mathrm{V}}=6$ and ${\mathsf{\Gamma }}_{\mathrm{E}\mathrm{V}}=6$ as an example, the prediction error time periods are set as $\tau$. If ${\tau }_{\mathrm{PV}}$ = 6, ${\tau }_{\mathrm{EV}}$ = 3, it means that the predicted values of PV in each service area have a 10% prediction error in 6 time periods, and the predicted values of EV loads have a 10% prediction error in 3 time periods. Through multiple (100 times) simulations using the Monte Carlo method, the average intra-day compensation under different prediction error time periods is calculated. The variation in intra-day compensation costs with the number of prediction error time periods $\tau$ under the two optimization methods is shown in Figure 11, and the comprehensive costs are shown in Figure 12. Several typical data points are selected and shown in

Table 8. Cost Calculations of Robust Optimization and Deterministic Optimization Under Different Prediction Error Time Periods.

$\mathbf{Prediction} \mathbf{Error} \mathbf{Time} \mathbf{Period} \mathit{\tau }$	$\mathbf{Robust} \mathbf{Optimization} \mathbf{with} {\mathsf{\Gamma }}_{\mathbf{P}\mathbf{V}}=6 \mathbf{a}\mathbf{n}\mathbf{d} {\mathsf{\Gamma }}_{\mathbf{E}\mathbf{V}}=6$			Deterministic Optimization
	System Day-Ahead Operation Cost (CNY)	Intra-Day Compensation Cost (CNY)	Comprehensive Cost (CNY)	System Day-Ahead Operation Cost (CNY)	Intra-Day Compensation Cost (CNY)	Comprehensive Cost (CNY)
${\tau }_{\mathrm{PV}}$ = 0, ${\tau }_{\mathrm{EV}}$ = 0	59,722	0	59,722	53,886	0	53,886
${\tau }_{\mathrm{PV}}$ = 6, ${\tau }_{\mathrm{EV}}$ = 6	59,722	2944	62,666	53,886	7121	61,007
${\tau }_{\mathrm{PV}}$ = 12, ${\tau }_{\mathrm{EV}}$ = 12	59,722	8282	68,004	53,886	14,332	68,218
${\tau }_{\mathrm{PV}}$ = 24, ${\tau }_{\mathrm{EV}}$ = 24	59,722	19,933	79,655	53,886	28,637	82,523

.

It can be seen that as the number of time periods with prediction errors increases, the intra-day compensation costs under both robust optimization and deterministic optimization increase. However, due to insufficient risk resistance capability, as the number of time periods with actual prediction errors increases, the deterministic optimization method requires more intra-day compensation, and its cost grows faster than that of the robust optimization method. In Figure 11, the intersection line of the two surfaces is shown as the green curve. When the prediction error $\tau$ is smaller than this curve, the intra-day compensation cost of deterministic optimization is lower. When the prediction error $\tau$ is larger than this curve, the intra-day compensation cost of robust optimization is lower. The same applies to the comprehensive costs in Figure 12. It can be seen that when there is significant uncertainty in the system (i.e., a large number of time periods with prediction errors), the comprehensive cost advantage of the robust optimization method becomes more prominent, demonstrating its effectiveness in dealing with the uncertainties of photovoltaic power generation and EV loads.

6. Discussion

This study addresses the operational optimization of the corridor-wide, multi–service-area photovoltaic -storage -charging (PESC) integrated system. A robust optimization framework is developed to account for both the corridor-wide service-area layout and the uncertainty of adjustable EV loads, and the problem is solved efficiently using the column-and-constraint generation (C&CG) algorithm.

First, unlike many existing studies that focus on a single site or an isolated microgrid for PV-storage-charging-station scheduling, this paper considers the coordinated operation problem of a highway corridor with multiple service areas and provides a unified modeling framework.

Second, a box uncertainty set is adopted to represent the uncertainties of PV generation and EV charging demand in a unified manner. An adjustable uncertainty parameter is further introduced to limit the frequency with which uncertain variables can reach their extreme values. In this way, the risk level can be effectively tuned, enabling a flexible trade-off between economic performance and robustness of the operating scheme.

Finally, the simulation results indicate that when the system faces larger uncertainty (i.e., more periods with forecast errors), the robust optimization approach shows a more pronounced advantage in terms of total operating cost. This demonstrates the effectiveness of the proposed framework in handling uncertainties in PV generation and EV charging loads.

This study still has several limitations. (i) Future work should incorporate periodic characteristics of EV demand (e.g., holidays and seasonal variations) as well as PV fluctuation patterns. (ii) The current work focuses on short-term optimization; future research will further investigate coordinated optimization across multiple time scales, such as yearly, monthly, day-ahead, intraday, and real-time operation.

7. Conclusions

This study addresses the operation optimization problem of the PV-energy storage-charging integrated system in highway service areas, constructs a robust optimization framework that considers both the layout of all service areas along the Highway and the uncertainties of adjustable EV loads, and solves it efficiently using the C&CG algorithm.

During model construction, considering the uncertainties of PV power generation and EV charging demands, a box uncertainty set is adopted to model the fluctuations of PV output and EV loads. The randomness of each service area is quantitatively characterized and flexibly controlled through uncertainty adjustment coefficients. A min-max-min robust optimization problem is established. This model comprehensively considers grid interaction costs, ESS degradation costs, and EV dispatch subsidy costs, enabling optimal economic performance of the system.

To solve this model, the original problem is decomposed into a master problem and a subproblem based on the C&CG algorithm, and solved iteratively until convergence. For the max-min structure of the subproblem, the strong duality theory and the Big-M method are utilized to convert it into a single-level maximization problem, significantly improving the solution efficiency.

Simulation results show that the algorithm can converge quickly and obtain the optimal operation dispatch scheme under the worst-case scenario. Moreover, by adjusting the uncertainty parameters, the conservativeness of the operation scheme can be flexibly adjusted. As the uncertainty parameters increase, the system’s robustness is enhanced, making it more capable of withstanding high compensation costs caused by real-time electricity price fluctuations. The system operation under robust optimization can achieve a good balance between economic efficiency and reliability.