Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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Two-Stage Robust Optimization for Multiple Microgrids with Varying 

Loads under Uncertainty Conditions 

 

Dr. Vijayalaxmi biradar1, Ikhar Avinash Khemraj2 

 1,2Department Of Electrical And Electronics Engineering, Kalinga University, Raipur, India 

 

Article History: 

Received: 28-08-2024 

Revised: 10-10-2024 

Accepted: 27-10-2024 

Abstract:  

The volatility in electricity consumption caused by the growing integration of Electric 

Vehicles (EV) might significantly impact the reliability of microgrids. Hence, this study 

aims to decrease the operational expenses of multi-microgrids and enhance the efficiency of 

solution algorithms for their enhanced electric power distribution. This is achieved by 

introducing an innovative two-stage robust optimization dispatch system that considers the 

uncertainties related to loads, renewable energy sources, and electric vehicle consumption.  

The multi-microgrid layer adjusts the limitations based on the number of EVsadded in real-

time and manages various energy units to achieve the minimum operational expenses in the 

most unfavorable situations. The EVAggregator (EVA) layer ensures minimal power 

outages and optimal charging operations by regulating charging power while preventing 

safety breaches. The enhanced uncertainty set is derived from a neural network that 

undergoes training using a substantial amount of past information, thereby eliminating 

unrealistic worst-case situations. The suggested method effectively captures the system's 

features, allowing for the substitution of a considerable amount of past information with 

system attributeswhen creating the uncertainty set. This approach ensures both high 

dependability and a substantial decrease in convergence time. 

Keywords: Robust Optimization, Microgrids, Uncertainty, Electrical Vehicles. 

 

1. Introduction toMicrogrids and Optimization 

Real-world decision-making issues often involve uncertainties and distinctive mathematical 

programming methods have been established to address these uncertainties. These methods include 

scenario-based or chance-constrained stochastic initiatives, Robust Optimization (RO), and 

distributionally robust optimization, designed to accommodate the particular features of different 

usages [1].  RO aims to find a solution that minimizes risk by carefully analyzing the potential 

negative impact of all possible outcomes of the unknown variable within an established range of 

uncertainty. It is particularly appealing when the decision-making needs more information on the 

probability densities of the unknown variables or when prioritizing the framework's viability over the 

whole uncertainty set. RO has become increasingly popular in recent centuries due to its significant 

benefits in modeling capacity, practicality, and mathematical tractability. It has been applied in 

various fields, such as process organizing, energy system planning and time management, and 

optimizing networks.The Enhanced Particle Swarm Optimization-driven Intrusion Detection and 

Secure Routing Algorithm (EPSO-IDSRA) is utilized in this study [2]. In network load , the 

Enhanced Particle Swarm Optimization (EPSO) method has enabled energy-efficient, secure, and 

confidence-based routing.  



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The adoption of microgrids has been growing due to their capacity to incorporate diverse distributed 

autonomous sources of electricity, such as Wind Turbines (WT), Fuel Cells (FCs), and Energy 

Sources (ESs), in addition to Renewable Energy Resources (RES) and photovoltaic (PV) energy 

resources, within a specific area [18]. The growing use of Electric Vehicles (EVs) in recent years has 

been a significant factor in this pattern, primarily because of the smart and easy power distribution 

facilitated by the Vehicle-To-Grid(V2G) method [3]. This mode allows for a two-way flow of 

electricity between EVs and microgrids. Nevertheless, the relationship between EVs and microgrids 

differs from traditional sent energy sources or electric loading. Many individual EVs in a particular 

region are connected to a unified Electrical Vehicle Aggregation (EVA) [4]. The EVA's primary goal 

is to maximize the V2G procedure by controlling EVs' charging and discharging activities. This 

optimization aims to provide economic benefits to EV users. Hence, the microgrid connectivity 

attributes of these EVs will be influenced by the unpredictability linked to the actions of individual 

EV consumers and the activities of intermediate EVAs. The rising number of multi-microgrid 

networks, especially those with complicated layouts, exacerbates the issue. These systems already 

need more certainty in demand and the addition of RES. However, the unpredictability of electric 

power requests caused by the growing use of EVs can significantly impact these multi-microgrid 

structures' financial viability and stability. 

An innovative two-stage robust optimization dispatching system uses the suggested uncertainty set.  

The robust dispatching model adjusts the limits based on the amount of real-time EV updates to 

achieve the minimum operational cost of multiple microgrids in the most unfavorable situations. The 

affordable EVA model has minimal power outages and optimal EV charging operations. 

The following sections are organized in the given pattern: section 2 discusses the literature survey 

about optimization in the EV domain. Section 3 proposes the robust energy optimization of EVs and 

RES. Section 4 discusses the simulation results of the proposed method in terms of power generation, 

load, etc. Section 5 concludes the research with the conclusion and findings of the study.  

2. Literature Survey and Analysis  

Prior research has used several techniques to gather historical EV load information, including 

mathematical equations, deep learning, and artificial neural networks.  Marzbnaiet al. used the 

Autoregressive Integrating Motion Averaging (ARIMA) approach and the Pattern Sequence-based 

Foreseeing (PSF) method to predict EV electrical consumption [5].  Dinget al. used two dynamic 

models to simultaneously predict the sales of plug-in electric vehicles and the daily pattern of 

recharging load in the United States from 2012 to 2020 [6].  

Yumikiet al. devised an innovative and optimal method for regulating the frequency and voltage of 

V2G operations [7]. This method relies on a group of EVs linked to a dispersed power system via a 

charging point network.  Multiple investigators have determined the probability density function of 

V2G technology to accurately predict the electricity demand from electric vehicles linked to the 

power grid. The functionalities of optimization, obfuscationutilities on the Android platform are 

compared and analysed in this paper [8]. 

 



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Adaikkappanet al. conducted a study on the multi-uncertainty collecting related to aggregations of 

EVs, precisely the uncertainty in the collecting findings regarding the State of Charge (SOC) for 

every EV at the moment of initial relationship [20].  They developed a two-layer Model Predicted 

Control (MPC) strategy for accurately forecasting the EV-imposing demand of a microgrid. 

González-González et al. suggested a universal time-varying storage model used in various markets 

and adaptable to diverse, flexible loads [9]. 

Model variables that reflect many EVs are easily combined by summing, and load prediction can be 

performed using autoregressive methods.  Haramet al. created an environmental management 

structure with two main parts [10]. The first part was an ARIMA system to forecast PV generation. 

The second part was a mixed-integer linear programming structure that efficiently distributed 

electricity to EVs to minimize charging expenses. 

Chaiet al. developed an effective two-stage unpredictable model to address the operational 

challenges of a microgrid with EVs [11]. The model takes into account the unpredictability 

associated with EVs. The first stage involves power dealing with the primary grid, while the 

subsequent phase focuses on optimizing the allocation of microgrid assets. This model is used to 

inform the decision-making procedure.   

Akhgarzarandyet al. aimed to solve the issue of over-conservatism in the RO results achieved when 

integrating the uncertainties related to EV demands by adding a dispatch spacing factor to control the 

conservative degree [12].  Zhanget al. defined RO as the problem of finding the shortest route in a 

probabilistic environment [13]. The goal function of this issue was a combination of the weighted 

usage of WT and the overall cost of EV recharging.   

Wuet al. developed a financially successful microgrid that facilitates trading energy and reserves via 

energy and resource markets [14]. This grid ensures an appropriate balance between the expenses 

and advantages of trading and buying energy characteristics.  The innovative periodic box ambiguity 

set restriction was used to include the uncertainties. Kumaret al. introduced a resilient model for 

enhancing the efficiency of V2G charging and discharging planning [15].  The dispersed RO 

methodology allowed V2G aggregation to effectively handle operational risks, such as customers' 

travel requests, even without having complete knowledge about the distribution information.The 

image classification models developed in this research were compared across five distinct datasets 

comprising different quantities of data [17]. In achieving this objective, four discrete pre-trained 

models—VGG16, InceptionV3, MobileNet, and DenseNet—are utilized. In addition, a novel model 

was proposed, which was trained using the provided datasets.A novel hierarchical non-linear 

machine learning technique is presented in this article [19]. It is proposed to utilize multivariate 

adaptive regression spline with genetic algorithm in order to manage the energy demand of smart 

grids.His research examines the feasibility of identifying a candidate for a heavy load solver through 

a heuristic approach involving the integration of multiple crossover methodologies into an artificial 

bee colony (ABC) [21].  

3. Proposed Robust Optimisation Algorithm for Microgrids 

The focus of this work is on grid-connected microgrids. The residual capacity of the microgrid is 

exchanged with the primary grid, and any excess power from the microgrid can also be bought from 



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the primary grid. The microgrid examined in this research encompasses conventional dispersed 

energy resources, sustainable dispersed power sources, batteries, and loads. Traditionalscattered 

power supplies include miniature gas turbines, whereas renewable distributed energy sources include 

wind turbines and photovoltaics. 

 

Figure 1. Grid-connected microgrid structure model 

Figure 1 depicts a schematic representation of the microgrid configuration. The energy monitoring 

center and the microgrid alone engage in data exchange.  The energy monitoring center uses previous 

information on power production from dispersed RES and historical use data of the demand to 

anticipate the future generation from the RES and the need for electricity during the following 

duration.  The energy managing center does RO computations using predictions to determine the 

ideal dispatching strategy for the microgrid. This scheme is designed to minimize the operating 

expenses of the microgrid. According to this system, the power managing center then carries out the 

optimum forecasting of the microgrid. 

3.1 Robust characterization of uncertain variables 

The uncertainty variables include the production of RES and the load demand. The research proposes 

a strong alternative description for renewable distributed power, which involves specifying 

photovoltaic energy output and wind power generating results. 

3.1.1 Robust representation of PV output 

The solar irradiance for a PV unit at point p and time duration t is modeled using a Beta dispersion 

function, characterized by its form and rate characteristics as defined in Equation (1). 

𝑃𝑥.𝑡
𝑃𝑉 = 𝛽(𝛼𝑥,𝑡, 𝛽𝑥,𝑡)         (1) 

To rectify the inaccuracy in the probability approach, it is necessary to include robust adjustment 

variables (𝛼𝑥,𝑡, 𝛽𝑥,𝑡) to change the system.  The built photovoltaic power supply is represented 

robustly in Equation (2), and the predicted PV power is expressed in Equation (3).  

𝑃𝑥.𝑡
𝑃𝑉 = 𝐶𝐷𝐹𝑥.𝑡

−1(�̂�𝑥.𝑡
𝑃𝑉 − 𝜌)        (2) 

�̂�𝑥.𝑡
𝑃𝑉 = ∫ 𝑃𝑥.𝑡

𝑃𝑉𝑑𝑡
𝐸𝑃𝑉,𝑡
−∞

         (3) 



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The predicted power is �̂�𝑥.𝑡
𝑃𝑉, and the percentage of 𝐸𝑃𝑉,𝑡is determined based on the forecasted 

performance of photovoltaic systems. 

3.1.2 Robust representation of WT output 

The wind speeds for a wind turbine unit at location u and time duration t are supposed to conform to 

a Weibull dispersion function, characterized by its form and scale variables as defined in Equation 

(4). 

𝑃𝑤,𝑡
𝑊𝑇 = 𝑊𝑒𝑖𝑏𝑢𝑙(𝐶𝑤,𝑡, 𝛿𝑤,𝑡)        (4) 

The robust depiction of WT (𝛿𝑤,𝑡), comparable to the robust corresponding depiction (𝐶𝑤,𝑡) of solar 

power production. The power of WT is expressed in Equation (5) and the predicted power is WT is 

expressed in Equation (6). 

𝑃𝑥.𝑡
𝑊𝑇 = 𝐶𝐷𝐹𝑥.𝑡

−1(�̂�𝑥.𝑡
𝑊𝑇 − 𝜌)        (5) 

�̂�𝑥.𝑡
𝑊𝑇 = ∫ 𝑃𝑥.𝑡

𝑊𝑇𝑑𝑡
𝐸𝑊𝑇,𝑡

−∞
         (6) 

The prediction variable is�̂�𝑥.𝑡
𝑊𝑇 for WT. The calculation of 𝐸𝑊𝑇,𝑡is based on the expected value of 

wind turbine production. 

3.1.3 Robust representation of load  

The active energy and reactive energy of the load request at bus x and interval t are taken to conform 

to a bivariate regular distributed function, as described in Equations (7) and (8). 

𝑃𝑥,𝑡
𝐷 , 𝑄𝑥,𝑡

𝐷 = 𝑁𝑤(𝛼, 𝑝𝑥,𝑡)        (7) 

𝛼 = [𝜇𝑥,𝑡
𝑃 , 𝜎𝑥,𝑡

𝑃 , 𝜇𝑥,𝑡
𝑄 , 𝜎𝑥,𝑡

𝑄 ]        (8) 

The load demand (𝑃𝑥,𝑡
𝐷 , 𝑄𝑥,𝑡

𝐷 ) robust comparable depiction (𝜇𝑥,𝑡
𝑃 , 𝜇𝑥,𝑡

𝑄
) must be modified using robust 

adjusting variables (𝜎𝑥,𝑡
𝑃 , 𝜎𝑥,𝑡

𝑄
).  The load requirement is represented robustly in Equation (9). 

𝑃𝑥,𝑡
𝐷 , 𝑄𝑥,𝑡

𝐷 = 𝐶𝐷𝐹𝑥.𝑡
−1(�̂�𝑥.𝑡

𝐷 + 𝜌, �̂�𝑥.𝑡
𝐷 + 𝜌)      (9) 

3.1.4 Robust adjustment variables 

The calculation method for the resilient adjusting variables discussed above appears in Equation 

(10): 

𝛿 = 𝐴 × 𝛽          (10) 

The parameter β has a value ranging between 0 and 1, exclusive. The computation method for 

variable A is presented in Equations (11) and (12): 

𝐴 = min{𝑃, 1 − 𝑃}         (11) 

𝑃 = max{𝑃𝑥,𝑡
𝐷 , 𝑄𝑥,𝑡

𝐷 , 𝑃𝑥,𝑡
𝑃𝑉, 𝑃𝑥,𝑡

𝑊𝑇}       (12) 

The power and quality demand are 𝑃𝑥,𝑡
𝐷 , 𝑄𝑥,𝑡

𝐷 , the PV and WT generated power are denoted 𝑃𝑥,𝑡
𝑃𝑉 , 𝑃𝑥,𝑡

𝑊𝑇.  

 



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3.2 Optimal scheduling  

The technique addresses the resilient dispatch issue by iteratively solving two Major Problems (MPs) 

and Sub-Problems (SPs). The MP is utilized to acquire a precise uncertainty situation, while the 

dualized SP is employed to ascertain the presence of more unfavorable situations inside the 

collection of uncertain situations. 

3.2.1 Cutting plane robust dispatchingmodel 

The technique applies the MP to resolve the robust dispatch system under perfect conditions without 

any ambiguity. The dualized SP is employed to verify when the solution achieved is resilient in the 

worst-case situation. McCormick masks are often utilized to address the issue of linking uncertainty 

parameters in the dual transformation procedure of the dualized SP. The answer obtained from the 

SP is submitted as input to the MP, and this iteration process is continued.  

The dispatch method relying on the usual tangential plane technique is defined as a two-stage RO 

approach using Equations (13) to (15). 

𝑀𝑃:min{𝐴𝑖} + 𝜑         (13) 

𝛿 = {
𝜑 > 𝐵𝑗, 𝐹𝑖 > 𝑓, 𝐺𝑗 > 𝑔

𝑈𝑗 + 𝑉𝑣 = 𝑘, 𝐶𝑖 + 𝑑𝑗 > 𝑑
       (14) 

𝜑 = {𝑣|𝑣𝑥 − 𝛼𝑥𝛽𝑥 < 𝑣𝑥 + 𝛼𝑥𝛽𝑥, ∑ 𝛼𝑥 < 𝜏𝑁
𝑥=0 }     (15) 

𝛼𝑥denotes the ratio of the most significant offset 𝛽𝑥, with 𝛼𝑥 being a number between 0 and 1. 

𝑣𝑥indicates the anticipated value of the x-th uncertainty parameter.  Likewise, the SP is defined in the 

same manner using Equations (16) and (17). 

𝑆𝑃:max{𝑔𝑇𝑐} − (𝐶𝑖 − 𝑙)𝑇𝛼 − (𝑉𝑣 − 𝑘)𝑇𝛾      (16) 

𝐵𝑇 − {𝐺𝑇𝑐 + 𝐷𝑇𝛼 + 𝑈𝑇𝛾} = 0       (17) 

Utilizing McCormick masks transforms the SP into a conventional linear issue. 

3.2.2 Enhanced robust dispatchingmodel 

The conventional model relies on a predetermined basic geometric structure. This structure considers 

all uncertainties in the most unfavorable situation, along with a particular structure for creating an 

organizing plan. The standard data-driven modelutilizes all previous data from a specific setting to 

identify the most unfavorable scenarios. Unlike dealing with continuum uncertain characteristics, the 

current study also encompasses binary uncertainties. This binary unpredictability pertains to the 

count of suddenly detached EVs, variables with integer values. These variables could introduce 

interdependence in the dualization procedure of the SPs. The data-driven approach avoids laborious 

processing procedures by utilizing particular situations from the indirect setting set as the initial point 

in the two-layer robust dispatch procedure. Despite a rise in analyzing complexities, this is a 

significant advantage of the proposed data-driven ambiguity set.  

An additional strategy to enhance the computational effectiveness of the two-stageRO dispatching 

technique is using a sub-uncertainty group Λ. In this set, just one situation is used in the initial 



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iteration.  The binary parameter derived from resolving the MP is sent into the SP, providing the 

worst-case possibilities. 

3.2.3 Optimal scheduling of the EVA layer 

To achieve the desired divergence of the uncontrolled parameter from the perfect 𝐺𝑥, it considers the 

charging power and SOC of an individual EV as separate factors. Using these variables, the systems 

create the transmitted energy matrix 𝐾𝑥and SOC matrix 𝐿𝑥.The approach of multi-objective planning 

is used, using deconstruction and weighing techniques. It derives the charge index scalar 𝑃𝑥and SOC 

series 𝑆𝑥of the ideal setting. Equation (18) shows𝐺𝑥. 

𝐺𝑥 = ∑ ∑ ∑
(𝑃𝐸𝑉)

2

2𝑃𝑥
+

(𝑆𝐸𝑉−𝑆𝐸𝑉(𝑥))
2

2𝑆𝑥

𝑁
𝑦=0

𝑁
𝑥=0

𝑇
𝑡=0       (18) 

The combined resolution of linear programming determines the last scheme's reasonableness. 

1. Acquire past information to train the Neural Network and then use it to determine the features of 

situations, identify the connection between various types of uncertainty, and construct the correlation 

architecture.  

2. Utilize the trained Neural Network to forecast likely situations in the following scheduling 

duration, then employ these situations to construct the polyhedral uncertainty collection.  

3. Calculate the variable 𝜏𝑥 based on the current total SOC and the number of EVs linked to the 

microgrids while considering the limitations. 

4. Begin the framework by assigning the higher limits 𝛹∗ as positive infinity, the smaller limits �̂�as 

negative infinity, and the iterative index x as 0. 

5. Use the MP technique to address the minimum cost in the most unfavorable situation of the sub-

uncertainty group Λ, determining𝑀𝑥
∗ and 𝐼𝑥

∗. 

6. Transfer the value of 𝐼𝑥
∗to the SP. 

7. The SP algorithm seeks to identify a collection of uncertainty possibilities, denoted as Ψ𝑥
∗ inside 

the enhanced uncertainty set. If the absolute difference between Ψ𝑥
∗ and M𝑥

∗ is less than or equal to ε, 

go to step (8). Alternatively, include the scene in Λ, increment the value of x by 1, and go to step (5). 

8. Utilize the MP algorithm to derive the ideal programming plan Ψ𝑥+1
∗  in the most dire case of Λ. If 

the absolute difference between the complex conjugates of Ψ𝑥+1
∗  and Ψ𝑥

∗  is less than or equal to ε, 

proceed to step (9); alternatively, go back to step (6)  

9. Determine the total power consumption (𝑃𝐸𝑉) and total energy consumption (𝑆𝐸𝑉) by considering 

the charging schedule of each distinct EV at the lowest EVA level. If it is not possible to satisfy the 

safety restrictions, then adjust 𝜏𝑖
𝑥+1 = 𝜏𝑖

𝑥 exp(1− 𝐺𝑥), and go back to step (4). 

 

4. Simulation Analysis and Outcomes 

The multi-microgrid evaluation system comprises four microgrids and their corresponding line 

connections. During the models, itconsiders a maximum variation in RES production of ±25% from 



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its usual value. The most significant variations in energy demands are ±10% from their average 

values. Microgrid (MG1) is characterized as a business region with high activity levels. The longest 

time for EVs to connect in EVA1 is 5 hours in MG2.  

MG3 is designated as a residential neighborhood, and the longest time for EVs to be connected in 

EVA2 is 7 hours in MG4. Each of the four microgrids implements a three-stage power pricing 

strategy. This policy applies low electricity rates during late night and dawn, ordinary electricity 

rates throughout the morning and this afternoon, and high energy prices at midday and evening.The 

numerical calculations were performed on a computer with an8-core 2.4 GHz CPU and 8 GB of 

RAM running the Windows 8 operating system. The YALMIP toolkit in MATLAB was used to 

create multilevel, two-stageRO dispatching systems [16].  The alternative models were used in four 

distinct cases, and their outcomes were contrasted to determine the specific advantages of the 

suggested strategy. 

 

Figure 2. Load analysis of different MGs 

Figure 2 depicts the varying load profiles of Microgrids (MG1, MG2, MG3, MG4) across a 24-hour 

duration. The average findings reveal diverse load patterns across the microgrids, with MG2 

regularly demonstrating larger loads. The fluctuation in load highlights the need for customized 

approaches in Microgrid functioning, which significantly affects the general efficiency and results of 

the Microgrid system. 



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Figure 3. Power generation analysis of different WTs 

Figure 3 illustrates the power production patterns of Wind Turbines (WT1, WT2, WT3, WT4) across 

24 hours. WT4 consistently exhibits superior power production, while WT2 has somewhat worse 

performance. The significance of these variances highlights the need for efficient management 

techniques, which influence the overall result and effectiveness of the Wind Turbine structure in the 

power production. 

 

Figure 4. Power stage analysis of different ESs 

Figure 4 depicts the fluctuations in power storage for Energy Storage systems (ES1, ES2, ES3, ES4) 

across 24 hours. ES3 regularly has superior power storage capabilities compared to other models, 

whereas ES2 shows comparatively poorer performance. The significance of these differences 

underscores the pivotal importance of efficient energy storage management, which directly affects 

the energy storage system's outcome and dependability within the power distribution and storage 

framework. 



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Figure 5. Supply and demand deviation analysis  

Figure 5 displays the percentage differences between the supply and demand of Microgrids (MG1, 

MG2, MG3, MG4) throughout 24 hours. MG2 often has minor variance, but MG1 and MG3 have 

balanced variability. The significance of these variances highlights the need for accurate control and 

management measures to guarantee effective alignment of supply and demand, hence affecting the 

overall dependability and consistency of the Microgrid system. 

The optimization findings indicate different load patterns in Microgrids (MG1-MG4) with variable 

average deviations. MG2 has the lowest deviation at 0.05%, while MG1 and MG3 have balanced 

variations about 0.42%. Among the Wind Turbines (WT1-WT4), WT4 constantly surpasses the 

others by generating an average of 69.71 kW. This highlights the effectiveness of its energy-

producing tactics. The dynamics of Energy Storage (ES1-ES4) show that ES3 consistently maintains 

the most significant average storage capacity at 21.84 kW. This highlights the critical function of 

ES3 in guaranteeing dependable power distribution and reducing variations.  

5. Conclusion and Findings 

This study aimed to enhance the cost-effectiveness and safety of multi-microgrid structures and 

increase the efficiency of algorithmic approaches in dealing with uncertainties related to loads, 

energy from renewable sources, and electric vehicle use. The achievement was made by introducing 

an innovative two-stage RO dispatching approach, including an upper-level RO dispatching 

approach for the multi-microgrid network and a lower-level EV aggregation forecasting system.  The 

operational limitations of the higher level are adjusted based on the real-time number of additions 

and the current charge level of every EVto achieve the lowest running cost in the most unfavorable 

situations. The system uses neural networks to analyze the pre-selected past information of the 

closely connected and unpredictable group to implement the effective planning method in the higher 

layer. The resulting many scenarios with various periods provide an enhanced collection of uncertain 

characteristics.  The bottom level of the system is built upon the innovative grid-connected design for 

electrically powered automobiles. To ensure the safe recharging for every EV, a multi-objective 

optimizing system is used to construct charging strategies.This approach ensures both high 

dependability and a substantial decrease in convergence time. 



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