Multi-Stage Centralized Energy Management for Interconnected Microgrids: Hybrid Forecasting, Climate-Resilient, and Sustainable Optimization

Abbreviations and Symbols

1  Introduction

1.1 Motivation

EMS play a critical role in the efficient operation of microgrids, which are distributed energy systems capable of operating autonomously or in coordination with the main power grid [1,2]. The primary objectives of an EMS are to optimize energy generation, distribution, and consumption while minimizing operational costs, reducing CO2 emissions, and maintaining system stability. This is achieved through the integration of diverse energy resources, including RES, ESS, and conventional generation units [3,4]. However, as RES penetration increases, EMS must tackle the growing complexity of managing the inherent variability and uncertainty associated with renewable generation [5–7]. Accurate short-term forecasting of renewable generation and load demand is a key enabler for effective EMS operation. Forecasting ensures energy balance, supports optimal dispatch strategies, reduces reliance on conventional energy sources, and maximizes the use of RES. Inaccurate predictions can lead to inefficient energy scheduling, higher operational costs, and reduced system reliability. Therefore, improving forecast accuracy directly improves the stability, efficiency, and sustainability of microgrid operations.

1.2 Literature Review

The rapid increase in carbon dioxide emissions and environmental pollution has made the transition to sustainable energy systems a global imperative [8,9]. Currently, advances in ICT are transforming modern societies, enabling smart applications in healthcare, transportation, finance, data centers, and, most importantly, SGs. Modern energy systems are expected to achieve high efficiency, resilience, and environmental friendliness by integrating advanced control, automation, and optimization technologies [10]. According to the International Energy Agency, intermittent RES is projected to account for 57% of global electricity generation by 2050, and certain regions experience instantaneous levels of penetration of RES of up to 100% during specific periods [11,12]. These trends exacerbate challenges such as poor power quality, complexity of scheduling, and reliability risks [12].

SGs supported by ICT infrastructure are a cornerstone of next-generation energy systems. They integrate communication and control technologies to achieve secure, efficient, and flexible operation of DERs [10]. Within this paradigm, MGs and IMM have emerged as promising decentralized energy systems that offer autonomy, self-containment, and improved reliability [10]. MGs and IMM improve the integration of renewable energy, reduce dependence on fossil fuels, and improve resilience, particularly in regions with limited or no connection to the utility grid [10]. Furthermore, MGs can be isolated (noninterconnected) or interconnected. Non-interconnected microgrids provide greater autonomy, but face several limitations, including energy imbalances when local generation cannot meet demand, higher operational costs due to fossil fuel dependency of backup, reduced reliability during interruptions, and under-utilization of RES due to limited local demand [8,9]. In contrast, IMM offer benefits such as optimized power flows, resource sharing, and increased reliability, but introduces additional challenges, including scalability issues, managing RES uncertainties, ensuring power flow stability, and maintaining fault tolerance to prevent cascading failures [10]. To address these challenges, Zhao et al. developed and discussed in [13] a new methodology for the energy management of IMM based on DRMPC to handle the variability of DERs. The proposed strategy reduced the daily operating cost of each microgrid by 5.3%, 8.9%, and 6.9%, respectively, compared with independent operation. However, the impact of forecast uncertainty was not considered in this work, and although the obtained results are promising, the proposed methodology remains computationally expensive for real-time implementation. In [14], the authors proposed a distributed energy trading framework for microgrids based on a game-theoretic model. Although the proposed approach is efficient, it remains limited in certain aspects, such as ensuring equilibrium among interconnected microgrids to achieve a truly optimal solution. Moreover, the real-time implementation of such computational methodologies remains costly and complex. The authors of [15] introduced a new method based on a simple fuzzy-logic technique. This methodology is efficient and achieves good accuracy; however, several important factors were not considered, including the distance between connected microgrids, power losses in transmission lines, and the forecast stage, which could significantly affect system performance. In [16], Moazzen et al. developed a methodology based on MIQP to provide an optimal solution for interconnected microgrids. The obtained results demonstrate significant improvements in energy management and coordination between interconnected systems. However, although the proposed method is accurate and effective, the forecasting aspect was not analyzed, which may influence the reliability of the results. Furthermore, the use of MIQP remains computationally demanding, especially for a large number of IMM, and issues related to scalability and robustness against disturbances were not considered.

Although these studies with other [15,17–19] have contributed significantly to advancing energy management strategies for IMM, most of them primarily focus on optimization and coordination aspects, often neglecting the critical role of accurate forecasting in overall system performance [20–22]. In practice, the effectiveness of any EMS strongly depends on the quality of short-term forecasts for renewable generation, load demand, and market prices. Without reliable forecasting, even the most advanced optimization algorithms may lead to suboptimal or unstable operation under uncertainty. Consequently, the integration of accurate forecasting models into EMS frameworks has become a crucial research direction for improving the reliability, cost-efficiency, and resilience of IMM operation.

Accurate short-term forecasting is essential for EMS performance. Conventional stochastic approaches (e.g., Monte Carlo simulations, probabilistic models, scenario-based methods) have been widely used to model uncertainty but often struggle to capture nonlinear temporal dependencies [1,6,7]. ML and DL approaches, such as SVR, RF, ANN, and LSTM networks [23–25], have shown significant promise in capturing complex patterns from historical data [26,27].

Hybrid forecasting models, which combine stochastic and ML/DL techniques, leverage the strengths of both approaches and have demonstrated superior performance in highly uncertain RES environments [28,29]. Recent works show that hybrid models improve EMS operation by providing more reliable forecasts for scheduling and optimization [30–33]. On the optimization side, conventional methods (LP, MILP, DP) offer mathematical rigor and computational efficiency, particularly in deterministic settings [6,34]. Heuristic/metaheuristic algorithms (GA, PSO, ACO, IARO, BWO, COATI) handle multi-objective and nonlinear problems but face convergence and scalability issues [6,35–37]. AI-based methods (Fuzzy logic, RL, DQN, ANN, LSTM) are adaptive and effective under dynamic [23–25], uncertain conditions but require large, high-quality datasets and computational resources [26,27,31,33].

1.3 Research Gaps

Despite extensive research on EMS for microgrids, several challenges remain. Real-time implementation is still difficult due to the inherent complexity of both the system and the optimization methods [16–18,45]. Developing a novel framework that ensures optimal solutions in both simulation and real-time execution [15,19,21,22], while reducing computational complexity, is therefore crucial for practical deployment in interconnected microgrids [4,20,46].

Most existing studies rely on deterministic models for power generation and load demand, even though real-time energy systems are subject to significant uncertainties, particularly from PV output, wind variability, and load fluctuations. Accurate short-term forecasting is essential to ensure efficient and reliable energy management under these volatile conditions. Incorporating stochastic elements into the EMS is necessary to better address such uncertainties. In addition, the adoption of advanced hybrid forecasting methods, which combine statistical models with machine learning or deep learning techniques, can significantly improve prediction accuracy. Choosing a robust uncertainty modeling and forecasting approach is therefore vital to optimize resource allocation, improve the balance between energy production and consumption, and enhance system resilience.

Furthermore, a new mathematical formulation of the energy management problem is needed to account for power flows and transmission losses between interconnected microgrids. While several optimization frameworks have been proposed for multi-objective energy scheduling in IMM systems, challenges remain regarding real-time implementation and the trade-off between competing objectives. A straightforward, computationally efficient formulation is needed to minimize operational costs while simultaneously reducing CO2 emissions and ensuring reliable energy management across interconnected microgrids.

1.4 Contributions

Table 1 provides a comparative overview of existing methodologies in the context of energy management for interconnected microgrids. It summarizes key aspects such as the number of interconnected micro-microgrids (Nb of IMM), the forecasting methods employed, the optimization techniques utilized, and whether the approaches target single-objective or multi-objective problems. The analysis highlights that most studies address both single- and multi-objective optimization problems using advanced methodologies that account for multiple constraints, thereby increasing the complexity of the problem. Compared to the existing literature, the proposed methodology introduces a comprehensive three-stage framework (Fig. 1) for EMS in interconnected microgrids. This framework integrates data analysis, forecasting, and optimized decision-making, leveraging the Prophet-BiLSTM hybrid model for load/production prediction and LP for optimization. The proposed approach facilitates efficient power exchange among microgrids while minimizing operational costs and CO2 emissions under uncertainty, demonstrating an improvement over previous methods in terms of both precision and operational efficiency.

•   A hybrid forecasting approach combines the stochastic Prophet model to capture seasonal patterns with a BiLSTM network for residual correction, improving PV generation and demand predictions [47–49].

•   A multi-objective LP-based optimization framework minimizes operational costs and CO2 emissions across IMM systems.

•   Power flow constraints are incorporated to ensure an accurate and balanced energy exchange within the IMM.

In this work, the Prophet model is selected as the baseline instead of more complex architectures such as TCN, Seq2Seq, or Transformer models. This choice is motivated by Prophet’s ability to explicitly decompose time series into trend and seasonality components, providing high interpretability and robustness for energy forecasting tasks dominated by such patterns. Unlike deep learning methods that require large datasets and substantial computational resources, Prophet performs reliably on smaller datasets while maintaining transparency. By combining Prophet with BiLSTM, we take advantage of Prophet’s strength in modeling deterministic components and BiLSTM’s ability to learn non-linear residuals, achieving both interpretability and precision in the resulting forecasts.

1.5 Paper Organization

The rest of this paper is structured as follows. Section 2 defines the formulation of the problem. Section 3 outlines the key tools and techniques used in this study. Section 4 describes the proposed methodology and offers an in-depth discussion. Section 5 presents the simulation results. Section 6 provides a dedicated discussion. Section 7 concludes the paper and outlines potential directions for future research.

2  Problem Formulation

This paper presents a novel energy management framework based on a LP approach for the day-to-day planning of IMM. In the IMM structure, the microgrids are connected to the main grid and are distributed in nearby and distant geographical areas, as illustrated in Fig. 2. These microgrids collaborate by sharing local generation and storage resources to reduce reliance on the main grid, minimize CO2 emissions, and lower overall operating costs. Each microgrid integrates renewable energy sources, primarily PV systems, with varying production levels, along with BESS of different capacities. These storage systems help mitigate uncertainties in both renewable generation and demand. This paper proposes a multi-objective optimization framework that simultaneously minimizes operating costs and CO2 emissions.

Figure 2: Structure of centralized EMS of interconnected microgrids

In the following, the architecture of the IMM, as illustrated in Fig. 2, will be used for the analysis and validation of the proposed methodology. The initial data used for the IMM are detailed in Table 2.

3  Mathematical Formulation of the Proposed Model

3.1 PV Model

3.1.1 Deterministic Model

The output active power of a PV system depends primarily on the size of the PV farm, the available solar irradiation, the system’s efficiency, and the ambient temperature. According to [50], the PV power generation PPV can be defined by Eq. (1).

PPV(t)=Si(t)∗Apv∗ηPV(1)

3.1.2 Uncertainty Model

To incorporate stochastic uncertainty-which represents inherent variability in the system or environment and cannot be reduced through data augmentation -randomness is introduced. This uncertainty stems from stochastic processes such as fluctuations in weather conditions (e.g., cloud cover, dust, atmospheric changes, and sudden irradiance variations), as well as sensor noise. As a result, random fluctuations are applied to obtain modified irradiance St′ and PV power (PPV′) values, as defined in Eqs. (2) and (4), respectively.

St′=max(0,St⋅ft)(2)

Here, ft is a fluctuation factor at time t, defined as a random multiplier in the range [0.7,1.3]. Its mathematical formulation is given by:

ft=1+signt⋅pt(3)

where signt∈{−1,1} is a random sign indicating the direction of fluctuation, and pt∼𝒰(0.1,0.3) is a random percentage drawn from a uniform distribution.

By applying this fluctuation to the irradiance dataset used for the validation of our methodology, we simulate real-time weather variability. The resulting fluctuated irradiance profile is illustrated in Fig. 3.

Figure 3: Direct irradiation on 21 April 2025—Original vs. Fluctuated (Cairo Area)

Consequently, the uncertain PV power model is updated as given in Eq. (4).

PPV′=St′⋅Apv⋅ηPV(4)

3.2 Battery Energy Storage System

The BESS model accounts for the charging/discharging power, the battery’s nominal capacity, and the charge/discharge efficiency. It can be expressed in both continuous and discrete forms as described in Eqs. (5) and (6).

•   Continuous form

SoC(t)=SoC(0)+1Enom∫0tηch/dis⋅Pbat(t)dt(5)

where, SoC(0) is the initial state of charge.

•   Discrete form

SoC(k)=SoC(k−1)+Pbat(k)⋅ηch/disEnom⋅Δt(6)

where, SoC(k) is the state of charge at time step k, Δt is the time step interval (in hours). Other variables are as defined above.

3.3 Transmission Line Model

For interconnected microgrids, the transmission line is modeled using the π-model, which includes series impedance and shunt admittances. This model provides a good balance between accuracy and simplicity for steady-state and quasi-dynamic studies.

Zline=R+jX,Ysh=jB(7)

The line current and voltage drop between nodes can be expressed using standard π-model relations:

Iline=Vs−VrZline+Vs2YshVr=Vs−Iline⋅Zline(8)

where, Vs and Vr are the sending and receiving end voltages, Zline is the series impedance, and Ysh is the shunt admittance.

3.4 Load Model

3.4.1 Deterministic Load Model

In the deterministic case, the load is assumed to follow a known and fixed demand profile over time:

Pload(t)=Pdet(t)(9)

This model neglects random fluctuations or forecasting errors.

3.4.2 Uncertainty-Based Load Model

To reflect the variability and unpredictability of real-time load profiles, we introduce a stochastic component to the load:

Pload(t)=Pdet(t)⋅(1+δt)(10)

where, δt∼𝒰(−α,α) is a random variable drawn from a uniform distribution (e.g., α=0.1 to reflect ±10% fluctuation), and Pdet(t) is the baseline deterministic load.

This uncertainty model is used to test the robustness of energy management algorithms under variable demand conditions.

3.5 Objective Functions

This study addresses the optimization problem as a multi-objective optimization task, aiming to minimize several conflicting objectives simultaneously. Since LP is traditionally applied to single-objective problems, the weighted sum method is adopted to transform the problem into a form compatible with LP. The total objective function, denoted as ftot, is defined as a weighted sum of two components: one representing the economic cost and the other representing the environmental impact (emissions). The combined objective function is given by:

ftot=w1f1+w2f2+w3f3(11)

where, f1, represents the total cost vector as:

f1=Cgridk⋅Egridkf2=λgrid(12)

Egridk represents the emission vector based on the grid’s CO2 emissions per kWh, and f3=λgrid⋅Egridk acts as a penalty function introduced to enhance battery participation. The weights assigned to each objective are w1=0.5 for cost, w2=0.25 for emissions, and w3=0.25 for battery participation, with the condition w1+w2+w3=1 ensuring normalization of the objective weights.

Notice that the weights in the multi-objective formulation were determined through a trial-and-error approach. Initially, equal weights were assigned to all objectives, and the optimization results were visualized to assess their influence. Subsequently, each weight was slightly adjusted to analyze its effect on the convergence of the solution. In some cases, these variations caused divergence, indicating that the selection of weights significantly impacts the convergence toward the optimal solution. Therefore, efforts were made to identify a suitable set of weights that yield stable and satisfactory results. Although the current approach relies on manual tuning, further improvements could be achieved through adaptive tuning using an optimization algorithm. Nevertheless, the results obtained are considered good and acceptable for the proposed framework.

4  Proposed Energy Management Framework

As illustrated in Fig. 2, the proposed methodology is structured into three main stages. The first stage involves data preprocessing. In our case, the raw data was extracted from the website [51] with an original sampling rate of 1 h. To enhance realism, the data were resampled at a frequency of 1 min, and random variations were introduced to account for various types of fluctuation, including weather-related and technical uncertainties, as detailed in section 3.1. Consequently, this section focuses solely on the prediction and optimization stages.

4.1 Power Prediction

With the increasing integration of RES, the rapid growth in power consumption, and the pressing impacts of climate change, accurately forecasting PV power production has become a critical yet complex task. This complexity arises from the stochastic nature of solar irradiance, weather variability, and other environmental factors influencing PV output. To address this challenge, this paper proposes a hybrid forecasting approach that integrates the Prophet model, well-suited for capturing trend and seasonality components-with the BiLSTM neural network, which excels at modeling nonlinear dependencies and temporal patterns in time series data. By leveraging both models’ strengths, the proposed method improves forecasting accuracy and robustness compared to individual models as given in Fig. 4. The framework uses key independent variables such as historical PV output, temperature, solar irradiance, and cloud cover to enhance prediction performance.

Figure 4: Solar power forecast provided by the proposed methodology and comparison with individual models

Residual Forecasting with Prophet

In the first stage, the Prophet model is applied to the cleaned dataset. It generates the fitted values for the training set (Ytrain) and forecasts future values (Ypredict) over a predefined prediction horizon (Ph). Additionally, the model computes residuals on the training set, defined as:

εtraining=Ypredict−Ytarget,(13)

where Ytarget represents the actual observed values.

In the second stage, a BiLSTM model is applied to the residuals generated by the Prophet model. The objective of the BiLSTM is to capture complex temporal dependencies in the residuals and to improve the overall forecasting performance by reducing the global error between the predicted and actual values. The final prediction result is obtained by combining the outputs of both the Prophet and BiLSTM models. Prophet models time series as [49,52]:

y(t)=g(t)+s(t)+h(t)+εt(14)

where, y(t) represents the target (e.g., solar irradiation), g(t) represents the trend function modeling changes that are not periodic, s(t) represents the seasonality (daily, weekly, yearly), and εt is the unpredictable noise and in this work will be considered as the error term.

Trend Function (Piecewise Linear)

g(t)=(k+∑j=1Jδj1(t≥sj))(t−t0)+(m+∑j=1Jγj1(t≥sj))(15)

where k represents the initial growth rate, m denotes the offset or intercept of the trend function, and t0 is the start time. The model incorporates change points at times sj, where each change point introduces an adjustment δj to the growth rate. As a result, the growth rate at any time t is given by: k(t)=k+∑j=1Jδj⋅1(t≥sj), where 1(t≥sj) is an indicator function that is equal to 1 if t≥sj and 0 otherwise. Each change in growth rate k must be accompanied by a corresponding adjustment to offset m, ensuring the continuity of the piecewise linear trend by smoothly connecting the endpoints of each segment.

Seasonal Component (Fourier Series)

As the forecasting task focuses on the short term, hourly seasonality is addressed using the Fourier series representation, as shown in Eq. (16). Additionally, due to the significant fluctuations in the weather data, daily, weekly, and yearly seasonalities are implicitly and automatically handled by the internal mechanisms of the Prophet model.

s(t)=∑n=1N[ancos⁡(2πntP)+bnsin⁡(2πntP)](16)

here, P denotes the seasonal period (equal to 1 for hourly seasonality), and N represents the number of Fourier terms—i.e., pairs of sine and cosine functions—used to model the seasonal pattern. In this context, a value of N=6 is used.

Holiday Component

In this work, to enhance forecasting performance using homogeneous variables, additional components such as holidays and external regressors are incorporated. For example, to improve the forecast of power generation, relevant external factors, such as wind speed and humidity, are included in the training process as regressors. The influence of these variables is modeled as:

h(t)=∑jβjXj(t)(17)

where βj denotes the impact coefficient learned from historical data for each regressor Xj(t). These regressors may include external factors such as wind speed, humidity, and other weather-related variables.

Residual Forecasting with BiLSTM

Due to the strong uncertainty associated with weather conditions, the standalone Prophet model is still insufficient to deliver highly accurate and robust forecasts. To overcome this limitation, an additional stage based on BiLSTM is introduced. The BiLSTM technique is particularly suited for time series forecasting as it can capture both past dependencies and future contextual information through its forward and backward recurrent structure.

In the proposed hybrid framework, the BiLSTM is used to model and correct residual errors resulting from the initial Prophet forecast. This approach enables the capture of complex temporal behaviors that Prophet, based on additive components and piecewise linear trend, might not fully represent. To give a mathematical formulation of BiLSTM, let us start with the formulation of simple LSTM as defined in Eq. (18).

hs(t)=f(Wxx(t)+Whshs(t−1)+b)(18)

where, hs(t) cen be considered as the hidden state at time t, x(t) presents the input at time t. Wx,Whs, and b are learnable weights, f represents activation functions (typically tanh and σ).

As the BiLSTM based on the past and future dependencies, so the simple LSTM computation should occurs in two direction as describe in Eq. (19).

hs(t)→=f(Wxx(t)+Whshs(t−1)+b)(Forward Pass)hs(t)←=f(Wxx(t)+Whshs(t+1)+b)(Backward Pass)(19)

The final BiLSTM output is:

ht=hs(t)→+hs(t)←(20)

Based on the output of the BiLSTM and the Prophet output, the corrected forecast becomes as expressed in Eq. (21).

Y^Combined(t)=YpredictProphet(t)+hsBiLSTM(t)(21)

4.2 Assessment Metrics

To rigorously assess the prediction performance of the proposed hybrid methodology, three widely recognized evaluation metrics are used: the RMSE, the MAE, and the coefficient of determination (R2). Based on these metrics, a comprehensive view of the accuracy and reliability of the forecasts can be defined as given in Eqs. (22)–(24). These metrics collectively allow for the quantitative comparison of forecasting models in terms of both absolute error and goodness-of-fit.

RMSE=1n∑t=1n(y(t)−y^(t))2(22)

MAE=1n∑t=1n|y(t)−y^(t)|(23)

R2=1−∑t=1n(y(t)−y^(t))2∑t=1n(y(t)−y¯)2(24)

where, y(t) can be considered as the actual value at time t, y^(t) presents the predicted value at time t, y¯ is the mean of actual values, n is the total number of observations.

Data description and analysis

The meteorological dataset used in this simulation was extracted from a publicly available web platform providing high-resolution environmental data. The available data include a variety of meteorological parameters measured or estimated over approximately three years, with a temporal resolution of five minutes. In total, 18 meteorological and environmental features were extracted, as summarized in Table 3. These variables include shortwave and direct radiation, diffuse and global tilted irradiance, air temperature, relative humidity, wind speed (at 10 and 100 m), dew point temperature, precipitation, and terrestrial radiation, among others. To more realistically simulate microgrid operation, uncertainty was incorporated into the dataset, reflecting the stochastic nature of solar irradiance, weather conditions, and measurement errors. Specifically, the hourly data were first resampled to one-minute intervals to capture high-resolution variability. For each day, a fluctuation vector was generated, representing relative changes in the measured variables. The fluctuations were randomly sampled from a uniform distribution between 10% and 30% and assigned a positive or negative sign. This vector was applied multiplicatively to all meteorological features, with radiation-related variables clipped to remain non-negative. This method introduces day-to-day stochastic variations while preserving the overall daily trend of the variables. The same fluctuation vector was reused for all days to maintain a controlled yet realistic level of variability. Following this, solar power output was computed based on panel area, efficiency, and the fluctuated direct radiation. All meteorological features were preprocessed to serve as potential predictors for solar power generation. This preprocessing included normalization using the Min-Max scaling method, transforming all variables into the [0, 1] range to improve numerical stability and convergence of the forecasting algorithms. A correlation analysis between the target variable (solar power) and each feature was performed to identify the most relevant predictors. The analysis revealed that variables such as shortwave radiation, direct irradiance, and temperature have the strongest relationships with solar power. In contrast, others, such as humidity and wind speed, have a weaker influence. Based on these results, the most significant features were retained for model training, reducing redundancy and enhancing the learning process. Furthermore, a sensitivity analysis was conducted to evaluate the impact of each selected feature on forecast accuracy. This step helps determine the optimal subset of features that balances model complexity with predictive reliability. The resulting feature configuration, combined with the stochastic dataset, is then used in the subsequent hybrid Prophet + BiLSTM forecasting framework.

From the results provide in Table 4, it is clear that radiation-related features are the most critical for accurate solar power prediction. Using only SWR provides a baseline R2 of 0.9573, and adding T2 slightly improves performance. The most substantial improvements occur when including DR, DRI, GTI, GTII, SWRI, TR, TRI, DNI, DNII, and DF/DF_I. These combinations achieve RMSE as low as 0.0070–0.0078 and R2 up to 0.9988, indicating highly accurate predictions. Adding additional meteorological variables such as WS100, DP2, RH2, or P does not significantly improve the metrics and sometimes slightly increases RMSE, suggesting a risk of overfitting, where the model begins capturing noise rather than meaningful patterns. Therefore, the best solution is to focus on the comprehensive set of radiation features, which provides excellent accuracy without unnecessary complexity, while caution should be taken when adding more meteorological features to avoid overfitting.

Optimization

Based on the result provided in Table 5 the proposed hybrid model demonstrates robust performance across typical solar generation days. The RMSE ranges from 0.0016 to 0.0094 and MAE remains below 0.01, while R2 consistently exceeds 0.99, indicating that the model captures almost all variability in the solar power data. These results confirm that BiLSTM effectively corrects the nonlinear residuals left by Prophet, providing highly accurate and consistent forecasts under normal irradiance conditions. The low standard deviations further highlight the model’s reliability and generalization ability for practical energy forecasting applications.

Table 6 presents a comparative evaluation of several forecasting models for solar PV prediction. The results clearly demonstrate that both the Prophet + BiLSTM hybrid and the BiLSTM-only models significantly outperform the standalone Prophet model. Specifically, the Prophet + BiLSTM hybrid achieves the lowest RMSE (0.007) and MAE (0.0043) while attaining a high R2 of 0.9988, corresponding to an improvement of 0.2122 in RMSE and 0.7126 in R2 over Prophet alone. The BiLSTM-only model shows comparable performance, confirming the effectiveness of deep learning in capturing nonlinearities in PV generation data. However, the slight advantage of the Prophet + BiLSTM hybrid suggests that decomposing the time series into trend and seasonal components before applying BiLSTM correction further enhances predictive accuracy. These improvements are particularly valuable for energy management systems, enabling more precise battery scheduling, reduced grid dependency, and optimized power dispatch.

Optimization

Based on the forecasting results obtained using the hybrid Prophet-BiLSTM method, an LP-based optimization approach will be implemented to manage the IMM system described in the previous section. This LP approach is designed to address a centralized energy management problem with multiple objectives, such as minimizing operational costs, optimizing power flows, and ensuring the reliability of the system. To this end, the standard formulation of the linear programming problem has been adapted accordingly. The equality constraints, which typically represent energy balance equations in interconnected microgrids, are defined by Eq. (27). Furthermore, inequality constraints, which capture system limitations such as generation capacities, storage limits, and grid exchange limits, are presented in Eq. (28). This formulation allows for integrating accurate load and generation forecasts into the decision-making process. By incorporating forecasts derived from the hybrid Prophet-BiLSTM method, the optimization framework can effectively reflect the microgrid system’s temporal dynamics and operational constraints. This ensures that the decisions made are grounded in realistic expectations of future energy demand and generation capabilities. In order to solve this problem, MATLAB’s linprog solver is utilized due to its efficiency in handling large-scale linear programming problems. The decision vector x is constructed to encapsulate all the controllable variables within the optimization horizon. Specifically, this includes the power exchanged with the grid (Pgrid,i), the power charged or discharged by the battery system (Pbatt,i), and the energy stored in the battery (Ebatt,i) for each microgrid unit i=1,2,3, across N discrete time intervals. The optimization goal is to minimize the total operational cost, represented by a linear cost function f⊤x, subject to system constraints. The LP problem is formulated as follows:

minxf⊤xs.t.Aeqx=beqAx≤b(25)

As in this paper three interconnected microgrids are used for the validation, the vector of decision variables x∈R9N is structured as given in Eq. (26).

x=[Pgrid1,Pgrid2,Pgrid3,Pbatt1,Pbatt2,Pbatt3,Ebatt1,Ebatt2,Ebatt3]∈R9N(26)

The matrices Aeq, beq, A, and b are defined, respectively in (27), (28).

This formulation ensures that the optimization model captures the temporal coupling of decisions, such as battery state-of-charge dynamics and inter-microgrid power exchanges, over the forecast horizon. Consequently, the LP-based strategy facilitates economically optimal and technically feasible operation of the interconnected microgrid system by leveraging accurate predictions from the hybrid Prophet-BiLSTM model.

[IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗KϕK∗K10K∗K0K∗K0K∗KIK∗K0K∗K0K∗KϕK∗K20K∗K0K∗K0K∗KIK∗K0K∗K0K∗KϕK∗K30K∗K0K∗KIK∗KξK∗K10K∗K0K∗K0K∗K0K∗KIK∗K0K∗KξK∗K20K∗K0K∗K0K∗KIK∗K0K∗K0K∗KξK∗K3]Tx=[Pload1(1:K)−Ppv1(1:K)−Pf12−Pf13Pload2(1:K)−Ppv2(1:K)+Pf21+Pf23Pload3(1:K)−Ppv3(1:K)+Pf31−Pf32Ebatt1(1)0K−11Ebatt2(1)0K−12Ebatt3(1)0K−13](27)

[0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗KIK∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K0K∗K−IK∗K]x≤[Pbat1max−Pbat1minPbat2max−Pbat2minPbat3max−Pbat3minEbat1max−Ebat1minEbat2max−Ebat2minEbat3max−Ebat3min](28)

with

ξ=Δt⋅[000⋯0100⋯0010⋯0⋮⋮⋮⋱⋮000⋯0],ϕ=[100⋯0−110⋯00−11⋯0⋮⋮⋮⋱0000⋯1].(29)

5  Simulation Results

To validate the proposed methodology, four scenarios are analyzed and implemented in three interconnected microgrids that integrate renewable energy generation and storage systems, as illustrated in Fig. 1. A comprehensive validation is carried out by evaluating power exchange between microgrids using three distinct load profiles. In addition, similar and varying production levels are considered to assess the adaptability of the system under different operating conditions. To account for fluctuations in renewable generation and load demand, uncertainty is systematically incorporated into the analysis. In addition, to quantify the impact of transmission lines on system performance, various line lengths are evaluated.

•   Scenario 1 (Reference Case). Three interconnected microgrids, each with identical generation and demand profiles, energy storage systems of equal size, and separated by a uniform distance of 5 km. Table 7 compares the proposed approach with the Heuristic, BiLSTM, and LP methods using four key performance indicators: (i) day-ahead operating cost, (ii) total grid energy purchased, (iii) total CO2 emissions, and (iv) line power losses between microgrids. Based on estimated PV profiles, the proposed methodology achieves lower operating costs, grid dependency, and emissions compared to Heuristic and BiLSTM methods, and closely approaches the performance of the optimal solution obtained using actual PV data. In particular, power losses are zero, indicating that there is no energy exchange among microgrids. As such, Scenario 1 serves as the reference case for validating subsequent scenarios. Moreover, as shown in Fig. 5, the participation of the battery under the heuristic method is minimal and suboptimal in microgrids, resulting in significantly higher operational costs. In the heuristic approach, the battery is continuously discharged until it reaches its minimum SOC threshold, without considering future variations in PV generation or load demand. Consequently, the battery does not actively contribute to improving or optimizing the overall performance of the system. In contrast, the proposed methodology, which combines forecasting techniques with the LP optimization framework, ensures an active and intelligent participation of the BESS. As illustrated in Fig. 6, the LP-based optimization enables the battery to charge during periods of PV production surplus and discharge when PV generation is insufficient. This dynamic and adaptive operation maximizes the utilization of renewable energy sources while minimizing reliance on the electrical grid. As a result, the system achieves a more balanced energy flow, reduced operational costs, and enhanced overall efficiency of the microgrid.

•   Scenario 2. In this scenario, the three microgrids have identical PV production and are interconnected via transmission lines of equal length (2.5 km). However, their load profiles differ to illustrate how the surplus energy of one microgrid can support another. Specifically, Microgrid 1 is assigned a fixed load of 350 kW plus a variable load; Microgrid 2 has a fixed load of 150 kW plus the same variable component; while Microgrid 3 is subjected only to the variable load. The results in Table 2 show that the proposed methodology outperforms both the Heuristic and BiLSTM-based methods, and its performance is nearly equivalent to the case where the exact PV profile is known. Compared to Scenario 1, the total cost savings increases to approximately 25%, demonstrating the benefit of energy sharing between micro-grids. However, this improvement comes at the expense of increased power losses, which confirms that active energy exchange is taking place across the network in a coordinated and beneficial manner.

•   Scenario 3. To assess the robustness of the proposed methodology, this scenario introduces a fault condition in Microgrid 3 by disconnecting both its battery and PV generation between time steps 25,000 and 50,000, as illustrated in Figs. 7 and 8. The transmission lines between the microgrids are 2.5 km in length. Similar to Scenario 2, the load profiles differ across the microgrids, allowing the system to demonstrate how surplus energy in one microgrid can compensate for deficits in another. This setup evaluates the system’s ability to maintain efficient and coordinated operation under partial failure conditions. It is observed that the total power exchange increases in this scenario, as the optimization algorithm seeks to compensate for the loss of renewable generation in Microgrid 3 while still meeting the defined objectives. The results show that the proposed methodology continues to outperform the competing approaches, confirming its robustness and adaptability in the presence of faults.

•   Scenario 4. This scenario evaluates the impact of the length of the transmission line on the global optimization problem. The microgrid parameters are identical to those in Scenario 3, with the exception that the transmission line length between each pair of microgrids is decreased to 2.5 km for Scenario 4.1 and increased from 5 to 15 km for scenario 4.2. The results indicate that power losses are significantly reduced compared to Scenarios 4.2 as given in Figs. 9 and 10. This suggests a decrease in power exchange between microgrids, as the longer transmission distance discourages energy sharing. In this case, the surplus energy, primarily generated in Microgrid 3, is not fully utilized by the neighboring microgrids. Due to the increased line length and the voltage limitation of 5 kV, a substantial portion of this surplus is instead injected directly into the main grid, which is assumed to be geographically closer to Microgrid 3.

Figure 5: Optimized energy management visualization (Scenario 1 Heuristic-based methodology)

Figure 6: Optimized energy management visualization (Scenario 1 proposed methodology)

Figure 7: Optimized energy management visualization under fault (Scenario 3 proposed methodology)

Figure 8: Interconnection line power losses with and without fault

Figure 9: Optimized energy management visualization (Scenario 4.1 proposed methodology)

Figure 10: Optimized energy management visualization (Scenario 4.2 proposed methodology)

6  Discussion

The integration of a hybrid Prophet–BiLSTM forecasting model with a Linear Programming (LP)-based energy management framework improves the operational efficiency and economic performance of interconnected microgrid systems. By leveraging accurate, high-resolution forecasts of renewable energy generation, the proposed architecture enables proactive and informed decision-making within the optimization horizon. Specifically, the LP optimizer dynamically coordinates power exchanges with the main grid, schedules battery charging and discharging cycles with high precision, and optimally manages inter-microgrid power flows. This is achieved while strictly adhering to system-level technical constraints, such as battery SoC thresholds, power flow capacity limits, and grid interconnection rules. The empirical results of Scenario 1, as summarized in Table 3, demonstrate that the LP-based control strategy consistently outperforms conventional heuristic approaches, achieving an average cost reduction of approximately 12%. In addition to economic benefits, the optimization-based strategy yields measurable reductions in grid energy consumption and associated CO2 emissions. These findings highlight the ability of the LP framework to capitalize on the increased accuracy provided by the Prophet–BiLSTM forecasts, enabling more intelligent and resource-efficient operation. Moreover, the LP model inherently captures the temporal interdependencies among decision variables, most notably in battery energy storage systems, by maintaining feasibility across multiple time steps. This time-coupled optimization ensures not only the reliability of system operations but also the preservation of long-term battery health and the mitigation of operational risks. Such features are particularly critical in multi-microgrid contexts, where distributed units must collaboratively balance fluctuating supply and demand conditions across varying temporal and spatial scales. The incorporation of inter-microgrid power exchanges further enhances the system’s flexibility by enabling dynamic, adaptive energy-sharing mechanisms between geographically distributed microgrids. This results in reduced transmission losses, improved local energy autonomy, and enhanced grid resilience, particularly under contingencies or peak demand conditions. The LP formulation is designed to be both scalable and adaptive, accommodating a wide range of operating scenarios, including diverse load profiles, stochastic renewable generation patterns, and time-varying network parameters. Collectively, these attributes underscore the practical viability and robustness of the proposed LP-based energy management strategy. Furthermore, the framework aligns with practical microgrid implementation standards such as IEEE 2030.7 (defining microgrid controller specifications) and IEEE 2030.8 (testing procedures for controller performance), and its core principles are consistent with ongoing validation efforts in experimental platforms such as NREL’s microgrid testbeds. By integrating machine learning–based forecasting with mathematical optimization, this framework offers a structured and forward-looking approach to multi-microgrid operation that advances economic efficiency, environmental sustainability, and operational resilience. As such, it provides a foundational step toward the realization of intelligent, autonomous energy systems capable of navigating the complexities of modern power networks.

7  Conclusion and Future Works

This paper presents a comprehensive three-stage energy management methodology that encompasses both prediction and decision-making. The first stage emphasizes data pre-processing, ensuring high-quality input for subsequent analysis. The second stage builds upon this foundation to achieve accurate forecasting, employing an iterative hybrid method that integrates BiLSTM and stochastic models. Finally, the third stage introduces an optimization framework based on a modified LP approach, validated for three interconnected microgrids.

The proposed LP-based multi-objective optimization strategy, applied to forecasts generated through the Prophet-BiLSTM hybrid model, demonstrates mathematical simplicity, computational efficiency, and superior performance compared to alternative methodologies. Nevertheless, integration of the ESS remains somewhat limited due to the centralized control structure. In some cases, battery operation control signals exhibit uniformity across microgrids, restricting operational flexibility. To address this, future enhancements could incorporate adaptive fuzzy logic within the LP framework, improving ESS responsiveness, reducing dependence on the main grid, and minimizing overall energy costs.

Looking forward, several avenues will be explored to further strengthen the methodology. The integration of battery degradation costs will be considered to better capture the economic impact of frequent charging and discharging cycles, enhancing model realism and providing a more reliable assessment of ESS operation. The framework will also be extended to larger networks of interconnected microgrids, focusing on coordination among multiple systems and the scalability of the optimization strategy. Advanced approaches, including fuzzy logic and intelligent optimization techniques, will be investigated to improve flexibility, robustness, and operational efficiency. Furthermore, transitioning from a centralized LP framework to distributed or ADMM-based optimization will be explored to efficiently manage clusters of microgrids while maintaining computational tractability. Finally, real-time implementation in practical microgrid testbeds will provide additional validation and insights into real-world performance.

Overall, the proposed methodology offers a robust foundation for predictive and optimization-based energy management, with clear pathways for enhancement and practical deployment in future research.

Acknowledgement: The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding their research work through the project number PSAU/2024/01/31821.

Funding Statement: The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding their research work through the project number PSAU/2024/01/31821.

Author Contributions: Mohamed Kouki, Nahid Osman, Mona Gafar, and Ragab A. El-Sehiemy confirm contribution to the paper as follows: Conceptualization, Mohamed Kouki and Nahid Osman; methodology, Mohamed Kouki; software, Mohamed Kouki; validation, Mohamed Kouki, Nahid Osman, Mona Gafar, and Ragab A. El-Sehiemy; formal analysis, Mohamed Kouki; investigation, Mohamed Kouki; resources, Mohamed Kouki; data curation, Mohamed Kouki; writing—original draft preparation, Mohamed Kouki; writing—review and editing, Mohamed Kouki; visualization, Mohamed Kouki and Ragab A. El-Sehiemy; supervision, Mohamed Kouki and Ragab A. El-Sehiemy; project administration, Mohamed Kouki; funding acquisition, Nahid Osman and Mona Gafar. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Data available on request from the authors.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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