Microgrid Optimization Technique Using Supervised Learning for Resiliency Enhancement in Power Systems

Abstract

This paper addresses key limitations in transmission–distribution (T&D) co-simulation for resiliency, including fragmented modeling, high complexity, synchronization issues, weak renewable control, and data access constraints. A unified co-simulation framework is proposed to optimize microgrid formation and operation in high-penetration renewable systems, improving resiliency while reducing costs and network losses. The developed co-simulation platform enables modular, conflict-free synchronization between transmission and distribution networks without additional handshake software, allowing independent data transfer and seamless co-optimization. The technique assists in transmission and distribution dynamic coordination, supports economic dispatch, and performs three-phase optimal power flow (OPF). An Adaptive Neuro-Fuzzy Inference System (ANFIS) is used for load forecasting and optimization modeling, enabling fast convergence and computational efficiency. The framework supports both grid-connected and islanded modes, including dynamic islanding, reconnection, and load prioritization. Case studies using IEEE 14-Bus transmission with 15-Bus and modified unbalanced 123-Bus distribution systems validate the approach. Results show up to a 68% reduction in operating costs and significant reductions in loss, demonstrating improved resilience, scalability, and secure data exchange for modern power systems.

1. Introduction

The increasing penetration of decentralized generation, electric vehicles, and flexible loads has introduced new dynamics and vulnerabilities into modern power systems [1]. At the same time, the growing frequency of extreme weather events and cyber threats has elevated the importance of grid resiliency as a primary operational objective [2]. While microgrids have been widely recognized for their potential in improving system survivability, current implementations often rely on fixed topologies or limited rule-based logic, lacking the ability to adapt dynamically to evolving grid conditions [3].

The operation of a microgrid in island mode is crucial for independently serving loads in the worst cases, when the traditional grid fails due to extreme weather or natural disasters. The increasing complexity of modern power systems, driven by high penetration of renewable energy resources, rapid electrification of transportation, and growing cyber-physical interdependencies, has elevated resilience as a critical requirement for both transmission and distribution networks [4,5]. Conventional centralized grid architectures are often ill-equipped to withstand and recover from natural disasters, cyber-attacks, or cascading failures, as shown in Figure 1.

According to a 2019 World Bank report, “overall, between 2000 and 2017, 54.8% of all recorded power outage events were caused by natural shocks, and 44.2% by non-natural causes (26.9% excluding vandalism)” [7]. In 2023, a Department of Energy (DOE) report established that power outages due to weather-related events account for about 80–83% of the major outages in the United States [8]. The high-impact low-frequency (HILF) events, according to [7,8], are becoming more frequent in recent times, and there is a need for proper mitigation against their undesirable effects.

The microgrid must be able to provide necessary backup for loads, especially critical loads, in the event of grid failure [9]. The state of the microgrid during this critical period must be ascertained to ensure proper control of nominal frequency and voltage [10]. Real power/frequency and reactive power/voltage droop controls are two important aspects of implementing these control techniques. Microgrids with a high share of renewable energy suffer from over-voltage due to the voltage/frequency controller deployed [10].

This research introduces hybridized multi-layer microgrid formation and supervised optimization techniques to improve the resiliency of T&D networks. Unlike previous approaches that focus on either the distribution system (DS) or transmission system (TS) network only, such as the use of electric buses (EBs) for the restoration of the DS network [11], a rolling integrated service restoration strategy using the Mobile Energy Storage System (MESS) to increase the resiliency of the distribution system [12]. Researchers in [13] proposed a multi-task learning framework and a safety layer that can effectively improve topology awareness in the DS network, while [14] proposed a formal approach to evaluate the operational resiliency of a power distribution system (PDS) and quantify the resiliency of a system using a code-based metric for the DS network. In [15], a modernized DS network restoration technique was introduced, geared towards improving the resiliency of the DS network against extreme weather. In [16], a method for defining and enabling the resilience of electric distribution systems with multiple microgrids was introduced.

Research in [17] developed a restoration algorithm to improve the DS network’s resiliency and reliability. In [18], a DS restoration approach was proposed, considering uncertain devices and associated asynchronous information, while [19] proposed a resiliency measurement scheme for transmission power systems using a decentralized Remedial Action Scheme (RAS). Thus, refs. [17,18,19] treat the transmission and distribution layers independently. In [20], a co-simulation platform was developed to enable the analysis of several simulators that suffer from synchronization and code compatibility issues. In [21], the importance of detailed DS network analysis for network stability is demonstrated using a co-simulation platform with a TS sub-transmission network. In [16], a resiliency metric for the DS network short-term planning is developed to assist distribution system operators. In [22,23], the importance of comprehensive, accurate, and computationally efficient modeling techniques was emphasized in the context of practical applications to high-impact, low-probability events, as well as the T&D co-simulation platform performance compared to only DS simulation in power systems. In [24], the authors deployed reinforcement learning and neural-based methods for real-time Energy Management System (EMS) optimization and distributed energy resource (DER) coordination using the mode selection and resource allocation (MSRA-TD3) approach to address a nonconvex optimization problem by learning efficient power and resource allocation policies.

The proposed framework employs a coordinated model with ANFIS-based learning, enabling intelligent, data-driven control across the entire grid hierarchy. The contributions include: (i) development of a scalable T&D co-simulation architecture that integrates transmission and distribution models using a modular and interoperable framework; (ii) implementation of resiliency-oriented analysis and optimization strategies that enable coordinated evaluation of disturbances, renewable variability, and load restoration scenarios; (iii) a real-time microgrid clustering model driven by topological and operational states; (iv) an ANFIS trained model to optimize control strategies under fault conditions; and (v) integrated optimal power flow (OPF) and voltage/reactive optimization routines tailored for T&D coordination. The framework is evaluated against standard resiliency benchmarks using IEEE network cases, demonstrating their robustness and operational value in emergency and recovery scenarios. Power system resiliency (PSR) is the ability of a power grid to withstand, respond to, and recover from major power disruptions. It is a power system’s ability to bounce back from high-impact low-probability (HILP) events, such as extreme weather, accidents on major grid facilities, or cyberattacks [25]. According to [26], Resiliency is defined as the power grid’s capability to withstand and recover quickly from severe incidents, react properly to changing conditions, and prevent future events.

The following are the characteristics of a resilient power system:

• A resilient power system can reduce the frequency of long-duration outages.
• It can limit the scope and impact of outages.
• It can be restored rapidly after outages.

However, to improve power system resiliency, one or more of the following steps have been considered in the past [11,12,13,14,15,16,17]:

(a)

Strategic deployment of renewable energy resources like solar, wind, water, geothermal, and bioenergy.

(b)

Enhancing power electronics (use of superior controllers in inverters and converter circuits).

(c)

Decentralizing the power supply.

(d)

Improving cybersecurity.

(e)

Developing resilience metrics.

(f)

Enhancing system design for resiliency.

(g)

Improving preparedness and mitigation measures (the system operator plays an important role in this regard).

(h)

Improving system response and recovery.

Analyzing and managing interdependencies, several research efforts have proposed resiliency metrics tailored to the case study under consideration that focus on the network [22]. In the wake of the growing concerns of unpredictable environmental conditions that impact both T&D networks, there is a need for a holistic solution that considers both the downstream and upstream sectors of the power network. In [27], an ANN-based short-term load forecasting model suffers from slow optimization speed and does not consider integration of DER mesh formation models.

The proposed solution achieves faster optimization times and incorporates DER’s integration into the solution formulation. The ANFIS-based solution reduced computational time at deployment while guaranteeing optimization accuracy. In [25], the initial assumption that transmission grids can no longer power distribution systems limits the scope of the study to microgrid formation and restoration optimization. However, the proposed solution considers both transmission (TS) and distribution (DS) networks for an appropriate T&D network solution before going into an island mode of operation with DER allocations. In [28], the transmission and distribution network interactions were left out of the reconfiguration procedure, limiting the robustness of the solution. The proposed solution is robust as it has a T&D co-simulation platform for the incorporation of the transmission network into the reconfiguration solution modeling. In [27], switching operations result in large frequency deviations that are outside normal ranges and cause the collapse of the microgrid. The Grid Friendly Appliances (GFAs) deployed in the grid formation are quite expensive to deploy on a larger power network [27]. Looking at the gaps, such as simulation time and accuracy, the ANFIS-based solution achieved improved accuracy and reduced computational burden. The proposed solution reduces the operational cost, which is cheaper to implement. The resiliency study here is based on simulations.

Among other benefits is a robust T&D co-simulation platform that handles the distribution and transmission network interaction with optimization capabilities, as compared to previous studies [29,30] that focus on the DS network, and [19] on TS. In the worst-case simulated HILF event, 79.84% of the network’s total active power demand is served. Thus, 70.39% of the total network reactive power demand was met, causing less computational burden in terms of simulation time and hardware requirements. With concentrated DERs (four-number DERs), the simulation time stands at 1.885045 s after numerical optimization using the developed ANFIS-enabled co-simulation platform. The 5.954774 s simulation time with 22-DERs distributed on the modified 123-Bus IEEE distribution network, which is faster compared to previous works [31,32]. The cost of operating the optimized network was reduced significantly as compared to conventional network operations, as indicated in the General Algebraic Modeling System (GAMS) numerical optimization model deployed. The scalability and adaptability of the proposed solution from a 15-Bus IEEE distribution network to the modified unbalanced 123-Bus IEEE distribution network yielded similar good results, and a successful simulation of abnormal operating conditions with excellent active power and grid frequency control in grid-tied and island modes of operation. A power serving capacity (PSC) of 75.12% was achieved in the face of the worst HILF event simulated. The remaining part of this paper is divided into sections: Section 2 presents Power Management Concepts’/Metrics’ definitions. Section 3 is the research methodology that details the procedure for the implementation of the proposed solution. Section 4 is the result analysis of the research findings. Section 5 provides the conclusion and recommendations.

2. Power Management Concepts/Metrics Definitions

This section provides a definition of important power management concepts and important resiliency metrics. Load shedding/peak shaving is a quick reduction in power consumption to avoid peak demand charges. This can be achieved by turning off high-energy equipment or adding a local energy source. This is otherwise known as ‘peak shaving’, where a consumer reduces power consumption (“load shedding”) quickly and for a short period of time to avoid a spike in consumption. This is either possible by temporarily scaling down production, activating an on-site power generation system, or relying on a battery. Load shifting is a short-term reduction in electricity consumption followed by an increase in production at a later. This is typically done to capitalize on lower power prices or reduced grid demand. Both load shedding and load shifting are demand-side management techniques that can be optimized with Battery Energy Storage Systems (BESSs). BESS can charge during off-peak hours and discharge during peak hours to smooth out demand spikes. Peak shaving can lead to significant energy cost savings, especially if electricity prices fluctuate widely or if you are a commercial customer with high demand charges. Several efforts have been made to provide robust solutions to power network failures in the face of HILF events, emphasizing the ability of transmission and distribution networks to withstand, adapt, and rapidly recover from disturbances. Foundational studies were led by Raoufi et al. [33] and Lin et al. [4]. The work in [4] defines resilience metrics that go beyond traditional reliability indices, thereby framing the withstand–respond–recover cycle. The National Laboratory of the Rockies (NLR), formally known as the National Renewable Energy Laboratory’s (NREL) resilience framework, further standardizes pre-event and post-event assessment using service continuity, energy not served, and recovery-time indicators [29].

Resiliency, Reliability, and Adequacy

These three terms are intertwined and geared towards the high dependability of the power network. Figure 2 presents the Venn diagram of resiliency, reliability, and adequacy interdependences.

Power network stability focuses on maintaining equilibrium under normal operating conditions and quickly recovering from small disturbances like load fluctuations. It is how the power system quickly adjusts to a sudden increase in load without experiencing voltage drops. Power network adequacy measures how sufficient resources are on the network, while power reliability is the ability of the power network to maintain a consistent power supply to the homes and businesses, even during minor equipment malfunctions. It is aimed at preventing small-scale failures and maintaining a steady power supply, resulting in minimal power supply interruptions. Power network resiliency emphasizes the ability to withstand and recover from major disruptions like extreme weather events (like hurricanes), cyber-attacks, or large-scale equipment failures.

3. Research Methodology

The research methodology is based on a microgrid planning scheme [34]. The microgrid planning procedure serves as the framework for developing this research methodology, which focuses on two key aspects: resource allocation and network reconfiguration. Resource allocation comprises Distributed Generators ($DGs$), Demand Response Systems ($DRSs$), and Energy Storage Systems ($ESSs$), while network reconfiguration ($NR$) involves the placement of tie-lines and the predefined and dynamic formation of microgrids. The research framework is shown in Figure 3.

Figure 4 represents the implementation path for the $ANFIS$-enabled optimization technique on the Unbalanced IEEE 123-Bus distribution network with four microgrids and four (4) IEEE 14-Bus sub-transmission networks, based on supervised learning training of the model using the network configuration, load, generation, and fault data forecast for decision-making that is geared towards enhancing the overall power network resilience. The health state of the distribution system ($DS$) and transmission system ($TS$) networks is initially monitored using OPF analysis, which utilizes data from the local control centers within the $DS\&TS$ networks. The two central control centers ($C1\&C2$) are virtually connected to improve decision-making and network profile restoration in the event of the loss of one center due to an $HILF$ event. The optimized results are used in contrast by local control centers for important $NR$. ANFIS acts as a supervisory, surrogate, and predictive model within the co-simulation platform. The supervisory decision layer involves BESS dispatch, load curtailment or prioritization. Predictive layer is made up of PV, wind, and load forecasting with smooth variability. ANFIS handles fragmented modeling challenges of complex power network with several subsystems modeled in different tools/language, making integration inconsistent. However, ANFIS provides a unified mapping creating a single nonlinear input–output mapping for multiple subsystems. Instead of passing full solver-level models between platforms, ANFIS replaces detailed subsystems with a trained function which reduces tool over dependency. This results in simplified co-simulation topology with fewer interfaces and less error-prone data exchange. Finally, ANFIS enables fast evaluation in optimization loops without full physics-level models. Since systems are carefully modeled in modular form with the DS restoration time limited often to 24 h, ANFIS proved very useful with medium data size.

3.1. ANFIS-Enabled Power Management and Optimization Theory

The Adaptive Neuro-Fuzzy Inference System ($ANFIS$) offers a convenient and flexible model that allows various decision-making criteria and their weights to be structured as a mixture of real values, interval values, or fuzzy values/sets. $ANFIS$ is based on a hybridization scheme of Artificial Neural Networks ($ANN$s) and Fuzzy Inference System ($FIS$). $ANFIS$ is a five-layer network with supervised learning, as shown in Figure 5.

Membership Layer: The choice of an appropriate membership function is based on the application. Two membership functions are deployed for this work, namely the Gaussian membership function ($gaussmf$) and the Generalized Bell membership function ($gbellmf$). According to [35,36], the Gaussian membership function is suitable for power generation and load forecasting, while the Generalized Bell membership functions are well-suited for load weight and priority modeling. ${\mu }_{P_{ij}}$ of this layer are Generalized Bell functions given by Equation (1) used for load weight and priority modeling.

The load criticality level is used to tag the DS and TS network load types where critical loads have weight of 1. Averagely critical loads are tagged 0.5. Non-critical loads are 0. The critical loads are first served during the restoration phase followed by averagely critical loads and lastly non-critical loads based on the available sources in the power network. The tags are used to select appropriate membership function and appropriate rule formulation (Generalized Bell function for weight allocation).

$${\mu }_{P_{ij}}={\displaystyle \frac{1}{1+{\left[{\left(\frac{x-c_{ij}}{a_{ij}}\right)}^2\right]}^{b_{ij}}}}\forall i,j\in \{1,2,3,\dots ,n\}.$$

(1)

where $\{a_{ij},b_{ij},c_{ij}\}$ is the parameter set regarded as ‘Premise Parameters’. The Gaussian membership function is given by the Equation (2) used for short-term power generation and load forecasting.

$$\mu \left(x\right)=e^{-}\left[{\displaystyle \frac{{(x-c)}^2}{2{\sigma }^2}}\right]$$

(2)

where $\mu \left(x\right)$ is the Gaussian function, ‘c’ is the mean, and ‘$\sigma$’ is the standard deviation. The function is smooth and symmetrical, and is controlled by the mean (‘c’) and standard deviation (‘$\sigma$’).

Fuzzification Layer: In this layer, every node is a fixed node labeled $\pi$. The node’s output is multiplied by the incoming values. For example,

$${\omega }_n={\mu }_{P_{i,j}}\left(x_n\right){\mu }_{P_{i,j}}\left(x_n\right),\forall i,j\in \{1,2,3,\dots ,n\}.$$

(3)

The nodal output represents firing strength (equivalent to fuzzy ‘$AND$’ operation), which is the minimum operation. Normalization Layer: Every node in this layer is a fixed node marked ‘N’. This is known as the normalization layer. The output of the node is the normalization firing strength (${\overline{\omega }}_i$) given by Equation (4):

$${\overline{\omega }}_i={\displaystyle \frac{{\omega }_i}{{\sum }_{k=1}^i{\omega }_i}},\forall i\in \{1,2,3,\dots ,n\}.$$

(4)

Defuzzification Layer: In this layer, every node is an adaptive node. The node output is given by Equation (5):

$$L_{O_{4,i}}={\overline{\omega }}_i{f}_i,$$

(5)

where the function ($f_i$) is given by Equation (6):

$$f_i=p_{i}x+q_{i}y+\cdots +r_i,$$

(6)

where $\{p_i,q_i,r_i\}$ is a parameter set. These parameters are known as ‘Consequent Parameters’.

Output Layer: In this layer, there is a single node, which is fixed and labeled as Σ. It computes the overall output by summing all incoming values. So, the output is given by Equation (7),

$$L_{O_{5,i}}=\sum _i{\overline{\omega }}_i{f}_i={\displaystyle \frac{{\sum }_i{\omega }_i{f}_i}{{\sum }_i{\omega }_i}}.$$

(7)

This is the way alternative adaptive neural networks function, much like a Sugeno-type FIS. This structure can be streamlined by merging layers three and four. Similarly, weight normalization can be applied at the output stage [37]. The ANFIS suffers from the curse of dimensionality if not well managed; that is, as the number of inputs (n) or the number of membership functions ($mf$) increases, the number of fuzzy rules ($FR$) also increases.

$$FR={\left(mf\right)}^n.$$

(8)

Although $ANFIS$ suffers from the curse of dimensionality, there is a need for precise and definite knowledge of the T&D power system under review, in terms of demand and generation profiles, which, when carefully studied, guides the choice of membership functions in the $ANFIS$ prediction algorithm to prevent unnecessary rule multiplicity and power system model complexity. The $FIS$ regulations and members for this study are well-defined to reflect the actual system’s behavior.

3.1.1. Advantages of ANFIS over Other Machine Learning Models

ANFIS is a rule-based ML model with transparent and explainable rules. However, other machine learning models, such as Graph-Based Learning (GBL) and LSTM, are based on a black box learning behavior, which makes them difficult to explain decision node-by-node. In power systems, especially Energy Management Systems (EMSs), interpretability is critical for system modeling and optimization. ANFIS performs well on small datasets and can incorporate expert knowledge [37]. LSTM and GBL require large, structured datasets and labeled data for training. This results in computational complexity with the high cost of implementation, as it requires a GPU for a larger network. ANFIS is computationally lightweight with fast training and inference, which makes it easier to deploy for real-time simulation on RTAC or SCADA. LSTM and GBL models are exposed to the risk of overfitting when dealing with small datasets due to several hyperparameter tuning requirements. ANFIS is more robust for small/medium datasets.

Table A2. Comparison summary of ANFIS and other machine learning models.

Feature	ANFIS	LSTM	ANN	GNN
Interpretability	Very high	Low	Low	Low–medium
Data requirement	Low	High	Medium	High
Training speed	Fast	Slow	Medium	Slow
Real-time use	Excellent	Limited	Good	Limited
Co-simulation efficiency	High	Medium–low	Medium	Low
Uncertainty handling	Excellent	Moderate	Moderate	Moderate
Rule-based control	Yes	No	No	No

presents a summarized comparison of ANFIS with other ML models.

3.1.2. ANFIS Training and Adaptation

The ANFIS modules are trained offline and updated online due to the ease of deployment for real-time applications. The ANFIS forecasting is modeled to minimize the forecast error using least-square error training objectives, as given by Equation (9):

$$\underset{{\Theta }_L}{min}{\displaystyle \sum _{i=1}^4}{\displaystyle \sum _{t=1}^{T_{tr}}}{\left(P_{i,t}^L-{\widehat{P}}_{i,t}^L\left({\Theta }_L\right)\right)}^2$$

(9)

where ${\Theta }_L$ contains the premise and the consequent parameters. $T_{tr}$ is the training horizon (which is a resolution of 1 min for this work). $P_{i,t}^L$ is the actual load demand in microgrid i at time t and ${\widehat{P}}_{i,t}^L$ is the forecast load. The ANFIS-enabled weight generation is trained against a target supervisory policy given by Equation (10)

$$\underset{{\Theta }_W}{min}{\displaystyle \sum _{i=1}^4}{\displaystyle \sum _{t=1}^{T_{tr}}}{∥w_{i,t}^{\mathrm{target}}-w_{i,t}^{\mathrm{ANFIS}}\left({\Theta }_W\right)∥}^2$$

(10)

where ${\Theta }_W$ is the parameter set of the ANFIS model for load-weight estimation. $w_{i,t}^{\mathrm{target}}$ is the desired load weight. $w_{i,t}^{\mathrm{ANFIS}}$ is the ANFIS-predicted load weight. ${∥.∥}^2$ is the squared error (Euclidean norm). $T_{tr}$ is the number of training time steps.

3.2. Mathematical Modeling of the Objective Functions

The following are three objective functions of this research:

• To minimize the outage/loss of load on both the distribution and transmission systems.
• To minimize the number of active network operations carried out to achieve the first objective. This objective function aims to minimize the restoration time of loads during $HILF$ events. The more active network operations carried out to restore the power supply, the more complex the restoration procedure and consequently the longer time to achieve the objective (1).
• Minimization of the costs of generation, switching, and load curtailment.

3.2.1. Modeling Objective Function I

$${\mathrm{OF}}_{\min .\mathrm{L}}={\mathrm{OF}}_{\min .\mathrm{DL}}+{\mathrm{OF}}_{\min .\mathrm{TL}},$$

(11)

$$\begin{array}{cc}\hfill {\mathrm{OF}}_{\min .\mathrm{DL}}& =\underset{P{L}_{i,t}^D,Q{L}_{i,t}^D}{min}\sum _{\forall i}\sum _{\forall t}{\mu }_i^D\left(P{L}_{i,t}^D+Q{L}_{i,t}^D-P{R}_{i,t}^D-Q{R}_{i,t}^D\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\mathrm{OF}}_{\min .\mathrm{TL}}=\underset{P{L}_{i,t}^T,Q{L}_{i,t}^T}{min}\sum _i\sum _t{\mu }_i^T(& P{L}_{i,t}^T+Q{L}_{i,t}^T-\sum _{k}P{R}_{k,i,t}^T\hfill \\ \hfill & -\sum _{k}Q{R}_{k,i,t}^T),\hfill \end{array}$$

(13)

where ${\mu }_i^T$ and ${\mu }_i^D$ are the $ANFIS$ weight coefficient allocations for the transmission and distribution loads, respectively. $P{L}_{i,t}^D$ is the $DS$ active load loss, $P{R}_{i,t}^D$ is the $DS$’s active load recovered. $Q{L}_{i,t}^D$ is the $DS$ reactive load loss, $Q{R}_{i,t}^D$ is the $DS$ reactive load recovered, $P{L}_{i,t}^T$ is the $TS$ active load loss, $P{R}_{i,t}^T$ is the $TS$’s active load recovered at the coupling point ‘k’, $Q{L}_{i,t}^T$ is the $TS$ reactive load loss, $Q{R}_{i,t}^T$ is the $TS$’s reactive load recovered at the coupling point ‘k’, $\forall$i represents the distribution nodes or transmission buses, and $\forall$k corresponds to the coupling points between the $TS$ and $DS$ network, where k = 1, 2, 3, 4. ${\mu }_i^T$ and ${\mu }_i^D$ are derived from the total power loss, reliability, and self-adequacy indices handled by the ANFIS-designed algorithm. The total power loss index (${\lambda }_{\mathrm{TPL}}$) is derived from Equation (14).

$${\lambda }_{\mathrm{TPL}}={\displaystyle \frac{\mathrm{Total}\mathrm{Power}\mathrm{Loss}}{\mathrm{Loss}\mathrm{Target}}}.$$

(14)

$$\mathrm{Total}\mathrm{Power}\mathrm{Loss}={\displaystyle \sum _{m=1}^M}{P}_{Loss}\left(m\right),\forall m\in M,$$

(15)

where M is the total number of microgrids formed in the network.

$$P_{Loss}\left(m\right)={\displaystyle \sum _{j=1}^M}{I}_{L_j}^2{R}_j,\forall j\in n,\forall m\in M,$$

(16)

where the bus current, except for the slack bus, is estimated as given by Equation (17).

$$I_i={\left({\displaystyle \frac{S_i}{V_i}}\right)}^{*},\forall i\ne 1.$$

(17)

$I_i$ is the current of bus i, $S_i$ is the complex power of bus i, and $V_i$ is the voltage of bus i. By applying Kirchhoff’s Current Law ($KCL$), in a backward sweep, the branch current between two buses is calculated using (18).

$$I_{L_j}={\displaystyle \sum _{\begin{array}{c}i\ne 1\end{array}}^n}{I}_i+I_{L_{j+1}},\forall j\in \{1,\dots ,n\}.$$

(18)

$I_{L_j}$ is the current of line j, and $I_i$ is the current of bus i. The suffix $j+1$ denotes the line next to j, which is used to apply the $KCL$ at the node. Bus voltage is estimated using Kirchhoff’s Voltage Law $KVL$, in a forward sweep, using Equation (19).

$$V_{i+1}=V_i-\left(I_{L_j}{Z}_j\right)\forall i\ne 1,\forall j\in \{1,\dots ,n\}.$$

(19)

$V_i$ is the voltage of bus ‘i’, $I_{L_j}$ is the current of line ‘j’, and $Z_j$ is the impedance of line ‘j’. Loss Target is the target value for power losses, calculated when the microgrid is formed, with loss minimization included only in the objective function. $P_{Loss}\left(m\right)$ is the power loss in microgrid ‘m’. ‘n’ is the total number of lines between buses in microgrid ‘m’; $I_{L_j}$ is the current flow, and $R_j$ is the resistance of line ‘j’ of microgrid ‘m’ [34].

$${\lambda }_{\mathrm{Adequacy}}={\displaystyle \frac{{\mathrm{Adequacy}}_{\mathrm{Real}}}{{\mathrm{Adequacy}}_{\mathrm{Target}}}}.$$

(20)

$${\mathrm{Adequacy}}_{\mathrm{Real}}={\displaystyle \sum _{m=1}^M}\sum _{i,j}\left(P_{G_{m,i}}-P_{L_{m,i}}-P_{{Loss}_{m,j}}\right).$$

(21)

${\mathrm{Adequacy}}_{\mathrm{Target}}$ is the target value of the Index Adequacy calculated when the microgrid is formed, with the adequacy index included only in the objective function. $P_{G_{m,i}}$ is the real power injection at node ‘i’; $P_{L_{m,i}}$ is the load connected at node ‘i’; $P_{{Loss}_{m,j}}$ is the power loss in line ‘j’ in microgrid ‘m’ and ‘j’ is the line between two nodes.

Reliability index (${\lambda }_{\mathrm{Reliability}}$) is given by Equation (22):

$${\lambda }_{\mathrm{Reliability}}={\displaystyle \frac{1}{{\lambda }_{\mathrm{ELL}}}}.$$

(22)

${\lambda }_{\mathrm{ELL}}$ is the Expected Loss Load (ELL) given by Equation (23):

$${\lambda }_{\mathrm{ELL}}={\displaystyle \sum _{i=1}^n}{P}_i\left(C_i-L_i\right).$$

(23)

$P_i(C_i-L_i)$ is the probability of loss of load on day ‘i’, $C_i$ is the availability capacity on day ‘i’, and $L_i$ is Forecast Peak Load on the day ‘i’ [34]. Thus, the ANFIS weight coefficients can be estimated using Equation (24):

$${\mu }_i={\lambda }_{\mathrm{TPL}}\left(i\right)+{\lambda }_{\mathrm{Adequacy}}\left(i\right)+{\lambda }_{\mathrm{Reliability}}\left(i\right).$$

(24)

3.2.2. Modeling Objective Function II

The following are considered the network active operations for this work:

• Line Switching operations ($LS$).
• $DER$ allocations ($DA$).
• Load shedding/peak shaving ($PS$).
• Physical Network Repair by linemen in cases where network power line re-energization could not be established via remote network re-configuration ($NR$).

$${\mathrm{OF}}_{\mathrm{Total}}={\mathrm{OF}}_{\min .\mathrm{D}}+{\mathrm{OF}}_{\min .\mathrm{T}},$$

(25)

where

$${\mathrm{OF}}_{min.D}=min\sum _{\forall i}\sum _{\forall t}{\beta }_i^D\left(L{S}_{i,t}^D+D{A}_{i,t}^D+P{S}_{i,t}^D+N{R}_{i,t}^D\right).$$

(26)

$${\mathrm{OF}}_{\min .\mathrm{T}}=min\sum _{\forall i}\sum _{\forall t}{\beta }_i^T\left(L{S}_{i,t}^T+D{A}_{i,t}^T+P{S}_{i,t}^T+N{R}_{i,t}^T\right).$$

(27)

${\beta }_i^D$ & ${\beta }_i^T$ are the $ANFIS$ weight coefficient allocations for the transmission and distribution operations, respectively. $LS$ and $DA$ are correlated (but modeled separately). $PS$ is based on historical data and the generation pattern of the $DS\&TS$. Additionally, control centers are strategically positioned to provide microgrid control from alternative locations in the event of abnormal operating conditions that may make one center inaccessible. The proposed hybridized multi-layer microgrid formation and optimization technique using $ANFIS$ aims to bridge this gap by leveraging adaptive learning and real-time optimization to enhance the grid’s ability to withstand and recover from disturbances.

3.2.3. Modeling Objective Function III

Minimization of generation, switching, and load curtailment costs.

$$\begin{array}{cc}\hfill min& \sum _{i\in \mathcal{G}}\left[C_{i,0}+C_{i,1}{P}_{g,i}+C_{i,2}{P}_{g,i}^2\right]u_{g,i}\hfill \\ \hfill & +\sum _{l\in \mathcal{L}}{C}_l{S}_l+\sum _{d\in \mathcal{D}}{\lambda }_{d}L{C}_d.\hfill \end{array}$$

(28)

The first part of Equation (28) minimizes generation cost using a quadratic cost function; the second part minimizes operation costs (switching and active operation costs); and the last part minimizes load curtailment cost. $P_{g,i}$ is the real power output of generator i; $C_{i,0},C_{i,1},C_{i,2}$ represent the generation quadratic cost coefficients; $u_{g,i}$ is the generator switching variable; $S_l$ is the switching decision; $C_l$ denotes the cost of switching; $L{C}_d$ is the amount of curtailed load; ${\lambda }_d$ represents penalty cost per unit of curtailed load.

3.2.4. Tie-Line Management and Dynamic Microgrid Design

A multi-microgrid dynamic economic dispatch case study of the modified IEEE 123-Bus network is presented. The mathematical modeling of the power systems involves using the GAMS V43.3.1 for numerical optimization of the network.

$$\begin{array}{cc}\hfill \underset{P_{i,t},P_{s,t}^{\mathrm{s}},P_{w,t}^{\mathrm{w}},{\mathrm{Tie}}_t^{a{a}^{*}}}{min}\mathrm{OF}& =\sum _t\sum _i\left(a_i{P}_{i,t}^2+b_i{P}_{i,t}\right)\hfill \\ \hfill & +\sum _t\sum _i{c}_i.\hfill \end{array}$$

(29)

$$P_i^{min}\le P_{i,t}\le P_i^{max}.$$

(30)

$$\begin{array}{cc}\hfill \sum _{s\in a}{P}_{s,t}+\sum _{w\in a}{P}_{w,t}+\sum _{i\in a}{P}_{i,t}& =\sum _{a^{*}}{\mathrm{Tie}}_t^{a{a}^{*}}+L_{e,t}^a,\hfill \\ \hfill & \forall i,s,w\in a,\forall t\hfill \end{array}.$$

(31)

$L_{e,t}^a$ is the electric load (kW) in the area ‘a’ in time (t).

$$\left|{\mathrm{Tie}}_t^{a{a}^{*}}\right|\le {\overline{\mathrm{Tie}}}^{a{a}^{*}}.$$

(32)

${\overline{\mathrm{Tie}}}^{a{a}^{*}}$ is the amount of transferable energy between area a and $a^{*}$.

$$0\le {\mathrm{Pw}}_{w,t}\le {\xi }_{w,t}{\overline{\mathrm{Pw}}}_w.$$

(33)

where ${\overline{\mathrm{Pw}}}_w$ are the capacities of the wind turbines in kW.

$$0\le {\mathrm{Ps}}_{s,t}\le {\xi }_{s,t}{\overline{\mathrm{Ps}}}_s.$$

(34)

where ${\overline{\mathrm{Ps}}}_s$ are the solar farms’ capacities in kW. ${\xi }_{w,t}$ and ${\xi }_{s,t}$ are wind and solar availability indices at time (t). The Distributed Energy Resource Management System employs the optimization modeling technique described above, ensuring the balance of power and the economic distribution of resources in the power network [38].

Figure 6 and Figure 7 illustrate the power network dynamic coordination, tie-line/$DERs$ management, and multiple microgrid formation on the modified IEEE 123-Bus distribution network, with dispatched resources operating in both grid-connected and island modes through T&D network handshake on the developed co-simulation platform. An overview of the expanded distribution power network in Figure 4 is shown in Figure 6.

3.3. Data Exchange in the T&D Co-Simulation

In [20], co-simulation is described as allowing simulators from different domains to interact by exchanging values during the simulation that define other simulators’ boundary conditions. Active power values, bus voltage magnitudes, switch status, dispatched resources, and curtailed load are exchanged at the feeder head through subscription and publication in MATLAB-GAMS (MATLAB v.R2025b, GAMS V43.3.1)data exchange protocols.

These variables are considered network physical variables that are exchanged at regular intervals. A feeder head is technically the coupling point between the transmission network and the distribution load via feeder lines, which transmit power to the loads in the distribution network area via a suitable transformer.

Figure 6 illustrates the T&D co-simulation network configuration, which involves the IEEE 14-Bus $TS$ sub-sections and the modified IEEE 123-Bus $DS$ network. Figure 7 is the summarized dynamic management of $DERs$ and tie-lines in the predefined 4-microgrid use case on the $DS$ network in Figure 6. The detailed network configuration parameters of Figure 7 are given in

Table A1. Network configuration details of Figure 7.

Symbol	Quantity	Quantity Values
$A1\_A2$	Tie-line inequality constraint between microgrid $A1$ and $A2$.	$-100\le P_{12}\le 100\left(\mathrm{kW}\right)$
$A1\_A3$	Tie-line inequality constraint between microgrid $A1$ and $A3$.	$-400\le P_{13}\le 400\left(\mathrm{kW}\right)$
$A1\_A4$	Tie-line inequality constraint between microgrid $A1$ and $A4$.	$-300\le P_{14}\le 300\left(\mathrm{kW}\right)$
$A2\_A3$	Tie-line inequality constraint between microgrid $A2$ and $A3$.	$-500\le P_{23}\le 500\left(\mathrm{kW}\right)$
$A2\_A4$	Tie-line inequality constraint between microgrid $A2$ and $A4$.	$-200\le P_{24}\le 200\left(\mathrm{kW}\right)$
$A3\_A4$	Tie-line inequality constraint between microgrid $A3$ and $A4$.	$-600\le P_{34}\le 600\left(\mathrm{kW}\right)$
${ESS}_1$	Battery Energy Storage System in microgrid area ($A1$).	200$\mathrm{kW}$
${ESS}_2$	Battery Energy Storage System in microgrid area ($A2$).	100$\mathrm{kW}$
${ESS}_3$	Battery Energy Storage System in microgrid area ($A3$).	200$\mathrm{kW}$
${ESS}_4$	Battery Energy Storage System in microgrid area ($A4$).	100$\mathrm{kW}$
$P_{g1}$	Convention Gen-1 constraint.	$20\le P_{g1}\le 150\left(\mathrm{kW}\right)$
$P_{g2}$	Convention Gen-2 constraint.	$40\le P_{g2}\le 200\left(\mathrm{kW}\right)$
$P_{g3}$	Convention Gen-3 constraint.	$30\le P_{g3}\le 300\left(\mathrm{kW}\right)$
$P_{g4}$	Convention Gen-4 constraint.	$30\le P_{g4}\le 350\left(\mathrm{kW}\right)$
$P_{g5}$	Convention Gen-5 constraint.	$10\le P_{g5}\le 70\left(\mathrm{kW}\right)$
$P_{g6}$	Convention Gen-6 constraint.	$20\le P_{g6}\le 80\left(\mathrm{kW}\right)$
$P_{g7}$	Convention Gen-7 constraint.	$40\le P_{g7}\le 450\left(\mathrm{kW}\right)$
$P_{g8}$	Convention Gen-8 constraint.	$50\le P_{g8}\le 130\left(\mathrm{kW}\right)$
$P_{g9}$	Convention Gen-9 constraint.	$100\le P_{g9}\le 340\left(\mathrm{kW}\right)$
$P_{g10}$	Convention Gen-10 constraint.	$40\le P_{g10}\le 130\left(\mathrm{kW}\right)$
$P_{g11}$	Convention Gen-11 constraint.	$40\le P_{g11}\le 650\left(\mathrm{kW}\right)$
$P_{g12}$	Convention Gen-12 constraint.	$30\le P_{g12}\le 350\left(\mathrm{kW}\right)$
$P_{g13}$	Convention Gen-13 constraint.	$15\le P_{g13}\le 125\left(\mathrm{kW}\right)$
$P_{g14}$	Convention Gen-14 constraint.	$40\le P_{g14}\le 650\left(\mathrm{kW}\right)$
$P_{s1}\&P_{s2}$	Solar Peak Power Ratings on microgrids $A2$ and $A4$.	$P_{s1}$ = 100 kW, $P_{s2}$ = 150 kW
$P_{w1}\&P_{w2}$	Wind Peak Power Ratings on microgrids $A1$ and $A3$.	$P_{w1}$ = 250 kW, $P_{w2}$ = 350 kW

. However, $DERs$ are gradually integrated into the distribution network to monitor the impact of each resource on the total operating cost of the DS, while operating in island mode based on the modified IEEE 123-Bus network configuration in Figure 6.

The block diagram in Figure 8 illustrates the machine learning component (ANFIS) enhancement of the GAMS numerical optimization engine by providing appropriate load and power generation weights ($W_L$ = Load Weight, and $W_G$ = Power Generator’s Weight). Optimized values are fed into the MATPOWER v8.0 environment for optimal power flow analysis for Simulink (MATLAB v.R2025b) model control and, finally, result visualization. Appropriate switch controls come from the GAMS numerically optimized output, ensuring proper network reconfiguration and appropriate resource allocation.

4. Result Analysis

This section is divided into three broad categories, namely:

• T&D co-simulation case studies;
• Multi-microgrid dynamic economic dispatch of $DERs$;
• Comparison of proposed ANFIS approach with LSTM and GNN learning models.

4.1. T&D Co-Simulation Case Studies

Case-1 is the co-simulation results between the IEEE 14-Bus $TS$ and the unbalanced IEEE 123-Bus $DS$ networks, shown in Figure 9, Figure 10 and Figure 11, with two different coupling points (Bus-15 and Bus-123 on the $DS$ network) using Bus-1 on the $TS$ network in both cases. These coupling points are determined after running an optimization subroutine that selects the coupling point with the lowest active power loss following power flow.

Figure 9 presents the bus voltages of both $TS$ and $DS$ networks before and after numerical optimization using the $GAMS$ numerical optimization engine, using Mixed Integer Linear Program (MILP) in $CPLEX$ solver, and Quadratic Constrained Program (QCP) for dynamic microgrid resources management on IPOPT solver. The results included two different coupling points used for the research work based on a systematic search for the best common coupling point to minimize load loss on the networks in an economically viable manner during the $HILF$ event, as shown in Figure 9. The $DS$ network bus voltages are significantly improved after optimization as compared to the $TS$ network. $DSN$ and $TSN$ are the distribution and transmission network bus voltages before optimization. $DSA15$ and $TSA1$ are the bus voltages using buses 15 and 1 between $DS$ and $TS$ networks respectively. $DSA123$ and $TSA{1}^{*}$ are network bus voltages after optimization using coupling points buses 123 and 1 between $DS$ and $TS$ networks. The objective function (operational cost) reduced from $\$25.76/$h to $\$9.24/$h after network optimization ($OPT$) of the distribution system. This is a 68% reduction in the distribution network operating cost ($\$/$h) as compared to normal $DS$ without the developed optimization scheme in Figure 10. The overall active and reactive power losses in the $DS$ network were reduced by 3.23% and 13.3%, respectively, compared with the benchmark IEEE 123-Bus $DS$ network at full load, as shown in Figure 11. The green and light green bar charts to the left are benchmarks, while the optimized results in red and orange bars are optimized results to the right of Figure 11.

From Figure 6, multiple sub-sections of the $TS$ network were used to restore power to the $DS$ microgrid networks (one at a time), ensuring maximum power transfer to the unserved loads with respect to tie-line constraints, ensuring the least active power loss on both networks. While coupling bus-1 of the $TS$ network to bus-15 of the $DS$ network, the measured bus voltages (blue profile in Figure 9) between bus-1 and 61 on the $DS$ network optimized close to 1 p.u. Further away from the point of common coupling, the bus voltages are below 1 p.u. but within the acceptable operating range. From Figure 9, establishing a coupling point between bus-1 $TS$ and bus-123 $DS$ gave the green bus voltage profile with voltages on bus-66 to 123 optimized around 1 p.u, as compared to buses before bus-66. This is in response to the unserved load location. Overall, the network bus voltages were better managed, and power losses were curtailed using the T&D co-simulation platform and proposed optimization solution.

Case-2 study involves the co-simulation results with the IEEE 14-Bus $TS$ and IEEE 15-Bus $DS$ network. The coupling point is between bus-1 on $TS$ and bus-15 on the $DS$ network. The bus voltage profile after network optimization in GAMS $OPF$ is shown in Figure 12.

Figure 12 shows the bus voltage profiles before and after optimization using GAMS $OPF$ and the co-simulation platform. $DSN$ is the distribution network bus voltage before network optimization using the developed co-simulation platform. $DSA15$ is the distribution network bus voltage after optimization. $TSN$ is the transmission network bus voltage before optimization. $TSA1$ is the transmission network bus voltage after optimization. $Min-Volt$ and $Max-Volt$ are the minimum and maximum bus voltage limits. The bus voltage violations observed before the optimization are eliminated after network optimization with the use of the proposed solution on both networks.

Figure 13 shows the $P-Q$ power loss bar charts in the $TS$ network, where the active and reactive power losses in the $TS$ network are reduced by 0.81% and 0.87%, respectively. A reduction of 3.23% in active power and 16.7% in reactive power loss is recorded in $DS$ after network optimization, as shown in Figure 14. Overall, the power network bus voltages and power flows are better managed in both networks with the proposed solution.

4.2. Multi-Microgrid Area Dynamic Economic Dispatch of DERs

Case-3 is the dynamic microgrid tie-line management and economic dispatch of $DERs$. Figure 7 models the dynamic microgrid formed in Figure 6.

Figure 15 shows the 24 h wind availability ratios of the two microgrid areas ($A1$ and $A3$) with wind turbines. The solar availability ratio is shown in Figure 16 on microgrid areas ($A2$ and $A4$).

Figure 17 presents the 24 h ANFIS-enabled multi-microgrid demand predictions to be adopted for the model under consideration, based on historical microgrid load profiles spanning 2 years.

Figure 18 is the multi-time-step 14-conventional generators dispatched on the distribution network (IEEE 123-Bus) after optimization. This is a dynamic economic dispatch of $DERs$ on the distribution network (in Figure 7), modeled after the modified IEEE 123-Bus network in Figure 7 in island mode of operation, while earlier results on T&D co-simulation are in grid-connected mode of operation.

The wind and solar resources dispatched to meet the set of optimization objective functions are shown in Figure 19 and Figure 20, respectively.

The state of charge of the Battery Energy Storage System ($BESS$) is given by Figure 21 after $OPT$, with a minimum $BESS\_SOC$ charging and discharging power set at 20% maximum $SOC$ and a minimum set at 0% of $SOC$ at any time (t) of simulation. This ensures that the $BESS$ remains in good operating condition without draining it completely or overcharging the system, which can reduce its lifespan.

From Figure 21, between hours 13 and 17, $BESS\_SOC$ is at the peak for microgrids corresponding to the time range of peak solar and wind power supply, during which period the $BESS$ is charging. The charging and discharging profiles of the $BESS$ are shown in Figure 22. As expected, the BESS charges during the low-load periods and discharges at peak load periods when supply can no longer meet the demand.

Figure 22 presents the efficient charging and discharging capability of the $GAMS$ optimization algorithm on the microgrids, with subsequent charging and discharging of the $BESS$ for a particular microgrid occurring at different hours as expected. ‘$BESS1c$, $BESS2c$, $BESS3c$, $BESS4c$’ are the hourly battery charging powers ($\mathrm{kW}$) requirement on microgrids 1, 2, 3, and 4, respectively. Major battery discharges occur at the 10th and 18th hours, corresponding to the lowest $BESS\_SOC$ power, as shown in Figure 21. The hours correspond to the peak-load hours. ‘$BESS1d$, $BESS2d$, $BESS3d$, $BESS4d$’ are the hourly battery discharged power ($\mathrm{kW}$) requirements to keep microgrid operation at optimal performance on microgrids 1, 2, 3, and 4, respectively.

The inter-microgrid power transfer is illustrated in Figure 23, where the optimal switching of the inter-microgrid tie-line results in economic and safe operation of the distribution network, with a reduction in operating cost when ANFIS-based numerical optimization is deployed. The negative power flows represent reverse power flows to microgrids, with insufficient power to sustain the maximum possible loads on the microgrid (e.g., tie-lines $A3\_A1$ and some sections of $A2\_A4$ at hours 10, 13, 14, 16–18, 21). Objective function reduced from $\$\mathrm{74,692.399}$ to $\$\mathrm{71,015.540}$ after optimization. The cost of operating the distribution network has reduced by 4.92% after improving resource availability through optimized tie-line switching operations without deploying additional resources.

Figure 24 presents the power serving capacity ($PSC$) of the modified IEEE 123-Bus distribution network, comparing the benchmark phase active–reactive power supplied to the $HILF$ event-impacted (curtailed load) case, where power lines are disconnected, and $DERs$ are eliminated from the network due to the impact of the $HILF$ event simulated. The first set of histograms, from left to right, is the benchmark measurements followed by the curtailed load measurements. A $PSC$ of 75.12% was achieved with the worst $HILF$ event simulated, using the proposed hybridized multi-layer microgrid optimization solution.

4.3. Solution Comparison of Proposed ANFIS Approach with LSTM and GNN Learning Models

This section presents solution comparisons between ANFIS-enabled network restoration, LSTM and GNN learning techniques. Since power network restoration is expected to be as fast as possible, a 24 h data window was selected with 14-Bus IEEE TS and modified unbalanced 123-Bus IEEE DS network for the comparison. A data sampling step of 3600 s is adopted in data capture.

The input data is 24 by 12 CSV file data set made up of 12 columns of power network parameters. The input variables are simulation time (s), transmission simulation time-step in seconds ($\Delta {\mathrm{T}}_T$), distribution simulation time-step ($\Delta {\mathrm{T}}_D$), total distribution load normalized (p.u.), battery state of charge (0–1), voltage at the coupling point between TS and DS network (p.u.), current at the coupling point between TS and DS network (p.u.), load weight/fuzzy load priority (0–1, used for $gbellmf$), targeted synchronized time-step output in seconds ($\Delta {\mathrm{T}}_{Sync}$), output weighting factor ($\alpha$, 0–1), voltage correction at coupling point between TS and DS network ($\Delta {\mathrm{V}}_{Corr}$, p.u.), and current correction at coupling point between TS and DS network ($\Delta {\mathrm{I}}_{Corr}$, p.u.). Total distribution load normalized $P_R^{\mathrm{norm}}$ is given by Equation (35).

$$P_R^{\mathrm{norm}}={\displaystyle \frac{P_R}{P_R^{\max }}}\in [0,1].$$

(35)

$P_R$ is given by Equation (36).

$$P_R={\displaystyle \frac{{\sum }_{i=1}^n{P}_i}{P_{\mathrm{base}}}}(\mathrm{p}.\mathrm{u}.),\forall i\in \{1,2,3,\dots ,n\},$$

(36)

where $P_{\mathrm{base}}$ is the base power of the DS network in $MW$, which is consistent for all the four microgrids in the DS network. A $P_{\mathrm{base}}$ of 5 MW is chosen for the analysis. $P_i$ is the load on bus ‘i’. ‘n’ is the total number of load bus in the DS network. $P_R$ is further normalized by maximum DS network load ($P_R^{\max }$) for ANFIS optimization purpose which needed load weights. The voltage correction at $PCC$ is given by Equation (37).

$$\Delta V_{\mathrm{corr}}={\displaystyle \frac{V_{\mathrm{desired}}-V_{\mathrm{measured}}}{V_{\mathrm{base}}}},$$

(37)

where $V_{\mathrm{desired}}$ is the target voltage at $PCC$ (volts). $V_{\mathrm{measured}}$ is the actual measured voltage at $PCC$ (volts) and $V_{\mathrm{base}}$ is the base voltage of the system (volts).

$$V_{\mathrm{base}}=V_{\mathrm{rated},\mathrm{line}\text{-}\mathrm{to}\text{-}\mathrm{neutral}}.$$

(38)

A $V_{\mathrm{base}}$ of 277V has been chosen for the analysis estimated from $V_{\mathrm{rated},\mathrm{line}\text{-}\mathrm{to}\text{-}\mathrm{line}}$ (0.48 kV) for the $DS$ network. From the estimated values, corrections are applied accordingly to ensure $DS$ network voltage stability as given by Equation (39).

$$\Delta V_{\mathrm{corr}}=V_{\mathrm{desired}}^{pu}-V_{\mathrm{PCC}}^{pu}.$$

(39)

A positive $\Delta V_{\mathrm{corr}}$ ($\Delta V_{\mathrm{corr}}$ > 0) indicates low voltage, requiring voltage increase through either reactive power injection, or transformer tap-up. A negative $\Delta V_{\mathrm{corr}}$ ($\Delta V_{\mathrm{corr}}$ < 0) is an indicator of too high voltage and requires reduction through load increase or generation curtailment.

The current correction at the $PCC$ in p.u. is given by Equation (40).

$$\Delta I_{\mathrm{corr}}=I_{\mathrm{desired}}^{pu}-I_{\mathrm{measured}}^{pu},$$

(40)

where $I_{\mathrm{desired}}$ is the target $PCC$ current (p.u.). $I_{\mathrm{measured}}$ is the actual/measured $PCC$ current (p.u.), and $I_{\mathrm{base}}$ is the base current of the DS system for p.u. normalization. $I_{\mathrm{base}}$ is given by Equation (41).

$$I_{\mathrm{base}}={\displaystyle \frac{S_{\mathrm{base}}}{\sqrt{3},V_{\mathrm{LL},\mathrm{base}}}},\left(\mathrm{for}\mathrm{three}\text{-}\mathrm{phase}\mathrm{system}\right),$$

(41)

where $S_{\mathrm{base}}$ is the base apparent power ($MVA$), $V_{\mathrm{LL},\mathrm{base}}$ is the base line-to-line voltage (kV). A positive $\Delta I_{\mathrm{corr}}$ indicates the need for more current injection, which can be achieved through an increase in $DER/BESS$ output. A negative $\Delta I_{\mathrm{corr}}$ means the need to reduce current via load or generation curtailment.

Alpha ($\alpha$) is generally an output weighting factor used in:

• ANFIS/fuzzy logic systems (as the fuzzy output scaling factor);
• Weighted aggregation of multiple variables like normalized load, renewable generation, and unserved energy;
• Multi-objective optimization such as balancing load priorities, cost, and energy not served.

Mathematically, for multiple outputs ($y_1$, $y_2$, …, $y_n$), the weighted output is given by Equation (42).

$$y_{\mathrm{weighted}}={\displaystyle \sum _{i=1}^n}{\alpha }_i{y}_i,$$

(42)

where $0\le {\alpha }_i\le 1$ and ${\sum }_i{\alpha }_i=1$ when normalized.

There are four output ($\Delta {\mathrm{I}}_{Corr}$, $\Delta {\mathrm{V}}_{Corr}$, $\alpha$, and $\Delta {\mathrm{T}}_{Sync}$) variables, making the learning models multi-input multi-output (MIMO) systems. The machine learning models are compared over the predicted output variable accuracy and training/simulation time. Figure 25 presents the $\Delta {\mathrm{I}}_{Corr}$ predictions on ANFIS, LSTM, and GNN learning models comparison results. The ANFIS model outperformed the LSTM and GNN models. ANFIS records an accuracy value of 99.99%, RMSE of 0 in 0.13 s. LSTM has an accuracy of 16.7%, RMSE of 0.01 in 3.84 s, while GNN has 0% accuracy, RMSE of 0.138 in 0.001 s. ANFIS has the best forecast on $\Delta {\mathrm{I}}_{Corr}$ with a short prediction window, which is suited for the DS restoration procedure.

Figure 26 presents the $\Delta {\mathrm{V}}_{Corr}$ predictions on the ANFIS, LSTM, and GNN learning models’ comparison results. The ANFIS model outperformed the LSTM and GNN models. ANFIS records an accuracy value of 100%, and an RMSE of 0 in 0.15 s. LSTM has an accuracy of 87.5%, RMSE of 0.032 in 3.48 s, while GNN has 0% accuracy, RMSE of 1.866 in 0.001 s. ANFIS has the best forecast on $\Delta {\mathrm{V}}_{Corr}$ with a short prediction window, which is suited for the DS restoration procedure.

Figure 27 presents the $\alpha$ forecasts on the ANFIS, LSTM, and GNN learning models’ comparison results. The ANFIS model outperformed the LSTM and GNN models. ANFIS records an accuracy value of 100%, RMSE of 0 in 0.18 s. LSTM has accuracy of 16.7%, RMSE of 0.005 in 3.48 s, while GNN has 0% accuracy, RMSE of 0.534 in 0.001 s. ANFIS has the best forecast on $\alpha$ with a short prediction window, which is suited for the DS restoration procedure.

Figure 28 presents the $\Delta {\mathrm{T}}_{Sync}$ forecasts on the ANFIS, LSTM, and GNN learning models’ comparison results. The ANFIS model outperformed the LSTM and GNN models. ANFIS records an accuracy value of 100%, RMSE of 0 in 0.17 s. LSTM has accuracy of 75.0%, RMSE of 0.033 in 2.65 s, while GNN has 0% accuracy, RMSE of 1.377 in 0.001 s. ANFIS has the best forecast on $\Delta {\mathrm{T}}_{Sync}$ with a short prediction window, which is suited for the DS network restoration procedure.

5. Discussion

The ANFIS-enabled Energy Management System developed integrates short-term load forecasting and adaptive weighting of energy resources into the developed multi-microgrid optimal dispatch model. The ANFIS forecasting layer estimates the future power demand of four microgrids, while the ANFIS weighing layer dynamically adjusts conventional generators and load priorities according to forecasted demands, renewable availability, battery SOC, electricity price, and operating mode. The resulting optimization minimizes operating cost, battery degradation, renewable curtailment, and weighted load shedding subject to generators, storage, network configuration, tie-lines, and point of common coupling (PCC) constraints in both grid-connected and islanded operation. The developed unified co-simulation framework goes beyond mere integration by algorithmically addressing synchronization and fragmentation issues. It employs adaptive hierarchical time-stepping, predictive state estimation, and centralized boundary reconciliation to ensure that T&D subsystems share consistent states and accurately track fast dynamics. Compared to conventional RTDS-OPAL-RT or FMI/FMU setups, this approach reduces temporal discrepancies, improves convergence, and maintains numerical fidelity, effectively overcoming the fundamental synchronization and fragmented modeling challenges inherent in T&D coupling.

While the focus of this work is on the integration and real-time operation of a co-simulation platform, it is fully compatible with advanced techniques such as blockchain or homomorphic encryption for privacy-preserving computations in future extensions. The research work establishes the following points:

• Multiple renewable energy resources improve the power system reliability and resiliency, as shown by the PSC of the networks after a simulated HILF event.
• The deployment of $DERs$ reduces the operational cost of the $DS$ network significantly compared to the $TS$ network.
• However, these benefits come at a cost of computational complexity, which machine learning aims to reduce.
• The developed co-simulation platform drastically reduces computation time.
• The developed solution effectively serves as a distributed energy resources management system (DERMS) platform where multiple resources can be managed and optimized.

6. Conclusions

This research comprises two major case studies of power networks: the 14-Bus IEEE transmission system with 15-Bus IEEE distribution networks (Case I) and the 14-Bus $TS$ with an unbalanced $3\text{-}\varphi$ 123-Bus IEEE (modified) $DS$ network (Case II), respectively. These two power systems were synthesized on MATLAB Simulink and optimized using a numerical optimization tool ($GAMS$). The systemic ANFIS-based optimization technique enhanced system resiliency while maximizing load. Unbalanced three-phase voltage analysis and use of $GAMS$, $MATPOWER$, and $MATLAB-Simulink$ targeted at making informed decisions and initiating appropriate controls. Optimal power flow analysis of the system ($OPF$), in conjunction with an Adaptive Neuro-Fuzzy Inference System ($ANFIS$) implemented in the allocation of weight factors to constraints considered in the definition of the objective function, achieved a 68% reduction in the operating cost of the unbalanced 123-Bus IEEE distribution network with the right network interaction when compared with the network running independent of the TS network and without optimization. Running the DS network in the island mode without the optimization algorithm developed will result in a higher operation cost.

In Case-1, there is a reduction in the estimated active and reactive power losses across the T&D network (0.81% and 0.87% reductions in the active and reactive power losses of the transmission network, while 3.23% and 16.7% reductions are achieved in the active and reactive power losses of the distribution network, respectively). In case II, active and reactive power losses were reduced by 3.23% and 13.3%, respectively, with the implementation of the proposed solution. The computational time was equally reduced as compared to the situation where the developed co-simulation platform was not used. The microgrid formation and allocation of Distributed Energy Resources ($DERs$) within the power network are governed by constraints, such as the grid frequency and voltage limitations, load type (critical loads), and the cost of generation, power line capacity, among others. The mix of these constraints is assigned a weight using the $ANFIS$ system, and the result is used by the Simulink model to optimally allocate required and available resources within the network.

The interaction between the distribution and transmission network is established using a robust co-simulation platform. Actual implementation was carried out on the MATLAB Simulink model of an IEEE-123 bus $DS$ network, where switches, tie-lines, power line segments, and $DERs$ are controlled for optimal results. $ANFIS$ is deployed in the prediction of load profiles for the research work. The research framework addressed, among others, the need for a robust T&D co-simulation platform, dynamic microgrid formation, network switching, operation cost minimization, Optimal Power Transfer between the networks, network loss curtailment, and load management.

Although ANFIS suffers from the curse of dimensionality if not well managed, the proposed solution handles the power system networks in a modular form, where memberships are carefully determined based on the expert knowledge of the system. The proposed solution is well-suited for short term restoration purposes.

Future research will focus on applying the proposed solution to larger power networks. Moreover, future deployment of the developed optimization solution on the real-time simulator (RT-Lab-based testbed, OPAL-RT, Hyper-Sim) will further evaluate real-time benefits of the proposed solution compared to the conventional methods.