An Automated Load Restoration Approach for Improving Load Serving Capabilities in Smart Urban Networks

Abstract

In this paper, a very fast and reliable strategy for load restoration utilizing optimal distribution feeder reconfiguration (DFR) is developed. The automated network configuration switches can improve the resilience of a microgrid (MG) equipped with a centralized and coordinated energy management system (EMS). The EMS has the authority to reconfigure the distribution network to fulfil high priority loads in the entire network, at the lowest cost, while maintaining the voltage at desirable bounds. In the case of islanded operation, the EMS is responsible for serving the high priority loads, including the establishment of new MGs, if necessary. This paper discusses the main functionality of the EMS in both grid-connected and islanded operation modes of MGs. The proposed model is developed based on a mixed-integer quadratically constrained program (MIQCP), including an optimal power flow (OPF) problem to minimize the power losses in normal operation and the load shedding in islanded operation, while keeping voltage and capacity constraints. The proposed framework is implemented on a modified IEEE 33-bus test system and the results show that the model is fast and accurate enough to be utilized in real-life situations without a loss of accuracy.

1. Introduction

1.1. Motivation

Increasing the penetration rate of renewable energy resources (RES) into the distribution networks introduces new challenges to the operation, control, and protection of smart grids (SGs) [1]. In this regard, the planning and operation entities are facing new challenges, and there are vast research projects that have been developed to address these challenges. Most of the practical issues in distribution network operation are loss reduction, voltage profile improvement, increasing the resilience of the network, and power quality issues. In the context of observability of the SGs, the automation of distribution networks has been considered in recent years. Introducing new technologies in the field of automation and human–machine interface (HMI), as well as the development of the communication infrastructure, provide the fundamental technology for network automation. In terms of power system apparatus, the installation of remotely operated switches and re-closers enhanced the flexibility of network reconfiguration in a reliable way, resulting in improved network resilience and optimal network operation under severe weather conditions.

The load restoration in the presence of local and onsite power generating units would be significantly enhanced. The main challenge in this paradigm is how to deal with the new reference units at each islanded microgrid (MG). The grid-forming units could provide the reference voltage in the islanded operation mode of the MG. In this case, assigning the grid-forming units to the MG with the main objective of restoring the critical loads as much as possible, while maintaining the technical constraints of the network operation. In this paper, the load restoration technique in the presence of the grid-forming units has been addressed as an optimization problem, including the power flow constraints in the radial distribution networks in the presence of multiple local power generators.

1.2. Literature Review

Network reconfiguration and changing the topology of the distribution network has been addressed in the literature focusing on different objective functions. Loss reduction [2], voltage profile enhancement [3], reliability improvement [4], congestion management [5], and cost reduction have been addressed as objective functions [6]. In terms of modeling, the reconfiguration of the network has been investigated as both deterministic [7] and as stochastic optimization problems [8]. In some cases, robust optimization has been developed to deal with the uncertainties of load and RES power generation [9]. Another relevant study case arises from the fact that the distribution network is vulnerable to extreme natural disasters, specifically in the networks with long feeders and distributed laterals [10].

Grid resilience improvement against such events has been introduced in the literature [11,12,13]. This ability can be attained through new control and management strategies. Increasing the penetration rate of new resources, with proper capability, along with the improvement of the network’s self-healing and self-sufficiency possibilities could provide more flexibility to the network operation and enhance the resilience of the power system [13]. Therefore, new restoration services have been introduced to cope with unexpected events and reduce the outage time, while improving the restoration rate of critical loads [14,15,16].

Additionally, there are recent research works focused on network restoration, resilience improvement and providing flexibility in the presence of distributed generation (DG) as well as electrical energy storage (EES) units. The optimal utilization of automatic and remote-controlled switches has been addressed in [17] for an automated active distribution system. A bi-level robust optimization has been developed to leverage the integrated resilience response for the automated network while providing the independent preventive and independent emergency responses by proper topology reconfiguration. The preventive response includes network reconfiguration and re-dispatching of DG units, while the emergency response considers load shedding as an extra option apart from preventive actions.

Maximizing the utility profit, as well as resilience enhancement, have been studied in [18] by introducing the reachability concept for both demand and generating units. The improvement of the system’s resilience via the optimal placement of new photovoltaic generating units and storage devices has also been addressed.

The notion of “aromatic networks” as an innovative framework for power distribution systems is presented in [19]. This research emphasizes the susceptibility of current power grids to natural disasters and suggests aromatic networks, modeled after chemical structures, to improve robustness, resilience, and self-healing capacities, thus reducing blackout durations. The configurations and reconfigurations of these networks have been assessed, illustrating their capacity for sustainable energy supply in smart grids and microgrids. Enhancing the resilience of distribution systems, particularly for essential loads following disasters, is a primary focus in [20]. This study presents a concept for a “Sustainable and resilient smart home”, utilizing plug-in hybrid electric vehicles (PHEVs) for emergency power generation through vehicle-to-grid (V2G) systems. The emphasis is on optimizing power injection into the main grid during short power outages, using solar photovoltaic (PV) systems and energy storage systems within intelligent home management, while taking into account resident comfort levels. Ref. [21] included an extensive analysis of EMS inside microgrids, highlighting its significance in the integration of renewable energy and the maintenance of optimal control for efficiency, dependability, and power quality. The research delineates MG elements, control strategies (primary, secondary, tertiary), aims of EMS (cost reduction, restoration, power quality), and evaluated protection mechanisms and policy ramifications. It emphasized the significance of microgrids in swiftly reinstating power systems following physical or cyberattacks. The planning views for smart distribution networks (SDNs) have been comprehensively investigated in [22]. This paper classified SDNs into categories such as microgrids, multi-microgrids, virtual power plants (VPPs), smart homes, and smart cities. It addressed the necessity for proactive planning, generation and load forecasts, resource distribution, and topology selection to improve dependability and resilience. The paper emphasized the conversion of traditional networks into active systems via automation and the integration of diverse distributed energy supplies. A novel restoration technique for microgrids during a blackout, addressing key loads, was proposed in [23]. This research work employed DG with black start capabilities and mixed-integer linear programming (MILP) to model the restoration process, with the objective of optimizing the restoration of important loads. This reference proposed approaches for evaluating electric load weight through multi-criteria decision making (MCDM) and testing the black start capability of DG, indicating effectiveness on the IEEE 39-bus test system.

Ref. [24] evaluated optimal resource allocation in hybrid microgrids, informed by demand response strategies. It outlined an approach to reduce energy consumption from the electrical grid by leveraging existing microgrid resources and demand control via a home automation system. The study led to substantial decreases in electricity usage, highlighting the opportunity for improved operational efficiency in university campus microgrids. A thorough examination of sustainable power system resilience against extreme weather events and climate change was presented in [25]. This paper addressed diverse weather management strategies, machine learning algorithms for vulnerability detection, and the essential function of microgrid technologies in bolstering resilience, especially during black start restoration processes. Ref. [26] provided a comprehensive analysis of distribution network reconfiguration (DNR) methodologies. It elucidated how DNR augments network sustainability by optimizing architecture to minimize power losses, enhance voltage profiles, equilibrate loads, and bolster dependability. The review classified methods into heuristic, metaheuristic, conventional, and modern approaches. It also tested their applicability for both static and dynamic reconfiguration, and emphasizing dynamic reconfiguration as a critical area for further investigation in smart networks. Ref. [27] presented a rule-based modular EMS designed for AC/DC hybrid microgrids. This system automatically adjusted to microgrid designs, representing various components as distinct entities with their own restrictions. The modular architecture facilitated versatile implementation in both basic and sophisticated microgrid configurations. This optimized operational costs and improved service reliability in grid-connected and islanded modes. Ref. [28] included the augmentation of smart microgrid resilience during natural disasters via VPP allocation utilizing the jellyfish search algorithm. It incorporated various distributed energy resources within interconnected microgrids to guarantee a continuous power supply to essential services. The research assessed the proposed method’s resilience, reliability, stability, and emission reduction efficacy via practical scenarios and a multi-objective function. An extensive distribution system restoration strategy has been examined in [29] considering the presence of large-scale electric buses with centralized coordination, to promote load serving. The mathematical problem was formulated as a mixed-integer programming problem to benefit from off-the-shelf commercial solvers. To provide the energy in severe conditions, an adjustable timing for rescheduling of the load served at electric buses has been established according to the flexibility that can be provided by the centralized electric transportation system.

The dynamic performance of the network with high penetrated DGs during the restoration process has been addressed in [30]. The model proposed a control strategy for critical load restoration while considering the impact of the frequency limits using dynamic stability analysis. Intentional islanding, as a fast and flexible tool, has been introduced in [31]. A two-stage optimization framework has been proposed to frame the optimal configuration of islands in the occurrence of high-impact low-probability (HILP) events. Graph theory has been implemented to select the best islands, with the local power generation in the first stage being the feasibility study performed in the second stage. A combination of depth-first search and particle swarm optimization methods has been adopted for the first stage, and the optimal power flow (OPF) problem is investigated using a commercial power system study package.

1.3. Contributions

Distribution feeder reconfiguration (DFR) to minimize total network loss has been extensively studied in previous research works. Proposing a fast, precise, and reliable model to deal with the functionality of the DFR problem has become an important aspect of energy management systems (EMS). Increasing the penetration rate of inverter-based (RES) and electrical energy storage (EES) units promoted carbon emissions reduction and overall loss reduction, and leveraged the resiliency and flexibility of network operation. The presence of such units imposes complex network operation and protection scenarios. In addition, in the presence of the EMS, the operator can maintain critical loads during the occurrence of HILP events, by optimally reconfiguring the network and imposing the islanding operation of MGs [32]. In such circumstances, each MG needs one reference node to determine the voltage and frequency reference within the islanded section. In a centralized EMS, the islanded MGs can fulfil the critical loads in both connected and islanded operations of the distribution network. In this paper, an optimization model is investigated to address the network reconfiguration in the presence of HILP events dealing with assigning the distributed generation (DG) units with grid-forming capabilities at each MG. The proposed model is benefiting from the linearized optimal power flow problem and the corresponding constraints in the centralized EMS platform. The main contributions of this paper are as follows:

• Proposing a fast and accurate MIQCP model for load restoration for distribution networks with multiple DGs;
• Optimal network reconfiguration to maximize critical load restoration by assigning the grid-forming units for each islanded MG;
• Improving the functionality of the EMS for both grid-connected and islanded mode operation by optimal network reconfiguration with minimum switching actions in the network.

1.4. Paper Organization

The rest of this paper is as follows. In Section 2, the concept behind the developed model for MG operation is addressed. The mathematical problem formulation proposed in this paper as a standard MIQCP optimization problem is investigated in Section 3. The simulation results on the modified IEEE 33-bus test system for both normal and contingency events are provided in Section 4, and some concluding remarks are provided in the last section of this paper.

2. Multi-Purpose Model for Optimal Operation of MGs

The optimal operation of MGs for both grid-connected and islanded mode operation is the main target of developing the proposed model in this paper. In addition, the developed tool has the capability to form the dynamic islanding operation of the MGs in the presence of the grid-forming units. In the case of planned or unplanned islanding operations, the developed tool is able to dispatch the controllable units at each MG to serve the critical loads as much as possible, while the technical constraints of the network operation should be met. The availability of the grid-forming is essential to form an independent MG for both planned and intentional islanding operations. The grid-forming units at each MG will determine the reference voltage and frequency in such a way that the MG can operate independently.

In the presence of other kinds of resources in the MG, they will be managed by the EMS to serve the loads as dependent resources. The main goal is to dispatch the resources in such a way that the load curtailment would be minimized at each MG level and overall distribution network. Therefore, forming the MGs will be performed to maximize the load restoration for the contingent events and intentional islanding operations. In this regard, a generalized tool is needed to manage and dispatch the resources for all possible scenarios. In the grid-connected operation mode, the main objective is to minimize the system operation cost, considering the operational cost of DGs and the utility. In the planned or unplanned islanding operation, the main target of the EMS is minimizing the load curtailment cost. Therefore, in this paper, a multi-purpose optimization model is developed for distribution networks for both grid-connected and islanded mode operation.

3. MG Operation Problem Formulation

This section provides the developed mathematical problem formulation for the optimal operation of the MGs in both normal and contingency events in the presence of the grid-forming units. The main objective of the optimization model in this study is to minimize the total operating cost of the distribution network, including the network losses, as well as the load shedding cost imposed by the contingency events in both connected and islanded operation modes.

The objective function of the developed model is expressed as minimization of the expected cost of imported power from the upstream grid and generating cost of the DG units, as the first term, and the load shedding cost, as the second term for each scenario:

$$\begin{array}{cc}Min\hfill & \hfill \\ \hfill & {\displaystyle \sum _{k\in {\Gamma }_K}}{\rho }_k\left(\left({\displaystyle \sum _{i\in {\Gamma }_S}}{P}_{i,k}{c}_i^S+{\displaystyle \sum _{i\in {\Gamma }_{DG}}}{P}_{i,k}{c}_i^{DG}\right)+\left({\displaystyle \sum _{i\in {\Gamma }_N}}{P}_{i,k}^{LS}{\gamma }_i^D{\alpha }_{i }^D+{Q_{i,k}^{LS}\gamma }_i^D{\beta }_i^D\right)\right)\hfill \end{array}$$

(1)

In this case, the normal and islanded operation modes are addressed with the corresponding probability ${\rho }_k$. For each scenario, $k$, the optimization problem will introduce the best operation strategy to fulfil the objective function. In this paper, partially serving loads is not possible in the case of load curtailment. It is supposed that each load is connected via a circuit breaker and, in the case of a problem to maintain the loads, the total load connected to the corresponding bus is curtailed. Therefore, a priority for each load is included in this formulation to minimize critical load curtailment. The corresponding coefficient is ${\gamma }_i^D$, and the cost of load curtailment for active and reactive power are introduced as ${\alpha }_{i }^D$ and ${\beta }_i^D$, respectively. The technical constraints of the optimization problem are described in the next paragraphs.

The active and reactive balance equations are provided in (2) and (3), respectively. The superscript $SQ$ has also been utilized to indicate the quadratic terms. The nodal power balance equations include the substation power injection, DG power injection, load curtailment terms, and power injected to the branches for both active and reactive powers. The active and reactive are modeled as $P_{i, k}^D$ and $Q_{i, k}^D$, respectively. In the case of having demand response programs, the corresponding firm terms could be different for normal and contingency events. In this notation, the direction of power from bus $i$ to bus $j$ is modeled as $ij$. It is expected not to have load curtailment in normal operation scenarios, while in contingency events, the possibility of load shedding is addressed in the load balance equations.

$$\begin{array}{cc}{P}_{i,k}^S+P_{i,k}^{DG}+P_{i,k}^{LS}+{\displaystyle \sum _{ji\in {\Gamma }_L}}{P}_{ji,k}-\hfill & {\displaystyle \sum _{ij\in {\Gamma }_L}}\left(P_{ij,k}+R_{ij}{I}_{ij, k}^{SQ}\right)=P_{i, k}^D\hfill \\ \hfill & \forall \left(i\in {\Gamma }_N, k\in {\Gamma }_K\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}{Q}_{i,k}^S+Q_{i,k}^{DG}+Q_{i,k}^{LS}+{\displaystyle \sum _{ji\in {\Gamma }_L}}{Q}_{ji,k}-\hfill & {\displaystyle \sum _{ij\in {\Gamma }_L}}\left(Q_{ij,k}+X_{ij}{I}_{ij, k}^{SQ}\right)=Q_{i, k}^D\hfill \\ \hfill & \forall \left(i\in {\Gamma }_N, k\in {\Gamma }_K\right)\hfill \end{array}$$

(3)

The real power as well as reactive power losses will be articulated as (4) and (5), respectively. The previously specified losses apply to each operational situation.

$${P_{ij, k}^{Loss}=R}_{ij}{I}_{ij, k}^{SQ} \forall \left(ij\in {\Gamma }_L, k\in {\Gamma }_K\right)$$

(4)

$${Q_{ij, k}^{Loss}=X}_{ij}{I}_{ij, k}^{SQ} \forall \left(ij\in {\Gamma }_L, k\in {\Gamma }_K\right)$$

(5)

The corresponding load shedding for active and reactive powers are introduced in (6) and (7), respectively. As previously mentioned, the load shedding possibility would be applied for the total loads connected to each bus. In this case, the associated binary variable, ${LS}_{i, k}^D$, is considered to model the status of the circuit breaker at each bus.

$$P_{i,k}^{LS}={LS}_{i, k}^D{P}_{i, k}^D \forall \left(i\in {\Gamma }_N,k\in {\Gamma }_K\right)$$

(6)

$$Q_{i,k}^{LS}={LS}_{i, k}^D{Q}_{i, k}^D \forall \left(i\in {\Gamma }_N,k\in {\Gamma }_K\right)$$

(7)

The voltage losses across the feeders are incorporated into two complementary constraints within the OPF model for the radial distribution system, allowing for the reconfiguration of feeders. To guarantee the functionality of this technical constraint, the Big-M method is applied to the original voltage drop equation. The voltage drop, as indicated in (8) and (9), can be applied to the connected branch ij. In the case of open switches, the Big-M could easily guarantee the functionality of this constraint. In the case of opened switches, $y_{ij,t}=0$, this set of constraints will be relaxed in the optimization problem. In the case of the connecting link, i.e., $y_{ij,t}=1$, the voltage drop is represented by the Kirchhoff voltage law (KVL):

$$V_{i,k}^{SQ}-V_{j,k}^{SQ}\ge \left[2\left(R_{ij}{P}_{ij,k}+X_{ij}{Q}_{ij,k}\right)+Z_{ij}^2{I}_{ij,k}^{SQ}\right]-\left[1-y_{ij,k}\right]M$$

(8)

$$V_{i,k}^{SQ}-V_{j,k}^{SQ}\le \left[2\left(R_{ij}{P}_{ij,k}+X_{ij}{Q}_{ij,k}\right)+Z_{ij}^2{I}_{ij,k}^{SQ}\right]+\left[1-y_{ij,k}\right]M$$

(9)

In these constraints, the resistance and reactance of the physical branch or switches connecting bus $i$ to bus $j$ are addressed by $R_{ij}$, $X_{ij}$, respectively, while $Z_{ij}^2=R_{ij}^2+X_{ij}^2$. In addition, the active and reactive powers transmitted through each of the branches are limited. The associated binary variable for the adjacent feeders is taken into account to ensure the viability of the power flow limits. The two-way flows of power through connected branches are represented in Equations (10) and (11) for real power and reactive power, respectively.

$${-y}_{ij,k}{\overline{P}}_{ij}\le P_{ij,k}\le y_{ij,k}{\overline{P}}_{ij} \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(10)

$${-y}_{ij,k}{\overline{Q}}_{ij}\le Q_{ij,k}\le y_{ij,k}{\overline{Q}}_{ij} \forall \left(ij\in {\Gamma }_L,t\in {\Gamma }_K\right)$$

(11)

The active and reactive powers can be provided by the upstream grid and by the local power generating units, which are modeled in (12)–(14). It is assumed that the power flow from the upstream network to the distribution grid is unidirectional and it is not possible to sell the surplus of local power generation to the upstream grid. Therefore, the active power is modeled as a positive variable bounded by the rated capacity of the local power generation units, DGs, and the rated power of the connecting transformer at the connection point of the distribution network to the upstream grid. For reactive power, generation and absorption by each unit are admitted. It is evident that the provided constraints are valid in the case of commitment of the mentioned units, i.e., $x_{i,k }=1$.

$$0\le P_{i,k}^{SQ}+Q_{i,k}^{SQ}\le x_{i,k }{\overline{S}}_i^{SQ} \forall \left(i\in \left\{{\Gamma }_S\cup {\Gamma }_{DG}\right\},k\in {\Gamma }_K\right)$$

(12)

$$0\le P_{i,k}\le x_{i,k }{\overline{P}}_{i,k} \forall \left(i\in \left\{{\Gamma }_S\cup {\Gamma }_{DG}\right\},k\in {\Gamma }_K\right)$$

(13)

$$x_{i,k }{{\displaystyle \underset{\_}{Q}}}_{i,k}\le Q_{i,k}\le x_{i,k }{\overline{Q}}_{i,k} \forall \left(i\in \left\{{\Gamma }_S\cup {\Gamma }_{DG}\right\},k\in {\Gamma }_K\right)$$

(14)

A primary issue in the power flow problem for distribution networks is ensuring the voltage profile complies with system requirements. The minimum and maximum permissible voltages have been formulated inside the MIQCP paradigm as indicated in (15). The assumption of the nodal voltage as a positive variable ensures that the quadratic representation of the bus voltage would not complicate the problem. The constraint on the current flow across the network branches is articulated in the quadratic expression in (16). Given the bidirectional current flow, the lower limit is expected to be zero.

$${{\displaystyle \underset{\_}{V}}}^{SQ}\le V_{i,k}^{SQ}\le {\overline{V}}^{SQ} \forall \left(i\in {\Gamma }_N,k\in {\Gamma }_K\right)$$

(15)

$$0\le I_{ij,k}^{SQ}\le {\overline{I}}_{ij}^{SQ}{y}_{ij,k} \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(16)

The optimization model for reconfiguration of the distribution network is developed as a mixed-integer optimization problem. There are several binary decision variables introduced for this regard. The binary variable representing the status of the external grid and local power generation units is addressed in (17). The status of the link between bus $i$ and bus $j$ is represented in (18). To model the load scheduling, the corresponding binary variable is provided in (19). It should be noticed that in the case of load curtailment, the total load connected to the target bus will be curtailed and partial load-serving is not possible in this case. In addition, there are some other binary variables introduced in this paper to model the direction of the power flow and to guarantee the radial topology of the distribution network. The direction of the power flow from one bus to another one is modeled by an associated binary variable introduced in (20).

$$x_{i,k}\in \left\{\mathrm{0,1}\right\} \forall \left(i\in \left\{{\Gamma }_S\cup {\Gamma }_{DG}\right\},k\in {\Gamma }_K\right)$$

(17)

$$y_{ij,k}\in \left\{\mathrm{0,1}\right\} \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(18)

$${\vartheta }_{ij,k}\in \left\{\mathrm{0,1}\right\} \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(19)

$${LS}_{i, k}^D\in \left\{\mathrm{0,1}\right\} \forall \left(i\in {\Gamma }_N,k\in {\Gamma }_K\right)$$

(20)

The distribution network must maintain a radial configuration for both grid-connected and stand-alone operational states. To ensure the radial configuration of the system, limitations (21)–(23) are proposed [33]. This set of restrictions establishes that when there is an active branch between nodes i and j, denoted by $y_{ij,t}=1$, the power flow direction is dictated by the corresponding binary variable. In both states, the slack node at each segment of the system cannot receive power from the remainder of the network; it is solely capable of supplying power to the network, as delineated in (22). This constraint is applicable solely to the reference node for each part of the system.

Consequently, in the stand-alone operation state, the reference for the respective island is required to be established. A supplementary constraint is introduced to prevent the formation of loops in the system configuration, as articulated in (23), which pertains to all connected feeders to every node. In other words, it is not possible to inject power from more than one link to each specific bus; however, it can inject power from one specific bus to more than one bus creating new laterals in the network.

$${{\vartheta }_{ij,k}+{\vartheta }_{ji,k}=y}_{ij,k} \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(21)

$$\sum _{j\in {\Gamma }_N}{\vartheta }_{ji,k}=0 \forall \left(i\in {\Gamma }_S,ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(22)

$$\sum _{j\in {\Gamma }_N}{\vartheta }_{ji,t}\le 1 \forall \left(ij\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(23)

Each island necessitates a minimum of one grid-forming unit to establish the voltage reference. The equations presented in (21)–(23) ensure the radial configuration for both grid-connected and stand-alone modes of the system. In the scenario of islanding, the reference voltage may be locally established by the grid-forming unit. In this regard, the node associated with the grid-forming unit would be tuned to minimize the power losses and voltage drops for the specific section of the network belonging to the new reference unit. Constraints (24) and (25) show that when designating the reference unit inside an island, the voltage may be established at the reference value. The positive slack variable, $v_{i,k}^{GF}$, has the potential to enhance the reference voltage of the grid-forming unit within the island. In the grid-connected operational mode, $v_{i,k}^{GF}=0$, as the upper limit of this variable is constrained to zero by the corresponding binary variable $y_{ji,k}$.

$$V_{i,k}^{SQ}+v_{i,k}^{GF}=V_i^{SQ,ref} \forall \left(i\in {\Gamma }_N,k\in {\Gamma }_K\right)$$

(24)

$$0\le v_{i,k}^{GF}\le \left({\overline{V}}^{SQ}-{{\displaystyle \underset{\_}{V}}}^{SQ}\right)\left(1-y_{ji,k}\right) \forall \left(ji\in {\Gamma }_L,k\in {\Gamma }_K\right)$$

(25)

4. Simulation Results

This section provides the simulation results of the proposed model for the normal and contingent events. The functionality of the model in making the possible MGs and imposing the intention islands is investigated. The test system selected for the simulation study is the modified IEEE 33-bus network. The system data for the base case, i.e., without the DGs and in normal operation strategy can be found in [23]. In this paper, the priority of the loads has been identified in the single line diagram illustrated in Figure 1.

The power flow models have been developed as a standard mixed-integer with quadratically constrained programming (MIQCP) approach, and the model has been solved using an embedded CPLEX (12.5.1) solver within the GAMS (24.1) environment. The functionality of the model has been tested and verified on the standard IEEE 33-bus system. Simulations were performed using an AMD R7 4800H laptop with 16 GB of RAM.

The total active and reactive loads are 3.715 MW and 2.3 MVAr, respectively. In this case, by considering the voltage at the reference bus as 1.0 pu, the total system power loss is 202 kW, while the minimum voltage is 0.913 pu at bus 18. The reconfiguration of the base network applying the proposed model in this paper resulted in 139.134 kW power loss, with the minimum voltage of 0.939 pu at bus 32. In this case, the switches 6–26, 2–19, 9–15, 18–33, 8–21, and 12–22 should be closed, while the switches 7–8, 14–15, 19–20, 25–29, and 32–33 should be opened to fulfil the radiality requirement for the distribution system. It is worth noting that this result is only valid in the case that the network operator is able to open each section of the network, i.e., each section of the network is supported as a switch. However, this assumption is a bit far from the concept of distribution networks. The obtained results are only valid for such assumptions and it only confirms the optimal switching actions for the given network, compared to the different methodologies and optimization models that have been addressed in the literature.

Figure 2 illustrates the voltage profile for the original network before and after reconfiguration. The mentioned results have been obtained by utilizing the commercial power flow tools reported in the literature [34]. However, the current model is developed based on the MIQCP model and the computational burden is less than the other MINLP solutions utilizing the ACOPF problem. The problem converged in 5 s in this case study considering the network reconfiguration possibility.

In the presence of the DGs, the OPF resulted in 99.622 kW without applying reconfiguration, while by considering the reconfiguration of the network in the presence of the DG units, the total loss would be 73.928 kW.

In this case, it is supposed that the voltage at the reference node is 1.0 pu.

Table 1. Techno-Economic Data of the Units.

Unit	Bus	${\overline{\mathit{P}}}_{\mathit{i}}$ (kW)	${\displaystyle \underset{¯}{{\mathit{Q}}_{\mathit{i}}}}$ (kVAr)	${\overline{\mathit{Q}}}_{\mathit{i}}$ (kVAr)	${\overline{\mathit{S}}}_{\mathit{i}}$ (kVA)	${\mathit{c}}_{\mathit{i}}$ ($/kWh)
Utility	1	4000	−3000	+3000	5000	0.010
DG 1	22	100	−50	50	100	0.005
DG 2	27	630	−450	450	630	0.005
DG 3	29	425	−300	300	425	0.005
DG 4	31	300	−220	220	300	0.005

provides the details of the DG units and upstream network considered in this study [35]. It should be noticed that the DGs installed at buses 27 and 31 have the grid-forming possibility.

After validation of the model for handling the ACOPF problem in the presence of multiple resources in the distribution network, considering the reconfiguration possibility and maintaining the radiality of the network, the presented model is investigated by HILP events to show the functionality of the model in the optimal management of the network. In this regard, multiple disaster events resulting in several feeder outages have been examined. It is supposed that the links 1–2, 3–23, 10–11, 15–16, and 30–31 are out of service. As branch 1–2 is out of service, there is no possibility to feed the demands by the utility. In this case, without the DGs, there is no possible way to serve the loads, but even by considering DGs, the installed capacity is not enough to fulfil the overall loads. Therefore, the EMS should manage the load-serving possibility to maintain high priority loads as much as possible. In addition, by optimal switching actions in the presence of grid-forming units, the EMS can impose intentional islands in the reconfigurable network. It should be noticed that the amount of high, medium, and low priority loads are 800 kW, 720 kW, and 2195 kW, respectively [35].

Figure 3 illustrates the optimal configuration of the studied network in the occurrence of multiple disasters, such as a storm or an earthquake. The islanded MGs, after the disaster, need at least two grid-forming units to maintain the supply of the critical loads as well as some other loads. As can be seen from the single line diagram, the critical load and some share of medium and even low priority loads have been supplied by the EMS. Since branches 15–16 and 30–31 are out of service and there is one grid-forming is available at bus 31, the formation of an island can be attained by closing switch S5 (18–33). In this case, DG4, with the capacity of 300 kVA, can serve some share of the loads at this MG. The critical load at bus 31, with a demand of 150 kW and 70 kVAr, will be served, as will the medium priority loads at buses 17 and 33, with individual demands of 60 kW and 20 kVAr and 60 kW and 40 kVAr, respectively. The operating point of DG4 is 207.104 kW and 130.128 kVAr, meaning that this unit is almost completely loaded (99.9%).

In the other MG, there are three DGs, DG1–DG3, with one grid-forming unit. The total capacity of these units is 1155 kVA. In this MG, five critical loads must be served by the EMS. The demand for the critical loads that would be served by this MG is 650 kW and 370 kVAr. In addition, there are four medium priority loads in this area, 5, 12, 24, and 27. The total demand for these buses is 600 kW and 290 kVAr, therefore, it is not possible to serve all critical and medium priority loads at this MG. It should be noticed that the demanded load at bus 24 is 420 kW and 200 kVAr, which is one of the biggest loads in the system. In this case, there is no possibility to serve this load and it should be curtailed. Indeed, it is possible to supply the critical load (650 kW and 370 kVAr) and some share of the medium priority load (180 kW and 90 kVAr). In addition, it is possible to maintain a share of the low priority loads in this case.

The optimal load-serving strategy to minimize the overall loss and loading of the feeders resulted in serving the loads on buses 13, 26, and 28. The total demand for these buses is 180 kW and 80 kVAr. The total served load at this MG is 1010 kW and 540 kVAr, which results in 1145.3 kVA, and it is almost close to the maximum loading of the available DGs at this MG. All in all, the intentional islands formed in this case study are able to serve all critical loads, as is expected, and the share of medium priority loads that could be served is considerable. In addition, the EMS can energize some share of low priority loads in this study. It should be noticed that partial load serving is not possible, and it would be 100% or 0% at the connection point of the circuit breakers.

Table 2. The operating points of DGs for the HILP event.

Unit	MG	${\mathit{P}}_{\mathit{i}}$ (kW)	${\mathit{Q}}_{\mathit{i}}$ (kVAr)	Loading (%)	${\sum }_{\mathit{i}}{\mathit{P}}_{\mathit{i}}^{\mathit{D}}$ (kW)	${\sum }_{\mathit{i}}{\mathit{Q}}_{\mathit{i}}^{\mathit{D}}$ (kVAr)
DG 1	1	86.690	49.836	100	1010	540
DG 2		555.956	296.300	100
DG 3		372.501	197.771	99.2
DG 4	2	270.104	130.128	99.9	270	130

summarizes the load restoration strategy and loadings of the DGs for the studied HILP event. The simulation run time is 13 s in this case.

5. Conclusions

This paper developed a fast and accurate MIQCP-based model for the automated load restoration in smart distribution systems. The model efficaciously addressed distribution feeder reconfiguration and optimal grid-forming distributed generators (DGs). This was carried out for the sake of improving the resilience of distribution systems, especially in the case of high-impact low-probability (HILP) events. A centralized energy management system (EMS) was also presented to coordinate grid-connected and islanded modes of operation, while addressing network radiality, operational feasibility, and efficient voltage regulation. The developed framework studied a number of operational scenarios and enforced binary-based load scheduling to thoroughly supply or shed loads, prioritizing critical demand during power outages. The results obtained from simulating the proposed framework on the modified IEEE 33-bus test system indicated that the model resulted in significant mitigations in system losses. It was also found that all critical loads were effectively restored utilizing optimal islanding and dispatch strategies. The model converged fast and fits well the real-time applications in practical EMS platforms.

The future works focus on enhancing the robustness the model as well as its applicability under real-world conditions. One research line will be addressing uncertainty due to intermittent renewable power generation and load forecasting using hybrid distributionally robust optimization (DRO) and chance-constrained programming. Further expanding the model to involve multi-energy systems, such as a coordinated operation of electric, thermal, and gas networks, will be another step forward. Developing a decentralized/peer-to-peer (P2P) EMS architecture will also be addressed to facilitate implementation in large urban networks. Finally, experimental validation using real-time digital simulation or hardware-in-the-loop (HIL) testbeds is planned to evaluate the practical feasibility of the proposed solution considering realistic grid conditions.