Service Restoration of AC–DC Distribution Grids Based on Multiple-Criteria Decision Analysis

Modern distribution grids are transforming due to the increasing amount of load and generation based on DC technologies, whose interconnection is facilitated by the upcoming AC-DC distribution networks. Their fault management is challenged by new issues related to the power transferred among AC and DC sub-networks. In light of this, the present paper proposes a novel Service Restoration (SR) algorithm, specifically tailored for AC-DC distribution grids, that optimizes the re-energization by establishing priorities of disconnected bus groups and computing, as candidate solutions, hierarchical combinations of normally open and normally closed switches. To overcome the limitations of existing approaches, a Multiple Criteria Decision Analysis (MCDA) is proposed to combine and objectively prioritize different SR goals (i.e., the use of telecontrolled switches, the minimization of power losses, and the applicability of the proposed solutions in a defined time horizon), and ultimately make the grid operator benefit from the possibility to flexibly tune different operational objectives. The proposed algorithm allows to effectively discriminating competing solutions, i.e., whence and how to re-energize disconnected buses, and complies with the time requirements for field implementation. The results prove the crucial role of DC sub-networks, associated to the control of power injection from AC-DC converters, in enhancing the SR by improving its targets and increasing the number of re-energized loads. Ultimately, the effect of MCDA comparison parameters on the SR outcomes is quantitatively investigated via global sensitivity analysis, whose adoption is recommended for supporting the grid operator in the algorithm implementation.


I. INTRODUCTION
The decentralization of energy management in AC distribution grids, together with the growth of electricity demand (e.g., in mobility sector and temperature control of buildings), toughens criticalities related to power quality, protection and control of power flow; the upgrade of networks with power electronics components, based on DC technology, could mitigate these issues [1], [2]. Particularly, several research activities propose the deployment of point-to-point or multi-terminal DC lines, linking DC-based generators and loads via DC-DC converters, to interconnect AC distribution grids [3], [4], [5]. To achieve this grid configuration, [6] and [7] analyze the conversion of AC lines to DC lines for Medium Voltage (MV) networks, enhancing the power distribution in different network conditions. The energy management of AC-DC distribution grids relies on the control of power converters, enabling the optimization of power transfer among AC and DC sub-networks as an additional controllability feature [8].
In deploying AC-DC distribution grids, further characterizations are needed for the protection schemes. Protection systems aim at enhancing the reliability of distribution grids, by minimizing the duration of electricity interruption and the affected geographical area [9]. The Distribution Management System (DMS) of modern distribution grids deploys automated self-healing functionalities commonly referred to as Fault Location, Isolation and Service Restoration (FLISR), which encompasses: the accurate determination of fault position, the operation of switching devices to isolate the minimum fault area and, then, the network reconfiguration to re-energize the restorable buses (located downstream the fault area) [10], [11].
In this paper the Service Restoration (SR) aspect of FLISR, for AC-DC networks, is addressed and an innovative algorithm is presented. Being applied to AC-DC distribution grids, the control of the converters, to optimize the power transfer among AC and DC sub-networks, has to be included in the SR process. Multiple research works have presented Optimal Power Flow (OPF) approaches for AC-DC grids, to determine operational setpoints of converters and Distributed Generators (DG). In [12] and [13] the mathematical models of AC-DC grids are expressed as OPF constraints and linearized; in [14], additionally, the detailed characterization of operating limits and losses for AC-DC converter station is included. The work presented in [15] proposes a decentralized approach, sub-dividing the optimization problem in AC sub-networks, DC sub-networks and AC-DC converters levels. In these studies, the proposed methods do not consider the SR use case, whereby network portions are electrically disconnected and the operating strategies have to be accordingly adapted; nonetheless the referred OPF formulations are integrated in the SR algorithm, applied to each reconfiguration solution.
Several works have addressed the design of SR algorithms for AC-DC grids. Prioritization of de-energized buses is of importance to compute the different reconfiguration topologies that involve the operation of Normally Open (NO) and Normally Closed (NC) switches. In this context, e.g., [16] proposes a graph theory approach to optimize the power transfer among AC and DC sub-networks in SR, though being designed uniquely for meshed HV grids without considering the radiality constraint and not applicable to MV distribution networks. The algorithm proposed in [17] determines the available restoration paths with a genetic algorithm, but the DC lines only interconnect AC-DC converters without modelling loads and DG; this aspect is particularly relevant since the future AC-DC distribution networks aim at directly connecting DC generation and DC consumption in the same sub-network, reducing the conversion stages. Reference [18] presents a SR method based on Mixed-Integer Second-Order Cone Programming (MISOCP) in which the de-energized loads are ranked according to their criticality and iteratively reconnected, although the opening of NC switches is not included in the restoration topologies hence limiting the set of candidate solutions.
In determining the candidate solutions for SR, the prioritization of the de-energized loads is of high importance, particularly in the case each bus cannot be reconnected. In [19] and [20] the SR computes the priority indices of every bus with respect to the hosted critical infrastructures (e.g. emergency facilities or communication networks); however this approach is expanded in the present work by computing the indices of bus groups and the nominal power of hosted DG and loads.
Moreover, the design of a suitable SR algorithm shall consider the time period between the fault clearance and the final network reparation; hence the applicability of the solution has to be verified, in addition to the actual time instant with the measured data, also in a specific time horizon by considering the forecast values as loads and generation profiles. References [21] and [22] present network reconfiguration algorithms that consider the short-term power forecasts in determining the switches to be operated, but this aspect is not extended among the objective criteria of the SR problem.
The deployment of SR with multiple criteria allows to enhance the algorithm performance and consider the different Distribution System Operator (DSO) objectives.
Reference [23] proposes a Mixed-Integer Quadratic Programming (MIQP) model to reconfigure the AC-DC distribution networks, by optimizing the combination of lost load, minimum DG curtailments and minimum network loss; whereas in [24] a hierarchical coordinated optimization solution method is applied, which considers the minimum active power loss and AC bus voltage offset of the distribution network as the optimization goals. These literature works deploy weighting factors in the optimization functions to combine the different criteria, introducing inconsistency and losing the control on each optimization goal. On the contrary, the algorithm proposed in this paper resorts to the Multiple Criteria Decision Analysis (MCDA) approach, in that it provides the user with the possibility to flexibly select specific SR criteria and define their priorities. Within MCDA, as first step, the Analytical Hierarchy Process (AHP) allows to compute the priorities of every criteria starting from a pair-wise comparison according to DSO objectives [25]. Then, as second step, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) technique is applied, in which the weights of criteria are combined with the set of candidate solutions. The goal is to compute the most optimal candidate solution, which is nearest to the positive ideal solution and farthest from the negative ideal solution [26].
To understand the implications that the relationships between diverse restoration criteria, i.e., the inputs to the MCDA approach, might have on its outcomes, Sensitivity Analysis (SA) comes into the fore. Traditional SA approaches mainly belong to the family of Local Sensitivity Analysis (LSA) [27], which is based on individual perturbations of the uncertain weights to analyze their effect on the outcome of the decision analysis process, hereafter referred to as Quantity of Interest (QoI). Graphical tools such as tornado diagrams, rainbow diagrams and strategy-region diagrams are usually adopted to visualize the LSA results [28]. A review of SA techniques for multi-objective decision making is provided in [29] and a comparison of a selected number of SA techniques (one-and two-way SA and regression analysis) is reported in [30]. However, LSA techniques-although being by far the dominant ones in the power system field, as shown in [31]-capture the effect of small perturbations on the QoI, hence being suitable to study only small changes in the close vicinity of the inputs operational/nominal values, unless the model is shown to have a linear behavior. As a consequence, LSA might be insufficiently informative and poorly performing in those situations whereby the studied model is characterized by large uncertainties of the inputs, nonlinearities or potentially even unknown degree of linearity (e.g., in the case of black-box models). To overcome these downsides, Global Sensitivity Analysis (GSA) is recommended, as it is able to explore (at least conceptually) the overall variability range of the inputs hence providing a more complete picture of the model sensitivity behavior. Different GSA techniques are available in the literature [31]: in this work, the choice is to adopt variance-based SA-which, to the best of authors' knowledge, has not been previously applied to study the sensitivity of MCDA approaches for SR purposes-, due to its easiness of interpretation, its ability to comprehensively account for all the uncertainty sources and its independence from any prior assumption on the model properties.
It is noteworthy that this paper builds on the work presented in [32], by extending the application of MCDA technique also in the prioritization of the disconnected bus groups and selecting different criteria in computing the near-optimal solution; additionally, a convexified formulation of OPF problem, based on the second order cone relaxation presented in [33] and adopted for meshed AC-DC grid in [14], and [18], has been implemented and GSA has been conducted. Table 1 summarizes the methodologies and features of the literature works on SR, including the comparison with the present paper.

A. CONTRIBUTIONS OF THE PAPER
The main contributions of this paper are: • The development of an SR algorithm for the computation of network reconfiguration solutions, including the opening of NC switches and accounting for the calculated priorities of the de-energized buses. The algorithm is specifically designed for AC-DC distribution networks, integrating a convexified OPF that enhances the SR performance by optimizing the operational control of DG and AC-DC converters.
• The implementation of the MCDA technique to select, among feasible candidates, the near-optimal SR solution. The adopted criteria consist of: the minimization of network power losses, the use of telecontrolled switches for the reconfiguration and the applicability of proposed network topologies at long term (in the time period beyond the fault occurrence).
• The quantitative analysis of the impact of the MCDA comparison parameters on the SR process via variancebased SA. By applying such GSA method, the effect of the subjective judgements on the computation of the near-optimal solution in the SR process is quantitatively retrieved to support the algorithm implementation (e.g., identifying which MCDA comparison parameters mostly affect the decision of the SR solution), ultimately enabling an informed decision making process under uncertainty.

B. STRUCTURE OF THE PAPER
This paper is structured as follows: Section II formulates the SR problem and describes the MCDA and OPF techniques; Section III describes the different steps of the proposed SR algorithm; Section IV describes the GSA methodology and the adopted variance-based SA technique; Section V presents and discusses the results obtained by running the proposed SR algorithm on three selected case studies; Section VI concludes the paper.

II. FORMULATION OF SERVICE RESTORATION PROBLEM
The occurrence of faults in electrical networks, as shortcircuits or series faults, requires the prompt intervention of the protection system, which detects the event and determines its location. The coordination of the switches to be operated (circuit breakers, reclosers, load break switches and disconnectors), in accordance with the selectivity plan, allows the minimization of the isolated fault area by opening the nearest switches upstream and downstream the fault location. The MV grids are normally operated with radial configuration and different feeders are interconnected by NO tieswitches. Hence the buses belonging to the faulty feeder but located downstream the fault area are indicated here as ''restorable'', i.e., although being de-energized, they can be reconnected by closing specific tie-switches. The goal of SR is to re-energize the maximum number of restorable buses according to specific criteria while ensuring the secure operation of the network. The topology yielded by the reconfiguration process is temporary, lasting several hours (on average, between 2 and 4 hours), and precedes the crew intervention to repair the faulty line [34]. The SR algorithm proposed in this paper starts as soon as the fault area is isolated by the open upstream and downstream switches; the flowchart in Fig. 1 depicts the main steps, which are described hereafter. a) Identify the restorable buses and, consequently, the alternative sets of switches commands (as candidate solutions) for the network reconfiguration. b) For each candidate solution: • Verify the feasibility according to grid security constraints.
• Deploy the OPF to compute setpoints for DG and AC-DC converters.
• Quantify the criteria using MCDA technique. c) Determine the near-optimal candidate solution to be implemented, by comparing the MCDA outcomes. d) In the case no candidate solution is feasible, repeat the process by excluding the least critical buses from the re-energization.
Each step of the SR process is hereafter described and their mathematical formulations are presented.

A. IDENTIFICATION OF RESTORABLE BUSES AND COMPUTATION OF CANDIDATE SOLUTIONS
As first step, the SR algorithm uses the information of the tripped switches to identify the de-energized buses outside the fault area. Depending upon the position of controllable switches, multiple power sources can re-energize the restorable buses.
The graph theory is applied to model the distribution network: each bus and line of the electrical grid corresponds to vertex c and edge e, respectively. Each bus has attributes associated to the possible presence of AC-DC converter station or High Voltage (HV) connection (consisting of HV/MV primary substation); the attributes of each edge specify the presence of a switch and, in case, the switch status (open or close) as well as the tripping condition (due to a fault, encompassing the fault area). Afterwards, the graph without edges corresponding to open switches (NO and tripped switches) is built; for each bus of the network, the connectivity, as a finite sequence of edges, toward each power source (HV/MV substation, also through AC-DC converter) is inspected: if a bus is not connected to any power source, it is identified as disconnected and added to the set C, which is the array of disconnected buses. Then, a new graph with edges corresponding to NO and NC switches but without tripped switches is built; for each disconnected bus, the connectivity toward each power source is verified: in the case a bus is connected to, at least, one power source, the former is identified as restorable; the array of restorable buses is indicated by C r . Within C r , some restorable buses are directly connected among them without interposed switches; since the SR algorithm includes the possibility to open the NC switches within the de-energized area, the buses not interconnected by any switch correspond to a ''bus group''. All the buses of a bus group will be re-energized simultaneously. The array C g r contains sub-arrays C g,b r that indicate the buses in the bus groups (with b = 1, 2 . . . n g and n g being the number of bus groups).
The sample 10-bus grid in Fig. 2 represents a radial AC grid with a DC sub-network; black and white squares indicate the NC and NO switches, respectively. A fault at bus 2 causes the tripping of switches S1 and S2 and, consequently, the de-energization of buses 3, 4 and 5. Since buses 3 and 4 are not interconnected by switches, they constitute a unique bus group. Hence the following bus arrays are obtained: [3,4] , [5]] If the computed SR solution causes the power flow to violate the network security constraints (e.g. voltage or ampacity limits), an SR solution shall be implemented that re-energizes only part of disconnected buses. Hence it is important to classify the disconnected bus groups, in order to prioritize the re-energization of the ''most important'' buses.

1) PRIORITIZATION OF BUS GROUPS WITH MCDA
The MCDA technique allows to mathematically determine the candidate solution, among multiple ones, that satisfies at best the objective criteria. In this specific application, MCDA is used to determine the order of priorities for bus groups considering the pre-determined criteria. To this scope, the index α is introduced and assigned by the DSO to each bus: the greater α, the higher the criticality of the infrastructure. The criteria considered by MCDA, based on which each bus group b is evaluated, are: 1) The highest criticality value for the bus group, indicated as α max . For the bus group C g,b r , it writes: 2) The total generated power P gen in the bus group, as sum of nominal power for each DG, if present, connected to the buses. For the bus group C g,b r , it writes: 3) The total absorbed power P load in the bus group, as sum of nominal power for each load connected to the buses. For the bus group C g,b r , it writes: The first step of the MCDA is the application of the AHP [25]. Pair-wise comparisons among the criteria (α max , P gen , P load ) are carried out and the symmetric comparison matrix (with dimensions d ×d, with d being the number of criteria) VOLUME 11, 2023 is obtained: w α max P gen w α max P load 1 w α max P gen 1 w P gen P load Each element w in indicates the pre-assigned comparison parameter between the two criteria indicated by the subscripts (α max , P gen or P load ), in the AHP scale, ranging discretely from 1/9 (i.e., the attribute of the first subscript is extremely irrelevant with respect to the second one) to 9 (i.e., the attribute of the first subscript is extremely important with respect to the second one). In the work described in this paper, the highest importance is assigned to α max criteria; moreover, for the same criticality index, the restoration of buses hosting DG is considered more important than the restoration of loads. The implemented values of comparison parameters are: w α max P gen = 4 w α max P load = 9 w P gen P load = 4 The weights of the criteria are the elements of the eigenvector v, according to: where λ max is the respective largest eigenvalue of matrix and I is the identity matrix with dimension d × d.
Afterwards, the TOPSIS (as part of MCDA) is applied [26]. The performance ratings matrix D is first created, having o rows as the number of alternatives and d columns as number of criteria: In this application, o and d correspond to the number of bus groups and the number of MCDA criteria, respectively; each element of D indicates the value of the criteria (α max , P gen or P load ) for the specific alternative (i.e., the bus group).
Then, the performance ratings are normalized and multiplied by the weights of criteria (as elements of the eigenvector v) computed with AHP: As next step, the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS) are calculated: where J 1 is the set of benefit criteria (i.e. the criteria that aim at being maximized) and J 2 is the set of cost criteria (i.e., the criteria that aim at being minimized), respectively. For the prioritization of bus groups every criteria belongs to J 1 . As next step, the Euclidean distance of each alternative from PIS (S + i ) and NIS (S − i ) is computed: Finally, the relative closeness of each alternative i to the ideal solution Q + i is computed: In particular, the larger Q + i , the better the performance of alternative i. The bus groups belonging to C g r are then sorted in descending order based on Q + i and constitute, as subarrays, the array of prioritized bus groups indicated as C g * r . It is noteworthy that the above-described MCDA technique is also used in a successive stage of the SR process (described in Section II-C) to determine the most optimal network reconfiguration topology to be implemented.

2) COMPUTATION OF CANDIDATE SOLUTIONS FOR NETWORK RECONFIGURATION
Starting from the restorable buses C r to be re-energized, the SR candidate solutions are computed in form of network reconfiguration topologies. By considering the graph of the network without the tripped switches, the most suitable path-if it exists-between each bus of C r and every HV/MV substation (also through AC-DC converters) is computed by implementing the Dijkstra's algorithm [35]: given two buses c 1 and c 2 in a graph G, it determines the path G ′ c 1 ,c 2 to connect c 1 and c 2 by minimizing a specific attribute of the edges. In the implemented application, the attribute to be minimized is the impedance of the lines, in order to minimize the power losses of the SR solution. The computed paths guarantee the radiality of the network and include, among the edges, the NO switches whose closure allows to re-energize the restorable buses. Multiple occurrences of the same NO switches are discarded and the array S NO is computed: it contains as many as s NO sub-arrays S v NO , which indicate combinations of NO switches to be operated in order to re-energize all the restorable buses: Moreover, the SR algorithm considers the opening of NC switches among the bus groups within the de-energized area. The computation of the shortest paths among each restorable bus and every HV/MV substation with Dijkstra's algorithm is repeated; however in this case the edges related to the opened NC switches are excluded from the graph. It is worth mentioning that, since the simultaneous opening of adjoining NC switches (i.e., among adjoining bus groups) could cause the isolation of restorable buses located in the middle, the location of NC switches leads to the definition of two categories: • The ''non-adjoining NC switches'', which are simultaneously opened (being them connected to different bus groups) and determine unique SR candidate solutions.
• The ''adjoining NC switches'', which are alternatively opened and determine different SR candidate solutions. The array S NC NO is then formulated, which contains s NO NC subarrays indicated by S NC,z NO . Each S NC,z NO sub-array is constituted by: 1) As initial elements, indicated by S NC,z , the combination of non-adjoining NC switches to be closed. 2) As successive elements, indicated by S * ,z NO , the combinations of NO switches to be closed. The order of NO switches reflects the order of prioritized bus groups in C g * r .
The array S NC NO assumes the following general formulation: Each S NC,z NO sub-array constitutes a different SR candidate solution to be further evaluated. It is worth to mention that, due to the opening of NC switches, the bus groups that are not encompassed by tie switches (as NO switches) cannot be re-energized and are hence removed from the array of restorable buses C r .
Considering the sample grid of Fig. 2 and assuming the prioritization of the bus groups (with MCDA technique) that leads to: it is, then, obtained: It is noteworthy that only one candidate solution as subarray S NC,z NO is present in S NC NO as the opening of NC switch S3 requires the closure of both NO switches S4 and S6 to re-energize every restorable bus. The candidate solutions of the SR process are constituted by the combinations of NO switches in each sub-array S v NO of S NO and, moreover, by the combinations of NC and NO switches in each sub-array S NC,z NO of S NC NO . Each specific candidate solution is indicated by K x , with K = S NO S NC NO being the whole set of candidate solutions.

B. OPTIMAL POWER FLOW FOR THE CANDIDATE SOLUTIONS
Each candidate solution (calculated as described in Section II-A2) indicates the NO switches to be closed and, eventually, the NC switches to be opened, determining the network topology that constitutes the input for the OPF computation. The deployment of OPF has two goals: to verify the applicability of the candidate solution with respect to the operational and security limits and, moreover, to optimize the power injections of DG and AC-DC converters (related to the power exchange between AC and DC sub-networks).
The model of AC-DC converters is developed according to Voltage Source Converters (VSC) technology and is represented in Fig. 3. In general, the DC sub-networks deploy a multi-terminal configuration (i.e., more than two electrically connected AC-DC converters) and in the operational strategy one AC-DC converter acts as DC slack bus, according to which the voltage of its corresponding bus n is set to 1 p.u. [14]. Moreover, considering the model in Fig. 3, the power losses of the converter station P loss C are defined as: with P AC C and P DC C being the active powers injected in the AC and DC sides, respectively, of the AC-DC converter. For the considered VSC station, P loss C aggregates the losses of each station component: transformer, filter, phase reactor, and converter. The power losses of the transformer, filter and phase reactor are related to the core losses and harmonic currents. The power losses of the AC-DC converter include both switching and conductance losses [36]. The power losses of the entire AC-DC converter station are approximated with a quadratic function of the phase current I C,m at bus m ∈ N C (with N C as set of converter buses) and the parameters a c , b c and c c : The OPF is formulated based on the branch flow model defined in [33], which describes a convex relaxation procedure for solving the OPF (normally nonconvex). The resulting objective function corresponds to the minimization of network power losses, indicated by P loss tot in (17), and it is VOLUME 11, 2023 defined as: (17) where r ij l ij represents the real power losses in the AC grid, composed by the real part of the line impedance, r ij , and the square of the branch current, l ij , for all i, j ∈ N AC , with N AC being the AC buses. The term r pq l pq represents the power losses in the DC grid, where r pq is the line resistance and l pq is the square of the branch current for all p, q ∈ N DC , with N DC being the DC buses.
The constraints for AC-DC distribution grids are: • Power balance constraints at AC buses i ∈ N AC : where P AC ji and Q AC ji represent the active and reactive power between the node i and the set of upstream neighboring nodes N up , whereas the term x ji is the imaginary part of the corresponding line impedance. The elements P ki and Q ki represent the active and reactive power between the node i and the set of downstream neighboring nodes N down . P AC gen,i and Q AC gen,i indicate the active and reactive power injected by generators connected to node i, P AC C,i and Q AC C,i indicate the active and reactive power injected by AC-DC converter connected to node i (equal to zero if the AC-DC converter is not present in node i) whereas P AC load,i and Q AC load,i indicate the active and reactive power absorbed by loads connected to node i.
• Power balance constraints at DC buses N DC : (20) where, based on (15), P DC C,p is substituted with: where P DC qp represents the active power between the node p and the set of upstream neighboring nodes N up , whereas P pr represents the active power between the node p and the set of downstream neighboring nodes N down . P DC gen,p indicates the active power injected by generators connected to node p, P DC C,p indicates the active power injected by AC-DC converter connected to node p (equal to zero if the AC-DC converter is not present in node p) whereas P DC load,p indicates the active power absorbed by loads connected to node p. • Power losses constraints at AC-DC converter; to comply with the convex OPF, the equation (16) is relaxed to the inequality: The set of constraints characterizing the AC-DC interface of the electrical system, namely (15), (18), (19), (20) and (22), are described in Figure 4, which shows how the interface is correctly modeled by the constraints. (23) where v is the square of the node voltage.
• DC voltage calculation at DC buses p ∈ N DC with respect to node q ∈ N DC up : where v is the square of the node voltage.
• AC current at AC-DC converter buses N C (m ∈ N C ): Given the quadratic inequality, this constraint is nonconvex. Therefore, following [33], the constraint is relaxed to inequality: where v m is the square of the node voltage.
• AC current at AC buses i ∈ N AC with respect to node j ∈ N AC up . From [33] the following relaxed constrained is obtained: • DC current at DC buses p ∈ N DC with respect to node p ∈ N DC up . From [33] the following relaxed constrained is obtained: • Voltage coupling between AC and DC converter sides (m and n buses, respectively, as in Fig. 3), considering the range of Pulse-Width Modulation (PWM) converter modulation index m C,m as [0,1], taken from [8]: This constraint is not applied to the DC slack converter, • AC current limit for AC-DC converters: • Reactive power limits for AC-DC converters (i.e., maximum values of absorption and injection), considering m B,m as the converter constraint factor and S nom C,m as the nominal apparent power of the converter. The lower limit is taken from [36]: whereas the upper limit, as described in [14], is represented by the upper limit on the bus voltage, which corresponds to the combination of (26) where S AC,max ij represents the maximum apparent power of the line between buses i and j, whereas P DC,max qp represents the maximum active power of the line between buses p and q.
• Voltage limits for AC and DC buses: The conic relaxation of the OPF would normally require the solution to satisfy the angle recovery conditions [33]. However, the radial characteristic of the AC side of the AC-DC distribution grid always guarantees the condition to be satisfied. The fulfillment of the angle recovery conditions for meshed grids is normally not trivial and could lead to an infeasible solution for the OPF. In a meshed DC network this problem does not occur, since the angles are always zero. Consequently the conic relaxation of the OPF always produces an optimal solution.
The OPF calculations are based on network parameters and real-time measurements. Additionally, the forecast values, as explained in Section II-C, are integrated to inspect the applicability of the proposed SR candidate solutions in the successive hours.

C. SELECTION OF RESTORATION SOLUTION WITH MCDA
Each candidate solution for which the OPF problem does not provide a solution (i.e., the operational and security limits are not respected) is discarded. The MCDA technique explained in II-A1 (to prioritize the disconnected bus groups) is applied in the SR algorithm to quantify the validity of each feasible candidate solution and select the near-optimal one.
The categories of criteria considered for the MCDA are detailed hereafter.
1) The parameter β corresponds to the number of switches, both NC and NO included in the considered candidate solution (as switches to be operated), that owns the telecontrolled feature (i.e., can be operated from the DSO control center). Indicating a switch as S and a telecontrolled switch as S t , for the candidate solution K x the parameter β corresponds to: The telecontrolled feature is worth to be considered: a telecontrolled switch, in fact, does not require the time-consuming intervention of DSO crew at the device location and reduces the duration of SR condition [37]. 2) The amount of power losses P of the AC-DC network represents the value of objective function of OPF, from (17), applied to the candidate solution: 3) The applicability of the candidate solution is verified with OPF using real-time data. Considering that the reparation of the permanent fault by the DSO crew can last multiple hours, it is worth to verify the applicability of the SR solution being implemented (e.g., for voltage, power and ampacity constraints) also in the successive hours. Hence the OPF is repeated by considering the power forecasts with time frames of 15 minutes (according to the typical time resolution of power forecasts). To this scope, four η parameters are introduced, i.e., η 1 , η 2 , η 3 and η 4 , covering a total verification period of four hours. Each of them records the number of applicable solutions for four 15 minuteframes (i.e., each η parameter accounts for one hour) and hence can assume discrete values from 0 to 4. Should no OPF solution for a specific time frame be found, the process is interrupted and the successive η parameters are set as zero. In formal terms, considering t as the actual time frame, the four η parameters write: Considering the six above-described criteria (β, P, η 1 , η 2 , η 3 and η 4 ) the symmetric comparison matrix of AHP process assumes the following configuration: Successively, the TOPSIS technique is deployed; the performance ratings matrix D has, in this application, n rows as the number of alternatives and six columns as the number of criteria. The criteria β, η 1 , η 2 , η 3 and η 4 aim at being maximised and, then, are considered in J 1 as benefit criteria, whereas P is a cost criteria in J 2 as its minimization is sought after. Among the candidate solutions inspected with MCDA process, the one having the larger value of Q + i is elected as the near-optimal to be implemented.
The presented MCDA criteria relate to different aspects of the service restoration: the operational obstacles in modifying the network topology, the efficiency of the computed solution (linked to its power losses and, consequently, to the economical costs) and its reliability in the following hours (while the grid remains faulted). Additional criteria can be easily added in the SR algorithm with limited required effort.

III. DESCRIPTION OF THE IMPLEMENTED SERVICE RESTORATION ALGORITHM
The different stages of the SR process, specifically described in the previous Sections, are combined together to deploy the SR algorithm. The DMS of the DSO, in which the SR algorithm is implemented, relies on different types of databases for the data that will be processed: static databases contain information regarding the topology and parameters of the network, whereas dynamic databases collect the data measurements and the logical statuses of field devices [10]. The proposed SR algorithm is designed as middleware, with the properties of being independent from the grid topology and electrical parameters; it is developed to directly interface the DMS databases and its field implementation requires the event-driven communication of fault occurrences from field devices. Both manual and telecontrolled switches can be used (with different impact in the MCDA); SR switching operations and calculated setpoints are sent to the Supervisory Control And Data Acquisition (SCADA) system over the data bus, for which the controllability of AC-DC converters and DG is required.
The flowchart of the implemented SR algorithm is shown in Fig. 5. The algorithm starts when, following a fault occurrence, the circuit breakers trip to interrupt the fault current and, eventually, additional switching devices perform operating sequences (opening and closing) to minimize the number of buses in the fault area. As initial step the algorithm determines the restorable buses C r and bus groups C b r to be reconnected. By means of MCDA, the buses groups are prioritized and the candidate solutions of the SR process are calculated, defining the arrays S NO and S NC NO . The algorithm inspects, with OPF, the feasibility of each candidate solution in a specific group of candidate solutions. If at least one feasible SR solution is found, the near-optimal solution is determined. Otherwise, due to the impossibility to find a solution respecting grid security constraints such as voltage, power and ampacity limits, the successive group of candidate solutions, in hierarchical order, is considered. The hierarchically ordered groups of candidate solutions, from the first one being inspected, are determined according to the following: 1) The re-energization of every restorable bus with NO switches, according to the array S NO . Each sub-array S v NO corresponds to a different candidate solution.
2) The inclusion of NC switches operation in the network reconfiguration, according to the array S NC NO . Each sub-array S NC,z NO corresponds to a different candidate solution.
3) The SR is applied, according to the OPF feasibility, only to the bus groups having the highest priorities. From each sub-array S NC,z NO the last element is removed (i.e., the NO switch, connected to the bus group with the lowest priority, is not operated) and re-inspected as candidate solution with OPF. The new array is indicated as S NC NO−1 . 4) The removal of the last NO switch in each S NC,z NO is repeated and re-inspection with OPF is carried out. If every NO switch is removed in a sub-array S NC,z NO , the specific sub-array is discarded from the candidate solutions. The process continues until all NO switches are removed by every sub-arrays, in which case the SR process is interrupted. Accordingly, the groups of candidate solutions are: with h being the maximum number of S NC,z NO reductions, as per point 4) of the previous paragraph. By inspecting each group of candidate solutions (from S NO to S NC NO−h ), the near-optimal solution, if suitable, is identified with the MCDA technique according to the criteria β, P, η 1 , η 2 , η 3 and η 4 . Consequently, the closing and opening commands are issued via the DMS data bus to the corresponding switching devices (for telecontrolled switches) or as information to the DSO crew (for manual switches); moreover, the controller setpoints, as obtained from the OPF results, are sent to AC-DC converters and DG. The SR process is then concluded.

IV. GLOBAL SENSITIVITY ANALYSIS FOR MCDA ASSESSMENT
In this work, SA is employed to quantitatively assess the uncertainty of the candidate solutions and apportion it to the variability of the MCDA comparison parameters. As the determination of the near-optimal SR solutions depends on the pair-wise comparison parameters among the MCDA criteria in the AHP matrix , SA is deployed to quantify the influence of the AHP comparison parameters on the SR solutions, ultimately providing precious information under the algorithm implementation viewpoint.

1) SENSITIVITY ANALYSIS
In this study the general methodology for running global SA is adopted [31], whose main steps are briefly summarized here.
1) The analysis goal is defined.
2) The QoI for the analyst is selected according to the goals of the analysis. 3) The uncertainty sources (also referred to as inputs), which might have potential influence on the selected QoI, are identified and the model elements which are not subject to the SA are set at their nominal values. 4) The uncertainty of the considered inputs is characterized in probabilistic terms, e.g., in terms of Probability Density Function (PDF) encapsulating the analyst's degree of knowledge of the inputs variability. 5) The SA method suitable for the problem at hand is selected.

6) The input samples of the N -dimensional Monte Carlo
Simulation (MCS) are drawn from the space of the plausible input combinations according to specific sampling strategies (e.g., random or quasi-random numbers, etc.). 7) The model response is collected by evaluating the model at the values of inputs combinations specified by each MCS sample. 8) The variability of the QoI is characterized via specific metrics of interest (e.g., mean, variance, etc.). 9) The specific sensitivity measures of the selected SA method are computed to identify the most influential uncertainty sources. 10) Graphical tools are employed to effectively visualize the results. 11) After interpreting the first round of SA results, successive iterations of the whole procedure might be done to explore different scenarios (e.g., to refine the inputs uncertainty, to consider different inputs, etc.).
In Table 2 the specific instantiation of the above-reported SA procedure is described, and the corresponding results are presented in Section V-A1.

2) VARIANCE-BASED SENSITIVITY ANALYSIS
Specifically, variance-based SA-widely acknowledged as the ''gold standard'' technique for analyzing the effect of VOLUME 11, 2023  different uncertainty sources on a specific QoI [38]-is adopted in this study given its easiness of interpretation, its ability to account for the whole variability range of the considered uncertainty sources and its model-free nature (i.e., not assuming any kind of a priori information about the model such as linearity or monotonicity).
Consider a generic mathematical model of the form: whereby Y is the QoI-assumed scalar for convenienceand X 1 , X 2 , . . . , X K are the K model inputs, the latter assumed to be independent random variables with a specific PDF. In the application under study, the QoI Y is Q + i of (12) for each feasible candidate solution and the model inputs X 1 , X 2 , . . . , X K are the 15 pair-wise comparison parameters in the AHP matrix of (45). Under the assumption of input independence, variance-based SA aims at decomposing the total variance Var(Y ) of the QoI into contributions of individual inputs and combinations thereof, according to [39]: Var ij + . . . + Var ij...K (47) where ) (49) and so on for higher order terms.
Specifically, the inner operator of (48), i.e., E X X X ∼i (Y |X i ), is the expected value E of Y taken over X X X ∼i , i.e., all possible values of the inputs except X i which is kept fixed, whereas the outside operator Var is the variance taken over all possible values of X i . Accordingly, the term Var X i (E X X X ∼i (Y |X i )) describes the variance reduction that would be obtained, on average, if X i could be fixed at its ''true'' (albeit unknown) value.
It is usually more convenient to normalise (48), hence leading to the definition of the Sobol' sensitivity indices: where S i (i = 1, 2, . . . , K ) are the first order Sobol' indices, which quantify the effect of the uncertainty of each input taken alone, whereas the higher order indices (e.g., S ij ) account for possible combined effects among inputs (e.g., among inputs X i and X j ). If the model is additive, i.e., without interactive effects among inputs, all terms of (50) higher than the first order are zero, yielding K i S i = 1. On the other hand, if interactions among inputs are present, K i S i < 1 and the quantity 1 − K i S i is an indicator of the overall amount of interactions present in the model.
In practice, instead of calculating all the sensitivity indices of (50) higher than the first order ones, the so-called total Sobol' index T i is computed to capture the overall contribution of the input X i [40]: where Var X X X ∼i (E X i (Y |X X X ∼i )) represents the variance reduction that would be obtained, on average, if all inputs except X i could be fixed at their ''true'' values. On the other hand, the residual variance E X X X ∼i (Var X i (Y |X X X ∼i )) represents the contribution to the QoI variance due to all terms of any orderin the decomposition formula of (50)-that include X i . Therefore, T i accounts for the overall contribution of input X i , including not only its first-order effect, but also all the other (higher-order) effects due to possible interactions with other inputs. For the sake of the example, for a model with K = 3 inputs, the total Sobol' index for the input X i is given by T 1 = S 1 + S 12 + S 13 + S 123 . In general, K i T i ≥ 1. Importantly, the quantity T i − S i signalizes how much the input X i is involved in interactions with other model inputs: a high value of this quantity highlights a high interactive role of X i in the model. Moreover, an input having T i ≈ 0 can be considered as non-influential and, hence, it can be fixed at any convenient value within its variation range without significantly causing loss of information, e.g., to remove it from further analysis and reduce the model dimensionality.
Notably, the set of all the S i s together with all the T i s provides the analyst with an overall and satisfactorily complete picture of the sensitivity behaviour of the model, with higher-order Sobol' indices being selectively computed only if further investigation is needed.

V. RESULTS OF CASE STUDIES
In this Section, the proposed SR algorithm is deployed on three different distribution networks, i.e., a 20 bus AC-DC grid, the correspondent AC version (to show the comparison due to the absence of the DC sub-network) and a 182 bus AC-DC grid to test the algorithm scalability. The results of selected test cases for these three case studies are presented in Sections V-A, V-B and V-C, respectively.
The programming language Python v3.8 is used to implement the algorithm and, specifically, the OPF applies the solver IPOPT v3.14 with the package Pyomo v6.4. The software platform is hosted onto a 64 bit Windows machine with an i5 -2.5 GHz processor and 8 GB RAM.
In the MV distribution grids of the case studies, the buses host loads (associated to the categories ''residential'' or ''industrial''), and DG from renewable energy sources (such as wind turbines and Photovoltaic (PV) units). The power absorbed by loads and maximum generated power by DG depend on the representative power profiles (as active power in per-unit scale) that consider the measurement time (with time frames of 15 minutes) in specific days and seasons of the year. In this work, two types of days (week-or weekend-day) and seasons (summer or winter) are considered, originating four scenarios for load and generation profiles: summerweek, summer-weekend, winter-week and winter-weekend. The profiles data, extracted from German historical database, are obtained from representative residential and industrial loads [41], averaged wind turbines generation in 2020 [42] and PV profiles with variable cloudiness levels created with the tool in [43]. The four scenarios of power profiles used in the case studies are shown in Fig. 6. These power profiles are combined with nominal power and power factor for each load and DG to compute the power injections inputs for the OPF, both in the actual time of fault occurrence (as field measurements) and in the successive time frames (as forecasts, to compute the values of criteria η 1 , η 2 , η 3 and η 4 ).

A. MODIFIED CIGRE AC-DC NETWORK
As first test grid the ''CIGRE MV Distribution Network Benchmark'' in the European configuration is used and modified by integrating a multi-terminal DC sub-network [44]. The grid is represented in Fig. 7; two AC feeders at 20 kV are connected to the HV/MV transformer stations, indicated by SS1 and SS2, and to the multi-terminal DC sub-network at 32.6 kV via the converter stations C1, C2 and C3. According to the standard EN 50160, the voltage limits in (37) and (38) are set at ±10%. The NO and NC switches, present only in the AC sub-network, are represented with white and black squares, respectively, and the subscript t indicates the telecontrolled feature. The line parameters as well as nominal power and power factors of loads (as residential or industrial) and DG are obtained from [44]; the wind turbine is present at bus 19, whereas the PV units are located in buses 3, 4, 5, 6, 8, 9, 10, 16, and 18. The parameters of each AC-DC converter station are taken from [8], and [36] and reported in Table 3.
The investigated test case considers the occurrence of a fault at bus 3. Accordingly, to accomplish the protection procedures, the upstream switch S1 trips and, to isolate the fault area, the downstream switches S2 and S3 are open. The array of disconnected buses C is: 3,7,8,9,10,11] As bus 3 is within the fault area, it cannot be re-energized with the SR process. Hence, it is excluded from the restorable buses, which are: 7,8,9,10,11] Considering the presence of NC switches S6 and S7 among the restorable buses, the array of restorable bus groups is: 7,8] , [9] , [10,11]] The MCDA technique described in Section II-A1, with the indicated comparison parameters w α max P gen , w α max P load and w P gen P load , is deployed. The highest criticality values, combined with the generated and consumed power at each restorable bus group, yield the array of prioritized bus groups C g * r : [10,11] , [7,8]] To re-energize these disconnected buses, the array S NO (with combinations of NO switches) is obtained: The closure of each single switch (S5, S8 or S9) restores every bus in C r and corresponds to a specific SR candidate solution. The NC switches interconnecting the bus groups are S6 and S7; since the bus groups are adjoining, both S6 and S7 are classified as ''adjoining NC switches'' and must be alternatively opened. The array S NC NO (with combinations of NO and NC switches) corresponds to: Each sub-array of S NC NO (to be considered in the case no subarray in S NO is applicable) re-energizes every bus in C r and corresponds to a different candidate solution. It is worth to mention that the order of NO switches reflects the prioritized bus groups in C g * r : in the case no sub-array in S NC NO is applicable, the last element of each array (i.e., NO switch) may be removed and only the buses with highest priorities will be re-energized.
In order to deploy the MCDA and determine the near-optimal candidate solution, the AHP comparison parameters are set for the criteria β, P, η 1 , η 2 , η 3 and η 4 , forming the comparison matrix . In the deployed test cases, two sets of AHP comparison parameters are considered, associated to the matrices 1 and 2 whose values are reported in Table 4. The comparison parameters in 1 prioritize the applicability of the candidate solutions in the successive hours (i.e., the criteria associated to η 1 , η 2 , η 3 and η 4 ), whereas the comparison  parameters in 2 prioritize the use of telecontrolled switches (i.e., the criteria associated to β); these aspects are reflected in the values of w βη 1 , w βη 2 , w βη 3 and w βη 4 .
The SR process is deployed considering the occurrence of the fault at the time frames equal to 15, 30, 45, 60, 75 and 90 for each scenario in Fig. 6 (i.e., summer-week, summerweekend, winter-week and winter-weekend). Moreover, the tests are carried out considering the absorbed power of loads equal to 100% and 120% of the nominal values. Each scenario is indicated by a label in the form of ''X-Y-t'': ''X'' indicates the season (Summer ''S'' or Winter ''W''), ''Y'' indicates the day (Week ''W'' or Weekend ''WE'') and ''t'' represents the 15-minute time frame.
In each test, for each feasible candidate solution, the criteria for the MCDA are computed. The spider plots of Fig. 8 show the values of the six criteria for four selected scenarios, i.e., S-WE-30 and W-W-15 with nominal loads values at 100% and 120%. In these scenarios, the candidate solutions correspond to the NO switches included in S NO : the three candidate solutions are applicable and the opening of NC switches in the de-energized area is not deployed. The different graphs show how the available candidate solutions differ with respect to each considered criteria. For the scenario S-WE-30, in both the loading conditions, the use of comparison parameters 1 or 2 determines the selection of different SR solutions to be implemented.  In every SR test, the relative closeness Q + i for each candidate solution i is computed: the highest value determines the near-optimal SR solution that is implemented. The heat maps in Fig. 9 show the closeness for each inspected scenario with the different configurations of AHP comparison parameters and nominal load values; consequently the SR topology being implemented is directly identified by the candidate solution with the highest Q + i in the specific row. The blue cells without the value indicate the unfeasibility of the corresponding candidate solutions in the specific scenarios; whereas the white cells imply that, in the specific scenarios, the corresponding candidate solutions have not been inspected, because feasible solutions are already identified in groups with higher hierarchical order (as described in Section III). For every test with nominal load values at 100% (panels (a) and (b) in Fig. 9), at least one S NO candidate solutions is feasible and, hence, the S NC NO candidate solutions are not inspected. The deployment of 1 or 2 determines different SR solutions in several cases, particularly in the central hours of the day. Whereas, in specific scenarios of nominal load values at 120% (panels (c) and (d) in Fig. 9 The test outcomes show the importance and effectiveness of MCDA technique in computing the SR solution to be implemented, particularly by quantifying the different Q + i values of the available candidate solutions. The determination of the SR solution without the proposed MCDA technique may lead to non-optimal network topologies and operating setpoints, causing worse energy distribution conditions (here, between 0.9% and 99.5% of closeness difference among feasible candidate solutions in the same scenario, in the TOPSIS scale) with respect to the considered criteria. Fig. 10 shows the active power setpoints for the AC sides of each AC-DC converter, injected at buses 2, 6 and 12 in every tested scenario with nominal loads values at 100% and 120%. The increases of power injected by C2 into the AC sub-network, accompanied by the reduction of power injected by C1 and C3, reflect the scenarios for which the closure of switch S8 corresponds to the candidate solution. In general, the power exchange among AC and DC sub-networks is greater in scenarios with nominal loads values at 120% than in those with 100%, denoting the importance of power converters in regulating the power flows while ensuring the respect of operating limits. The power injected by C2 with nominal loads values at 100% is greater than the one with nominal loads values at 120% in specific scenarios, i.e., when the candidate solution corresponds to the closure of switch S8 only in the 100% loads condition.

1) SENSITIVITY ANALYSIS RESULTS
The SA methodology described in Table 2 is employed to assess and quantify the impact of the subjective judgements (i.e., the assigned w comparison parameters) on the calculation of the near-optimal solution in the SR process resulting from the utilized MCDA approach. Considering the AHP method presented in Section II-A1, the comparison parameters range discretely between 1/9 and 9. Hence, SA is applied by considering comparison parameters over the whole AHP range, unlike the specific values of 1 and 2 of Table 4. The MCDA for each of the 2000 MCS samples is run and the variability of the collected Q + i values is then evaluated. In Figure 11, the variability of the Q + i values is reported in terms of box plots for each of the three candidate solutions over the four considered scenarios (W-W-30, W-W-60, W-WE-30,W-WE-60). As it can be observed, the variability of S5 is almost negligible and its Q + S5 values are around zero, meaning that S5 is never the near-optimal solution. On the other hand, the variability of the values of Q + S8 and Q + S9 has a greater magnitude (e.g., for S8 σ = 0.12, 0.15, 0.15, 0.15 over the four scenarios, respectively). Moreover, it is worth analyzing the number of times that each candidate solution is ranked first among the three (i.e., the one with the highest value of Q + i in every MCS). These frequencies are reported in Table 5: it can be seen that S9 is very often the best candidate solution, with S8 being though ranked first in a not negligible number of times (especially for the three last scenarios).
In addition, it is worth evaluating which of the 15 comparison parameters is mostly responsible for the high variability of the Q + i values of S8 and S9. This is done by computing the first order and total Sobol' sensitivity indices. The results for the candidate solution S8 are shown in Figure 12 by means of heat maps to efficiently visualize the sensitivity indices values for each of the four scenarios under study (similar results, not shown in the figure, are obtained for S9). From Figure 12, it emerges that the uncertainty of the first five comparison parameters (i.e., w βP , w βη 1 , w βη 2 , w βη 3 , w βη 4 ) is responsible of almost the whole variability of the Q + i values in each of the four scenarios under study. Moreover, the remaining ten comparison parameters have an almost negligible effect on the variability of the Q + i values, since their total Sobol' sensitivity indices assume values close to zero. This signalizes that the comparison parameters from w Pη 1 to w η 3 η 4 can be set at any given value within their variation range without significantly affecting the variability of the Q + i values. On the other hand, the high values of the first order Sobol' indices for the comparison parameters w βη 3 and w βη 4 reveal that the variability of each of them is responsible, on average, of almost 20% of the variability of the Q + i values in all the four scenarios. Hence, if the DSO wants to reduce the Q + i variability the most, the best candidates for the title of 'most influential comparison parameters' would be w βη 3 and w βη 4 . Such information is of considerable help for supporting the algorithm implementation; practically speaking, the DSO deploying the SR algorithm has to consider that the choice of w βη 3 and w βη 4 (i.e., the two comparison parameters among the related criteria) assumes great importance on the decision of the SR solution. Additionally, since the quantity 1− K i S i is around 0.15 in all the four scenarios, the total amount of interactive effects among inputs is a worth 15%.
In view of these results, it is interesting to iterate the whole analysis with the same variability range of all the comparison parameters except w βη 3 and w βη 4 , each of which is instead fixed at two specific values, namely 1 3 and 4 for w βη 3 and 1 2 and 5 for w βη 4 (the values already employed for the previous simulations from the 1 and 2 matrices in Table 4). Two distinct situations arise. If w βη 3 and w βη 4 are alternatively fixed at the values of 1 , an overall increase of the times the candidate solution S9 is near-optimal can be highlighted, signalizing that w βη 3 = 1 3 and w βη 4 = 1 2 are ''adequate'' values for S9 to be clearly superior than S8 in being the near-optimal candidate solution. Moreover, the combined effect of simultaneously setting w βη 3 and w βη 4 at the ''adequate'' values of 1 is higher than fixing them separately and makes S9 be near-optimal in almost 90% of the times, as shown in Table 6. On the other hand, when alternatively fixing w βη 3 and w βη 4 at the values of 2 , a decrease in the number of times that S9 reveals to be near-optimal is observed, with S8 being the near-optimal candidate solution even more often than the situation depicted in Table 5 (i.e., with w βη 3 and w βη 4 left free to vary within their variability range). Also in this case, such tendency is even more stressed when simultaneously setting w βη 3 and w βη 4 at the values of 2 (whose results are shown in Table 7). In practical terms, once SA has allowed to identify which comparison parameters are the most influential ones in affecting the variability of the Q + i -e.g., by analyzing Figure 12-, the final outcome still depends on the specific values assigned to them (here, w βη 3 and w βη 4 ) for the specific scenario under study. For example, as it is clear from the comparison of Table 5 and 7, choosing a specific value of the most influential comparison parameters might also lead to a less clear distinction among two scenarios in terms of near-optimality, even up to a situation whereby S8 turns to be as near-optimal as S9, e.g., as for the scenarios W-W-60, W-WE-30 and W-WE-60.

B. MODIFIED CIGRE NETWORK WITHOUT DC SUB-NETWORK
Further test cases are carried out by excluding the DC sub-network from the modified CIGRE AC-DC test grid. In order to deploy the comparison with the AC-DC test grid used in Section V-A, the DC sub-network is replaced with  AC lines and buses having the same network parameters and nominal values of loads and DG. The modified CIGRE AC network is represented in Fig. 13. As in Section V-A, the fault occurrence is considered at bus 3, with the same arrays of prioritized bus groups C g * r and the candidate solutions in S NO and S NC NO . Moreover, the load and generation profiles shown in Fig. 6 and the AHP comparison parameters of 1 and 2 are used. Fig. 14 shows, for the scenarios S-WE-30 and W-W-15 combined with nominal loads weights at 100% and 120%, the values of the six MCDA criteria for the S NO feasible solutions. With respect to Fig. 8, it can be observed a decrease of the η criteria and an average increase of the power losses (criterion P) in multiple candidate solutions and scenarios. The comparison between each criteria from the tests on AC-DC and AC networks, for every scenarios (season, day of the week and time frame) is performed. The average variations of the criteria values for every candidate solution are reported in Table 8. Only the candidate solutions that are feasible in both tests with AC-DC and DC networks are considered; moreover, since the telecontrolled features of the switches remain unchanged, the analysis focuses only on the criteria P, η 1 , η 2 , η 3 and η 4 . The outcomes show that, on average, the power losses are higher in the AC network than AC-DC one; on the other hand, the η values are higher for the tests in the AC-DC network. Since P is a criterion that aims at being minimized and η are criteria that aim at being maximized, the results indicate, for each criteria, a better performance of AC-DC grid than that of the AC one and, particularly, a better suitability for AC-DC grid in deploying the SR process. The difference among AC-DC and AC grids performances is wider when the nominal load values increase from 100% to 120%.  Using the AHP comparison parameters in 1 and 2 , the SR process is deployed and its results are shown in Fig. 15. With respect to the available candidate solutions it is worth to notice that, in the case each sub-array of S NO is discarded due to unfeasibility, no scenario exists in which the candidate solutions of S NC NO are applicable and, in these cases, the near-optimal solution to be implemented belongs to S NC NO−1 array; this situation occurs at S-W-30 and W-W-45 for nominal load values at 120%. Moreover, in the same loading condition, for the scenarios with faults occurring at S-WE-45, W-W-75, W-WE-45 and W-WE-75 no feasible SR solution exists. With respect to the results shown from AC-DC grid test in Fig. 9, more scenarios in the AC test grid admits S9 as the only feasible candidate solution. This aspect is in line with the worsening of criteria values shown in Table 8, as the absence of DC sub-networks curtails the power distribution among the two feeders of SS1 and SS2.

C. MODIFIED MV OBERRHEIN AC-DC NETWORK
In order to test the scalability of the SR algorithm, assessment tests are conducted on a larger distribution network. The MV Oberrhein network is a generic AC distribution network at 20 kV developed within the Pandapower framework [45]. Two HV/MV substations (SS1 and SS2) feed 179 buses grouped in four feeders that are interconnected by six NO tie-switches; the complete network parameters are available in [46]. The network has been modified for this work with the addition of a point-to-point DC sub-network, with three DC buses, at 32.6 kV. The modified MV Oberrhein network is represented in Fig. 16, in which the fault occurrence and the tripped switches are highlighted. The network portion bounded by buses 35, 69, 140 and 159 (indicated in Fig. 16) is the area relevant for SR process and is depicted in Fig. 17.
The occurrence of the fault at bus 144 causes the tripping of the upstream switch S4 and, in order to isolate the smallest fault area, the switches S3 and S5 are opened. The array of restorable buses (shown with the yellow circle in Fig. 17) is composed by two sub-arrays, since these buses must be separately re-energized:   Hence, the arrays S NO and S NC NO are calculated, indicating the candidate solutions for the SR being tested: It is worth to specify that S NO contains only one sub-array since both NO switches S1 and S6 must be closed in order to re-energize every restorable load. Moreover, by applying the sub-array of S NC NO as candidate solution, the buses included in the last bus group of C g * r are excluded from the re-energization.
Similarly to Section V-A, the tests are conducted with the load and generation profiles represented in Fig. 6, for the scenarios in each season, day of the week and time frames equal to 15, 30, 45, 60, 75 and 90. The AHP comparison parameters considered in these tests corresponds to 1 . The values of MCDA criteria in the central time frames (30,60 and 75), for each inspected combination of day and season, are shown in Fig. 18. The different values of criteria η indicate the importance of performing OPF analysis in the future time frames with forecast values, in order to prevent the deployment of SR solutions that become unfeasible before the fault reparation. In the performed tests, the feasibility of S NO candidate solution is verified in each scenario (and, hence, applied as SR solution) except for the scenarios W-W-75 and W-WE-75: in these cases, the candidate solution in S NC NO is tested as feasible and applied as definitive SR solution.
The average computation time for the SR algorithm applied to the modified MV Oberrhein network, measured since the tripping of switches downstream the fault until the issuing of computed switches commands and operating setpoints to the SCADA data bus, corresponds to 15.3 s, having a standard deviation equal to 1.4 s. The data are obtained by repeating 100 times the SR process in the different scenarios. As presented in [47], in modern DSOs that deploy advanced automated services, the standard duration of FLISR process has an estimation of 5 minutes and the duration of SR itself is evaluated at 3 minutes. Hence the SR algorithm proposed in this paper, with respect to the indicated time range, proves to be suitable for a field implementation.
Finally, it is noteworthy that the SR method proposed in this paper showcases a wider degree of applicability as compared to those surveyed in Section I. In particular the integration of NC switches into the network reconfiguration allows to augment the set of possible candidate solutions. Moreover, considering the MCDA-based prioritization of bus groups to be reconnected offers a way to reflect network criticalities into the SR. Last, the inclusion of load and DG forecasts guarantees a higher robustness of the computed solution, hence alleviating the DSO burden in the network operations until fault reparation.

VI. CONCLUSION
In this paper a SR algorithm for the re-energization of AC-DC distribution grids, which specifically considers the direct interconnection of DC-based loads and generators in the same sub-network, is presented. The field implementation of the proposed algorithm relies on the presence of DC sub-network in the distribution grid as well as the controllability of field devices (AC-DC converters, DG, and switches) with commands and power setpoints via the SCADA system. As opposed to other approaches available in the literature, MCDA has allowed to flexibly combine and objectively prioritize different functional aspects while maintaining their mathematical consistency. The considered features, relevant for the grid operator, are the operational limits in timely modifying the grid topology, the energy efficiency (in relation to the dispatch costs) and the necessary control measures in the hours preceding the fault reparation. The importance of carefully choosing the relative weights of the operational objectives has also been highlighted via SA, which effectively supports the grid operator in tuning the SR in the DMS.
The proposed algorithm has been tested in multiple scenarios, related to different time frames, days of the week and seasons. When more than one candidate solution was feasible (even with differences of the closeness indices down to 5%), the MCDA has resulted crucial to determine the near-optimal one among the re-energization from either the AC-DC converter or an HV-MV substation, hence proving that DC sub-networks expand the set of SR options. As the loads values increased by 20%, the candidate solutions involving just the closing of NO switches have not always guaranteed the respect of operational security limits and, hence, the opening of NC switches was necessary. Additionally, the role of DC sub-networks has proven to be pivotal for the SR: in fact, MCDA criteria showcased an improvement of up to almost 30% when the power injected by AC-DC converters was optimized, determining a higher number of re-energized buses and more feasible SR candidate solutions. This tendency has revealed to be even more pronounced with higher nominal load values. Moreover, the scalability of the algorithm has been tested in a 182-bus distribution network: the computational time was, on average, 12 times faster than the SR estimated duration, thus being compatible with field deployment requirements.
Further extensions include the development of an ontology-based data exchange platform, which is foreseen to smooth the integration of field devices and alleviate the communication burden. Also, future developments may envisage the investigation of the role of energy storage systems and encompass the expansion of MCDA criteria by including the costs of energy not supplied and power not generated, to consider the monetary compensations to the grid operator.