Improving supply quality in distribution power networks: A game-theoretic planning approach

Electric Power Systems Research 213 (2022) 108666 Available online 9 August 2022 0378-7796/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Improving supply quality in distribution power networks: A game-theoretic planning approach Francisco Javier Murosa,b,∗, Daniel Sarachob, José María Maestrea aDepartment of Systems and Automation Engineering, University of Seville, Spain be-Distribution Digital Networks, Endesa, Enel Iberia, Spain ARTICLE INFO Keywords: Distribution power networks Supply quality indicators Local devices (LDs) Remote control devices (RCDs) Cooperative game theory Shapley value ABSTRACT Electric companies need to plan carefully the updates in their distribution power networks, assessing not only the expected impact on supply quality indicators, but also the cost of the different possibilities considered. To this end, we propose a planning method to improve the quality of supply in distribution power networks by upgrading devices so that they can be remotely operated. In the first place, our method identifies the lines with the lowest performance in an area of interest. Then, in each of these lines, the set of devices that can be upgraded is used to define the set of players of a cooperative game whose characteristic function provides a measure of the expected performance and economic impact of the updates in the infrastructure. Finally, the best locations to install remote control devices are determined by computing the Shapley value of this cooperative game, which, in addition, offers very valuable insight regarding each location considered. A case study based on a real substation in Spain is used to illustrate our approach and compare its performance with respect to the greedy-based method. 1. Introduction An effective reliability assessment is essential in the design and planning of distribution power networks so that they can operate in an economical manner with minimal interruption times for customer loads [1,2]. For example, recent approaches study reliability under cyber attacks [3] and consider time-of-use strategies [4]. In general, the notion of reliability is directly associated with the concept of electricity supply quality, which in turn comprises: (i) supply continuity, related to the number and duration of interruptions; (ii) power quality, which studies voltage and frequency deviations from its reference; and (iii) service quality, which deals with the interaction between the supplier and the customer [5]. Focusing on the continuity of supply, there are several reliability indices utilized worldwide, e.g., the System Average Interruption Frequency Index (SAIFI), which measures how often the average customer experiences a sustained interruption; the System Average Interruption Duration Index (SAIDI), which computes the average time that customers’ service is interrupted; the Customer Average Interruption Duration Index (CAIDI), which provides the average time required to restore the service to the average customer per sustained interruption; and the Average Service Availability Index (ASAI), which is given by the fraction of time that a customer receives power during a certain interval. Further information is given in [6,7], where several indices can be found, e.g., based on load ∗Corresponding author at: Department of Systems and Automation Engineering, University of Seville, Spain. E-mail addresses: [email protected] (F.J. Muros), [email protected] (D. Saracho), [email protected] (J.M. Maestre). rather than customers, or including only customers who actually experience an interruption. Other alternative performance indicators are the Total Power Quality Index (TPQI) [8], which is calculated using neural networks, and the Interruption Time Equivalent to the Installed Capacity, known by its Spanish acronym TIEPI [9], which will be considered in this article because it is the load-based legal reliability index used in Spain for distribution power lines. In particular, TIEPI measures the time of supply interruptions in terms of interrupted power. Power interruptions are mainly caused by scheduled works to improve the electric infrastructure, which are planned to minimize the number of affected customers, and power failures or faults, which are unpredictable shortages of supply due to, e.g., material degradation, electric storms, wind, birds, and vandalism. When a fault affects a distribution line, the recloser located at its beginning isolates the electric facility, and a process starts to find its location. In particular, sections of the power line are sequentially energized to locate the fault, which will be isolated to be repaired. Several fault detection methods can be found in the literature, dealing, e.g., with transmission lines and distribution systems [10], pipeline faults [11], commercial buildings [12], and photovoltaic systems [13]. Regardless of the method used, different devices are required to control, protect, and isolate electrical equipment during fault location/repair in a power line. Depending on whether human intervention is required at the device location, these can be https://doi.org/10.1016/j.epsr.2022.108666 Received 6 December 2021; Received in revised form 12 July 2022; Accepted 22 July 2022 Electric Power Systems Research 213 (2022) 108666 2 F.J. Muros et al. classified either as local devices (LDs) or remote control devices (RCDs). While the former are more economic, they need to be operated by human maintenance teams, which is time-inefficient. On the contrary, a fully remotely controlled line is expensive for the company, but it allows for a fast isolation of faulty sections. Therefore, a balance between LDs and RCDs should be attained. In this context, it is clear that companies need to plan the updates in their power infrastructure, evaluating how they affect the previously mentioned supply quality indicators, but also the economic impact of the different options considered. In the literature there can be found some recent works in this line. For example, reclosers are allocated in radial nonreturn lines in [14], where cross-entropy methods [15] are used to reduce the occurrence of voltage sags. Also, [16] places load switches and reclosers in theoretical bus test systems to reduce nonsupply energy, by using mixed integer programming (MIP) techniques. A similar mathematical approach is followed in [17], where fault indicators (FIs), reclosers and switch-disconnectors are inserted/relocated based on devices and crew costs; [18], which seeks to reduce the isolation times of faults by placing switch-disconnectors and FIs in strategic locations; and [19], which focuses on installing FIs with the aim of balancing costs due to customer interruptions and updates on the power lines. Likewise, a metaheuristic approach based on multi-objective particle swarm optimization is employed in [20,21] to minimize power outages and company costs by finding the number and optimal placement of switch-disconnectors and reclosers/sectionalizers, respectively, in electric distribution feeders. In [22], the set of switchdisconnectors to be upgraded is determined by formulating a weighted set cover problem considering both functional and economic requirements. Furthermore, in [23] an algorithm is provided to determine the optimal number of load switches assuming that all outage sections can be restored by supplying power from other feeders. In general, the previously mentioned articles lead to optimization problems that can provide an allocation for the new or upgraded resources to be included in the power network. However, these approaches may suffer from the curse of dimensionality in large-scale networks and do not provide as much insight regarding all the existing elements in the network. To cover this gap, we address the same problem from a different perspective. In particular, we propose a planning method to improve supply quality in distribution power networks based on the replacement of LDs by RCDs without reconfiguring the network, which is less costly for the company. To this end, we consider a TIEPI- based index to find which distribution lines have more potential to increase performance. Then, we enhance the preliminary results of [24] by setting a cooperative game in which players are LDs that can be upgraded to RCDs. Note that transformation and distribution centers will also be considered as LDs, generalizing previous approaches that only use overhead devices in their analysis. Our proposal leverages cooperative game theory to define a coalitional game where every possible subset of upgraded elements is assessed by a weighted measure of expected performance and economic impact [25–27]. In this way, it is possible to evaluate the benefit/cost of upgrading each of the LDs considered using tools such as the Shapley value [28], which allows us to rank the LDs according to their average impact, providing a means to find the best locations to install RCDs, hence reducing the non-supply intervals in a cost-efficient manner. Unlike many of the previously reported works, the proposed method does not imply solving an MIP problem. Instead, it averages the marginal contribution of the players for each possible coalition they can form and can be approximated in polynomial time [29]. Therefore, the major contribution of our work is to provide a gametheoretic method that assesses and ranks every element in the power network according to its average impact on a multi-criteria function, providing a superior insight regarding the different alternatives available. The benefits of this method are illustrated with a case study based on a real Spanish substation. Indeed, other related fields have Table 1 List of abbreviations. LD Local device RCD Remote control device 𝛹Set of lines in the area under study Set of players (LDs) in a given line 𝑙 Coalition of players or topology Set of topologies 𝑡o()Overall time spent in restoring normal service 𝑡a()Actuation time spent in locating the fault 𝑡rRepair time 𝑛Number of stages 𝑡𝑗()Duration of stage 𝑗 𝑡REM Average time spent in remote maneuvers 𝑡1 MAN (𝑡MAN) Average time spent in the first (the rest of) manual maneuvers 𝑡REC Average reclosing time 𝑃𝑙Power capacity of line 𝑙 TIEPI -𝛾𝑙Interruption time equivalent to the installed capacity of line 𝑙 NIEPI -𝜎𝑙Number of interruptions equivalent to the installed capacity of line 𝑙 𝛿𝑙Relative TIEPI of line 𝑙with respect to set 𝛹 𝛼𝑙Overhead section length of line 𝑙 𝛽𝑙Underground section length of line 𝑙 𝐶𝑙Number of transformation and distribution centers in line 𝑙 𝜌𝑙Expected performance index of line 𝑙 𝜂𝑙Distribution network performance index of line 𝑙 𝑐ins RCD installation average cost 𝑐man RCD yearly maintenance average cost 𝑐ene Energy cost per kVA 𝑟Annual average rate of faults per km 𝑇cAmortization horizon 𝑇pInvestment cycle 𝐽ins()Installation cost of topology  𝐽man()Maintenance cost of topology  𝐸loc()Amount of non-supply energy during fault location of topology  𝐽loc()Average power failure location cost of topology  𝐸rep Amount of non-supply energy during power failure repair 𝐽rep Average power failure repair cost 𝐽()Full cost of topology  (,𝑱)Cooperative game 𝜙𝑖(,𝑱)Shapley value of player 𝑖∈for game (,𝑱) 𝐌𝑁= [𝑚𝑖]Shapley standard matrix for a game with 𝑁players 𝐿Number of samples in probability distributions 𝜇𝜙𝑖Mean value of the Shapley value of player 𝑖 already approaches making use of tools from game theory, e.g., electrical and power applications in the context of solar energy [30,31], cellular networks [32], renewable energy communities [33,34], circuit theory [35], microgrids [36,37], and energy trading [38–40]. The rest of the article is organized as follows. In Section 2, the formulation of the problem is stated and some common devices are described. In Section 3, a reliability index based on the Spanish indicator TIEPI is presented to classify the distribution lines by their performance. Section 4presents the methodology employed in this work, introducing a cooperative game based on supply quality and economic indicators, and computing its Shapley value to find the most appropriate LDs to be upgraded to RCDs. Next, in Section 5, a scheme representing a Spanish electrical substation is considered as a case study to test the feasibility of the proposed approach. Finally, conclusions and lines of future research are given in Section 6. A list with the key abbreviations employed along the text is detailed in Table 1. 2. Problem statement We focus on distribution power networks, which are composed of electrical substations from where distribution power lines supply energy to end-users. Our goal is to improve the performance – in terms of continuity of supply – of the lines in an area under study, e.g., substations, geographical territories, etc. To this end, let us represent the different distribution lines of the area by the set 𝛹= {𝑙1, 𝑙2,…, 𝑙ℎ}.(1) Electric Power Systems Research 213 (2022) 108666 3 F.J. Muros et al. Fig. 1. Distribution line 𝑙∈𝛹with an initial configuration of 𝑁= 3 LDs, represented by hexagons, which leads to a set = {0,1,…,7}composed of 23= 8 topologies. In general, each line 𝑙∈𝛹is composed by sections of overhead/underground cables that connect a set of switchgear, which can be manipulated either remotely or manually to locate, isolate, and repair a power failure. Although an exhaustive analysis of switchgear properties is beyond the scope of this article,1a brief description of common devices is presented below: •Disconnectors guarantee an isolation distance between electrical equipment (LD). •Load switches make/break currents under nominal load, and also break temporary fault currents (LD/RCD). •Switch-disconnectors combine the two previously mentioned features (LD/RCD). •Cut-outs blow in fault conditions and must be replaced after the power failure is repaired (LD). •Fault Indicators (FIs) identify the path followed by the fault current (LD). •Reclosers detect and interrupt fault currents and have the ability to automatically restore power in temporary fault situations (LD/RCD). •Sectionalizers automatically isolate a faulted section after a preselected number of operations of an interrupting device that feeds the circuit, such as a recloser, have happened. It must be closed manually after the fault is repaired (LD). •Distribution centers are installations that share power between several branches (LD/RCD). •Transformation centers reduce the distribution lines’ voltage to supply the end-users (LD/RCD). Devices such as disconnectors, cut-outs, FIs, and sectionalizers cannot be normally remotely controlled in distribution lines, whereas the rest can be controlled, either locally or remotely. The elements in the latter set, highlighted in bold font in the above list, will be associated with the players in this work. Hence, let us introduce set = {1,2,…, 𝑁}, which denotes all LDs in a distribution line 𝑙∈𝛹that can be upgraded to RCDs. Therefore, there are 2𝑁different scenarios or topologies depending on the specific subset ⊆of LDs proposed to be replaced by RCDs. The full set of topologies is symbolized here by = {0,1,…,2𝑁−1}, where 0= ∅ and 2𝑁−1 =correspond, respectively, to the configurations of no and full upgrade. The cardinality of , symbolized by 𝑆=||, gives us the number of new RCDs. For instance, for a line with three LDs, the eight different topologies and their corresponding elements are depicted in Fig. 1. The overall time 𝑡othat takes place since the fault occurs until service is restored can be divided into the actuation time 𝑡a, i.e., the time spent 1The interested reader can find several specialized references in the literature, e.g., [41,42]. Fig. 2. Actuation and repair time parameters involved in presence of a power failure. by remote operators at the control center and the local maintenance crew to locate the fault, and the repair time 𝑡r, or time to repair damages caused by the fault 𝑡o() = 𝑡a() + 𝑡r.(2) Note that the actuation time depends on topology ⊆, for it establishes the number and position of LDs/RCDs in the line object of study. In particular, it can be defined as 𝑡a() = 𝑛−1 ∑ 𝑗=1 𝑡𝑗(),(3) where 𝑛refers to the number of stages, i.e., time periods where the number of customers affected by the power outage remains constant. Once the fault has been located, the last stage, say 𝑡𝑛, corresponds to the repair time 𝑡𝑟, i.e., 𝑡𝑛=𝑡𝑟. Several time parameters come into play in the computation of 𝑡a(). In particular, we will distinguish between the time consumed in remote maneuvers executed in the control center, denoted by 𝑡REM, the time used by the maintenance team in performing the first manual maneuver, which includes the time spent to move from the industrial unit to the fault area, and it is symbolized by 𝑡1 MAN, and the time utilized in manual maneuvers (except the first one) denoted by 𝑡MAN. Finally, a time 𝑡REC will be included to consider the reclosing time, i.e., the recloser opening/closing cycle at the beginning of the fault location process. Note that the aforementioned times strongly depend on the line length, its location, e.g., urban/city areas, the time when the fault occurs for it affects the number of people available, unexpected events as storms, traffic jams or car breakdowns, etc. These aspects make the duration of these times unpredictable. For this reason, we have averaged them considering the opinion of many network operation experts. An scheme that illustrates these concepts is depicted in Fig. 2. Finally, we consider the following assumptions: Assumption 1. The RCDs installed on a given line 𝑙∈𝛹always work correctly and require a maintenance cost per year. Assumption 2. The assessment of the lines in 𝛹and the decision to install new RCDs are performed cyclically by the company in investment cycles. Energy, installation, and maintenance costs will remain constant along each cycle. An average consumer price index (CPI) will be used to adjust these costs at the end of the cycle as a function of the expected inflation. Assumption 3. A standard sequence of maneuvers, which depend on the exact location of the power failure and the expertise of the remote network operators, will be considered to isolate the affected section, independently of the number and position of LDs/RCDs. Electric Power Systems Research 213 (2022) 108666 4 F.J. Muros et al. Fig. 3. Historical data for faults origin in distribution power networks in the South of Spain. Power failures can be due to: overhead power lines, which include overhead devices, underground power lines, small transformers inside transformation centers, distribution and transformation centers not including transformers, and substations. 3. Supply quality indicators for distribution power networks The notion of supply quality consists in the distribution of electricity maintaining its voltage and frequency values under desirable limits, while minimizing the number and duration of electric interruptions, and without neglecting the interaction between supplier and customer. In particular, the most critical aspect from the consumer’ viewpoint is the supply continuity. In this regard, the Spanish law defines two reliability indices to measure the impact of a power failure in a distribution line 𝑙∈𝛹, which are known by their Spanish acronyms as TIEPI (𝛾𝑙) and NIEPI (𝜎𝑙), and respectively refer to the Interruption Time and the Number of Interruptions Equivalent to the Installed Capacity [9]. These indices are respectively defined as 𝛾𝑙= 𝑛 ∑ 𝑗=1(𝑃𝑗×𝑡𝑗()) 𝑃𝑙 ,(4) 𝜎𝑙= 𝑛 ∑ 𝑗=1 𝑃𝑗 𝑃𝑙 ,(5) where 𝑃𝑙is the full power capacity of line 𝑙in kVA and 𝑛refers to the number of stages in the fault location/repair as commented before. Finally, terms 𝑃𝑗and 𝑡𝑗(), with 𝑃𝑗≤𝑃𝑙, denote respectively the nonsupply power capacity in kVA during stage 𝑗and its duration in hours, which will be computed for each fault and topology as a function of the average times introduced in the previous section. Note that index 𝛾𝑙 is measured in hours while 𝜎𝑙is dimensionless. Indeed, TIEPI is more restrictive from a legal perspective, given that it averages the power affected by periods of supply disruption. On the contrary, the duration of power interruptions is not considered in NIEPI. For this reason, the results provided in this work will focus on the former indicator.2Note that both definitions (4) and (5) can be easily extended to a set of faults involved in the chosen line in any time interval, typically a year, simply by aggregating the individual indices due to each fault. Also, the indices related to a full area, e.g., a substation, can be obtained by summing the individual indices of its lines and transformers. 3.1. A refined supply quality indicator We are interested in improving the performance of the distribution power lines in 𝛹. To this end, the first step is to detect which of those 2Power failures whose cause is external to the company, e.g., due to private facilities or extreme weather conditions, are not legally considered for the computation of TIEPI and NIEPI. Also, there are some legally established annual upper bounds for these indices, which depend on the line configuration: urban, rural, etc. If exceeded, they may result in a penalty for the company. lines need more improvement from a supply-continuity viewpoint. Since 𝛾𝑙does not consider the relative power capacity of line 𝑙with respect to 𝛹, we define the following weighted measure of all TIEPI indices in an area of interest regarding their power capacity: 𝛿𝑙[%] = 100 × 𝛾𝑙 ∑ 𝑠∈𝛹 𝛾𝑠 ×𝑃𝑙 ∑ 𝑠∈𝛹 𝑃𝑠 ,(6) where 𝑃𝑙is the power capacity of line 𝑙in kVA and 𝛾𝑙is given by (4). Hence, by index 𝛿𝑙, we give more relevance to lines with higher contracted power, normally related to a large number of customers. Lines 𝑙∈𝛹can be classified regarding their current performance using (6), but it is also interesting to consider their expected performance. To this end, historical average fault data in distribution power lines obtained from a Spanish electric company will be used (see Fig. 3). Considering this information, it is possible to compute the expected performance index for each line 𝑙as the following weighted summation over 𝛹: 𝜌𝑙[%] = 53.2𝛼𝑙 ∑ 𝑠∈𝛹 𝛼𝑠 +19.5𝛽𝑙 ∑ 𝑠∈𝛹 𝛽𝑠 +9.0𝐶𝑙 ∑ 𝑠∈𝛹 𝐶𝑠 +10.2𝑃𝑙 ∑ 𝑠∈𝛹 𝑃𝑠 +8.1 |𝛹|,(7) where 𝛼𝑙and 𝛽𝑙respectively correspond to the overhead and underground section lengths in km, 𝐶𝑙is the number of transformation and distribution centers, 𝑃𝑙gives the power capacity in kVA, and |𝛹|=ℎ represents the cardinality of 𝛹. Finally, with the aim of ranking lines 𝑙∈𝛹regarding the relation between their current and expected performance, the distribution network performance index is introduced as 𝜂𝑙=𝜌𝑙 𝛿𝑙 .(8) Eq. (8) considers not only the continuity of supply, but also the relative power capacity inside the area of interest and the physical and electrical features of the lines involved. This index connects the TIEPI impact of a distribution line relative to a reference scope with its expected TIEPI based on historical data. The larger 𝜂𝑙is, the better the continuity of supply of line 𝑙, achieving a value of 1 if both the current and expected performance coincide. For this reason, index (8) can be used to perform an efficient ranking of power lines inside 𝛹. Example 1. Let the area under study be described by the scheme shown in Fig. 4, where set 𝛹is composed of ℎ= 3 distribution power lines that have experienced one power failure each during a year, with information per stage detailed in Table 2. Note the short actuation times in the first stage of the fault in line 2 due to the RCD installed on the line, symbolized in this work by a double edge. The aggregated values for 𝛾𝑙are given in Table 3, achieving a full area TIEPI index of 𝛾=𝛾1+𝛾2+𝛾3= 174.11 min. The parameters 𝛼𝑙,𝛽𝑙,𝐶𝑙,𝑃𝑙, which allow computing indices 𝛿𝑙,𝜌𝑙and hence 𝜂𝑙, following (6),(7) and (8), respectively, are also shown in Table 3. When analyzing the results, note that line 2, with the lowest TIEPI, is, however, the one with the worst performance regarding 𝜂𝑙. Similarly, it is remarkable the high performance index of line 3independently of its TIEPI. Indeed, the relative power capacity with respect to area 𝛹penalizes line 2and fosters line 3in terms of performance. Also, the length of the latter line contributes to its high performance index. Summing up, by exploring 𝜂𝑙, it can be concluded that first line 2, and secondly line 1, should be improved, while line 3presents satisfactory results. 4. Improving distribution lines performance by game theory Sections 2and 3have respectively introduced the problem statement and some performance metrics. Here, we present the methodology employed to determine the upgrades of devices in the power network. In particular, we leverage the cooperative game theory framework to model the impact of the upgrades through the characteristic function Electric Power Systems Research 213 (2022) 108666 5 F.J. Muros et al. Fig. 4. Power network diagram of Example 1. Table 2 TIEPI index 𝛾𝑙for lines in Example 1. Stages/fault Line 1 Line 2 Line 3 𝑃1= 335 𝑃2= 675 𝑃3= 175 1 2 3 1 2 3 1 2 𝑃𝑗[kVA] 335 260 100 675 225 125 175 100 𝑡𝑗()[min] 39 12 71 6 27 183 38 31 Table 3 Line parameters needed to compute index 𝜂𝑙in Example 1. 𝛼𝑙𝛽𝑙𝐶𝑙𝑃𝑙𝛾𝑙[min] 𝜌𝑙𝛿𝑙𝜂𝑙 Line 1 4.1 4.5 3 335 69.51 30.43 35.26 0.86 Line 2 3.1 4.7 4 675 48.89 30.99 49.97 0.62 Line 3 7.3 3.7 2 175 55.71 38.58 14.76 2.61 of the game, which is a weighted measure of the expected performance given by the TIEPI index (recall Eq. (4)) and other relevant dimensions for the company such as economic viability and sustainability over time. By finding the Shapley value [28] of the cooperative game, it is possible to rank the alternatives considered according to their average impact, therefore providing a means to select which parts of the network should be upgraded. 4.1. Measuring expected performance and impact While replacing all LDs by RCDs provides the best results for the customer, it also generates the highest costs for the company. Therefore, given a line 𝑙∈𝛹, the best locations for new RCDs should be determined balancing their both needs. In particular, given a topology , the company investment will depend on installation and maintenance costs 𝐽ins() = 𝑐ins 𝑆, 𝐽man() = 𝑐man 𝑆(𝑇c− 1),(9) where 𝑐ins and 𝑐man are respectively the average costs of installation and yearly maintenance of RCDs, and 𝑇ccorresponds to a reasonable amortization horizon. In contrast, customer satisfaction is related to location and repair costs of the average power failure, which can be expressed as 𝐽loc() = 𝑐ene 𝑟(𝛼𝑙+𝛽𝑙)𝑇c𝐸loc(), 𝐽rep =𝑐ene 𝑟(𝛼𝑙+𝛽𝑙)𝑇c𝐸rep,(10) where 𝑐ene is the energy cost per kVA×h, 𝑟measures the annual average rate of faults per km, and the overhead and underground lengths are given respectively by 𝛼𝑙and 𝛽𝑙, as seen in Section 3. The amount of non-supply energy in kVA during the power failure location and repair can be calculated as 𝐸loc() = 𝑛−1 ∑ 𝑗=1(𝑃𝑗×𝑡𝑗()), 𝐸rep =𝑃𝑛×𝑡r,(11) where 𝑛is the number of stages, 𝑃𝑗provides the non-supply power in kVA during stage 𝑗, and 𝑡𝑗()measures the duration in hours of stage 𝑗, which depends on topology . Note that this term is closely related to the time spent in completing the fault location and repair maneuvers, as commented in Section 2. Finally, 𝑡ris the average time to repair the fault during last stage 𝑛. To evaluate the different possibilities for LDs to be upgraded to RCDs, i.e., the topologies ⊆, we define the following cost function: 𝐽() = 𝐽ins() + 𝐽man() + 𝐽loc() + 𝐽rep.(12) Eq. (12) goes beyond the affected power capacity and the duration of interruptions included in TIEPI, for it considers installation, maintenance, and operating costs as well. Therefore, it can be used to balance the customers’ and electric company’s goals. A cooperative game based on (12) is introduced below. 4.2. Cooperative game theory tools applied to power networks A cooperative game is defined by pair (,𝑱), where is the set of players containing the possible locations for new RCDs in a given distribution line, and vector 𝑱= [𝐽(0), 𝐽(1),…, 𝐽(2𝑁−1)]T evaluates (12) for every topology. In other words, 𝑱provides us with the cost of each of the 2𝑁possible configurations as a function of the upgraded locations in the line of study. In order to find the average contribution of the 𝑁players in this large amount of information, we will consider the Shapley value [28], the most used payoff rule in cooperative game theory. This value assigns to game (,𝑱)payoff vector 𝝓(,𝑱), defined for all 𝑖∈as 𝜙𝑖(,𝑱) = ∑ ⊆∶𝑖∉ 𝑆!(𝑁−𝑆− 1)! 𝑁![𝐽(∪ {𝑖}) − 𝐽()],(13) that is, the marginal contribution of each player in is averaged for all the possible permutations it can be part of, with weights depending on the size of each coalition by term 𝑆!(𝑁−𝑆−1)! 𝑁!. In our context, the Shapley value of a location corresponds to the average impact of its upgrade from LD to RCD. Therefore, the best locations to become RCDs will be associated with the lowest payoffs. Since we deal with the cost game given by (12),negative Shapley values will be related to locations that are beneficial to be remotely controlled, both from the viewpoint of the company and the customers. From an axiomatic perspective, the Electric Power Systems Research 213 (2022) 108666 6 F.J. Muros et al. Fig. 5. Game-theoretic planning method overview. Shapley value is formally characterized by the properties of linearity, efficiency,dummy player, and symmetric players [28]. In particular, the efficiency property states that the value of full set is totally distributed among the players ∑ 𝑖 𝜙𝑖(,𝑱) = 𝐽().(14) We will consider an alternative matrix formulation of the Shapley value that allows for a fast and simple computation. Following [43], it is possible to rewrite (13) as 𝝓(,𝑱) = ⎡⎢⎢⎢⎢⎣ 𝜙1 𝜙2 ⋮ 𝜙𝑁 ⎤⎥⎥⎥⎥⎦ =𝐌𝑁⎡⎢⎢⎢⎢⎢⎣ 𝐽(0) 𝐽(1) 𝐽(2) ⋮ 𝐽(2𝑁−1) ⎤⎥⎥⎥⎥⎥⎦ =𝐌𝑁𝑱,(15) where 𝐌𝑁∈R𝑁×2𝑁is known as the Shapley standard matrix [43], with its rows corresponding to players 𝑖∈, and its columns to the different topologies ⊆. Each element of this matrix is defined as 𝑚𝑖={(𝑆−1)!(𝑁−𝑆)! 𝑁!, 𝑖 ∈, −𝑆!(𝑁−𝑆−1)! 𝑁!, 𝑖 ∉.(16) For instance, for a game with three players, as that of Fig. 1, the Shapley standard matrix becomes 𝐌3=⎡⎢⎢⎢⎣ −1 3 1 3−1 6−1 6 1 6 1 6−1 3 1 3 −1 3−1 6 1 3−1 6 1 6−1 3 1 6 1 3 −1 3−1 6−1 6 1 3−1 3 1 6 1 6 1 3 ⎤⎥⎥⎥⎦ .(17) Remark 1. To formally consider pair (,𝑱)as a cooperative game, the cost of the empty set should be zero, that is, 𝐽(∅) = 0. This fact can be achieved by considering a new game (,𝑱′)that corresponds to a simple redefinition of cost function (12) by 𝐽′() = 𝐽() − 𝐽(∅).(18) The Shapley value remains unchanged because both games only differ in a constant term, i.e., 𝝓(,𝑱) = 𝝓(,𝑱′). 4.3. Probability distributions to manage varying parameters When evaluating (12), parameters are assumed fixed so that the characteristic function only depends on topology . Nevertheless, parameters can vary to generalize the results. To this end, we will consider here that both the average rate of faults 𝑟and the energy cost 𝑐ene can change in a prespecified interval. Therefore, the characteristic function would be re-formulated as 𝐽(, 𝑟, 𝑐ene), and the Shapley value could be computed by averaging a large number 𝐿of these parameters’ samples. Following a Monte Carlo approach [44], discrete probability distributions for the Shapley value can be obtained, with their respective mean values 𝜇𝜙𝑖=1 𝐿 𝐿 ∑ 𝑝=1 𝜙𝑝 𝑖(,𝑱),(19) representing a measure of each location’s performance. 4.4. Optimization procedure To conclude this section, we summarize the steps to apply the full game-theoretic planning method to improve the quality of the supply of distribution power networks: 1. Rank the lines inside 𝛹using the distribution network performance index 𝜂𝑙presented in Section 3.1, which measures their continuity of supply, and choose those below a considered threshold 𝜂∗. 2. Take the lines resulting from step 1 and perform a qualitative analysis to discard those whose faults can be solved by simple maintenance works (e.g., faults generated by trees touching power lines, high salinity concentration, and cables in bad conditions). 3. For each remaining line, consider the LDs to form the set of players of a cooperative game. 4. To define the characteristic function of the game, choose an amortization cost 𝑇c, and evaluate Eq. (12) for every possible subset of players (and a statistically relevant sample size 𝐿of parameter values in case the parameters are not fixed). 5. Compute 𝝓(,𝑱)and upgrade those LDs that verify 𝜙𝑖(,𝑱)< 0(𝜇𝜙𝑖<0in case the parameters are not fixed). 6. Rank the remaining LDs in increasing order with respect to the Shapley values and evaluate one by one performing a cost– benefit analysis regarding the possible upgrade. 7. The procedure stops once the cost–benefit analysis reveals that one LD should not be upgraded. In a nutshell, the method seeks distribution lines where RCDs can substantially reduce supply shortage, which could be particularly interesting in lines that require large actuation times for manual maneuvers, e.g., those in areas that are hard to access. All LDs with a negative Shapley value will be proposed to be replaced by RCDs because this upgrade implies a net benefit for both the company and the customers. Additionally, extra LDs can be promoted to RCDs depending on the company’s investment budget for the next 𝑇pyears. Note that this analysis can be repeated cyclically to deal with the customers’ evolving needs, and hence the installed infrastructure can change dynamically. A full diagram that illustrates the proposed method is given in Fig. 5. Electric Power Systems Research 213 (2022) 108666 7 F.J. Muros et al. Remark 2. In the first step of the method, a search of low-performance lines is performed inside 𝛹, which is not computationally expensive because it scales linearly with the number of lines ℎ. Later, a game over a set of 𝑁players is defined for some of the selected lines. The computational complexity related to the evaluation of 𝐽(𝑆)for every topology ⊆grows exponentially with 2𝑁, which limits the applicability of our method as it is to medium-scale lines with around 15–20 LDs. Nevertheless, randomized methods such as those proposed in [29,45] can be implemented to mitigate this issue. Basically, a number 𝑞of random orderings from the full set of permutations are taken for estimating the Shapley value of each player, with the estimation error being bounded as a function of 𝑞. Some large-scale network applications of this method are given in [46], which deals with the PageRank computation, and [47], which partitions a drinking water network. Therefore, it could be possible to use the proposed method in larger networks. Also, as the method is implemented offline for each investment cycle, there are no critical time restrictions to compute the solution. For example, it is acceptable if it is found after several days of computation. 5. Case study To evaluate the method introduced in the previous section, we consider the electric scheme for the substation and the distribution lines depicted in Fig. 7, which is based on the orthogonal scheme of a real substation in the South of Spain. More specifically, three-phase circuits are represented by a single-line diagram for ease of display, and different line sections are not to scale. In particular, the substation – detailed in Fig. 6 – consists of two 66/20 kV, 20 MVA power transformers, which feed five distribution lines each, through their corresponding busbars B1 and B2. Other elements inside the substation are the coupling, which provides an alternative feeder for the customers when their transformer is unavailable; the capacitor banks, which mainly regulate the reactive power; and the ancillary services, which give energy to control, measurement, protection, and supervision equipment. Focusing on distribution lines, we can broadly distinguish between urban, rural, mixed, and private lines, which in this particular case correspond to photovoltaic facilities. Regarding the network configuration, each line may have an alternative feeder coming from another line (in the same or other substation) or even from the same line. Two iterations of the procedure in Section 4.4 have been performed considering an investment cycle of 𝑇p= 3 years and a performance threshold of 𝜂∗= 0.9. The simulations were implemented using the numpy [48,49] package of Python®, Microsoft Excel®, and the tool Crystal Ball [50] by Oracle®in a 3.2 GHz Intel®CoreTM i3-6100T/8 GB RAM computer. 5.1. Performance index The first step is to compute the performance indices 𝜂𝑙for every line in the substation. For this computation, the private photovoltaic lines are not taken into account because their management does not correspond to the electric company. The resulting values for the indices are given in Table 4, iteration 1, where fault events in a full year have been taken into account. It can be seen that lines UTRERA, ROQUETAS and J.S.TOMAS should be improved, as they verify 𝜂𝑙< 𝜂∗. Nevertheless, it is detected that rural line UTRERA has many recurrent faults related to several cypress trees touching lines, which can be solved by tree pruning tasks, while urban line J.S.TOMAS has most of its faults located in a very damaged cable, which should be replaced. By contrast, the faults in line ROQUETAS are very diverse. Also, large actuation times for manual maneuvers are required in this line due to its length. Hence, it is an interesting option to upgrade to RCDs some of its LDs. Fig. 6. Detail of the electrical substation from the full scheme depicted in Fig. 7. 5.2. Ranking devices by the Shapley value To define a cooperative game in line ROQUETAS, whose initial configuration is represented in Fig. 8a, we consider an amortization horizon of 𝑇c= 15 years, with 𝑐ins = 14200 eand 𝑐man = 500 ebeing the RCDs installation and yearly maintenance average costs, respectively. Also, a fault repair is assumed to be done in 𝑡r= 175 min. Regarding actuation time 𝑡a, different scenarios depending on topology have been considered for the computation of 𝑡𝑗(), assuming 𝑡REM = 1 min, 𝑡1 MAN = 37.5min, 𝑡MAN = 14.5min, and 𝑡REC = 2.8min. These values were obtained averaging the estimates given by more than 50 network operation experts. Also, unlike [24], note that faults in the devices – and not only in the cables – are also considered in this work. To implement the Monte Carlo method commented in Section 4.3, we assume that both the annual average rate of faults 𝑟and the energy cost 𝑐ene are normally distributed, with mean 0.19 and standard deviation 0.003 in the former case and mean 0.83 e ∕kVA×hand standard deviation 0.0005 e ∕kVA×hfor the latter. Taking this information into account, which was estimated based on data from a Spanish electric company, a game in line ROQUETAS has been considered, with its 10 LDs acting as the players, which correspond to the seven transformation centers indicated in Table 4 and three switch-disconnectors. Therefore, cost function 𝐽(, 𝑟, 𝑐ene)for the 210 topologies has been evaluated by (12) for 𝐿= 105samples, and the average results considering all the fault locations scenarios have been used in the computation of the fault location and repair costs. Terms 𝐽ins() + 𝐽man(), concerning the company, and 𝐽loc() + 𝐽rep, related to the customers, have been represented for every topology in Fig. 9a. As expected, the former increases with cardinality S. By contrast, the latter strongly depends on the sequence of fault location maneuvers and the location of RCDs, being reduced when relevant locations – indeed those with lowest Shapley values – are remotely controlled. At this point, the Shapley standard matrix 𝐌10 has been utilized to compute (15), obtaining for a sample size 𝐿of parameters 𝑟and 𝑐ene the probability distributions of the Shapley values, which are drawn in terms of standard deviation bands in Fig. 10a. Their mean values have been calculated by (19), achieving the following results: 𝝁𝜙= 103 × [19.15 18.45 −92.35 21.70 2.47 15.02 −16.51 16.88 20.26 21.70]T, (20) where the values corresponding to LD3and LD7are negative. Therefore, it is economically interesting for the company and also beneficial for the customer to install RCDs in these locations, reaching the configuration shown in Fig. 8b. This upgrade makes sense from a topological viewpoint, for these locations are respectively situated in the line bifurcation point and in the urban center alternative feeder. Finally, at the end of this investment cycle, no more LDs are considered to be upgraded by the company. Electric Power Systems Research 213 (2022) 108666 8 F.J. Muros et al. Fig. 7. Electrical substation and their distribution lines analyzed in the case study. Electric Power Systems Research 213 (2022) 108666 9 F.J. Muros et al. Table 4 Computation of index 𝜂𝑙for lines in the substation under study. Indices that verify 𝜂𝑙< 𝜂∗are highlighted in bold font, meaning that these lines should be improved. From this set, those with 𝜂𝑙≥𝜂∗in iteration 2 are underlined. Busbar Line Faults/yr 𝛼𝑙𝛽𝑙𝐶𝑙𝑃𝑙𝛾𝑙[min] 𝜌𝑙𝛿𝑙𝜂𝑙 Iteration 1 B1 UTRERA 4 22.9 0 16 1180 44.84 11.69 18.38 0.64 PONCE 9 35.8 5.6 10 585 68.65 16.70 13.95 1.20 REINA 7 43.7 4.3 7 560 72.31 18.92 14.07 1.34 HESPERIDES 3 12.6 5 9 1480 15.87 8.70 8.16 1.07 CRUCES 1 2.1 22.5 15 2900 8.12 10.77 8.18 1.32 B2 MOLINO 4 14.3 4 13 660 16.40 8.90 3.76 2.37 ROQUETAS 3 16 18.3 7 2490 23.72 13.68 20.52 0.67 J.S.TOMAS 3 0 20.3 14 4730 7.89 10.65 12.97 0.82 Iteration 2 B1 UTRERA 2 26.3 0 17 1230 22.95 12.54 13.27 0.95 PONCE 7 35.8 10.1 11 685 57.27 16.95 18.44 0.92 REINA 8 43.7 8.4 8 585 69.10 18.95 19.00 1.00 HESPERIDES 6 12.6 5.9 9 1480 18.03 8.50 12.54 0.68 CRUCES 2 3.4 22.5 15 2900 6.42 10.43 8.75 1.19 B2 MOLINO 5 14.3 7.2 14 710 14.94 9.28 4.99 1.86 ROQUETAS 2 19.1 18.3 7 2490 13.80 13.87 16.15 0.86 J.S.TOMAS 1 0 20.3 13 4100 3.56 9.48 6.86 1.38 Fig. 8. Configuration of line ROQUETAS at the beginning of first (left) and second (right) iteration. Note that a symbolic power capacity of 1 kVA has been considered in switch-disconnectors to model their driving motors and ancillary services. 5.3. Second iteration As commented, it is expected that lines experiment changes through the investment cycle of 𝑇p= 3 years. Consequently, these changes will alter the needs of the distribution network, which in turn will modify the sorted list of LDs to be upgraded to RCDs. The new specifications and the corresponding performance indices for iteration 2 are also detailed in Table 4, where it can be seen that with the exception of line HESPERIDES, the TIEPI indices for every line have been reduced. It is usual that this index gradually decreases over the years due to the more efficient power network management. Note also that the three lines that were improved in the first iteration have accordingly increased their performance index 𝜂𝑙. Nevertheless, as can be seen, the improvement of line ROQUETAS is unsatisfactory considering threshold 𝜂∗, and hence a new cooperative game will be defined in this line. Also, line HESPERIDES does not satisfy our requirements, but most of their faults are condensed in a near-the-sea area, whose isolators require immediate cleaning and/or replacement. Hence, a corrective maintenance is the best choice for improving this line. The new game implemented in line ROQUETAS contains nine players corresponding to the eight LDs that were not remotely controlled in the first iteration and a new switch-disconnector LDAthat has been inserted in the line, providing an alternative feeder, as can be seen in Fig. 8b. That is, actuation times in RCD3and RCD7, which are no longer players, correspond to 𝑡REM = 1 min for every topology, which in turn will consequently reduce the costs and hence, in general, the Fig. 9. Cost function terms due to company and customers: Evolution with topology . mean values obtained, as shown below. The same parameters used in the first iteration are considered, but with a 3% CPI 3-years increment in the costs, i.e., 𝑐ins = 14626 e,𝑐man = 515 eand 𝑐ene =0.8549 e ∕kVA×h. The new cost function 𝐽(, 𝑟, 𝑐ene)computed for each topology is shown