Managing power supply interruptions: A bottom-up spatial (frontier) model with an application to a Spanish electricity network

Argüelles, Pablo; Orea, Luis Working Paper Managing power supply interruptions: A bottom-up spatial (frontier) model with an application to a Spanish electricity network Working paper, No. 7-2020 Provided in Cooperation with: Department of Economics, Copenhagen Business School (CBS) Suggested Citation: Argüelles, Pablo; Orea, Luis (2020) : Managing power supply interruptions: A bottom-up spatial (frontier) model with an application to a Spanish electricity network, Working paper, No. 7-2020, Copenhagen Business School (CBS), Department of Economics, Frederiksberg, https://hdl.handle.net/10398/9806 This Version is available at: https://hdl.handle.net/10419/222883 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Department of Economics Copenhagen Business School Working paper 7-2020 Department of Economics – Porcelænshaven 16A, 1. DK-2000 Frederiksberg Managing Power Supply Interruptions: A Bottom-Up Spatial (Frontier) Model with an Application to a Spanish Electricity Network Pablo Argüelles Luis Orea WORKING PAPER Copenhagen School of Energy Infrastructure | CSEI Pablo Argüelles Luis Orea Managing Power Supply Interruptions: A Bottom- Up Spatial (Frontier) Model with an Application to a Spanish Electricity Network CSEI Working Paper 2020-0 7 CBS Department of Economics 7-202 0 1 Managing power supply interruptions: A bottom-up spatial (frontier) model with an application to a Spanish electricity network Pablo Argüelles EDP España and University of Oviedo Luis Orea a University of Oviedo and Oviedo Efficiency Group Abstract In December 2013 a new electricity law was approved in Spain as part of an electricity market reform including a new remuneration scheme for distribution companies. This remuneration scheme was updated in December 2019 and the new regulatory framework introduced a series of relevant modifications that aim to encourage the regulated firms to reduce their power supply interruptions using a benchmarking approach. While some managerial decisions can prevent electricity power supply interruptions, other managerial decisions are more oriented to mitigate the consequences of these interruptions. This paper examines the second type of decisions using a unique dataset on the power supply interruptions of a Spanish distribution company network between 2013 and 2019. We focus on the effect of grid automatization on the restoration times, the relative efficiency of the maintenance staff, and the importance of its location. We combine a bottom-up spatial model and a stochastic frontier model to examine respectively external and internal power supply interruptions at municipal level. This model resembles the conventional spatial autoregressive models but differs from them in several important aspects. Keywords: electricity distribution, power supply interruptions, spatial econometrics, frontier models. _______________________ a Corresponding author. Email: [email protected]. Address: School of Business and Economics, Department of Economics, Avda. Del Cristo, s/n. 33006, Oviedo, Spain. 2 Managing power supply interruptions: A bottom-up spatial (frontier) model with an application to a Spanish electricity network Pablo Argüelles EDP España and University of Oviedo Luis Orea University of Oviedo and Oviedo Efficiency Group 1. Introduction The National Commission on Markets and Competition (CNMC), the Spanish regulator of the electricity sector approved last year the Circular 6/2019, which was published in the Official State Gazette on 5th December 2019, establishing a new methodology for calculating the remuneration of the electricity distribution companies for the years to come. The new regulatory framework introduced a series of relevant modifications to current regulation that aim to encourage the regulated firms to improve their performance, reducing not only distribution costs but also power supply interruptions. The previous regulatory framework (see Royal Decree 1048/2013), based on intertemporal comparisons, was not able to give the market the appropriate signal to achieve these targets. The new incentive scheme resembles a benchmarking (or yardstick) regulation as the incentives for quality of service improvements rely on the performance of other firms. In particular, the incentive reward/penalty will be computed by comparing the performance (variation) of a standard reliability indicator of each electricity distribution company with average sector performance in comparable urban and rural areas of supply. It is expected that the proposed approach will be more rewarding for the regulated firms if they improve the quality of service in the distribution network related to the interruption time of the electricity supply to their clients. Therefore, they will likely pay more attention to this important topic of the electricity distribution activity. Although a large percentage of Power Supply Interruptions (hereafter PSI) are beyond the management control of utilities (e.g. due to weather conditions or external human manipulations), managerial decisions such as vegetation management, asset investments and maintenance strategies can contribute to improvements of quality of service. While some of these decisions are more oriented to prevent electricity PSI caused by severe environmental conditions (e.g. reducing the number and length of overhead lines or improving grid assets capabilities, etc.), other managerial decisions are more oriented to mitigate the consequences of these interruptions (e.g. increasing the number of maintenance crews, improving their location, etc.). While some of these decisions require additional capital costs, other decisions require increasing firm’s operation and maintenance costs. It is not clear in the literature which strategy is better. As pointed out by Giannakis et al. (2005) and Jamasb et al (2012), the electricity distribution firms might adopt different strategies to combine capital and operating costs to improve their quality of service. Other literature has shown that weather conditions influence quality of service in electricity distribution networks. While Coelho et al. (2003), Domijan et al. (2003) and Zhou et al.(2006) find a significant correlation between PSI and rain, wind and temperatures, Yu et al. (2009) find that such factors often do not have a significant economic and statistical effect on the overall performance of the UK utilities. Wang and Billington (2002) show that severe weather conditions do not only increase the frequency of PSI, but also the restoration time. 3 Other researchers have examined the effect of the network characteristics on quality of service. For instance, Kjølle et al. (2003) have found that the number and duration of interruptions are significantly higher in overhead networks compared to cable networks. From a remedy perspective, the emergence of grid digitalization, which allows faster and more accurate detection of damaged equipment and its location, helps to restore electricity supply and fix faults more quickly. For a matter of time, previous literature has not studied the impact of such technological development on the duration of interruptions. This issue can be now studied as the electricity distribution firms in many countries have been installing remote control on transformers and switching and protection circuits in the last decade. On the other hand, faster restoration in most outages requires efficient and well-located maintenance crews. As the previous literature uses firm level data or aggregate geographical data, they do not allow examining the effect of the location of the maintenance crews on restoration times, as well as the relative efficiency of each maintenance crew. This paper uses a unique data set on the PSI in 91 Spanish municipalities of a Spanish distribution company network between 2013 and 2019 to identify the main technological and managerial drivers of the duration of power outages. As our dataset is very detailed, we know the source municipality of each PSI as well as, if any, the subsequent municipalities affected by the same PSI. In this sense, we are able to distinguish between “internal” PSI where the origin is located in the source municipality and “external” PSI caused by outages located in other municipalities that are physically connected through the distribution network. Distinguishing between the two types of supply interruptions is not a semantic issue because while the quick restoration of supply in an “internal” PSI requires one of the maintenance crews to be close and/or the existence of remote controlled switching and protection systems located in the source municipality, the restoration time in an “external” PSI has nothing to do with its own factors but with the location and network characteristics of the source municipality firstly affected by the outage. For this reason, we use two different approaches to examine “external” and “internal” PSI. The duration of the external PSI is modelled using a spatial non-frontier econometric model. A spatial non-frontier model is used here because the external PSI only appear if there are PSI in neighboring and connected municipalities. As we know the sequence of the individual PSI across municipalities, we can develop a spatial model from scratch, i.e. using the engineering or physical information of the PSI that occurred in each municipality. Therefore, our model can be viewed as a Bottom-Up (BoU) spatial model. A frontier specification is not used here because the managerial decisions aiming to restore the electricity supply in these municipalities fix network faults located in other municipalities (i.e. in the municipality firstly affected by the outages). For this reason, inefficient performance is only examined using internal PSI. The duration of the internal PSI is modelled using a standard nonspatial frontier model. It does not make sense to use here a spatial specification because the faults that have triggered these PSI took place within the municipality. The paper is structured as follows. Section 2 develops several spatial and frontier models that aim to identify the main technological and managerial drivers of the restoration times of internal and external PSI. The bottom-up nature of our models is not only apparent in that we can distinguish between internal and external PSI, but also in that we aggregate our engineering-based (outage) data on monthly and municipality basis in order to mimic the traditional spatial econometric models. Section 3 discusses the data used in the empirical analysis and its sources. Section 4 provides the parameter estimates and discuss the main results. Finally, Section 5 presents the conclusions. 4 2. Methodology 2.1. Preliminaries Let us first explain briefly how we use the individual information of each outage to define our dependent variable(s) and, in particular, to compute an engineering-based spatially lagged dependent variable, which differs from the one often used in the standard spatial econometric models. As customary in the Spanish electricity distribution system, we use the so-called TIEPI as a reliability indicator to measure (lack of) quality of service. This indicator, which is defined further in Section 3, is a power-adjusted measure of the restoration times as it considers the active power loss due to the outage, relative to the whole system power. As we are interested in the duration and not the frequency of PSI, our final sample does not include months without outages.1 The duration of all power outages occurred in a municipality 𝑚𝑚= 1, … , 𝑀𝑀 during month 𝑡𝑡= 1, … , 𝑇𝑇 will be expressed as 𝑌𝑌 𝑚𝑚𝑚𝑚. In a standard Spatial Autoregressive Model (SAR), it is assumed that the dependent variable 𝑌𝑌 𝑚𝑚𝑚𝑚 is spatially correlated, and this correlation is modelled as follows: 𝑌𝑌 𝑚𝑚𝑚𝑚 =𝜆𝜆𝑊𝑊𝑌𝑌 𝑚𝑚𝑚𝑚 +𝜀𝜀𝑚𝑚𝑚𝑚 (1) where 𝜀𝜀𝑚𝑚𝑚𝑚 is an error term, 𝑊𝑊𝑌𝑌 𝑚𝑚𝑚𝑚 =𝑊𝑊 𝑚𝑚𝑌𝑌 𝑚𝑚=∑𝑊𝑊 𝑚𝑚𝑚𝑚𝑌𝑌 𝑚𝑚𝑚𝑚 𝑀𝑀 𝑚𝑚=1 stands for the endogenous spatial lag of the dependent variable, 𝑌𝑌 𝑚𝑚= (𝑌𝑌 1𝑚𝑚,𝑌𝑌 2𝑚𝑚,…,𝑌𝑌 𝑀𝑀𝑚𝑚) is an 𝑀𝑀𝑀𝑀1 vector of the dependent variables, and 𝑊𝑊 𝑚𝑚= (𝑊𝑊𝑚𝑚1,𝑊𝑊𝑚𝑚2,…,𝑊𝑊𝑚𝑚𝑀𝑀) is a spatial weight vector where the weights (𝑊𝑊 𝑚𝑚𝑚𝑚) equal one for adjacent units and zero for non-bordering units. Finally, the 𝜆𝜆 parameter is the spatial autoregressive coefficient that measures the degree of spatial correlation between units. Notice that 𝑊𝑊𝑌𝑌 𝑚𝑚𝑚𝑚 can be viewed as a weighted measure of the duration of all PSI that occurred in adjacent municipalities to the municipality 𝑚𝑚 in period 𝑡𝑡, even if the outages in neighboring municipalities have nothing to do with the PSI in municipality 𝑚𝑚. In some cases, 𝑊𝑊𝑌𝑌 𝑚𝑚𝑚𝑚 include common PSI. However, as this spatially lagged dependent variable ignores the true sequence of the PSI across municipalities, it surely includes the duration of PSI of subsequent municipalities affected by the same outage. If so, equation (1) would wrongly suggest that the outage in a preceding municipality is caused by the outage of a subsequent municipality. In other words, the traditional spatially lagged dependent variable will provide biased results because it ignores the true physical propagation (i.e. true causality) of PSI in a real electricity distribution network. Like in a dyadic-type data setting where it is clearly possible to distinguish an origin unit from a destination unit (see e.g. Neumayer and Plümper, 2010), we know the source municipality of each PSI as well as, if any, the subsequent municipalities affected by the same outage. Therefore, our dataset allows us to model properly contagion, diffusion or spillover effects across municipalities that are physically connected through the distribution network. To achieve this objective, let us decompose the total duration of the PSI (𝑌𝑌 𝑚𝑚𝑚𝑚) into two sets: internal and external PSI. That is: 𝑌𝑌 𝑚𝑚𝑚𝑚 =𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 +𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 =∑𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚 𝐼𝐼𝑚𝑚𝑚𝑚 𝑖𝑖=1 +∑𝑒𝑒𝑗𝑗𝑚𝑚𝑚𝑚 𝐽𝐽𝑚𝑚𝑚𝑚 𝑗𝑗=1 (2) 1 Analysing the frequency of PSI is also more challenging as we should cope with an excessive number of zero values in our data. The development and estimation of zero-inflated econometric models in non-frontier settings have become widespread. See Yang et al. (2017) for a comparison of methods. Our models, however, allow examining whether the number of outages matters when estimating the coefficients in which we are interested. 5 where 𝑖𝑖= 1, … , 𝑌𝑌𝑚𝑚𝑚𝑚 stands for outages started in municipality 𝑚𝑚 in period 𝑡𝑡, and 𝑗𝑗= 1, … , 𝐽𝐽𝑚𝑚𝑚𝑚 stands for PSI in municipality 𝑚𝑚 in period 𝑡𝑡 caused by outages that have started before in other municipalities physically connected through the distribution network.2 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚 and 𝑒𝑒𝑗𝑗𝑚𝑚𝑚𝑚 are respectively the duration of the internal and external PSI from the perspective of the municipality 𝑚𝑚. Therefore, while 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 measures the duration of all internal PSI occurred in municipality 𝑚𝑚 in period 𝑡𝑡, 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 measures the duration of all external PSI that affect municipality 𝑚𝑚 in period 𝑡𝑡 but that have been ‘imported’ from other municipalities. In other words, our dataset allows us to know the true outcome (i.e. 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚) of the spatial spillovers we are able to capture using a spatial econometric model. In a standard spatial application, only the aggregate effect of both internal and external PSI (i.e. 𝑌𝑌 𝑚𝑚𝑚𝑚) is observed by the econometrician. Thus, we have a kind of quasi-natural experiment to test what spatial spillovers are being captured by the most common spatial econometric models. On the other hand, as we have perfect information about the characteristics of the electricity distribution network and all connections across municipalities, we know the true W matrix. This matrix should be computed in our application considering the number of connections between two municipalities and the capacity of these connections. Otherwise, our spatial specification of a physical phenomenon such as the propagation of PSI would not make sense. Moreover, unlike most spatial models in regional economics, contiguity in our case is a necessary but not sufficient condition to be affected by other municipality. It also requires being connected with the preceding municipalities.3 Another attractive feature of our engineeringbased W matrix is that its elements can be treated as exogenous spatial weights when estimating the model due to the physical nature of the network connections. This is very useful from an econometric view because it avoids the need to address challenging endogeneity issues associated with W. 2.2. Modelling external PSI As aforementioned, explaining 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 and 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 requires a different model because while 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 emanates from municipality 𝑚𝑚 and the network equipment that should be fixed is located in this municipality, 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 emanates from other municipalities and thus there is nothing to fix in municipality 𝑚𝑚. In other words, 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 is a pure spatial spillover or contagion effect that disappears if municipality 𝑚𝑚 in period 𝑡𝑡 is not involved in multi-municipality PSI that had previously started in another municipality. For this reason, we propose using the following SAR specification to explain the duration of each external interruption: 𝑒𝑒𝑗𝑗𝑚𝑚𝑚𝑚 =γ𝑚𝑚Ω𝑗𝑗𝑚𝑚𝑌𝑌 𝑗𝑗=γ𝑚𝑚∑Ω𝑗𝑗𝑚𝑚𝑚𝑚𝑌𝑌 𝑗𝑗𝑚𝑚 𝑁𝑁𝑗𝑗 𝑚𝑚=1 (3) where 𝑁𝑁 𝑗𝑗 is the number of municipalities involved in outage 𝑗𝑗, 𝑌𝑌 𝑗𝑗= (𝑌𝑌 𝑗𝑗1,𝑌𝑌 𝑗𝑗2,…,𝑌𝑌 𝑗𝑗𝑁𝑁𝑗𝑗) is the 𝑁𝑁 𝑗𝑗𝑀𝑀1 vector of PSI durations involved in outage 𝑗𝑗, and Ω𝑗𝑗𝑚𝑚 = (Ω𝑗𝑗𝑚𝑚1,Ω𝑗𝑗𝑚𝑚2,…,Ω𝑗𝑗𝑚𝑚𝑁𝑁𝑗𝑗) is a 𝑁𝑁 𝑗𝑗𝑀𝑀1 spatial weight vector where the weights equal one for first-order preceding municipalities that are connected through the electricity distribution network and have been affected immediately before than municipality 𝑚𝑚 in outage 𝑗𝑗. Otherwise, the elements in Ω𝑗𝑗𝑚𝑚 are equal to zero. Finally, the γ𝑚𝑚 parameter is the spatial autoregressive coefficient that measures the contagion 2 The time lag between outages cannot be perceived by humans as it usually lasts milliseconds. 3 In this sense, our model looks like a multilevel or hierarchical SAR model, which is becoming increasingly popular in social sciences. See Corrado and Fingleton (2016) for a summary of these models. It is assumed in this models that there exist a number of well-defined groups organized within a hierarchical structure, such as class within schools. Much of the multilevel literature assumes that inter-individual interaction is restricted to within group boundaries. From a spatial perspective, this implies that the inter-individual interactions are restricted spatially in a similar fashion that our bottom-up spatial model. 6 degree of municipality 𝑚𝑚 from preceding municipalities. We expect that this degree depends on the number of connections of municipality 𝑚𝑚 with other municipalities, as well as the capacity of these connections. If we next plug (3) into 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 =∑𝑒𝑒𝑗𝑗𝑚𝑚𝑚𝑚 𝐽𝐽𝑚𝑚𝑚𝑚 𝑗𝑗=1 , and add the traditional noise term, we get the following bottom-up SAR model:4 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 =γ𝑚𝑚Ω𝑌𝑌 𝑚𝑚𝑚𝑚 +𝜔𝜔𝑚𝑚𝑚𝑚 (4) where Ω𝑌𝑌 𝑚𝑚𝑚𝑚 =∑ ∑ Ω𝑗𝑗𝑚𝑚𝑚𝑚𝑌𝑌 𝑗𝑗𝑚𝑚 𝑁𝑁𝑗𝑗 𝑚𝑚=1 𝐽𝐽𝑚𝑚𝑡𝑡 𝑗𝑗=1 , and 𝜔𝜔𝑚𝑚𝑚𝑚 is a symmetric and normally distributed noise term. One might be tempted to extend this model with municipality-specific factors, i.e. adding the traditional 𝛽𝛽𝑋𝑋𝑚𝑚𝑚𝑚 term in a conventional SAR model. However, this might yield the unsound result that municipality 𝑚𝑚 has been affected by outages emanated in other municipalities even when municipality 𝑚𝑚 was not involved in common outages, i.e. when either Ω𝑗𝑗𝑚𝑚𝑚𝑚 = 0 or 𝑌𝑌 𝑗𝑗𝑚𝑚 = 0. Equation (4) can be estimated using a simple Ordinary Least Squares (OLS) estimator5 given that Ω𝑌𝑌 𝑚𝑚𝑚𝑚 is by definition an exogenous variable because it has been computed using past values although our notation in equation (4) does not explicitly indicate this recursive nature of Ω𝑌𝑌 𝑚𝑚𝑚𝑚. Therefore, equation (4) can be viewed as a time-space recursive model where the dependent variable is lagged in both space and time dimensions (see e.g. Elhorst, 2010).6 Instead of using variables in levels, it is also possible to estimate a SAR model in per outage terms. That is, we can replace respectively 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 and Ω𝑌𝑌 𝑚𝑚𝑚𝑚 with 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚/𝐽𝐽𝑚𝑚𝑚𝑚 and Ω𝑌𝑌 𝑚𝑚𝑚𝑚/𝐽𝐽𝑚𝑚𝑡𝑡 in (4). Compare to a model in levels, the per outage specification reduces heteroskedasticity issues and improves goodness-of-fit.7 As it is customary in both spatial and frontier literatures, the SAR model can also be estimated using variables in natural logarithms. The logarithm specification allows us to prevent the existence of extreme values for the heteroskedastic autoregressive parameter in our spatial models. In addition, using logs allows us to reduce convergence issues when estimating our frontier models. The γ𝑚𝑚 parameter that measures the contagion degree of municipality 𝑚𝑚 from preceding municipalities likely depends on the number of connections of municipality 𝑚𝑚 with other municipalities, as well as the average capacity of these connections. Indeed, we expect larger spatial spillovers when the number and capacity of the connections with preceding 4 Notice that equation (4) cannot be estimated in a conventional spatial econometric application because only, say, the whole GDP in one region is observed, and not the portion of such GDP that actually depends on the GDP of neighbouring regions. 5 Equation (4) can also be estimated using Maximum Likelihood (ML) techniques if we assume that 𝜔𝜔𝑚𝑚𝑚𝑚 follows a normal distribution, i.e. 𝜔𝜔𝑚𝑚𝑚𝑚~𝑁𝑁(0, 𝜎𝜎𝜔𝜔=𝑒𝑒𝜏𝜏0). 6 In this sense, Skevas (2019) shows that the endogeneity issues that exist in conventional SAR models such as 𝑌𝑌 𝑚𝑚=γΩ𝑌𝑌 𝑚𝑚+𝜔𝜔𝑚𝑚 are caused by the fact that the dependent variable of any individual appears both on left and the right-hand side of the spatial autoregressive model, after replacing 𝑌𝑌 𝑚𝑚 on the right-hand side of the above equation with 𝑌𝑌 𝑚𝑚=γΩ𝑌𝑌 𝑚𝑚+𝜔𝜔𝑚𝑚. A time-space recursive model such as 𝑌𝑌 𝑚𝑚=γΩ𝑌𝑌 𝑚𝑚−1 +𝜔𝜔𝑚𝑚 overcomes such a bias because for a particular observation, while 𝑌𝑌 𝑚𝑚 appears on the left-hand-side, the right-hand-side contains 𝑌𝑌 𝑚𝑚−2 after replacing 𝑌𝑌 𝑚𝑚−1with 𝑌𝑌 𝑚𝑚−1 =γΩ𝑌𝑌 𝑚𝑚−2 +𝜔𝜔𝑚𝑚−1, and the endogeneity issue is wiped out. 7 Notice that the total duration of external PSI in (4) can be decomposed into the number of external PSI that affect municipality 𝑚𝑚 in period 𝑡𝑡 (𝐽𝐽𝑚𝑚𝑚𝑚) and the average duration of these interruptions (𝑒𝑒𝑚𝑚𝑚𝑚 =∑𝑒𝑒𝑗𝑗𝑚𝑚𝑚𝑚 𝐽𝐽𝑚𝑚𝑚𝑚 𝑗𝑗=1 /𝐽𝐽𝑚𝑚𝑚𝑚). That is, 𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 =𝐽𝐽𝑚𝑚𝑚𝑚 ·𝑒𝑒𝑚𝑚𝑚𝑚. This decomposition suggests that using a per outage specification of (4) allows us to focus on outages’ duration (i.e. on 𝑒𝑒𝑚𝑚𝑚𝑚) and not on the number of such outages (i.e. on 𝐽𝐽𝑚𝑚𝑚𝑚). 13 the number of connections of municipality 𝑚𝑚 with other municipalities (𝑙𝑙𝑙𝑙𝑁𝑁𝑃𝑃) and their average capacity (𝑙𝑙𝑙𝑙𝑃𝑃𝑃𝑃) capture the effect of municipality 𝑚𝑚’s connectivity with nonpreceding municipalities, i.e. subsequent municipalities involved in common outages plus municipalities that are connected with municipality 𝑚𝑚 but not involved in common outages. The parameter estimates in Table 4 confirm our expectations. The contagion degree increases with the average capacity of the connections with preceding municipalities, as expected. The relationship with the number of connections with preceding municipalities is not statistically significant. We also find the expected result that γ𝑚𝑚 decreases with 𝑙𝑙𝑙𝑙𝑁𝑁𝑃𝑃 and 𝑙𝑙𝑙𝑙𝑃𝑃𝑃𝑃 because other sources of supply tend to attenuate the impact of PSI imported from preceding municipalities. 4.2. Frontier models The duration of the internal PSI is modelled using standard non-spatial stochastic frontier (SF) models, where the dependent variable is now the duration of all internal PSI occurred in municipality 𝑚𝑚 in period 𝑡𝑡. As we have found convergence problems when the dependent variable is not logged, we discuss the frontier results using 𝑙𝑙𝑙𝑙𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚 and 𝑙𝑙𝑙𝑙(𝑌𝑌𝑌𝑌𝑚𝑚𝑚𝑚/𝑌𝑌𝑚𝑚𝑚𝑚). Table 5 shows the parameter estimates of our two SF models. In general, both specifications of equation (5) yield very similar results, except for 𝑙𝑙𝑙𝑙𝑁𝑁𝑙𝑙. The coefficient of this variable is positive and statistically significant when restoration times are in logs, but not when they are expressed in per-outage terms. Both results together seem to suggest that the monthly interruptions increase with both network length. On the other hand, the fact that we do not find a significant effect on per-outage restoration times indicates that, in general, the differences in accessibility between municipalities do not matter to restoring power supply.14 The coefficient of 𝑙𝑙𝑙𝑙𝑃𝑃𝑃𝑃𝑇𝑇 is positive in both specifications of the SF model. This is an expected result because our dependent variable is a power-adjusted measure of the restoration times in each municipality and the installed capacity varies notably across municipalities. However, the positive effect found in both specifications might also indicate that the severity of the PSI increases with the number of transformers of the network. [Table 5 here] The percentage of distribution stations or step-down transformers fitted with a remotecontrol system (𝐷𝐷𝑌𝑌𝐷𝐷𝑇𝑇) allows us to measure the effect of grid automatization on the restoration times. We find a negative and statistically significant effect on restoration times in both specifications. Therefore, grid automatization does reduce the duration of all PSI and the restoration times of each outage. If the percentage of digital transformers increases in 10% points, the restoration times will be reduced about 8%. On the other hand, we do not find a significant effect of underground transformers (𝑃𝑃𝑇𝑇) on restoration times. This is an expected result because the reference transformers are located indoor, and there are no differences in restoration times with the underground transformers from an engineering point of view. The proportion of outdoor transformers (𝑂𝑂𝑇𝑇) has a negative and significant effect on restoration times when the restoration times are not measured in per-outage terms. We originally expected a positive effect because firm’ engineers believe that fixing an outdoor transformer normally requires more time than an indoor or underground transformer. Therefore, the negative coefficient is likely to do with the fact that most of the outdoor 14 If the lack of accessibility has been extraordinarily relevant in occasional outages, its effect should be captured by the corresponding inefficiency score. 14 transformers are located in rural areas where the installed capacity tends to be small. In this way, the percentage of outdoor transformers might have a negative effect on a power-adjusted measure of restoration times. Similar comments can be made for the negative coefficient found for the proportion of overhead lines (𝑂𝑂𝑙𝑙). The overhead lines are mainly located in rural areas. Therefore, the overhead lines tend to supply power to a smaller number of customers per network kilometer. However, the coefficient of the 𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃𝑙𝑙 dummy variable is positive and statistically significant in both specifications. As the proportion of overhead lines in rural municipalities (85%) is higher than in more urban municipalities (78%), this dummy variable thus might be capturing part of the effect of 𝑂𝑂𝑙𝑙 on restoration times. The two variables included in the model to measure the effect of the distance to the nearest maintenance crews (𝑙𝑙𝑙𝑙𝐷𝐷𝑌𝑌𝑙𝑙) on outage durations have positive coefficients and in general are statistically significant. This result seems to confirm that the location of maintenance crews is a relevant factor to managing PSI, especially if the municipality affected for the outage is more urbanized. The larger effect for urban municipalities has likely to do with the fact that the installed capacity in these municipalities tend to be larger than in urban areas. In addition, we do not find an increasingly effect of the distance on restoration times given the lack of significance of the coefficient of the quadratic distance term. We have tried to capture the effect of weather conditions on restoration times through the 𝑊𝑊𝑌𝑌𝑃𝑃𝑇𝑇𝐻𝐻𝑌𝑌𝑅𝑅 variable, that measures the proportion of outages caused by weather-related issues, plus a set of seasonal dummy variables. The larger the proportion of outages caused by weather-related issues, the larger the restoration times are in both specifications. The estimated coefficient is however much smaller when we use a per-outage measure for the duration of the PSI. This result therefore seems to indicate that the weather conditions not only increase restoration time of each outage, but also the frequency of outages, as found in previous literature. Unlike the 𝑊𝑊𝑌𝑌𝑃𝑃𝑇𝑇𝐻𝐻𝑌𝑌𝑅𝑅 variable, the three seasonal dummy variables included as frontier determinants do not provide additional information about the severity of the PSI. Finally, and regarding the set of regional dummy variables, we have found highly significant coefficients, indicating that the above set of variables were not able to control for all differences in the electricity distribution network between Spanish provinces. It is worth highlighting the large values of the coefficients of 𝐻𝐻𝑃𝑃𝑌𝑌𝑙𝑙𝑃𝑃𝑃𝑃 and 𝑍𝑍𝑃𝑃𝑅𝑅𝑃𝑃𝐷𝐷𝑂𝑂𝑍𝑍𝑃𝑃, i.e. the two provinces of Aragón region. However, both coefficients are quite different because the network of both provinces differ notably.15 Similar comments apply to the two coefficients estimated for Alicante and Valencia provinces. Figure 2 shows the average efficiency scores by provinces. They have been computed using the per-outage specification of our model because the goodness-of-fit is much better in this model, and in addition because the estimated efficiency scores do not depend on the number of power interruption, which is far from the control of the maintenance crews. Our results show a relatively good performance of the maintenance crews located in each province because the average value in all of them is larger than 92%. This figure also provides a very interesting result regarding the performance of the maintenance crews. The distribution of the inefficiency scores in Asturias and Huesca provinces are much more skewed than in other provinces. This is again a somewhat expected outcome because the municipalities of these two provinces are more rural than in other provinces and their population is widely scattered in 15 For instance, while 95% of the network in Huesca province is made up with overhead lines and its population is widely scattered in a wide area, the overhead lines in Zaragoza province only represents a 35% in Zaragoza province due it is a much more urbanized area. 15 wider areas. The large number of extremely low values for the efficiency scores in Asturias and Huesca reveals the existence of frequent difficult-to-restore PSI in these two provinces. [Figure 2 here] We depict the annual evolution of the efficiency scores in Figure 3. Our SF model yields much lower efficiency scores in 2019 than in previous years. This is something that the distribution company must examine using the outage-level information. The Spanish regulator (CNMC) justifies the new regulatory framework on the poor quality of services performance of many Spanish electricity distribution utilities. The observed deterioration in quality of service seems thus to corroborate such decision. [Figure 3 here] 5. Conclusions and final remarks This paper uses a unique dataset to identify the main technological and managerial drivers of the duration of power outages. Unlike previous literature we focus our analysis on two issues: the emergence of grid digitalization and the location (and inefficiency) of maintenance staff. As we know the sequence of the individual PSI across municipalities, we use two different approaches to examine external and internal PSI. In a standard spatial application, only the aggregate effect of both PSI is observed by the econometrician. The duration of the external PSI is modelled using a bottom-up SAR model that is developed using the engineering information of the PSI occurred in each municipality. Unlike the standard SAR model, our bottom-up model can be estimated using a simple OLS estimator because of the recursive nature of the spatially lagged variable. We show that the standard SAR model is seriously biased because it ignores the true sequence of the PSI across municipalities. In contrast, the duration of the internal PSI is modelled using a standard frontier model because the equipment that must be fixed is in the municipality initially affected by the outages. For this reason, the inefficient performance of the maintenance crews is only examined using internal PSI. The estimated standard SAR models provide very poor goodness-of-fit as well as unsound autoregressive parameters. The logged specifications of our heteroskedastic BoU spatial models show that there are not one-to-one contagion effects in most municipalities, and that the degree of municipality contagion increases with the number of connections, and their average capacity, with preceding municipalities involved in common outages. However, the analysis also confirms that radial grid areas with few alternative routes to supply energy to end customers, often located in rural municipalities, are more affected by external outages than more interconnected areas, often in urban municipalities. Regarding the internal PSI, our SF models show that both the weather conditions and the network characteristics influence quality of service in electricity distribution networks, as in previous literature. Unlike previous papers, we find a negative effect of transformers fitted with a remote-control system on restoration times in both specifications. Therefore, grid automatization does reduce the duration of all PSI and the restoration times of each outage. We finally find that the performance of the maintenance crews is quite good because their average efficiency is larger than 92%. However, our efficiency analysis also reveals that there are numerous difficult-to-restore PSI in Asturias and Huesca provinces, an outcome that the firm might examine in detail in order to improve its quality performance in these two Spanish provinces. Another result that the firm should pay attention to is the observed deterioration in the estimated efficiency scores in 2019. 16 From a distribution company point of view, the analyzes carried out in this paper are useful to reinforce the idea that grid digitalization or remote control of grid assets is a must in order to reduce the impact of outages in terms of active power loss. Our results seem to confirm that the location of maintenance crews is also a relevant factor to managing PSI. Whether increasing capital expenditure by installing remote control systems in distribution stations or step-down transformers is more efficient than increasing operational costs in maintenance crews is an issue that we will try to explore in the future. We will also examine in the future whether the installation of a second line to increase the supply capacity to a neighborhood or a new commercial area is better than reinforcing the capacity of the existing line, or it is preferable to install underground assets. 17 Acknowledgements The authors thank Charles Howell for providing helpful comments and suggestions on an earlier version of this paper. Compliance with Ethical Standards Funding: Luis Orea thanks the “Salvador de Madariaga” grant obtained from the Spanish Ministry of Science, Innovation and Universities (Grant PRX19/00596) and the financial support from the Ministry of Education and Research of Madrid Region and FEDER funds (Grant H2019JHUM- 5761). Conflict of Interest: Luis Orea and Pablo Argüelles declare that they have no conflict of interest. 18 References Coelho, J., Nassar, S. M., Gauche, E., Ricardo, V. W., Queiroz, H. L., de Lima, M., and Lourenço, M. C. (2003, June). 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Min Max YE Restoration times of external PSI seconds 1516 62.876 487.467 0.001 10252 J Number of internal PSI number 1516 6.658 8.463 1 77 WY Restoration times of preceding municipalities seconds 1516 69.711 420.295 0.001 10570 WY Restoration times of connected municipalities seconds 1516 1025.473 9953.262 0.001 215927 UN Number of connections with neighboring municipalities number 1516 21.090 20.138 1 88 CA Capacity of connections with neighboring municipalities KW 1516 931.677 1092.167 2 3962 PNU Number of connections with preceding municipalities number 1516 6.660 7.879 1 41 PCA Capacity of connections with preceding municipalities KW 1516 269.725 463.876 2 2530 21 Table 2. Descriptive statistics. SF models Variable Definition Units Obs. Mean Std. Dev. Min Max YI Restoration times of internal PSI seconds 3739 217.349 3958.901 0 215154 I Number of internal PSI number 3739 6.126 7.880 1 82 NL Network length km 3739 395 431 2 2125 OL Proportion of overhead lines proportion 3739 0.807 0.259 0 1.00 DIGT Proportion of digital transformers proportion 3739 0.073 0.082 0 0.50 UT Proportion of underground transformers proportion 3739 0.025 0.095 0 0.79 OT Proportion of outdoor transformers proportion 3739 0.562 0.296 0 1.00 WEATHER Proportion of outages caused by weather issues proportion 3739 0.013 0.092 0 1.00 DIS Distance to the nearest maintenance crew km 3739 13.902 10.895 0 51 RURAL Rural municipality dummy 3739 0.362 0.481 0 1 URBAN Urban municipality dummy 3739 0.638 0.481 0 1 AUTUMN Season of the year dummy 3739 0.252 0.434 0 1 WINTER Season of the year dummy 3739 0.244 0.430 0 1 SPRING Season of the year dummy 3739 0.249 0.432 0 1 SUMMER Season of the year dummy 3739 0.255 0.436 0 1 ASTURIAS Spanish province dummy 3739 0.857 0.350 0 1 HUESCA Spanish province dummy 3739 0.059 0.236 0 1 ZARAGOZA Spanish province dummy 3739 0.010 0.100 0 1 ALICANTE Spanish province dummy 3739 0.014 0.116 0 1 PALENCIA Spanish province dummy 3739 0.043 0.203 0 1 MADRID Spanish province dummy 3739 0.017 0.128 0 1 2013 Year dummy 3739 0.145 0.352 0 1 2014 Year dummy 3739 0.148 0.355 0 1 2015 Year dummy 3739 0.153 0.360 0 1 2016 Year dummy 3739 0.144 0.351 0 1 2017 Year dummy 3739 0.146 0.353 0 1 2018 Year dummy 3739 0.136 0.343 0 1 2019 Year dummy 3739 0.128 0.334 0 1 22 Table 3. Linear homoscedastic SAR models BoU Standard In levels Coef. t-ratio Coef. t-ratio Intercept 8.995 0.95 58.885 *** 4.69 Autoregressive parameter (γ) 0.773 *** 34.78 0.004 *** 3.10 R-squared 0.444 0.006 Per outage Coef. t-ratio Coef. t-ratio Intercept -4.578 -0.89 24.678 *** 3.17 Autoregressive parameter ( γ ) 1.029 *** 44.73 0.003 *** 2.73 R-squared 0.569 0.005 In logs Coef. t-ratio Coef. t-ratio Intercept 0.037 0.89 -0.680 *** -7.24 Autoregressive parameter (γ) 0.724 *** 43.08 0.462 *** 23.59 R-squared 0.551 0.269 Per outage in logs Coef. t-ratio Coef. t-ratio Intercept 0.003 0.10 -0.511 *** -9.21 Autoregressive parameter (γ) 0.745 *** 49.98 0.437 *** 29.46 R-squared 0.623 0.364 Obs. 1516 1516 Table 4. BoU heteroscedastic SAR models In logs Per outage in logs Coef. s.e. t-ratio Coef. s.e. t-ratio 𝛼𝛼 0.149 *** 0.043 3.51 0.055 * 0.029 1.88 𝜆𝜆0 -0.483 *** 0.034 -14.37 -0.453 *** 0.036 -12.76 𝑙𝑙𝑙𝑙𝑁𝑁𝑃𝑃 (𝜆𝜆1) -0.125 *** 0.031 -4.05 -0.066 ** 0.031 -2.17 𝑙𝑙𝑙𝑙𝑃𝑃𝑃𝑃 (𝜆𝜆2) -0.108 *** 0.035 -3.07 -0.114 *** 0.036 -3.17 𝑙𝑙𝑙𝑙𝑇𝑇𝑁𝑁𝑃𝑃 (𝜆𝜆3) 0.013 0.037 0.34 -0.059 0.038 -1.56 𝑙𝑙𝑙𝑙𝑇𝑇𝑃𝑃𝑃𝑃 (𝜆𝜆4) 0.050 * 0.028 1.76 0.077 *** 0.029 2.68 R-squared 0.582 0.640 Average SAR parameter 0.626 0.643 Obs. 1516 1516