Bayesian Optimization and Hierarchical Forecasting of Non-Weather-Related Electric Power Outages

Abstract

Power outage prediction is important for planning electric power system response, restoration, and maintenance efforts. It is important for utility managers to understand the impact of outages on the local distribution infrastructure in order to develop appropriate maintenance and resilience measures. Power outage prediction models in literature are often limited in scope, typically tailored to model extreme weather related outage events. While these models are sufficient in predicting widespread outages from adverse weather events, they may fail to capture more frequent, non-weather related outages (NWO). In this study, we explore time series models of NWO by incorporating state-of-the-art techniques that leverage the Prophet model in Bayesian optimization and hierarchical forecasting. After defining a robust metric for NWO (non-weather outage count index, NWOCI), time series forecasting models that leverage advanced preprocessing and forecasting techniques in Kats and Prophet, respectively, were built and tested using six years of daily state- and county-level outage data in Massachusetts (MA). We develop a Prophet model with Bayesian True Parzen Estimator optimization (Prophet-TPE) using state-level outage data and a hierarchical Prophet-Bottom-Up model using county-level data. We find that these forecasting models outperform other Bayesian and hierarchical model combinations of Prophet and Seasonal Autoregressive Integrated Moving Average (SARIMA) models in predicting NWOCI at both county and state levels. Our time series trend decomposition reveals a concerning trend in the growth of NWO in MA. We conclude with a discussion of these observations and possible recommendations for mitigating NWO.

1. Introduction

Power outages demonstrate a failure in the proper functioning of an electrical distribution system [1,2,3,4]. These outage events can result in substantial financial losses [5,6,7], such as food spoilage [8,9] or a serious health emergency in a health facility [10], especially when back-up sources of generation fail [11]. Developing a robust outage risk mitigation strategy [12,13] is important for utility managers to formulate protective and preventive measures that reduce the occurrence of electric outages. Outage forecasting models are a useful tool for uncovering historical and future trends of outage events and can, therefore, guide outage preventive and mitigation strategies [14].

The cause of an outage influences its magnitude or severity. Outages as a result of extreme weather events (Extreme Weather Outages, EWO), such as a storm or high winds, may result in a loss of power for numerous customers at a time [15,16,17]. Outages unassociated with adverse weather events (Non-extreme weather outages or non-weather related outages, NWO) often affect a smaller proportion of electricity customers at a time, such as those emanating from the ecology domain as a result of an animal disturbance [18,19,20]. While in most cases (and as shown in

Table 1. Comparison of Extreme Weather outages (EWO) and Non-Extreme Weather outages (NWO) in MA based on the outage data from 2013–2018. Source: statistics are based on data obtained from MA department of public utilities [21].

	EWO	NWO
Average Number of Customers Affected	144	127
Median Number of Customers Affected	14	13
Average Outage Duration (Hours)	13.60	6.85
Median Outage Duration (Hours)	3.57	1.78
Total Number of Outages	61,173	75,671
Total Duration of Outages (Days)	834,513	518,644
Total Number of Customers Affected	8,810,039	9,680,001

), the average and median values of the number of customers affected and the duration of outages from EWO often exceed those of NWO, these NWOs have a significant cumulative effect on the electrical distribution grid. As can be seen in the summary statistics in Table 1, the total number of outages and total number of customers affected as a result of NWO in MA exceed those of EWO within the studied 6-year period from 2013–2018. However, it is important to note that the statistics presented in Table 1 are only valid for the study location and period. Large EWO are less frequent and therefore short—term comparison (as in Table 1) gives only partial information about the long term effect of EWO relative to NWO.

The majority of outage forecasting models in literature have focused on EWO [14,16,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] while neglecting the impact of NWO whose long-term cumulative effects are substantial. There have been a few NWO studies that have focused on subsets of these outages, such as animal and vegetation related outages. For example, the prediction and analysis of animal related outages have been explored using a combination of statistical and advance machine learning techniques [18,20,38]. Also, similar predictive techniques for vegetation related outages (as a result of tree growth) have been explored [39].

This study explores NWO and aims to understand trends in these outages that may be useful in improving electrical distribution grid reliability. The specific contribution of this study include: (1) development of a quantitative metric for NWO that can aid the understanding and quantification of these outages, (2) development of a time series forecasting framework that exposes critical trends and seasonal patterns in NWO, and (3) improvement of the state-of-the-art prediction performance using proposed Bayesian and hierarchical methods. Based on the study methodology (Figure 1), we develop a robust time series forecasting framework for NWO at the state and county level. Our case study is based on the state of Massachusetts (MA) with outage data from 2013–2018 [21]. After deriving a quantitative count-based metric (Non-weather Outage Count Index, NWOCI) for NWO and leveraging advanced data preprocessing tools in Kats [40], a Bayesian optimization method with a Prophet model [41] is used for the state level forecast and a hierarchical Prophet bottom-up approach is used to produce the county level forecasts. Bayesian optimization helps to improve model performance and reduce computation time [42], while hierarchical models are the optimal choice for multi—time series forecasting, especially for grouped or subdivided data [43] (in this case, county level forecasting for multiple counties in the study region—MA). A comparative analysis evaluates the performance of the proposed models relative to other SARIMA and Prophet Bayesian and hierarchical models. The forecast trend and seasonality are also analysed to better understand observed historical and future NWO trends. The complete end-to-end methodology used in this study is describe in Figure 1.

2. Methods

2.1. Data Collection

2.1.1. Outage Data

The state of MA was selected for this study based on the availability of outage reports across the state between 2013–2018. These reports are a component of the Emergency Response plan filed by each of the three major utilities in MA—National Grid, Eversource Energy, and UNITIL corporation—and contain information on the date, time, location, and cause of the outage [21].

Data preprocessing was done to ensure data integrity and develop consistent daily time series. First, data verification procedures were used to address problems such as typographical errors or mismatch in the location reported for the outage. Details of these procedures can be found in the Appendix A.1. Next, steps were taken to obtain a consistent, ready-to-use, daily time series. First, days with no outages in the aggregated time series are replaced with zeros. Then outlier values were treated using the outlier detector algorithm in the Kats toolkit [40] using the standard 1.5 × (IQR) outlier designation and replaced with the interpolated value.

2.1.2. Weather Data

Wind speed data (including gust speed) between 2013–2018 (corresponding to the range of the outage data), was collected from all 23 operating weather stations in MA (Figure 2) at hourly level from the National Oceanic and Atmospheric Administration (NOAA) database via the climate data online local climatological data repository [44]. The wind gusts data are only reported when there is a short-term wind speed (typically 20 s or less) that both exceeds 16 knots (about 18 mph) and exceeds the average wind speed by at least 9–10 knots (10–11 mph) [45]. Whenever there is a gust speed, we replace the average wind speed value at that hour with the gust speed to account for the influence of these short term extreme winds. The hourly wind speed were matched with the raw outage data corresponding to the starting time (hour) and location (shortest euclidean distance) of the outage event and then used to delineate extreme weather outage instances.

2.2. Non-Weather Outage Count Index (NWOCI)

We classify an outage as an NWO if the outage did not occur during instances of high wind threat and other adverse weather conditions including precipitation, lightning, thunderstorm, and snow. First, we begin with classifying Extreme Weather Outages (EWO) based on two indicators:

(1)

Categorical classification of the outage event—adverse weather conditions (which captures all other extreme weather events including Wind, Precipitation, Snow, Lightning, and Thunderstorm)

(2)

Wind threat threshold: If the wind speed exceeds 40 mph—a threshold above which it poses a high or extreme threat, according the National Oceanic and Atmospheric Administration (NOAA) as seen in

Table 2. Wind threat threshold as defined by NOAA. Source: Adapted from NOAA [46].

Threat Level	Description	Wind Speed (mph)
Extreme	Damaging high wind: An Extreme Threat to Life and Property from High Wind.	>58
High	High wind: A High Threat to Life and Property from High Wind.	40–57
Moderate	Very windy: A Moderate Threat to Life and Property from High Wind.	26–39
Low	Breezy to windy: A Very Low Threat to Life and Property from High Wind.	21–25
Very low	Breezy: No Discernable Threat to Life and Property from High Wind.	-

[46], the outage is categorized as an EWO. We use this wind threat to further validate our classification of EWO while capturing potential mis-classified outages.

We then define the daily Non-extreme Weather Outage Count Index (NWOCI) as the total number of NWOs each day in the state and each county, respectively. This is calculated by subtracting the number of outages classified as EWO from the total number of reported outages. It is important to note that each NWO is counted equally, regardless of outage duration and the number of customers affected. As seen in Table 1, the duration and number of affected customers is typically modest for NWO relative to EWO. The focus of this study is on the frequency of NWO and not the magnitude of these events.

2.3. Prophet Model

Prophet [41], an open-source software released by Facebook’s Core Data Science team, was used for time series forecasting. Prophet uses a decomposable model architecture including the following model elements: trend g, seasonality s, and holidays h.

$$y\left(t\right)=g\left(t\right)+s\left(t\right)+h\left(t\right)+{\epsilon }_t$$

(1)

In this study, we use the linear trend model specification in Prophet given as:

$$g\left(t\right)=kt+m$$

(2)

where k is the growth rate and m is the offset parameter; details on the Prophet model can be found in [47].

2.4. Bayesian Optimization Using True Parzen Estimator (TPE)

In this study, we incorporated Bayesian optimization into the Prophet time series model, using the True Parzen Estimator (TPE) [48]. TPE is a type of Sequential Model Based Optimization (SMBO) [49] technique that uses the Bayesian approach in an effort to reduce computational time. The implementation of TPE Bayesian optimization in this study was carried out using the hyperopt software package [48] in the Python programming language. We call this integrated model, that uses the Prophet-based objective function with TPE, Prophet-TPE. The steps in developing Prophet-TPE (as illustrated in Figure 3) include:

• Search domain: First, the domain over which the hyperparameter search will be conducted is defined. For the first iteration, a random combination of hyperparameters within the ranges set forth in

  Table A1. Hyperparameter search domain for the Prophet and SARIMA models.

  Model	Search Domain
  Prophet	‘changepoint_prior_scale’: [0.0001, 0.0005, 0.001, 0.005], ‘changepoint_range’: [0.8, 0.85, 0.9, 0.95], ‘seasonality_mode’: [’additive’, ’multiplicative’], ‘seasonality_prior_scale’: [0.01, 0.1, 1.0, 10.0], ‘yearly_seasonality’: [True, False]
  SARIMA	‘d’: [1, 2], ‘p’: [1, 2, 3, 4, 5], ‘q’: [1, 2, 3, 4, 5], ‘seasonal_order’: [(1, 0, 1, 7), (1, 0, 2, 7), (2, 0, 1, 7), (2, 0, 2, 7), (1, 1, 1, 7), (0, 1, 1, 7)], ‘trend’: [“n”, “c”, “t”, “ct”]

  , is used. In each subsequent iteration, the combination of hyperparameters is adjusted using the probability distribution based on the performance of other combinations used in previous iterations.
• Objective function: The objective function takes in a combination of hyperparameters and output the 5-fold expanding window cross-validated root mean squared error (RMSE) to be minimized over the Prophet model one-year ahead forecast. Unlike a rolling window that moves the upper and lower bound with each time step, the expanding window fixes the lower bound; thus, the amount of data considered incrementally increases (expands) with each time step. The cross validation was performed using the built-in cross validation diagnostic in Prophet [41]. A model is then built to evaluate the objective function. This model is called the surrogate model.
• History: In Prophet-TPE each iteration forms the history. This set of historical information on the performance of a set of hyperparameters on the actual objective (minimizing the Prophet model’s error) is used to construct the probability distributions.
• Probability distribution: This is a mapping of the probability of error, y, for a combination of hyperparameters, x.
• Evaluation criteria: This is the method for obtaining the next best set of hyperparameters. The evaluation criteria is called the Expected Improvement (EI) which is given as [48]:

  $$E{I}_{y*}\left(x\right)={\int }_{-\infty }^y(y^{*}-y)P\left(y\right|x)dy$$

  (3)

  $$P\left(y\right|x)=\frac{P\left(x\right|y)}{P}\left(y\right)P\left(x\right)$$

  (4)

  $$P\left(x\right|y)=\left\{\begin{array}{c}l\left(x\right),ify<y*\hfill \\ g\left(x\right),ify\ge y*\hfill \end{array}\right.$$

  (5)

  where $g\left(x\right)$ and $l\left(x\right)$ are probability distributions and y* is the current best set of model hyperparameters in the history (lowest RMSE). The aim is to choose the next hyperparameter combinations that maximizes EI.
• Update history: The set of distributions developed from historical information forms the basis of a series of iterative improvement for the surrogate model until the maximum number of iterations ($n=10$) is reached.

2.5. Hierarchical Forecasting Model with Bottom-Up Approach

Hierarchical forecasting is a way to obtain disaggregated, multi-time series forecast at finer scales that must add up to the whole [50]. This approach may improve forecast performance while reducing dis-aggregation error. In the bottom up approach, forecasts are obtained for lower levels in the hierarchy with upward aggregation to reconcile the forecast with the upper levels of the hierarchy. We represent the outage data as a hierarchical tree, with a focus on the county-level forecasts and reconciliation at the state level, as seen in Figure 4. We then propose a Prophet model with bottom up approach (Prophet-BU) for forecasting county-level NWOCI. The Prophet-BU forecasting approach was implemented in this study using the scikit-hts software package [51] in the Python programming language.

2.6. Performance Evaluation

The performance of the proposed models is evaluated using the RMSE to assess the difference between the daily forecasted prediction of NWOCI and the actual NWOCI at the state level for Prophet-TPE and the county level for Prophet-BU. Since we have time series data, the data are divided sequentially into train and test with 2013–2017 NWOCI used for training and 2018 data used for testing for both the Prophet-TPE (state level forecast) and the Prophet-BU (county level forecast). The default interval width of 0.8, representing the 80% confidence interval, was used to understand the forecast uncertainty of the proposed models.

The performance of the proposed models, Prophet-TPE and Prophet-BU, is compared with other Bayesian and hierarchical model combinations. For the state-level forecast, the proposed Prophet-TPE method is compared with other Seasonal Autoregressive Integrated Moving Average (SARIMA) and Prophet Bayesian models. These include: SARIMA-TPE, SARIMA-Anneal (SARIMA model with simulated annealing), and Prophet-Anneal (Prophet with simulated annealing). For the county-level forecast, the proposed Prophet-BU model is compared with other Prophet and SARIMA heirachical forecasting models. These include the Prophet and SARIMA model in combination with each of the following: AHP (Average Historical Proportion—top down), OLS (Ordinary Least Square revision), WLSS (Structurally Weighted Least Squares revision), FP (Forecasted Proportions—top down), and PHA (Proportion of Historical Averages—top down). A complete list of the models used in the performance evaluation is shown in Figure 5.

3. Results and Finding

The result for the state level forecast (

Table 3. Performance comparison of proposed Prophet-TPE model (state level) with other Prophet and SARIMA Bayesian optimized forecasting models.

	RMSE
Prophet-TPE (Proposed Model)	18.21
Prophet-Anneal	18.59
SARIMA-TPE	25.64
SARIMA-Anneal	22.25

) shows the performance of the Prophet-TPE in comparison with other models. The optimal hyperparameter values found for the Prophet-TPE and other models are given in

Table A2. Hyperparameter results for state-level NWOCI forecast showing the optimal hyperparameter values for each of the four Bayesian Prophet and SARIMA models.

Model	Hyperparameters
Prophet-TPE	‘changepoint_prior_scale’: 0.0001,
	‘changepoint_range’: 0.9,
	‘seasonality_mode’: ‘additive’,
	‘seasonality_prior_scale’: 10.0,
	‘yearly_seasonality’: True
Prophet-Anneal	‘changepoint_prior_scale’: 0.005,
	‘changepoint_range’: 0.85,
	‘seasonality_mode’: ‘multiplicative’,
	‘seasonality_prior_scale’: 0.1,
	‘yearly_seasonality’: True
SARIMA-TPE	‘d’: 1, ‘p’: 1, ‘q’: 3,
	‘seasonal_order’: (2, 0, 2, 7), ‘trend’: ‘t’
SARIMA-Anneal	‘d’: 1, ‘p’: 1, ‘q’: 5,
	‘seasonal_order’: (1, 0, 1, 7), ‘trend’: ‘t’

. We observed that the proposed Prophet-TPE gives the best prediction for NWOCI at the state level. The Prophet-TPE forecast results can be seen in Figure 6. A decomposition of the forecast Prophet-TPE trend (Figure 7) reveals a positive linear growth trend in average NWOCI representing a 100% increase from 2013–2019 (from 17 in 2013 to 34 in 2019). The seasonal decomposition plot of the Prophet-TPE model forecast shows the temporal patterns of NWOCI with a peak in the summer months (July peak).

For the county level Prophet-BU model, we observed an improved performance in the proposed model relative to other hierarchical and Prophet forecasting models (

Table 4. Performance comparison of proposed Prophet-BU hierarchical model (county level) with other Prophet and SARIMA hierarchical forecasting models.

	RMSE
Prophet-BU (Proposed Model)	2.40
Prophet-PHA	2.55
Prophet-AHP	2.57
Prophet-WLSS	2.43
Prophet-OLS	2.47
Prophet-WLSV	2.59
Prophet-FP	3.81
SARIMA-PHA	3.80
SARIMA-AHP	3.80
SARIMA-WLSS	3.80
SARIMA-BU	3.81
SARIMA-OLS	3.84
SARIMA-WLSV	3.81
SARIMA-FP	3.81

), indicating that the proposed Prophet-BU model better captures the temporal patterns of NWOCI at the county level. As an example, the forecast result for Middlesex county using Prophet-BU model is shown in Figure 8. In the trend decomposition plot for Prophet-BU’s Middlesex county forecast, we also observe a linear growth in NWOCI in the county in 2013–2018 (Figure 9). The seasonal decomposition plot also shows a summer peak in NWO in Middlesex county. The forecast and trend for all remaining 13 counties is shown in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12 and Figure A13. Note that the performance of the Prophet-BU model (like any other time series model) varies depending on the amount of data available. Some counties as shown in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12 and Figure A13 have very sparse data while some have more data points (e.g., Middlesex county) and, therefore, give more robust forecasts. A table of the performance of the Prophet-BU model (RMSE) across counties is given in

Table A3. Performance of the Proposed Prophet BU model for each county in MA.

	RMSE
Barnstable	3.63
Berkshire	2.44
Bristol	14.10
Dukes	2.44
Berkshire	0.41
Essex	3.18
Franklin	1.98
Hampden	3.97
Hampshire	1.57
Middlesex	59.65
Nantucket	0.22
Norfolk	3.86
Plymouth	4.04
Suffolk	5.56
Worcester	4.65

.

4. Discussion

As shown in Figure 10, the three main causes of NWO are: (1) failed equipment, (2) tree contact (in normal weather condition), and (3) animals.

Failed equipment accounts for 33% of all NWOs in MA (Figure 10). This is similar to national estimates. According to a US Department of Energy study, 30% of all non-weather related outages in the United States are a result of failed equipment [52]. Equipment failure is a common problem in electrical distribution systems and can emanate from several electrical components e.g., conductor lines, load breakers, transformers, etc. Much of these outages equipment failure can be attributed to the aging electric infrastructure [53]. The summer peak observed in the NWOCI seasonal decomposition plot (Figure 7) may be attributed to high demand for air conditioning in the summer. The increased electrical load and stress on the grid may ultimately lead to more electrical equipment failures.

Tree contact as a result of vegetation growth is also a major problem in the electrical distribution grid. While tree contact is most prevalent in adverse weather outage events [54,55], there are tree contacts that occur even in normal weather conditions [56]. During the study period, 28% of NWOs in MA were a result of tree contact (Figure 10).

Animal-caused outages represent about 17% of NWO in MA (Figure 10). Squirrels and other animals chew electrical conductors and can cause a short-circuit leading to electric power outages [18,57,58]. Birds are a leading cause of overhead distribution outages in the United States as a result of their roosting, nesting, and breeding activities [59,60]. Animal interference may attribute to the observed summer peak in NWOCI (Figure 7) since there is an increased presence of migratory animals and the summer weather is more conducive for increased animal activity.

5. Conclusions

Unlike previous studies that focus on extreme weather outage (EWO) prediction, we highlighted the importance of capturing non-extreme weather outages (NWO), which occur more frequently and thus have substantial cumulative effect on the electrical distribution grid. We introduced a metric to quantify NWO (Non-extreme Weather Outage Count Index, NWOCI) and then proposed forecasting models to predict NWOCI at both state and county levels. We show that our proposed models outperform other Bayesian and hierarchical forecasting models in predicting NWOCI. Our proposed state-level model (Prophet-TPE) and county-level model (Prophet-BU) outperformed other models by a reduction in RMSE between 2–22% and 6–59%, respectively.

The time series trend decomposition for both proposed models shows an upward trend in NWOCI indicating an increasing frequency of NWOs in MA. This reinforces the importance of developing preventive measures by stakeholders and utility managers that capture not just the EWO but also those occurring from non-extreme weather events, such as those caused by failed equipment, tree contact in normal weather conditions, and animals. These measures could include:

• Efforts to curb equipment failure (the leading cause of non-weather outage) by identifying weaknesses in the distribution grid and high-risk targets for predictive maintenance [52].
• Prediction models that can forecast trends in vegetation growth and thus enable strategic tree trimming measures to be put in place.
• Exploring the feasibility of underground power line solutions. Undergrounding electrical wires could be a solution especially for NWO which originate from animal interference. However, more research is needed on the cost-benefit tradeoff of this strategy.
• Monitoring systems (which may be put in place by utilities) to track the rate of growth of NWOs. Such systems may reveal underlying causes of the problem and aid the development of manageable short and long term mitigation plans.