# Towards Improved Understanding of the Applicability of Uncertainty Forecasts in the Electric Power Industry

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## Abstract

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## 1. Introduction

- Atmospheric unpredictability,
- Uncertainty of (observational) data interpretation,
- Uncertainty when composing the forecast, and
- Forecast interpretation.

## 2. Definition of Forecast Uncertainty

- Uncertainty estimation should be conditional to a set of explanatory variables, like forecasted wind speed, wind direction, expected value of the generation level, etc. Meteorologists call this flow dependent uncertainty (e.g., [21,22]). The collection of historical forecast errors and construction of the empirical distribution should not be named the forecast intervals or identified as the forecast uncertainty.

## 3. Review of Weather Uncertainty Forecasting Approaches and Methods

- Initial condition uncertainties.
- Physical approximation uncertainties.
- Boundary condition uncertainties (at the surface and, for limited area models, at the lateral boundaries from the driving models).

#### 3.1. NWP Ensemble Generation

- Quantification of initial conditions uncertainties:
- Singular vector methodsSingular Vectors (SV) identify the directions of initial uncertainty that are responsible for the largest forecast uncertainty of a model state at a given time in the future. In their review, Diaconescu and Laprise [37] describe SVs to provide optimal information about the probability density function of model states at a future time. The fact that SVs capture the dynamically most unstable perturbations means that they identify the directions of initial uncertainty that are responsible for the largest forecast uncertainty. This property made the SV a good candidate in producing uncertainty from the initial conditions with reasonable dispersion and very suitable for medium range ensemble forecasting. In recent developments, SV are used to detect regions of large sensitivity to small perturbations in order to generate adaptive observations. Such adaptive observations have a large influence on enhancing weather forecasts and producing a better representation of uncertainty [38].
- Breeding methodsThe bred-vectors can be understood as difference fields between two non-linear fields. In order to obtain a realistic development of growing errors in the analysis cycle, the difference between the current analysis fields and the previous ensemble member fields are applied. The bred vectors are constructed as superpositions of leading local time-dependent Lyapunov vectors (named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system, and are being used in predictability analysis and as initial perturbations for ensemble forecasting in NWP.) of the atmosphere. When implemented back in 1994, it was important that the structure of the leading Lyapunov vectors assumed a transient period, which for large-scale atmospheric processes are 3 days, fitting the target time frame for the uncertainties in the medium range forecasting [39]. Bred vectors grow slower than singular vectors and therefore have been shown in comparisons to the singular vectors to lack spread in the first 3–6 days , but then approach the SV results [40].
- Ensemble Kalman Filter techniquesEnKF techniques are today mostly used to provide initial conditions for Ensemble prediction systems (EPS). In that sense, the EnKF is a variant of the initial conditions perturbations. The EnKF was first introduced in 1994 by [29] and has, according to Evensen gained popularity, because of its conceptual formulation and relative ease of implementation. The work in [41] states that it requires no derivation of a tangent linear operator or adjoint equations, and no integrations backward in time, such that the computational requirements are affordable and comparable with other popular sophisticated assimilation methods. The most known methods applied in the generation of initial conditions perturbations are:
- –
- Stochastic and Deterministic Ensemble Kalman Filter (EnKF)The assumption in the stochastic EnKF is that in order to generate a consistent analysis field, the observations need to be treated as random variables with stochastic features. To keep the filter computationally reasonable, a Gaussian error distribution of the observation matrix is assumed [42].The deterministic EnKF does not treat the observations as random, because it is assumed that the small, but spurious correlations between the ensembles of backgrounds and observations could lead to a degradation of analysis quality. The disadvantage of the deterministic EnKF is that the ensemble often becomes under-dispersive and needs covariance inflation or relaxation procedures to compensate for the missing spread [42]. Variants of the deterministic EnKF are:
- (a)
- Ensemble square root filter (EnSRF)
- (b)
- Ensemble adjustment Kalman filter (EAKF)
- (c)
- Ensemble transform Kalman filter (ETKF)

All these variants follow in large the same principles. - –
- Sequential and Local Ensemble Kalman Transform Filter (EnKTF)The sequential EnKF is a data assimilation method that applies an ensemble of model states to represent the error statistics of the model estimate, it applies ensemble integrations to predict the error statistics forward in time, and it uses an analysis scheme which operates directly on the ensemble of model states when observations are assimilated (EnKF: The ensemble Kalman filter home page: enkf.nersc.no).In the Local EnKF (LETKF) the computational heavy part is moved to a smaller, local scale. This practice has the advantage that more observations can be handled with parallel computing techniques and thereby exponential increase in required computational resources is avoided. The EnKF takes the analysis resulting from the data assimilation to lie in the same subspace as the expected forecast error. The LETKF makes use of the hypothesis that the dimension of the subspace corresponding to local regions is low and therefore allows operations only on relatively low-dimensional matrices in a local space. In this way all local matrices are used to compute the global states for advancement to the next forecast time [43]. The main difference between the algorithms are that the sequential algorithm assimilates observations in a sequence of small batches, while in the local algorithm the spatial domain is split into a number of local areas, where the analysis is solved independently ([42]).

Newest developments also use the EnKF method for data assimilation and perturbations in order to overcome the spacial scales issues in the global models [44]. Hybrids of EnKF, LETKF and Physical parameterization schemes, multi-models or other physics perturbations are applied in a number of meteorological centers in order to capture both initial condition and model uncertainties, even though many issues remain and no optimal solution has been found as yet [42].

- Quantification of uncertainties due to physical approximations:
- Stochastic physics (perturbations of tendencies in the physics)The principle of stochastic physics is to represent uncertainty in the model physics, or in other words the uncertainty arising from assumptions made to solve the physical equations of motion is used to perturb physical tendencies such as wind components, temperature and humidity. The first stochastic physics perturbation scheme was introduced by [45] in order to make ensemble forecasts using only perturbations in the initial conditions more reliable. In fact, it had been found that uncertainties in the model and the model’s parameterization schemes to solve sub-grid scale problems have an equally large effect on the uncertainty of weather predictions as uncertainties in the initial conditions. The first versions of the stochastic physics contained multiplicative noise perturbations on the net parametrized physics tendencies of the wind components, temperature and humidity.Since then, other, more refined or complementary methods have been developed. One of them is the so-called Stochastically Perturbed Parameterization Tendencies (SPPT) method that uses perturbations collinear to the unperturbed tendencies [46]. The main difference is that the multivariate distribution of the original scheme is replaced by an univariate distribution in order to achieve perturbations that are more consistent with the model physics. The so-called Backscatter scheme is a complementary scheme and describes aspects of structural uncertainty in the dynamic parametrization that is missing in conventional parameterization schemes [47].In summary, these types of perturbations try to simulate the uncertainty related to the assumptions taken in the NWP models physical and dynamical equations due to non-solvable sub-grid scale processes.
- Multi-Scheme approach (application of different physics schemes)The multi-scheme approach is characterized by using one NWP model, where entire parameterization schemes are exchanged in order to form new members. In that way, perturbations can be added effectively in the dynamics and physics of the model, e.g., convection, cloud and micro physics, horizontal and vertical diffusion, radiation and surface roughness. This approach, even though computationally relatively expensive, has the advantage over a multi-model approach that it is computationally easier to handle, due to the common model kernel. Additionally, the differences of the individual model results are well-defined. The main advantage of the multi-scheme approach is that the uncertainty has no time-dependence and is valid in every time step of a forecast. This means, if the amount of members is large enough and the chosen parameterization schemes reflect the uncertainty of the variables of interest, there is no post-processing or calibration required. The use of multiple physical parameterizations also permits the sampling of different possible closure assumptions in deep convection and in boundary layer processes [42], which are the main driver of uncertainty in the context of renewable energy forecasting. Here, the ideal multi-scheme ensemble is one, where the schemes are targeted to the fast physical processes in the boundary layer since these processes at the surface are the driver for uncertainty. If the uncertainties are covered by the EPS member’s parameterization schemes, the uncertainty will be well described by the ensemble in every time step of the forecast. An example of such a dedicated system for the power industry is the Multi-Scheme Ensemble Prediction System (MSEPS) run by weather ensemble service provider WEPROG (Weather & Energy PROGnoses) [48,49].

- Quantification of surface boundary condition uncertainties:
- Perturbation of surface parametersAlthough given a lower weight compared to the impact of initial conditions perturbations and model perturbations on the forecast uncertainty in the literature so far ([50]), several centers apply perturbations on surface boundary conditions. The main objective here is to account for the effects of uncertainties in the surface energy budget and the surface roughness which significantly determine latent and sensible heat fluxes and turbulences in the planetary boundary layer. Parameters that are most commonly varied include roughness length, soil moisture, snow cover, surface albedo, vegetation properties, sea surface temperature. The variations are introduced e.g., by time constant perturbation fields [50], variations relative to the climatological analysis fields ([51]), or additive, multiplicative and correlated stochastic perturbations [52]. Their impact have been shown to be significant on the quality of forecasts in particular near the surface [52,53].

- Methods that implicitly combine several sources of uncertainties:
- Multi-model approachProbably the easiest way to create an ensemble is simply to combine forecasts from several NWP models [42,54]. The underlying assumption is that the combination of different model systems is a good representation of model error. Depending on the nature of the combined models (deterministic and/or including initial condition perturbations), the resulting ensemble may also include effects of uncertain initial conditions. The interpretation of the outcome again depends on the characteristics of the ensemble. The members can be taken as random, equally likely draws from a probability distribution or as scenarios with assigned probability of occurrence [55]. Based on these assumptions, probabilistic products can be derived. This approach is particularly attractive for (energy-) meteorological services without running an own operational NWP model system. However, the sources of the resulting variability in the forecast are not necessarily transparent and therefore the statistical assumptions on the resulting distributions are error-prone. In addition, optimization of the forecast distribution can only be done statistically as a post-processing step (see later) rather than by optimizing the ensemble properties themselves. Forming a multi-model EPS with deterministic NWP models are often prone to being under-dispersive, because deterministic models usually suppress extremes, which are desired in an EPS for reliability purposes.

#### 3.2. Reliable Weather Forecast Products through Statistical Post-Processing

#### 3.2.1. Deterministic or Probabilistic Weather Input

#### 3.2.2. Application-Specific Weather Forecast Products

#### 3.3. Overview of NWP Ensemble Methods and Their Applicability in the Power Industry

## 4. From Weather to Wind Power Uncertainty

- Statistical methods based on deterministic NWP forecastsUncertainty is determined by a statistical approach using deterministic NWP input
- Methods based on NWP ensemble forecastsUncertainty is derived by applying a weather-to-power conversion method on each NWP ensemble member or by considering input derived by a reduction method applied on the ensembles

#### 4.1. Statistical Methods Based on Deterministic NWP Forecasts

#### 4.1.1. Parametric and Non-Parametric Probabilistic Forecasts

#### 4.1.2. Statistically-Based Ensembles

#### 4.2. Methods Based on NWP Ensembles

#### 4.2.1. Physical Methods

#### 4.2.2. Statistical Methods

#### 4.3. Regional Aggregation

- The capacity of the aggregation may change as new wind farms can be installed or old ones decommissioned.
- Historical data of the new wind farms are not necessarily available to retrain the forecast models on the new configuration.
- Maintenance operations and down-regulation of the wind farms production due to grid issues corrupt the measured production time series and impact the forecast performance.

## 5. Communication of Uncertainty

#### 5.1. Communication of Weather Uncertainty

- Promote enhanced decision making;
- Manage user expectations;
- Promote user confidence;
- Reflect the state of the science.

#### 5.2. Communication of Wind Power Uncertainty

- Probabilistic forecasting;
- Scenario or ensemble forecasting;
- Skill forecast;
- Ramp forecasting.

- Magnitude $\left(\right)$: wind power variation observed during the event;
- Duration $\left(\right)$: time period associated to the large variation event;
- Ramp rate: ramp intensity, which is the ratio between magnitude and duration $\left(\right)$;
- Timing $\left(\right)$: time instant related with the ramp occurrence, which can be the start or central time instant of the event;
- Direction: increase or decrease in wind power.

## 6. Mapping Uncertainty Representations and End-User Requirements

#### 6.1. Reserve Requirements and Unit Commitment

#### 6.2. Participation in Electricity Markets

- Uses wind speed measurements, power curves of the wind turbines and/or physical models of the turbines to estimate in real-time the reference power (i.e., called AAP);
- Delivered amount of reserve power is calculated in real-time as the difference between the reference and the actual feed-in of the wind farm.

#### 6.3. Predictive Grid Management

#### 6.4. Maintenance Scheduling of Wind Power Plants

#### 6.5. Long-Term Portfolio Planning

- Policy makers and regulatory authorities: Annual and decadal forecasts can be utilized to investigate the associated future risks of wind resource volatility for a specific region and country. Annual to decadal forecasts help the policy makers to understand the changes in energy mix.
- Energy investors and wind power operators: Seasonal forecasts can be used to plan operation and trading of portfolios with mixed wind power and other generation capacity.
- Power transmission and distribution system operators: Weekly, monthly, seasonal, and annual wind power forecasts can be used to plan energy balance, UC/ED, and future power grid capacity expansion / reinforcement purposes for the target power system. Generation cost models simulate the UC and ED operations of the specific power system in the short to long term forecast horizon (one week to a year) with 5 min to hourly resolution.

#### 6.6. Summary of Links between Uncertainty Representations and End-User Requirements

## 7. Pitfalls in Decision-Aid Methods

#### 7.1. Uncertainty Modeling and Integration in the Decision-Making Phase

#### 7.2. Decision Quality Evaluation

#### 7.3. The Role of Short-Term Forecasting Uncertainty in Longer Term Decisions

## 8. Recommendations and Best Practices

- ERCOT added a “Reliability Risk Desk” in January 2017 into its control room for RES forecasting and extreme weather monitoring. This desk makes forecast adjustments during icing and other extreme weather events and monitors the adequacy of scheduled resources to cover forecast errors and net-load ramps [170].
- The Compagnie Nationale du Rhône (CNR) included meteorological feedback in the wind power forecasting system by exploring expert charts (maps) from different NWPs available over France. This information is then used to examine spatial-temporal trajectories when forecast uncertainty is high, and to use expert knowledge to provide different weightings to different NWP models [171].
- The Hawaiian Electric Company (HECO), system operator in Hawaii, is in the process of implementing uncertainty forecasts, a ramp warning system, Automatic Generation Control, reserve monitoring and unit commitment into the control room. The implementation is part of the Distributed Resource Energy Analysis and Management System) project (DREAMS)[172] that supports the development of the next generation, integrated energy management infrastructure able to incorporate advance visualization of behind-the-meter distributed technology performance information and probabilistic renewable energy generation forecasts to inform operational decisions [9].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AAP | Available Active Power |

aFRR | Automatic Frequency Restoration Reserve |

AnEn | Analog Ensemble |

BMA | Bayesian Model Averaging |

CIM | Common Information Model |

CLT | Central Limit Theorem |

CMC | Canadian Meteorological Center |

CRPS | Continuous Ranked Probability Score |

EAKF | Ensemble Adjustment Kalman Filter |

ECC | Ensemble Copula Coupling |

ECMWF | European Centre for Medium-Range Weather Forecasts |

ED | Economic Dispatch |

EMOS | Ensemble Model Output Statistics |

EnKF | Ensemble Kalman Filter |

EnSRF | Ensemble Square Root Filter |

EPS | Ensemble Prediction Systems |

ERCOT | Electric Reliability Council of Texas |

ETKF | Ensemble Transform Kalman Filter |

IEA | International Energy Agency |

IEC | International Electrotechnical Commission |

LETKF | Local Ensemble Kalman Filter |

MAE | Mean Absolute Error |

MOS | Model Output Statistics |

MTD | Mass Transportation Distance |

MSEPS | Multi-Scheme Ensemble Prediction System |

NCEP | National Centers for Environmental Prediction |

NWP | Numerical Weather Predictions |

Probability Density Function | |

RMSE | Root Mean Square Error |

RTO | Regional Transmission Operators |

SD | Stochastic–Dynamic |

SPPT | Stochastically Perturbed Parameterization Tendencies |

SV | Singular Vector |

UC | Unit Commitment |

VPP | Virtual Power Plant |

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**Figure 1.**Example spaghetti plot of 75 normalized wind power forecasts converted from wind speed forecasts at 10 m from a multi-scheme ensemble prediction system (MSEPS) from weather ensemble service provider WEPROG at a near-coastal wind farm in the North of Spain.

**Figure 2.**Same plot as Figure 1, in this case with 300 normalized wind power from wind speeds at 10 m, 40 m, 100 m and 150 m, illustrating that an entire wind profile may be required to capture the uncertainty of the power generation.

**Figure 3.**Same as Figure 1, in this case with 75 normalized wind power from wind speeds 40 m, illustrating that in this specific case the effective wind speed at the turbines, where the MSEPS captured the entire uncertainty was at that model level.

**Figure 4.**Illustration of links between input data statistical methodologies and forecast types. Dashed lines indicate that methods relate to individual members of ensemble NWP which require calibration.

**Figure 8.**Seventy-five wind power forecasts from 75 ensemble members at 4 different model levels at 100 m for a 48 h-ahead time horizon.

Method | Type | Member Differences | Perturbation Application | Number of Members | Member Differences | Expense (Technically) | Forecast Horizon | Target Time Horizon |
---|---|---|---|---|---|---|---|---|

Monte-Carlo method | statistical | Statistical perturbations | random “white noise” perturbations from climatology | large (≥800) | random and not unique | expensive, due to large number of members | any | to be defined, e.g., 72 h |

Breeding Vector method | statistical | Statistical perturbations from non-linearized Lyapunov (Bred) vectors | differences generated from perturbation of initial conditions (analysis) | limited (typically 10–50) | can be defined statistically, but are not unique | reasonable, because only 1 NWP model is required | medium range (3-10 days) | to be defined, e.g., 96 h |

Singular vector method | statistical | Statistical perturbations from linearized equations— singular vectors | differences are generated from perturbation of the initial conditions (analysis) | limited (typically ca. 50) | can be defined statistically, but are not unique | reasonable, because only 1 NWP model is required | medium range (3–10 days) | to be defined, e.g., 96 h |

Ensemble Kalman Filter | statistical | Random statistical perturbations of the initial conditions (analysis) | differences from filtered “white noise”—basic concept from Monte Carlo Simulations | unlimited (typically 15–25) | all differences are random and not unique | low, pure statistical approach | any | to be defined, e.g., 96 h |

Stochastic Physics approach | physical | sub-grid physics parameterizations | parameterization differences | limited only to available parameters | yes, well-defined parameter changes | low, mostly used in combination with method 1–4 | short-range (0–3 days) | every time step |

Multi-Scheme approach | physical | computation of different processes inside one NWP model kernel | use of different physical and/or dynamic parameterization schemes inside NWP model | only limited to schemes & combinations (typically 25–100) | yes, well-defined by difference in parameterization schemes | manageable, maintenance is limited to 1 full NWP model | short-range and medium range (0–10 days) | every time step |

Surface perturbations | physical | well- defined | roughness length, surface albedo, snow cover, soil moisture, SST, veget. properties | limited to parameters to be perturbed | yes, well-defined parameter changes | low, mostly used in combination with method 1–4 | short-range (0–3 days) | every time step |

Multi-Model approach | physical | Every member is an individual NWP model | Models as well as initial conditions are different | unlimited | no, differences can have a technical or physical reason | large, maintenance of many models is required | short-range and medium range (0–10 days) | every time step |

Decision-Making Problem | Objectives | Uncertainty Product(s) | Performance Verification | Use of Information | End-User Requirements |
---|---|---|---|---|---|

Operating reserve (System Operator) | Setting the operating reserve requirements considering wind power variability and uncertainty. Decision strategies like setting an acceptable risk level or trade-off between economic issues and risk | (i) Conditional quantiles (ii) Conditional PDF (iii) Skill forecasts | Calibration, sharpness, quantile score and CRPS | (i) and (ii) Convolution of forecasted marginal distribution functions (iii) Mix between skill forecast and empirical rules to set reserve requirements | Sharp forecasts → low reserve requirements High calibration → Adequate estimation of tails quantiles leads to accurate risk quantification Resolution → dynamic reserve requirement according to power system operating conditions |

Unit commitment (UC) and economic dispatch (ED) (System Operator) | UC: scheduling the generation units for minimizing the cost of supplying the load with a set of operation constraints. ED: for UC result, computes the generation levels of each unit with min cost | (i) Conditional quantiles (ii) Temporal or spatial-temporal ensembles | (i) Calibration, sharpness, quantile score (ii) p-variogram score, mass transportation distance | (i) Quantile-based reserve rule; (ii) Integrate ensembles in stochastic optimization | High calibration to achieve cost savings; Too sharp ensembles do not adequately describe the uncertainty; High event-detection skill (ramp-up, ramp-down); Temporal and spatial dependence structure of forecast uncertainty |

Market bidding (Market Player) | Forecast/bid optimization to minimize imbalance costs/maximize returns; ability to participate as price maker | Conditional quantiles/PDF | Calibration, sharpness, quantile score and CRPS | (i) Maximize expected profit (optimal quantile) (ii) Risk-expected profit trade-off | Perfect calibration since optimal bid corresponds to a quantile value; Low sharpness → low uncertainty (reduced imbalance cost) |

Virtual power plant operation (Market Player) | Provision of reserve capacity; Coordination of multiple generation sources and storage units | (i) Conditional quantiles (ii) Temporal or spatial-temporal ensembles (e.g., if storage available) | (i) Calibration, sharpness, quantile score and CRPS (ii) p-variogram score, mass transportation distance, energy score | (i) Choose quantile according to system operator’s reliability requirements (ii) Stochastic multi-period optimization | (i) Perfect calibration → perfect reserve reliability estimation; high sharpness → high reserve margin (ii) temporal or spatial dependence structure of forecast uncertainty |

Predictive grid management (System Operator) | Solve and detect technical constraints violations in transmission and/or distribution grids | Spatial-temporal ensembles | p-variogram score, mass transportation distance, energy score | Stochastic optimal power flow | Spatial-temporal dependence structure; Perfect calibration → avoid false and overlooked alarms; High sharpness → low operating costs; low computational requirements |

Maintenance scheduling (Wind Farm Operator/ Market Player) | Find access windows for safe wind power plant maintenance | Simultaneous forecast intervals; ensembles | Calibration and sharpness | Cost-loss model under uncertainty | Temporal dependence structure for low wind speed periods, co-dependence with wave height in offshore environment; Perfect calibration → accurate financial risk estimation |

Long-term portfolio planning (Market Players, System Operator, Wind Farm Operator) | Portfolio management and find the optimal size of the investment; Mid-term O&M planning; Investment in storage capacity, generation and network expansions | Wind speed PDF (Gaussian), probabilistic multi-category events for 10m height | Calibration/rank and sharpness diagrams, CRPS | Business cases are still needed, but tools like stochastic UC/ED can be used for long-term analysis | Calibration is a critical requirement from the end-user perspective (trustworthiness) |

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**MDPI and ACS Style**

Bessa, R.J.; Möhrlen, C.; Fundel, V.; Siefert, M.; Browell, J.; Haglund El Gaidi, S.; Hodge, B.-M.; Cali, U.; Kariniotakis, G.
Towards Improved Understanding of the Applicability of Uncertainty Forecasts in the Electric Power Industry. *Energies* **2017**, *10*, 1402.
https://doi.org/10.3390/en10091402

**AMA Style**

Bessa RJ, Möhrlen C, Fundel V, Siefert M, Browell J, Haglund El Gaidi S, Hodge B-M, Cali U, Kariniotakis G.
Towards Improved Understanding of the Applicability of Uncertainty Forecasts in the Electric Power Industry. *Energies*. 2017; 10(9):1402.
https://doi.org/10.3390/en10091402

**Chicago/Turabian Style**

Bessa, Ricardo J., Corinna Möhrlen, Vanessa Fundel, Malte Siefert, Jethro Browell, Sebastian Haglund El Gaidi, Bri-Mathias Hodge, Umit Cali, and George Kariniotakis.
2017. "Towards Improved Understanding of the Applicability of Uncertainty Forecasts in the Electric Power Industry" *Energies* 10, no. 9: 1402.
https://doi.org/10.3390/en10091402