Uncertainty Modeling Approach Research Articles

Optimizing software release decisions: a TFN-based uncertainty modeling approach

Optimizing software release decisions: a TFN-based uncertainty modeling approach

Exploring the Impact of Petrophysical Uncertainties on Recoverable Reserves: A Case Study

Reliable estimation of critical parameters such as hydrocarbon pore volume, water saturation, and recovery factor are essential for accurate reserve assessment. The inherent uncertainties associated with these parameters encompass a reasonable range of estimated recoverable volumes for single accumulations or projects. Incorporating this uncertainty range allows for a comprehensive understanding of potential outcomes and associated risks. In this study, we focus on the oil field located in the northern part of Iraq and employ a Monte Carlo based petrophysical uncertainty modeling approach. This method systematically considers various sources of error and utilizes effective interpretation techniques. Leveraging the current state of available data, our approach generates a wide range of theoretically possible results. Furthermore, establishing a set of probabilities to indicate the likelihood of each possible outcome is of utmost importance. By implementing this approach, we aim to enhance reserve assessments by accounting for petrophysical uncertainties, thereby providing decision makers with valuable insights into the range of possible outcomes and associated risks. This study contributes to a more robust understanding of recoverable reserves and supports informed decision making in the oil and gas industry.

Profit-based unit commitment models with price-responsive decision-dependent uncertainty

Highlighting the increasing importance of demand elasticity in electricity markets and its impact on the revenues of power generating companies, this paper proposes new profit-based unit commitment models that effectively capture the uncertainty in the willingness-to-pay the price set for the elastic demand. To develop a new revenue scheme for power generating companies, we use a coupling function to model the willingness-to-pay response of the elastic demand as a decision-dependent source of uncertainty. The coupling function reflects how power generating companies’ pricing decisions may influence the market appeal (i.e., the buyer’s willingness-to-pay) and how it affects their revenues. The optimization models are stochastic mixed-integer nonlinear problems with nonconvex continuous relaxations and are not amenable to a numerical solution in their original forms. We devise a convexification reformulation method and derive valid inequalities to strengthen the formulation. We propose a learning framework to parameterize the willingness-to-pay functions and the concept of the value of the decision-dependent solution to quantify the value of the uncertainty modeling approach. Numerical tests on power systems of various sizes, demand portfolios, and price elasticity levels show (i) how the valid inequalities speed up the solution process, (ii) the benefits of properly modeling decision-dependent uncertainty and demand elasticity, and (iii) how the incorporation of decision-dependent uncertainty in demand elasticity can change the power generating companies’ decisions and revenue estimation.

LSTM-MLP BASED UNCERTAINTY MODELLING APPROACH FOR COMPLEX HUMAN INDOOR TRAJECTORY

Abstract. Modelling the movement uncertainty of human indoor trajectory consist of an essential part in promoting the performance of smart city related applications. At this stage, the existing uncertainty modelling algorithms usually take the constant sampling error or measurement error into consideration and cannot adapt well to the changeable human motion modes and complex handheld modes of smartphones. To fill this gap, this paper applied the Long Short-Term Memory (LSTM) network for continuous prediction of uncertainty error of human indoor trajectory with complex motion modes and detected indoor landmark points. The human motion information including handheld modes, walking speed, and heading information in extracted and fused with detected landmark points for reconstruction of human indoor trajectory under large-scale areas using Gradient Descent (GD) algorithm. In addition, the hybrid LSTM and Multilayer Perceptron (MLP) network is adopted for uncertainty error prediction, by considering both sampling error and measurement error in a specific time period, and the reconstructed trajectory with human motion features are modelled as the input vector for model training with the ground-truth uncertainty error as reference. Comprehensive experiments on real-world collected dataset indicate that the proposed LSTM-assisted uncertainty modelling algorithm has robust outperformance in uncertainty error prediction and uncertainty region definition compared with traditional uncertainty modelling approaches.

Uncertainty parameters of battery energy storage integrated grid and their modeling approaches: A review and future research directions

The continuously growing population and urban growth rates are responsible for the sharp rise in energy consumption, which leads to increased CO2 emissions and demand-supply imbalances. The power sector is switching to alternative energy sources, including renewable energy resources (RES) such as Photovoltaic (PV) and wind power (WP) and battery energy storage systems (BESS), among others, due to an increase in the use of fossil fuels and their shortage. Since the power generation of these resources is uncertain due to climatic fluctuations and the direct integration of these resources into the power grid is very complex due to the issues such as; voltage and frequency regulation, overloading of active transmission lines, and supply-demand disparity, the research on system uncertainties is receiving increasing attention. This study provides a comprehensive analysis of the several parameters of uncertainty, approaches for dealing with the uncertainty in battery energy storage (BES)-based RES integrated grid, and the advantages and disadvantages of each method. Moreover, various analytical and numerical approaches were developed for integrating RES and BESS into the power grid, including probabilistic methods, possibilistic methods, robust optimization-based techniques, and machine learning algorithms. The comparative analysis of these approaches highlights their relative strengths and weaknesses, providing a valuable resource for researchers and utility planners. Additionally, this review paper identifies several issues and challenges associated with the integration of RES and BESS into the power grid, such as power quality, economical effect, battery aging effect, and environmental effect. Furthermore, the paper suggests a few future research directions, including the development of novel models for analyzing uncertainty in power systems, coordination of uncertainty parameters, integration of BESS into RES and grid, power electronics integration, and environmental factor. Overall, this article's novel contributions include a comprehensive analysis of uncertainty parameters, a comparative analysis of uncertainty modeling approaches, an identification of critical issues and challenges, and the suggestion of future research directions to promote a sustainable and reliable power system. This article will aid in defining the requirements and specifications for novel models for analyzing uncertainty in power systems. The discussion and analysis will assist researchers and utility planners in selecting a suitable uncertainty modeling approach with significant penetrations of distributed RESs, which can lead to achieving a reliable and sustainable power system.

Survey on Modeling of Temporally and Spatially Interdependent Uncertainties in Renewable Power Systems

Constructing a renewable energy-based power system has become an important development path for the power industry’s low-carbon transformation. However, as the proportion of renewable energy generation (REG) increases, the power grid gradually changes to uncertainty. Technologies to address this issue have been introduced. However, the majority of existing reviews focus on specific uncertainty modeling approaches and applications, lacking the consideration of temporal and spatial interdependence. Therefore, this paper provides a comprehensive review of the uncertainty modeling of temporal and spatial interdependence. It includes the discrete and continuous stochastic process-based methods to address temporal interdependence, the correlation coefficient and copula functions in modeling spatial interdependence, and the Itô process and random fields theory to describe temporal and spatial interdependence. Finally, their applications in power system stability, control, and economic scheduling are summarized.

A Systematic Review of Uncertainty Handling Approaches for Electric Grids Considering Electrical Vehicles

This paper systematically reviews the techniques and dynamics to study uncertainty modelling in the electric grids considering electric vehicles with vehicle-to-grid integration. Uncertainty types and the most frequent uncertainty modelling approaches for electric vehicles are outlined. The modelling approaches discussed in this paper are Monte Carlo, probabilistic scenarios, stochastic, point estimate method and robust optimisation. Then, Scopus is used to search for articles, and according to these categories, data from articles are extracted. The findings suggest that the probabilistic techniques are the most widely applied, with Monte Carlo and scenario analysis leading. In particular, 19% of the cases benefit from Monte Carlo, 15% from scenario analysis, and 10% each from robust optimisation and the stochastic approach, respectively. Early articles consider robust optimisation relatively more frequent, possibly due to the lack of historical data, while more recent articles adopt the Monte Carlo simulation approach. The uncertainty handling techniques depend on the uncertainty type and human resource availability in aggregate but are unrelated to the generation type. Finally, future directions are given.

A scalable planning framework of energy storage systems under frequency dynamics constraints

As the penetration of renewables increases in power systems, the declining system inertia can cause frequency stability issues. Battery energy storage systems (BESSs) respond fast and therefore can relieve the low inertia difficulty but need to be appropriately sized considering the associated cost. This paper presents a novel stochastic optimization model for economically planning BESS capacity while considering the spatial–temporal correlation of wind generation and generator outages under frequency stability constraints, which include the rate-of-change of frequency (RoCoF), frequency nadir (FN), and quasi-steady-state (QSS) frequency. A set of new FN constraints that can be easily linearized is developed. To account for renewable uncertainties, a realistic uncertainty modeling approach, Random Field, is adopted to generate wind generation scenarios by considering both spatial and temporal evolutions of wind speed profiles. The ESS sizing is formulated as a mixed-integer linear programming problem and solved by using a scalable decomposition-and-coordination approach, Surrogate Absolute Value Lagrangian Relaxation (SAVLR). To further improve the scalability and reduce computational burdens, a rolling-horizon-based update is developed and incorporated into SAVLR for providing a practical solution to the long-term planning of very large-scale power systems. A modified IEEE 118-bus system and the Polish system are used to validate the effectiveness and scalability of the model and solution methodology.© 2017 Elsevier Inc. All rights reserved.

Quantification of data and production uncertainties for tire design parameters in the frame of robustness evaluation

The design process of products and structures such as tires constitutes a multi-objective optimization to enhance the performance for a specific usage. However, the specified geometry and material parameters as well as the boundary conditions, which are basis of the performance prediction resulting from a numerical simulation, cannot be guaranteed to coincide with the real parameters of a final product in use. Most parameters are subjected to variation in production, use and service, which can lead to significant variation in performance.The objective of this contribution is to introduce a scheme of the procedure for design optimization with consideration of uncertainty, including uncertainty modeling, uncertainty analysis and robustness evaluation. Thus, the choice of the design parameters will not be solely based on the predicted value, but also on the reliability of the performance. The focus lies on a detailed description of modeling uncertainties of tire design parameters with regard to production variation as well as to non-precise data, to enable uncertainty analysis and evaluation for tire performances.For choosing appropriate uncertainty models, a distinction is made for the two types aleatoric and epistemic uncertainty with respect to origin as well as consequence. Probabilistic, possibilistic and polymorphic uncertainty modeling approaches are selected for several geometry and material parameters, based on the situation of available data as well as their sensitivity. The utilized approaches include random variables, fuzzy variables and probability-boxes (p-boxes).

Quantifying the uncertainty of precipitation forecasting using probabilistic deep learning

Abstract. Precipitation forecasting is an important mission in weather science. In recent years, data-driven precipitation forecasting techniques could complement numerical prediction, such as precipitation nowcasting, monthly precipitation projection and extreme precipitation event identification. In data-driven precipitation forecasting, the predictive uncertainty arises mainly from data and model uncertainties. Current deep learning forecasting methods could model the parametric uncertainty by random sampling from the parameters. However, the data uncertainty is usually ignored in the forecasting process and the derivation of predictive uncertainty is incomplete. In this study, the input data uncertainty, target data uncertainty and model uncertainty are jointly modeled in a deep learning precipitation forecasting framework to estimate the predictive uncertainty. Specifically, the data uncertainty is estimated a priori and the input uncertainty is propagated forward through model weights according to the law of error propagation. The model uncertainty is considered by sampling from the parameters and is coupled with input and target data uncertainties in the objective function during the training process. Finally, the predictive uncertainty is produced by propagating the input uncertainty in the testing process. The experimental results indicate that the proposed joint uncertainty modeling framework for precipitation forecasting exhibits better forecasting accuracy (improving RMSE by 1 %–2 % and R2 by 1 %–7 % on average) relative to several existing methods, and could reduce the predictive uncertainty by ∼28 % relative to the approach of Loquercio et al. (2020). The incorporation of data uncertainty in the objective function changes the distributions of model weights of the forecasting model and the proposed method can slightly smooth the model weights, leading to the reduction of predictive uncertainty relative to the method of Loquercio et al. (2020). The predictive accuracy is improved in the proposed method by incorporating the target data uncertainty and reducing the forecasting error of extreme precipitation. The developed joint uncertainty modeling method can be regarded as a general uncertainty modeling approach to estimate predictive uncertainty from data and model in forecasting applications.

Overview and applications of Robust optimization in the avant-garde energy grid infrastructure: A systematic review

Over the past decade, escalating trends toward renewable energy resources, electrified transportation, magnifying load demand, erratic energy reserve capacities, dynamic demand response programs, and fast ubiquitous connectivity models have amplified the complexity of energy predicaments owing to the acute intensification of operational and technical uncertainties in the energy grid’s evolutionary phases, one of the pressing challenges that need to be properly addressed. For risk hedging and circumventing the impact of ambiguous parameters, several experts and decision-makers have developed uncertainty modeling approaches for optimization problems under uncertainty. This article first offers a generic overview of traditional uncertainty modeling techniques (such as probabilistic techniques, possibilistic techniques, hybrid probabilistic–possibilistic methods, information gap decision theory, and interval-based analysis) to highlight the significance of robust optimization (RO) method, a state-of-the-art deterministic set-based uncertainty methodology used to optimize a system having uncertain inputs. For this reason – a most popular and pertinent uncertainty modeling technique, the RO approach is precisely introduced to study and recapitulate its remarkable features, decisive modules, and shortcomings. Next, the preceding research on the RO’s contributions in the domain of the power grid is reviewed over three key enablers: Decentralization, Decarbonization, and Digitalization. More specifically, the literature on microgrid, virtual power plant, co/trigeneration, renewable energy with storage units, electric mobility, demand response, and security threats are covered in this survey. Finally, a rigorous discussion on foremost prospects, research gaps, and future directions is quantified. This strategic study can be utilized by researchers, engineers, and power industrialists to anticipate open research areas and unprecedented opportunities related to the application of the RO uncertainty handling method in the futuristic power grid.

A home energy management model considering energy storage and smart flexible appliances: A modified time-driven prospect theory approach

Smart home is a small but an important energy segment that has a significant potential to implement authentic energy policies, where human is a major decision-maker in the home energy management dilemma. Therefore, humans’ emotions and tendencies plays a vital role as the End-User's daily decisions. In this paper, we develop a behavioral home energy management model based on time-driven prospect theory incorporating energy storage devices, distributed energy resources, and smart flexible home appliances. The conventional prospect theory is then modified by considering the psychological time discount effect in the value function. The non-flexible load and photovoltaic generation uncertainties are demonstrated by a scenario-based uncertainty modeling approach. The proposed methodology is a Mixed Integer Linear Programming (MILP) model which is solved by the CPLEX solver in the GAMS software environment. The proposed methodology is verified by comparing the results from conventional and behavioral perspectives. In addition, the role of electrical energy storage and smart flexible home appliances are investigated clearly. The obtained results of the current study are compared with previous conventional home energy management studies to show the effectiveness of the proposed methodology. The results of the simulation studies show that the End-User's satisfaction in the proposed home energy management behavioral structure will be increased substantially compared with the conventional monetary home energy management models.

Multi-energy microgrid optimal operation with integrated power to gas technology considering uncertainties

In recent years, multi-energy microgrids (MEMGs) have emerged as an invaluable framework for enabling the use of clean and efficient electro-thermal resources as well as the integration of multi-energy storage facilities. Uncertainties modelling in such systems is a challenge because of the heterogeneity of the resources and consumers involved. This paper tackles this issue by proposing a hybrid robust energy management tool for MEMGs encompassing electric, heat, hydrogen and gas sub-networks. The variety of uncertainties brought by unpredictable demand and renewable generation are managed using adequate techniques. This way, renewable generation is modelled using the Hong 2m + 1 approach, the electrical and heat demands are managed using the information gap decision theory and the fuel-cell electric vehicles refueling demand is modelled via scenarios. The novel methodology is validated on a benchmark case study, in which extensive simulations are performed. The obtained results demonstrate the accurateness of the novel proposal and its effectiveness to manage a wide variety of uncertainties. The evidence for accurateness is that the difference in the objective function with the Monte Carlo and Hong 2m + 1 uncertainty modelling approaches only differs by ∼0.2%. Moreover, the new proposal is computationally competitive with the Monte Carlo simulation, improving its computation time by 2–3 times.

A two-stage resilience improvement planning for power distribution systems against hurricanes

This paper presents a novel planning strategy for distribution system planners (DSPs) to enhance the resilience of distribution systems confronting emergencies. The problem is formulated as a two-stage stochastic programming model under emergency and normal scenarios. The decisions on line hardening, distributed generation (DG) placement, mobile emergency generators (MEG) allocation, and tie switch placement are made in the first stage to maximize the system resilience. In the second stage, the operation costs of the system pertaining to the DSP power purchase from the upstream network, DG power production, and forced load shedding in emergency conditions are minimized to achieve a techno-economic compromise of investment costs and enhanced operation/resilience benefits over both planning and operation scales. Since access to dependable distribution functions for probabilistic approaches is a notable challenge in resilience studies, an uncertainty modeling approach is presented based on the thresholds for the line damage in the worst-case event. The proposed approach can drastically decrease the number of line damage scenarios for resilience studies. The efficiency and effectiveness of the new approach are validated on two distribution system testbeds with 33 and 118 nodes.

Debottlenecking cogeneration systems under process variations: Multi-dimensional bottleneck tree analysis with neural network ensemble

Due to lucrative economics and energy policies, cogeneration systems have blossomed in many existing industries and became their backbone technology for energy generation. With ever-increasing energy demands, the required capacity of cogeneration gradually grows yearly. This situation unveils a crawling problem in the background where many existing cogeneration systems require more energy output than their allocated design capacity. To debottleneck cogeneration systems, this work extends the bottleneck tree analysis (BOTA) towards multi-dimensional problems with novel consideration of data-driven uncertainty modelling and multi-criteria planning approaches. First, cogeneration systems were modelled using an ensemble neural network with mass and energy balance to quantify the system uncertainty while assessing energy, environment, and economic indicators in the system. These indicators are then evaluated using a multi-criteria decision making (MCDM) method to perform bottleneck tree analysis (BOTA), which identifies optimal pathways to plan for debottlenecking projects in a multi-train cogeneration plant case study. With zero initial investment and only reinvestments with profits, the method achieved 54.2 % improvement in carbon emission per unit power production, 46.3 % improvement in operating expenditure, 59.0 % improvement in heat energy production, and 58.9 % improvement in power production with a shortest average payback period of 93.9 weeks.

Stratigraphic uncertainty modelling with random field approach

Due to the complexity and inherent spatial variability of the strata and the limited availability of borehole information, the subsurface stratigraphic configuration at a site is hard to characterize accurately, which gives rise to stratigraphic uncertainty. This paper presents a random field-based approach for modelling the stratigraphic uncertainty, in which, the spatial correlation of the stratum existence between different subsurface elements is characterized by an autocorrelation function, and the probability of the existence of a particular stratum in a given non-borehole element is determined according to the derived spatial correlations. With the existence probabilities calculated, Monte Carlo simulation (MCS) is used for sampling the possible realizations of the stratigraphic configuration. The number of the MCS samples is determined based upon an information entropy analysis. To illustrate the effectiveness of this new approach, a set of hypothetical examples (with different stratigraphic settings) are studied; and, the advantages of this new approach over the existing stratigraphic uncertainty modelling approaches are depicted through a comparative analysis. Finally, this new approach is applied to Majiagou Landslide for a probabilistic stability analysis. The influences of the stratigraphic uncertainty on the stability of this landslide and the location of the slip surface are revealed.

Mapping the spatial sources of atmospheric dust using GLUE and Monte Carlo simulation

Atmospheric dust has many negative impacts within different ecosystems and it is therefore beneficial to assemble reliable evidence on the key sources of the dust problem. In this study, for first time, two different source modelling approaches comprising generalized likelihood uncertainty estimation (GLUE) and Monte Carlo simulation were applied to map spatial source contributions to atmospheric dust samples collected in Ahvaz, Khuzestan province, Iran. A total of 264 surficial soil samples were collected from five potential spatial dust sources. Additionally, nine dust samples were collected in February 2015. The performance of both GLUE and Monte Carlo simulation for quantifying uncertainty associated with the source contributions predicted using an un-mixing model were assessed and compared using mean absolute fit (MAF) and goodness-of-fit (GOF) estimators as well as 14 virtual sediment mixtures (VSM). Finally, the erodible fraction (EF) of topsoils and HYSPLIT model were used as further tests for validating the results of the GLUE and Monte Caro simulation. Based on both uncertainty modelling approaches, the loamy sand soil texture was recognized as the main spatial source of the target dust samples. Silty clay soils were estimated to be the least important spatial source of the target dust samples using the two modelling approaches. Both GLUE and Monte Carlo simulation returned MAF and GOF estimates >80%, with Monte Carlo performing slightly better. Based on the virtual mixture tests, the RMSE and MAE of the Monte Carlo simulation (<13.5% and <11%, respectively) was better than for GLUE (<20% and <16.3%, respectively). Spatial source maps generated using both GLUE and Monte Carlo simulation were consistent with the EF map generated using multiple regression (MR) and with routes dust transportation detected by HYSPLIT. Therefore, we recommend that future research into to the sources of atmospheric dust pollution integrates modelling approaches, VSM, EF and HYSPLIT model to quantify and map dust provenance reliably.

Optimal thermal and electrical operation of the hybrid energy system using interval optimization approach

Hybrid energy system is an appropriate solution to meet the local electrical and thermal loads using different energy resources. The operator of hybrid energy system decides on optimal scheduling of its resources and trading power with the main grid in an optimal way. The price of trading power with the main grid as an uncertain parameter has an important impact of the decisions of the hybrid energy system’s operator. The novelty of this paper is to apply interval optimization approach for obtaining a risk-constrained operation of photovoltaic/fuel cell/battery hybrid energy system considering the uncertainty of energy price, which is a powerful uncertainty modeling approach. Using the interval optimization approach the objective function of the problem is transformed into a multi-objective optimization one in which there are two conflicting objectives consist of the average and the deviation costs of operation problem of hybrid energy system. This problem is minimized using ε-constraint and fuzzy decision making approaches. Moreover, demand response program has been used to help the operator of hybrid energy system to reduce its total operation cost. The total increase of operation cost with demand response program is 5.52% less in comparison with the case no demand response program is implemented while cost deviation with demand response program is 13.28% less in comparison with no demand response program case.

Orthogonal Uncertainty Modeling in the Engineering of Cyber-Physical Systems

Software-intensive cyber-physical systems (CPS) perform essential tasks such as controlling automated production processes in industrial production plants. The required levels of autonomy, openness, and self-adaptation, as well as the dynamic nature of the context of such CPS, result in challenging tasks for their engineering. During operation, unexpected situations in which the system has insufficient knowledge about the current state of the system itself as well as its context may occur. Engineering CPS, e.g., for industrial production sites, must account for such uncertainties the system will have to cope with during its lifetime in a structured and systematic way. Since the development of CPS requires consideration of different system perspectives, current uncertainty modeling approaches cannot be applied right away, as they do not explicitly consider uncertainty aspects that affect different artifacts. To aid the engineering of CPS, this article presents a model-based approach to document uncertainty. We propose “Orthogonal Uncertainty Models,” which closely integrate with other engineering artifacts from different perspectives, as a means for capturing a dedicated uncertainty viewpoint. Our approach has been evaluated in the industry automation domain. The application shows that the idea of regarding uncertainty within a dedicated perspective is highly beneficial. Particularly, our approach helps to uncover and document uncertainties related to behavioral, functional, and structural properties of a system, as well as uncertainties related to business models that would otherwise possibly remain covert. Note to Practitioners-Identifying and documenting uncertainties, which may occur during operation of a system, is a common problem in engineering processes. Such uncertainties may lead to severe damage, and thus need to be mitigated appropriately. It is crucial to account for these uncertainties during engineering, especially in the early phases. Depending on the specific project characteristics, a multitude of different diagram types are used to model a system. Uncertainties thus reflect in many artifacts, which leads to: 1) redundancies in the specified uncertainty attached to diagram elements and 2) uncertainty information (e.g., about the cause or effect of uncertainty) that is spread across different diagrams. The latter makes it difficult to structure uncertainty information and trace it throughout the engineering process so that uncertainty can be systematically considered. Our approach provides a graphical modeling language that employs a dedicated perspective on uncertainty in separate diagrams that can be linked to any engineering artifact.

Stochastic PV model for power system planning applications

Planning photovoltaic (PV) power systems integration into the grid necessitates accurate modelling of renewable power generation. Global solar irradiance, weather temperature and PV power losses due to overheating specifically in hot regimes are major factors contributing to PV power generation uncertainty. This study targets demonstrating the effectiveness of deploying advanced five parameter probabilistic distribution ‘Wakeby’ for modelling PV uncertain power generation, measured as a function of such factors, in power system planning applications. The impact of different approaches for incorporating weather temperature on PV energy estimation is studied. Wakeby-Monte Carlo Simulation for PV power data training with an emphasis on MCS stopping criteria for such advanced distribution is presented. The model is tested and verified in 31-bus distribution system to demonstrate its effectiveness over other literature uncertainty modelling approaches when planning integration of PV systems' integration into the grid to minimise the grid losses cost. Real PV power measurements are utilised as benchmark verifying the accuracy and suitability of the presented uncertainty modelling approach. Simulation results demonstrate a small error of $4.7 in the expected annual cost of grid losses when deploying Wakeby model compared to the benchmark case and that error can vary significantly when deploying other PV models.