Because of environmental and economical reasons, in Sweden and the rest of Europe, both personal and goods transports on railway are increasing. Therefore great railway infrastructure investments are expected to come. An important part of this infrastructure is the railway power supply system. Exactly how much, when and where the traffic will increase is not known for sure. This means investment planning for an uncertain future. The more uncertain parameters, such as traffic density and weight of trains, and the further future considered, the greater the inevitable amount of cases that have to be considered. When doing simulations concerning a tremendous amount of cases, each part of the simulation model has to be computationally fast - in real life this means approximations. The two most important issues to estimate given a certain power system configuration, when planning for an electric traction system, are the energy consumption of the and and the train delays that a too weak system would cause. In this paper, some modeling suggestions of the energy consumption and the maximal train velocities are presented. Two linear, and one nonlinear model are presented and compared. The comparisons regard both computer speed and representability. The independent variables of these models are a selection of parameters describing the power system, i.e.: power system technology used on each section, and traffic intensity.

This paper presents the main findings of a Master's Thesis project carried out in cooperation between Transrail and Royal Institute of Technology (KTH). The main objective was to create a plugin for checking the electric power system feasibility of a train traffic plan with an associated driving strategy created by TRAINS a Transrail software product. Secondary aims with the project was to study power system feasibilities during converter station outages, and to which extent optimal operation of the locomotive converters' reactive power assure power system feasibilities. In the developed optimal reactive power strategies, the main priority was to fulfill the desired traffic plans, whereas the secondary priority was to minimize railway power system power consumption. The case studies are applied on representative traffic scenarios and power system models representing the northern part of the Iron Ore line in Northern Sweden. The focus of the study is set on the IORE locomotives and the iron ore trains they haul. The optimized locomotive reactive power regards IORE, so also the investigated power system feasibilities of the traffic plans.

This paper presents and proposes an optimization model for railway power supply system simulations. It includes detailed power systems modeling train movements in discretized time considering running resistance and other mechanical constraints, and the voltage-drop-induced reduction of possible train tractive forces. The model has a xed number of stationary power system nodes. The proposed model uses SOS2 (special ordered sets of type 2) variables to distribute the train loads to the two most adjacent power system nodes available. The impact of the number of power system nodes along the contact line and the discretized time step length impacts on model accuracy and computation times are investigated. The program is implemented in GAMS (General Algebraic Modeling System). Experiences from various solver choices are also presented. The train traveling times are minimized in the example. Other studies could, e.g. consider energy consumption minimization. The numerical example is representative for a Swedish non-centralized, rotary-converter fed railway power supply system. The proposed concept is however generalizable and could be applied for all kinds of moving load power system studies.

The objective of this paper is initially to present a basic modeling of the railway traction system. This model includes the basic technologies used today. The voltage dependencies of the maximal possible power consumption as well as the maximal velocity of the common Re-locomotives are included. The latter is very crucial for the studies of time table sensitivity, which is of our immediate interest. Moreover, a method is presented, that estimates the expected train delay time for a given feeding technology. The reference timetable assumes the same train and surrounding conditions, but no voltage drops. In the numerical example where the developed model is applied to a realistic test system, a set of possible amounts of railway traffic are treated as uncertainties. Mainly, the contributions of this paper are three: compiling and connecting already accepted models, the development of a method for numerical calculations using this model compilation, and an example to apply this model on.

Because of environmental and economical reasons, in Sweden and the rest of Europe, both personal and goods transports on railway are increasing. Therefore great railway infrastructure investments are expected to come. An important part of this infrastructure is the railway power supply system. Exactly how much, when and where the traffic will increase is not known for sure. This means investment planning for an uncertain future. The more uncertain parameters, such as traffic density and weight of trains, and the further future considered, the greater the inevitable amount of cases that have to be considered. When doing simulations concerning a tremendous amount of cases, each part of the simulation model has to be computationally fast – in real life this means approximations. The two most important issues to estimate given a certain power system configuration, when planning for an electric traction system, are the energy consumption of the grid and the train delays that a too weak system would cause. In this paper, some modeling suggestions of the energy consumption and the maximal train velocities are presented. Two linear models, and one nonlinear model are presented and compared. The comparisons regard both computer speed and representability. The independent variables of these models are a selection of parameters describing the power system, i.e.: power system technology used on each section, and traffic intensity.

Transports on rail are increasing and major railway infrastructure investments are expected. An important part of this infrastructure is the railway power supply system. The future railway power demands are naturally not known for certain. This means investment planning for an uncertain future. The more remote the uncertain future, the greater the amount of scenarios that have to be considered. Large numbers of scenarios make time demanding (some tens of minutes, each) simulations less attractive and simplifications more so. The aim of this paper is to present a fast approximator that uses aggregated traction system information as inputs and outputs. This facilitates studies of many future railway power system loading scenarios, combined with different power system configurations, for investment planning analysis. Since the electrical and mechanical relations governing an electric traction system are quite intricate, an approximator based on neural networks (NN), is applied. This paper presents a design suggestion for a NN estimating power system caused limits on active and reactive power load, i.e., limits on the levels of train traffic.

Transports via rail are increasing, and major railway infrastructure investments are expected. An important part of this infrastructure is the railway power supply system (RPSS). Future railway power demands are not known. The more distant the uncertain future, the greater the number of scenarios that have to be considered. Large numbers of scenarios make time-demanding (some minutes, each) full simulations of electric railway power systems less attractive and simplifications more so. The aim, and main contribution, of this paper is to propose a fast approximator that uses aggregated traction system information as inputs and outputs. This approximator can be used as an investment planning constraint in the optimization. It considers that there is a limit on the intensity of the train traffic, depending on the strength of the power system. This approximator approach has not previously been encountered in the literature. In the numerical example of this paper, the approximator inputs are the power system configuration; the distance between a connection from contact line to the public grid, to another connection, or to the end of the contact line; the average values and the standard deviations of the inclinations of the railway; the average number of trains; and their average velocity for that distance. The output is the maximal attainable average velocity of an added train for the described railway power system section. The approximator facilitates studies of many future railway power system loading scenarios, combined with different power system configurations, for investment planning analysis. The approximator is based on neural networks. An additional value of the approximator is that it provides an understanding of the relations between power system configuration and train traffic performance.

Transports on rail are increasing and major investments in the railway infrastructure, including the Railway Power Supply System (RPSS), are expected. The future railway power demands are naturally not known for certain. The more remote the uncertain future, the greater the number of scenarios that have to be considered. Large numbers of scenarios make time demanding simulations unattractive. The aim of this paper is to present a fast approximator that uses aggregated RPSS information. Since the electrical and mechanical relations governing an RPSS are quite intricate, an approximator based on Neural Networks (NN), is applied. This paper presents a design suggestion for an NN estimating the power and energy flows through each converter station, given RPSS data and levels of train traffic. Even if the future usage of the NN is investment planning, the modeling of such an approximator has a value in itself concerning the understanding of the relations between RPSS and train traffic.

Transports on rail are increasing and major investments in the railway infrastructure, including the railway power supply system (RPSS), are expected. The future railway power demands are naturally not known for certain. The more remote the uncertain future, the greater the number of scenarios that have to be considered. Large numbers of scenarios make time-demanding simulations unattractive. The aim of this paper is to present a fast approximator that uses aggregated RPSS information. Since the electrical and mechanical relations governing an RPSS are quite intricate, an approximator based on neural networks (NN) is applied. This paper presents a design suggestion for an NN estimating the power and energy flows through each converter station, given RPSS data and levels of train traffic. Even if the future usage of the NN is investment planning, the modelling of such an approximator has a value in itself concerning the understanding of the relations between RPSS and train traffic.

The objective of this paper is initially to present a basic modeling of the railway traction system. This model includes the basic technologies used today. The voltage dependencies of the maximal possible power consumption as well as the maximal velocity of the common Rc-locomotives are included. The latter is very crucial for the studies of time table sensitivity, which is of our immediate interest. Moreover, a method is presented that estimates the expected train delay time for a given feeding technology. The reference timetable assumes the same train and surrounding conditions, but no voltage drops. In the numerical example where the developed model is applied to a realistic test system, a set of possible amounts of railway traffic are treated as uncertainties. Mainly, the contributions of this paper are three: compiling and connecting already accepted models, the development of a method for numerical calculations using this model compilation, and an example to apply this model on.

Transports on rail are increasing and major railway infrastructure investments are expected. An important part of this infrastructure is the railway power supply system. The future railway power demands are naturally not known for certain. The more distant the uncertain future is, the greater the number of scenarios that have to be considered. Large numbers of scenarios make time demanding simulations unattractive. Therefore a fast approximator that uses aggregated railway power supply system information has been developed. In particular the approximator studies the impacts of voltage drops on the traffic flow. The weaker the power system and the heavier the traffic, the greater the voltage drops. And the greater the voltage drops, the more limited the maximal attainable tractive force on the locomotives. That approximator is in this paper used as a constraint in the embryo of a railway power supply system investment planning program, where investment decisions are assumed to be realized immediately, and there is no preexisting power supply system to consider. The traffic forecasts are in this first approach assumed to be perfect. This stepwise creation of the planning program makes evaluating it easier. The basic investment planning model presented here constitutes the foundation for further improvements.

The aim, and main contribution, of this paper is to propose a fine-tuned fast approximator, based on neural networks, that uses aggregated traction system information as inputs and outputs. This approximator can be used as an investment planning constraint in the optimization. It considers that there is a limit on the intensity of the train traffic, depending on the strength of the power system. In the numerical examples of this paper, the approximator inputs are the power system configuration, the distance between a connection from contact line to the public grid to another connection, and the average number of trains for that distance. The output is the maximal attainable average velocity of trains of a specific kind for the by the inputs described railway power system section. An alternative output – the traveling time is also presented. The main emphasis of this paper is on the example section, since the contribution of this paper is mainly to show on the improved simplicity and reality compliance. The applicative contribution is twofold, an improved TPSA as a planning/decision making program constraint, whereas it also can be used as a scientifically developed rule of thumb for a planner active in the field. The aim is not primarily to show that the idea works, or to motivate the principal idea, since that is done earlier. The approximator facilitates studies of many railway power system loading scenarios, combined with different power system configurations, for investment planning analysis. The approximator is based on neural networks. An additional value of the approximator is that it provides an understanding of the relations between power system configuration and train traffic performance.

This paper presents an optimization model for simulations of railway power supply systems. It includes detailed power systems modeling, train movements in discretized time considering running resistance and other mechanical constraints, and the voltage-drop-induced reduction of possible train tractive forces. The model has a fixed number of stationary power system nodes, which alleviates optimized operation overtime. The proposed model uses SOS2 (Special Ordered Sets of type 2) variables to distribute the train loads to the two most adjacent power system nodes available. The impacts of the number of power system nodes along the contact line and the discretized time step length on model accuracy and computation times are investigated. The program is implemented in GAMS. Experiences from various solver choices are also discussed. The train traveling times are minimized in the example. Other studies could e.g. consider energy consumption minimization. The numerical example is representative for a Swedish decentralized, rotary-converter fed railway power supply system. The proposed concept is however generalizable and could be applied for all kinds of moving load power system studies.

In this paper a railway power system design based on an HVDC feeder is suggested. The converter stations between the public grid and the HVDC feeder can be sparsely distributed, in the range of 100 km or more, whereas the converters connecting the HVDC feeder to the catenary are distributed with a much closer spacing. The ratings of the catenary-connected ones can be lower than substation transformers or rotary converters, since the power conversion can be fully controlled. Simulations of the proposed solution show clear advantages regarding transmission losses and voltages compared to conventional systems, especially for cases with long catenary sections, and when there are substantial shares of regeneration from the trains.

In this paper an alternative railway power systems design based on an HVDC feeder is studied. The HVDC feeder is connected to the catenary by converters. Such an HVDC line is also appropriate for DC-fed railways and AC-fed railways working at public frequency.

A unit commitment optimal power flow model has been developed and is applied on a test system. In this paper, the model is presented in detail. The model, in the form of an MINLP program, uses unified AC-DC power flow to minimize the entire railway power system losses.

Simulations of the proposed solution show clear advantages regarding transmission losses and voltages compared to conventional systems, especially for cases with long distances between feeding points to the catenary, and when there are substantial amounts of regeneration from the trains.

In this paper an alternative railway power systems design based on an HVDC feeder is studied. The HVDC feeder is connected to the catenary by converters. Such an HVDC line is also appropriate for DC-fed railways and AC-fed railways working at public frequency. A unit commitment optimal power flow model has been developed and is applied on a test system. In this paper, the model is presented in detail. The model, in the form of an MINLP program, uses unified AC-DC power flow to minimize the entire railway power system losses. Simulations of the proposed solution show clear advantages regarding transmission losses and voltages compared to conventional systems, especially for cases with long distances between feeding points to the catenary, and when there are substantial amounts of regeneration from the trains.

Distributed generation (DG) has attracted a lot of attention recently and might become more important in future power generation systems. As different definitions are used worldwide, the paper briefly discusses the definition of DG. The future development of DG, however, will, to a not insignificant part, depend on the legal framework. As the legal framework can vary significantly for different competitive electricity markets, this paper briefly identifies and analyses some variations in the regulatory approaches, e.g. for power exchanges, balance services and ancillary services, in different countries. It also illustrates the influence of market regulations on the development of distributed power generation. Based on this analysis, it can be concluded that regulatory aspects might decisively influence the development of distributed power generation

The aim of the power plants in a power system is to supply the load in an economical, reliable and environmentally acceptable way. Different power plants can fulfil these requirements in different ways. In order to select the right sources it is important to compare the value of the different sources using an objective approach. The aim of this chapter is describe the different needs of a power system and how these needs can be met with wind power, that is, the value of wind power in a certain system. The values are operating cost value, capacity value, control value, grid loss reduction value and grid investment value. The values can be calculated for different types of power plants, they can be both positive and negative, and they can be calculated both as a physical cost value and a market value.

The increase in hydrogen production to support the energy transition in different sectors, such as the steel industry, leads to the utilization of large scale electrolyzers. These electrolyzers have the ability to become a fundamental tool for grid stability providing grid services, especially frequency regulation, for power grids with a high share of renewable energy sources. Alkaline electrolyzers (AELs) have low cost and long lifetime, but their slow dynamics make them unsuitable for fast frequency regulation, especially in case of contingencies. Proton Exchange Membrane electrolyzers (PEMELs) have fast dynamic response to provide grid services, but they have higher costs. This paper proposes a dynamic power allocation control strategy for hybrid electrolyzer systems to provide frequency regulation with reduced cost, making use of advantages of AELs and PEMELs. Simulations and experiments are conducted to verify the proposed control strategy.

Hydrogen electrolyzers are promising tools for frequency regulation of future power systems with high penetration of renewable energies and low inertia. This is due to both the increasing demand for hydrogen and their flexibility as controllable load. The two main electrolyzer technologies are Alkaline Electrolyzers (AELs) and Proton Exchange Membrane Electrolyzers (PEMELs). However, they have trade-offs: dynamic response speed for AELs, and cost for PEMELs. This paper proposes the combination of both technologies into a Hybrid Hydrogen Electrolyzer System (HHES) to obtain a fast response for frequency regulation with reduced costs. A decentralized dynamic power sharing control strategy is proposed where PEMELs respond to the fast component of the frequency deviation, and AELs respond to the slow component, without the requirement of communication. The proposed decentralized approach facilitates a high reliability and scalability of the system, what is essential for expansion of hydrogen production. The effectiveness of the proposed strategy is validated in simulations and experimental results.

Measures such as energy curtailment or gridreinforcement are required to integrate the upcoming windgeneration in parts of the power system with existing transmissionbottlenecks. In order to choose between these two measurespotential wind energy curtailments and its costs need to becarefully evaluated. The paper analyzes the effect of wakeconsideration on the overall energy curtailment cost. For thispurpose detailed wake model was used taking into account partialand multiple shading of wind turbines. It is shown that not onlywind speed but also wind direction of the incoming wind affects theamount of energy produced by a wind farm. A comparison ofcurtailment cost with cost for grid reinforcement in areas withlimited transmission capacity was carried out with and withoutconsideration of wake effect. The effect on curtailment cost due toavailability of wind turbines is also investigated both with andwithout wake effect consideration. The results have proven thatwith consideration of wake effect and availability potential windenergy curtailments are reduced and hence curtailment costs arelowered, making curtailment a cheaper option than gridreinforcement. The method illustrated in the paper can be used inpre-feasibility study to compare the costs of wind curtailment withthe costs of grid reinforcement in order to make sound economicdecision. The method can also be applied in wind farm energy yield estimation.

In distribution system planning and operation, accurate assessment of reliability performance is essential for making informed decisions. Also, performance-based regulation, accompanied by quality regulation, increases the need to understand and quantify differences in reliability performance between networks. Distribution system reliability performance indices exhibit stochastic behavior due to the impact of severe weather. In this paper, a new reliability model is presented which incorporates the stochastic nature of the severe weather intensity and duration to model variations in failure rate and restoration time. The model considers the impact of high winds and lightning and can be expanded to account for more types of severe weather. Furthermore, the modeling approach considers when severe weather is likely to occur during the year by using a nonhomogeneous Poisson process (NHPP). The proposed model is validated and applied to a test system to estimate reliability indices. Results show that the stochasticity in weather has a great impact on the variance in the reliability indices.

In power system planning and operation, accurate assessment of reliability worth is essential for making informed decisions. The accuracy of the reliability worth estimation is directly affected by the interruption cost model used in the analysis. Residential interruption costs vary with season, day of week and time of day, and can be difficult to handle because of their intangible characteristics. This paper develops a cost model for residential customers that includes the timing of the outage by modeling the underlying factors that give rise to the temporal variations in residential interruption costs. By considering the stochastic nature of the underlying factors, as for example outdoor temperature, the proposed model makes it possible to estimate the costs for an event that is extreme in other senses than having a long duration. Time sequential Monte Carlo simulations were applied to a test system in order to assess reliability worth. The results show that the commonly used customer damage function overestimates the reliability worth. By accounting for the timing of the outages a more realistic estimation of the interruption costs can be obtained.

In power system planning and operation, accurate assessment of reliability worth is essential for making informed decisions. One common simplification when modeling power system reliability is assuming constant failure rates and non time-varying restoration times. However, historical outages show differently; failure rates and restoration times for especially overhead lines are dependent upon time-varying factors as, for instance, weather conditions. When modeling this time dependence a two or three-state weather model is often used. The reliability model proposed in this paper does in contrast use the stochastic nature of the severe weather intensity and duration to model variations in failure rate and restoration time. Further, the model also considers when severe weather is likely to occur during the year by using a non-homogeneous Poisson process (NHPP). A time-sequential Monte Carlo technique is applied to a radial distribution system. By combining the proposed reliability model with a time-dependent interruption cost model, the effect of the inclusion of time-varying failure rates and restoration times is investigated and found to be of importance when assessing reliability worth.

This paper develops an interruption cost model for households that, as well as outage duration uses activity patterns, outdoor temperature and daylight to describe the impact of different electrical power outages. For households the interruption costs usually measure the inconvenience associated with interrupted activities and uncomfortable indoor temperature due to the outage. Further, the model also captures the large variations in interruption costs for identical outages among households. The model is applied to a test system, and using a Monte Carlo technique the total interruption cost is studied. The results imply that both the time of occurrence and the distributed nature of residential interruption costs have a significant impact on ECOST.

Customer interruption costs are functions of many different factors such as interruption duration, timing and customer sector. Various cost models with different number of outage and customer characteristics included have been proposed during the years. This paper compares the customer interruption cost assessments of seven different cost models in a case study.Time sequential Monte Carlo simulations are used to simulate the customers’ benefits of increased reliability in a test system. The investigated cost models’ estimations of the Expected Customer Interruption Cost (ECOST) are compared and used in a costbenefitanalysis. Results show that the ECOST results are so different that the cost model choice affects the outcome of the cost-benefit analysis. Only when using some of the cost models the investigated reinvestments are estimated to be socioeconomically beneficial.

To find an acceptable level of reliability in distribution systems, cost-benefit analysis using customer interruption costs can be applied. In a case study of a test distribution system, investment in cables instead of overhead lines, aimed to increase reliability, is investigated. In addition to considering average values of reliability indices, tools for risk analysis in the financial industry, value-at-risk (VaR) and conditional value-at-risk (CVaR), are also used for the evaluation. Applying these tools allows for extreme events to be given more weight in the investment decision-making process. Even though these kind of events are very infrequent, the consequences are devastating and extreme cases should be included in cost-benefit analysis. By the help of VaR and CVaR the case study shows that cables can cause higher customer interruption costs for some load points in the system during extreme years.

Designing a quality regulation that results in an adequate level of reliability in a distribution system is indeed a challenging task for the regulator. If the regulation is not well designed a socioeconomically beneficial reinvestment project is not beneficial for the DSO, and hence is not selected. This paper proposes an evaluation method for quality regulation designs. The proposed method is applied in a case study to evaluate what incentives for investments in distribution system reliability two different quality regulation designs give. One design is similar to the Swedish quality regulation that will apply from 2012 and the other design is similar to the current Norwegian quality regulation. The effect on network investment decisions when the two designs are modified to give optimal incentives for reliability on system level is also investigated. The case study result shows that even though the quality regulation on system level is designed to give incentives for socioeconomically beneficial investments, these investments may not be beneficial for the regulated DSO if the reward/penalty on the system level is capped too low.

In the reregulated electricity market, performance-based regulations accompanied by quality regulations are gaining ground. The quality regulation results in new financial risks for the distribution system operators (DSOs) which calls for risk assessment methods that can simulate what costs a certain regulation implies for the DSO. When, for example, guaranteed standards for worst-served customers is combined with a reward and penalty scheme the methods must be able to predict both customer and system reliability. This paper presents a new risk assessment methodology based on time sequential Monte Carlo simulations that can handle both of these levels of reliability to simulate the total regulation cost due to an arbitrary quality regulation. Since most quality regulations are corrected ex-post for each year, variations in yearly reliability can cause large variations in the total regulation cost. Instead of only considering the average total regulation cost the developed methodology uses risk tools from the financial industry to also measure the costs of more extreme years. Doing so gives the DSOs a better understanding of the financial risks they are exposed to. The developed risk assessment methodology is used to evaluate different investment alternatives in a case study.

Performance-based regulations accompanied by quality regulations are gaining ground. Quality regulations imply new financial risks for the distribution system operator (DSO). In fact, the development of the regulatory model has been identified as a key factor in operations planning for a DSO. Lifetimes of distribution system components are very long and how the quality regulation might develop in the future is unknown. This paper develops a method - the regulation impact method - that can be used to investigate how changes in the quality regulation parameters affect the economic performance of an investment strategy. The proposed regulation impact method is based on net present value calculations of the total reliability cost. The new method is applied to the current Swedish quality regulation in a case study. In the case study, possible future parameter changes and their effect on the DSO's financial risk when adopting different investment strategies are investigated. With the new method it is possible to analyze how robust an investment strategy is to changes in quality regulation design.

In the reregulated electricity market there is a growing interest in performance-based regulation accompanied by quality regulation for electric distribution networks. This paper develops a new risk-based method for reliability investment decisions when the distribution system operator (DSO) is exposed to financial risks defined by a quality regulation. As quality regulation design becomes more complex, more detailed risk management methods are needed in order to adequately capture the financial risk the DSO is exposed to. The proposed method applies a Monte Carlo simulation technique in order to assess the risks of the considered reinvestment projects. By using the proposed method the impacts that different risk strategies (risk-neutral/risk-averse) and risk models (non-time-varying/time-varying) have on which reinvestment project is selected is investigated in a case study. This is investigated for two different quality regulation designs. The result show that primarily the quality regulation design but also the risk model formulation and risk strategy have a major impact on which reinvestment project is selected.

This paper develops a new risk-based cost-benefit analysis method for distribution system reliability applications. In the conventional cost-benefit analysis, decisions are based on expected values which correspond to assuming that society is risk-neutral. Furthermore, input variables are assumed to be uncorrelated. In contrast to previous work this paper incorporates risk into the analysis by using time-sequential Monte Carlo simulations. By using the proposed method the impact that different risk strategies (risk-neutral/risk-averse) and risk models (non-time-varying/time-varying) have on the result of a cost-benefit analysis is investigated in a case study. Results show that when incorporating time-dependent failure rates, restoration times, customer interruption costs, and loads (correlated input data) a different reinvestment project is selected compared to when these time dependencies are ignored. This result holds regardless if decisions are made based on a risk-averse or a risk-neutral strategy.

Long-term planning of a power system requires that the electricity market can be simulated. One important aspect that should be simulated is that the owners of hydro power plants with reservoirs (generally referred to as dispatchable hydro power) are forced to base their scheduling on uncertain forecasts on precipitation and load. It is inevitable that this lack of perfect information has an impact on the operation costs and reliability of the system. Hence, a simulation of an electricity market should include this property of dispatchable hydro, power. This paper shows how the consequences of hydro power planning based on uncertain forecasts can be incorporated in a Monte Carlo model by using random water value errors