Stochastic models for power generation system production costs

Production costing models are widely used in the electric power industry for the purpose of generation capacity expansion planning, fuel management, and operational planning. These models account for the load variation over time and generator outages. A widely used model, due to Balériaux and Booth, yields a prediction of the expected production costs and is based on the load duration curve and forced outage rate of the generating units. This paper highlights the fact that, in order to obtain a more detailed characterization of the probability distribution costs beyond the expected value, a model involving the stochastic processes underlying the generator outages is necessary. A stochastic model is considered as an enhancement to the traditional Balériaux model. It is shown that Monte Carlo simulation can be routinely used on the enhanced model to provide answers concerning the distribution of production costs. Monte Carlo methods avoid the problems associated with the complexity of the analytical methods. Numerical examples are given using the enhanced model where load is considered to be either a deterministic or stochastic time-varying function. An example is given using decision analysis where a possible use of the more detailed information on the probability distribution of production costs in generation system planning is illustrated.     

Monte Carlo computation of power generation production costs underoperating constraints

This paper highlights the need for considering the stochastic processes associated with the frequency and duration of generating unit outages for assessing the mean and variance of production costs under operating constraints. A numerical example based on a Markov model is given to show that Monte Carlo estimates of these quantities may be incorrect if only the forced outage rates are used in place of the stochastic parameters underlying the outage frequency and duration. Additionally it describes a variance reduction procedure whereby the Monte Carlo estimates can be obtained with a much smaller sample size than would be required otherwise. A numerical example is given for a small system     

Monte Carlo stochastic simulation of electric power generation system production costs under time-dependent constraints

Probabilistic production costing models are widely used in the electric power industry to forecast the cost of producing electricity. A widely used model due to Balériaux and Booth provides an analytical formula for the expected production costs using the load duration curve (LDC) in place of chronological sequence of loads and the forced outage rates of the generating units. Since the chronological information is lost in the LDC, it cannot accurately simulate those aspects of production cost that are time dependent in nature. The paper points out that, in addition to the need for a chronological simulation of load to capture the time-dependent constraints, it is also necessary to model the frequency and duration of the generation outages. Monte Carlo results are given for a Markovian model for the frequency and duration of the outages where several unit commitment constraints are considered. It is shown that the mean and variance of the production costs may differ significantly if the failure and the repair rates of the generating units are changed although the respective forced outage rates remain unaltered. The paper also highlights the simplicity of using continuous-time simulation in the Markov model.     

An analytical formula for the mean and variance of marginal costsfor a power generation system

In the upcoming deregulated environment, the marginal cost of power generation will play an important role. This paper provides an analytical formula for estimating the expected value and variance of the average marginal cost over a given study horizon using an enhanced stochastic version of the Baleriaux model. For the numerical example considered, this procedure provides accurate estimates in much less computer time as compared to a Monte Carlo simulation     

基于风险和条件风险方法的光伏电站并网极限容量计算

大规模光伏电站并网会影响电网的安全稳定运行,因此其并网极限容量受到限制。提出了风险和条件风险的随机性优化方法,建立了并网光伏电站极限容量计算的条件风险随机优化模型,该模型以光伏电站并网极限容量为优化目标,同时考虑了极端情况的影响。为了解决条件风险函数直接计算困难的问题,采用蒙特卡罗模拟和解析法将条件风险函数变换和离散化。仿真结果表明,该模型适用于考虑了光照强度随机变化情况下的并网光伏电站极限容量的计算,计算量小,准确性较高。     

Probabilistic production cost model for a power system with storage

The conventional production cost model uses the power system load for an average day with no consideration of the uncertainty that is entailed in this load. A probabilistic production cost model is presented that considers the stochastic nature of the load which is used for a power system with energy storage operating on a daily or weekly cycle. The optimum energy storage device, in the sense of minimizing the expected cost of meeting demand, is found. The model minimizes the statistical data and permits a separation of the fuel cost from it, so that a change in fuel cost would not require a great deal of recomputation. Finally, the deterministic approach using an average load is compared with the stochastic one, and it is shown that the deterministic approach leads to significant error in the optimum storage capacity.     

市场竞价下梯级水电站短期优化调度研究

针对梯级水电站短期优化调度的特点，通过对电力市场中发电厂商竞价策略特点的分析，提出了市场竞价下梯级水电站的竞价运行策略，构建了市场竞价下梯级水电站优化调度模型。模型采用概率分布来模拟发电厂商报价策略，并利用蒙特卡洛方法对电力市场出清过程进行随机模拟，而且在最大效益子模型中考虑了梯级各电站发电成本不同的因素。最后通过实际算例，利用改进蚂蚁算法和遗传算法对硕多岗河梯级水电站在市场竞价条件下的整体调度过程进行了模拟优化求解，得出了较优的市场出清结果和梯级优化运行方案，并通过对优化结果的详细分析，指出了模型的优点和缺陷。     

Security-constrained expansion planning of fast-response units for wind integration

This paper proposes a stochastic expansion planning of fast-response thermal units for the large-scale integration of wind generation (WG). The paper assumes that the WG integration level is given and considers the short-term thermal constraints and the volatility of wind units in the planning of fast-response thermal units. The new fast-response units are proposed by market participants. The security-constrained expansion planning approach will be used by an ISO or a regulatory body to secure the optimal planning of the participants’ proposed fast-response units with the WG integration. Random outages of generating units and transmission lines as well as hourly load and wind speed forecast errors are modeled in Monte Carlo scenarios. The Monte Carlo simplification methods are introduced to handle large-scale stochastic expansion planning as a tradeoff between the solution accuracy and the calculation time. The effectiveness of the proposed approach is demonstrated through numerical simulations.     

大型电力系统可靠性评估中的马尔可夫链蒙特卡洛方法

提出大型电力系统可靠性评估的一种新的蒙特卡洛模拟方法-马尔可夫链蒙特卡洛方法(Markov chain Monte Carlo, MCMC)。MCMC方法是一种特殊的蒙特卡洛方法,它将随机过程中的马尔可夫过程引入到蒙特卡洛模拟中,实现动态蒙特卡洛模拟。该方法通过重复抽样,建立一个平稳分布与系统概率分布相同的马尔可夫链,从而得到系统的状态样本。由于MCMC方法考虑了系统各个状态间的相互影响,相比于随机采样的蒙特卡洛方法所得到的独立样本序列,更准确模拟了电力系统运行实际情况。IEEE-RTS 24节点算例表明,该算法可快速收敛,节省计算时间,提高计算速度。同时,由于每条马尔可夫链均收敛于同一个分布,即所谓平稳分布,所以算法具有良好的稳定性。对西北330 kV电网的可靠性评估再次表明了该方法的正确性和有效性以及该方法用于大型电力系统的可靠性评估的优越性和潜力。     

Three-dimensional optimal multi-level Monte–Carlo approximation of the stochastic drift–diffusion–Poisson system in nanoscale devices

The three-dimensional stochastic drift–diffusion–Poisson system is used to model charge transport through nanoscale devices in a random environment. Applications include nanoscale transistors and sensors such as nanowire field-effect bio- and gas sensors. Variations between the devices and uncertainty in the response of the devices arise from the random distributions of dopant atoms, from the diffusion of target molecules near the sensor surface, and from the stochastic association and dissociation processes at the sensor surface. Furthermore, we couple the system of stochastic partial differential equations to a random-walk-based model for the association and dissociation of target molecules. In order to make the computational effort tractable, an optimal multi-level Monte–Carlo method is applied to three-dimensional solutions of the deterministic system. The whole algorithm is optimal in the sense that the total computational cost is minimized for prescribed total errors. This comprehensive and efficient model makes it possible to study the effect of design parameters such as applied voltages and the geometry of the devices on the expected value of the current.     

基于时序和随机风速模型的含风电场发电系统充裕性评估

建立基于蒙特卡洛模拟(MCS)的含风电场发电系统可靠性评估模型,包含时序和随机风速模型的建立及序贯和非序贯蒙特卡洛模型的建立,并把该模型应用到RBTS测试系统中。由不同风速模型和不同MCS方法构成4种组合可靠性评估模型,本文分析4种组合模型的系统缺电概率LOLP、缺供电时间期望LOLE、期望缺供电量EENS、可靠性指标,并确定时序和随机风速模型及序贯和非序贯MCS方法对系统充裕性的影响,同时分析了风电场装机容量、系统峰值负荷等因素对系统充裕性指标的影响。     

基于机会约束规划的电力系统安全成本优化计算

提出了一种基于机会约束规划计算电力系统安全成本的新方法。安全成本模型由发电侧备用成本、可中断负荷成本、再调度成本和期望失稳损失等因素构成。在求解该模型的过程中,应用遗传算法和蒙特卡罗仿真技术,将发电侧备用容量和可中断负荷作为随机变量并用概率的形式表示约束条件,从而得到最优安全成本费用。最后通过IEEE30节点测试系统验证了该方法的有效性。     

Estimating the variance of production cost using a stochastic loadmodel

The purpose of this paper is to provide a realistic load variation model to be used in short-term (one to three years) planning studies. A stochastic model is proposed, and this model is used to quantify the variation of the estimated production cost that is directly affected by the load uncertainty. The paper presents a method of estimating the variation of production cost. This is the first paper to use a Gauss-Markov stochastic model of load with a chronological production simulation model. This load model captures the stochastic load variation behavior and the correlation between weekly peak demand and weekly energy. A weekly Gauss-Markov sampling scheme is incorporated in the proposed approach to model load variation. This stochastic load model is used to generate sample chronological load profiles that represent the annual load variation in weekly detail. These load profiles are then used in annual Monte Carlo production simulation. Case studies illustrate the implementation of this stochastic load variation modeling. These case studies illustrate that load uncertainty has a significantly larger effect on cost uncertainty than does uncertainty in unit availability     

Features that influence composite power system reliability worthassessment

Reliability worth assessment using customer interruption costs is an important element in electric power system planning and operation. This paper deals with two features that affect the composite generation-transmission system reliability worth assessment. One feature is the incorporation of temporal variations in the cost of interruption. This paper illustrates the effect on the expected annual system outage cost of temporal variation in the interruption costs for the residential, agricultural, industrial, commercial and large user sectors. The other aspect considered in this paper is using a probability distribution approach to represent the cost of interruption model. The conventional customer damage function approach utilizes average customer costs while the probability distribution approach recognizes the dispersed nature of the customer outage data. These two methods of cost evaluation are applied to reliability worth assessment in this paper. A sequential Monte Carlo approach incorporating time varying loads is used to conduct all the studies. Case studies performed on two composite test systems show that incorporating time varying costs of interruption for the industrial sector resulted in a significant reduction in the expected outage cost. A comparison of the reliability worth obtained using the customer damage function method (CDF) with the probability distribution approach suggests that using the CDF method may significantly undervalue the reliability worth by a factor of three to four     

GENCO's Risk-Based Maintenance Outage Scheduling

This paper presents a stochastic model for the optimal risk-based generation maintenance outage scheduling based on hourly price-based unit commitment in a generation company (GENCO). Such maintenance outage schedules will be submitted by GENCOs to the ISO for approval before implementation. The objective of a GENCO is to consider financial risks when scheduling its midterm maintenance outages. The GENCO also coordinates its proposed outage scheduling with short-term unit commitment for maximizing payoffs. The proposed model is a stochastic mixed integer linear program in which random hourly prices of energy, ancillary services, and fuel are modeled as scenarios in the Monte Carlo method. Financial risks associated with price uncertainty are considered by applying expected downside risks which are incorporated explicitly as constraints. This paper shows that GENCOs could decrease financial risks by adjusting expected payoffs. Illustrative examples show the calculation of GENCO's midterm generation maintenance schedule, risk level, hourly unit commitment, and hourly dispatch for bidding into energy and ancillary services markets.     

基于MCMC法进行电压跌落随机预估方法的研究

针对电压跌落随机预估中蒙特卡洛法（MC法）静态性、计算效率低、耗时长的缺陷,提出基于马尔可夫链蒙特卡罗法（Markov chain Monte Carlo,MCMC）对电压跌落进行随机预估。研究建立了电压跌落故障状态变量的数学模型,并利用Matlab建立了IEEE-9节点测试系统模型,使用Gibbs抽样法得到故障模型的状态变量。通过分析电压跌落幅值的概率分布,并用MCMC法和MC法分别对电压跌落指标进行仿真计算。仿真结果表明,该方法较蒙特卡罗法稳定性好,收敛速度快,计算时间短。     

计及风险的发电厂商机组检修规划模型

根据发电厂商的风险偏好,提出了基于效用函数的机组检修规划模型,用以规避市场电价预测不确定性及机组突然故障带来的风险。首先,分析了影响发电厂商经济效益的相关不确定性因素,采用MonteCado法及SBR（Simuhaneous Backward Reduction）降阶技术模拟可能存在的场景以评估其影响水平。然后,基于日前电能市场交易模式及发电厂商面临的风险．分析了规划期内的经济效益,主要包括各时段的售电利润、更新费用及检修费用,并根据发电厂商的自身风险偏好,构建了基于期望一风险的检修效用函数,在其经济效益期望与风险之间取得平衡。基于此,以规划期内发电厂商的效用函数最大为目标,考虑相关约束后,利用遗传算法和线性规划方法确定各机组检修时段,并用简单算例进行了定量验证。与常规模型相比,本文充分考虑了不确定性因素及发电厂商的风险偏好对检修规划、经济效益及机组出力的影响,规避了相关风险损失。     

Risk-constrained optimal strategy for retailer forward contract portfolio

In the medium term planning, the objective of an electricity retailer is to configure its forward contract portfolio and to determine the selling price offered to its clients. To procure the electricity energy to be sold to the clients, a retailer has to face by two major challenges. Firstly, at buying electricity energy, it must cope with uncertain pool prices and sign forward contracts at higher average prices. Secondly, at selling electricity, it should handle the demand uncertainty and consider this fact that customers might choose a different retailer if the selling price is not competitive enough. In this paper the financial risk associated with the market price uncertainty is modeled using expected downside risk, which is incorporated explicitly as a constraint in the mixed-integer stochastic optimization problem. Roulette wheel mechanism and Lattice Monte Carlo Simulation (LMCS) are employed for random scenario generation wherein the stochastic optimization problem is converted into its respective deterministic equivalents. The proposed optimization problem is solved by a decomposition technique using Benders decomposition algorithm. A realistic case study is implemented to demonstrate the capability of the proposed method.     

基于改进交叉熵算法的电力系统随机生产模拟

蒙特卡洛模拟的计算效率与系统的可靠性密切相关，在其用于高可靠性系统的随机模拟时存在计算效率偏低的问题。为此，提出了一种基于多层交叉熵与对偶变数抽样技术相结合的随机模拟算法。首先使用多层交叉熵构造零方差概率密度函数的近似函数，提高小概率失效事件的抽取概率；其次基于已构造的近似概率函数，采用对偶变数抽样法进行抽样，进一步提高抽样的收敛速度。以IEEE RTS修改系统为例进行了算例分析，算例结果验证了所提出的基于改进交叉熵的电力系统随机生产模拟算法的有效性。     

水源地地下水综合补给量频率分析的方法

如何解决水源地地下水综合补给量的频率分析计算问题 ,采用解析法、数值法、物理实验以及经验频率都存在着各种困难 ,运用蒙特卡罗方法会使此类问题迎刃而解。蒙特卡罗方法是用数学的方式在属性概率模型控制下 ,产生足够多的伪随机变量 ,构成大容量样本 ,形成分布来求解的。经过理论和实践说明蒙特卡罗方法能在水资源量评价中发挥重要的作用。