Development of stochastic dynamic Nash game model for reservoir operation. I. The symmetric stochastic model with perfect information

Increasing water demands, higher standards of living, depletion of resources of acceptable quality and excessive water pollution due to agricultural and industrial expansions have caused intense social and political predicaments, and conflicting issues among water consumers. The available techniques commonly used in reservoir optimization/operation do not consider interaction, behavior and preferences of water users, reservoir operator and their associated modeling procedures, within the stochastic modeling framework. In this paper, game theory is used to present the associated conflicts among different consumers due to limited water. Considering the game theory fundamentals, the Stochastic Dynamic Nash Game with perfect information (PSDNG) model is developed, which assumes that the decision maker has sufficient (perfect) information regarding the associated randomness of reservoir operation parameters. The simulated annealing approach (SA) is applied as a part of the proposed stochastic framework, which makes the PSDNG solution conceivable. As a case study, the proposed model is applied to the Zayandeh-Rud river basin in Iran with conflicting demands. The results are compared with alternative reservoir operation models, i.e., Bayesian stochastic dynamic programming (BSDP), sequential genetic algorithm (SGA) and classical dynamic programming regression (DPR). Results show that the proposed model has the ability to generate reservoir operating policies, considering interactions of water users, reservoir operator and their preferences.     

A BME solution of the stochastic three-dimensional Laplace equation representing a geothermal field subject to site-specific information

This work develops a model of the geothermal field in the Nea Kessani region (Greece) by means of the Bayesian maximum entropy (BME) method, which describes the temperature variations across space in the underground geological formations. The geothermal field is formed by a thermal reservoir consisting of arcosic sandstones. The temperature distribution vs depth was first investigated by the Greek Institute of Geology and Mineral Exploration (IGME) using measurements in a set of vertical drill holes. These measurements showed that hot fluids rising from the deep enter the reservoir in a restricted area of the field and flow towards local thermal springs. The field modelling, which was based on the powerful BME concept, involves the solution of a stochastic partial differential equation that assimilates important site-specific information. The stochastic three-dimensional steady-state Laplace equation was considered as general knowledge and the drilling exploration data were used to construct the specificatory knowledge base in the BME terminology. The produced map is more informative and, in general, it gives higher temperature estimates compared to previous studies of the same region. This is also in agreement with the quartz geothermometry analysis carried out by IGME.     

On a power series solution to the Boussinesq equation

For certain initial and boundary conditions the Boussinesq equation, a nonlinear partial differential equation describing the flow of water in unconfined aquifers, can be reduced to a boundary value problem for a nonlinear ordinary differential equation. Using Song et al.'s (2007) [7] approach, we show that for zero head initial condition and power-law flux boundary condition at the inlet boundary, the solution in the form of power series can be obtained with Barenblatt's (1990) [2] rescaling procedure applied to the power series solution obtained in Song et al. (2007) [7] for the power-law head boundary condition. Polynomial approximations can then be obtained by taking terms from the power series. Although for a small number of terms the newly obtained approximations may be worse than polynomial approximations obtained by other techniques, any desired accuracy can be achieved by taking more terms from the power series.     

龙滩水库蓄水后库区中小地震震源机制

本文在对龙滩库区2006年9月30日至2013年5月26日发生的3682次地震进行精定位的基础上,利用FOCMEC方法和改进的格点尝试法反演了ML2.0以上地震震源机制及区域构造应力场,并综合龙滩库区地层岩性、断裂构造和渗透条件等资料,探讨了地震活动类型与库区蓄水过程的关系,获得以下认识:(1)龙滩水库蓄水后的地震活动主要丛集在罗妥、八茂、拉浪、坝首和布柳河5个深水区,地震类型以逆断为主,正断和走滑也占有一定比例.蓄水初期,库区地震类型呈现多样性,蓄水约4年3个月后,地震主要发生在浅部地层中,并大多为逆断型地震.(2)5个地震丛构造应力场最大主应力方向以NW-SE为主,倾角均较小,中等和最小主应力分布较凌乱,在此构造应力环境中,龙滩库区主要断裂整体上呈现走滑运动的性质,局部伴生不同程度的逆倾滑或正倾滑运动分量.(3)库区深、浅部地震活动水平和地震性质之所以会随蓄水过程发生变化,可能与深、浅部构造应力环境、岩体力学性质和渗透性能的差异有关.     

On the integral equation of the continuous-state stochastic reservoir with seasonally varying autocorrelated inflows

The paper deals with continuous-state reservoirs in discrete time, in terms of the (continuous) Moran transform U_t = Z_t + X_t of the (discontinuous) storage levels Z_t. In the non-seasonal case, if the inflows X_t have a first-order Markov structure, the modified storage levels U_t have a second-order Markov structure. It is shown that, for each value of u, the limit p(u,v) as t → ∞ of the joint density p_{t + 1} (u, v) of U_{t+ 1} at u and U_t at v satisfies a certain non-standard first-order linear integral equation of the form

or, equivalently, a standard second-order equation of the form

In the seasonal case, with two seasons, the corresponding bivariate density p(u,v) for consecutive seasons satisfies a second-order integral equation of the form

the kernel being a function of the season-to-season inflow transition densities.     

水平界面上P-SV转换波转换点的精确解

转换波共转换点的叠加和道集选取都需要准确地计算转换点的位置.Tessmer和Behle、Taylor分别给出了水平单层介质中P-SV转换波转换点坐标的解析解,由于其表达式的复杂性,在应用中几乎不被采用.在本文中,运用Snell定律重新建立了在水平反射界面上反射的P-SV转换波的转换点坐标的四次方程,并严格地推导出与纵波速度、横波速度、炮检距和反射深度有关的转换点坐标的解析解,确定了惟一的解析表达式.将这一结果应用于P-SV转换波的速度分析和叠加处理中.简化的公式有较好的应用价值.     

Robust numerical solution of the reservoir routing equation

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson–Pilgrim method, (2) the fourth-order Runge–Kutta method, and (3) the fixed order Cash–Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.     

Evaluation of rainfall-runoff performance using the stochastic integral equation method

The stochastic integral equation method (S.I.E.M.) is used to evaluate the relative performance of a set of both calibrated and uncalibrated rainfall-runoff models with respect to prediction errors. The S.I.E.M. is also used to estimate confidence (prediction) interval values of a runoff criterion variable, given a prescribed rainfall-runoff model, and a similarity measure used to condition the storms that are utilized for model calibration purposes.Because of the increasing attention given to the issue of uncertainty in rainfall-runoff modeling estimates, the S.I.E.M. provides a promising tool for the hydrologist to consider in both research and design.     

Evaluation of rainfall-runoff performance using the stochastic integral equation method

The stochastic integral equation method (S.I.E.M.) is used to evaluate the relative performance of a set of both calibrated and uncalibrated rainfall-runoff models with respect to prediction errors. The S.I.E.M. is also used to estimate confidence (prediction) interval values of a runoff criterion variable, given a prescribed rainfall-runoff model, and a similarity measure used to condition the storms that are utilized for model calibration purposes.Because of the increasing attention given to the issue of uncertainty in rainfall-runoff modeling estimates, the S.I.E.M. provides a promising tool for the hydrologist to consider in both research and design.     

南亚热带典型调水型水库——广东大镜山水库的富营养化特征分析

调水型水库是一种以抽水入库为主要来水水源的水体,是沿海地区重要的供水水源地.为了解这类水库的富营养化特点,于2005年全年每月2次对地处我国南亚热带地区(广东珠海市)的大镜山水库进行采样和监测.监测和测定指标主要包括氮、磷营养盐浓度、水温、透明度及叶绿素a浓度等,结合水库水文数据对水库富营养化特征和主要的影响因素进行分析.结果表明,2005年,大镜山水库的富营养化状态TSI_M指数在45-53之间,水库处于中富营养状态,多数时间处于富营养状态.水体富营养化主要参数表现出明显的季节变化,即叶绿素a浓度和富营养化状态指数在早春和晚秋出现两个峰值,明显地与温带富营养化水体在夏季出现单个峰值的特征不同.调水入库增加了水库营养盐负荷的同时,也在很大程度上影响了水库水动力学过程,与夏季的集中强降雨一起成为影响该水库富营养化的关键因素,这些因素改变了浮游植物群落对营养盐的直接响应,导致叶绿素a浓度与总磷、总氮浓度之间呈弱相关关系,降水和调水量在时间上相对配置重要性决定了叶绿素a浓度与营养盐浓度的关系.     

近20年来三峡水库泥沙淤积及其对库区的影响

泥沙问题是三峡工程建设与运行中的关键技术问题之一,只有妥善处理好泥沙问题,才能保证三峡工程长期有效使用,维持水库功能的全面发挥。本文首先结合实测水文、河道地形观测资料,对三峡水库运行近20年来的泥沙淤积特性及水库排沙比进行了较为全面的分析研究,并与已有研究成果进行了对比;其次,围绕防洪、航运以及坝前段的泥沙淤积等方面,进一步分析了水库淤积产生的影响。结果表明:三峡水库蓄水以来,在不考虑区间来沙的情况下,三峡水库共淤积泥沙20.484亿t,近似年均淤积1.102亿t,水库排沙比为23.6%,水库年均淤积量为原论证预测值的33%。其中,库区干流段累计淤积泥沙17.835亿m³(变动回水区冲刷0.694亿m³;常年回水区淤积18.529亿m³),淤积在水库防洪库容内的泥沙为1.648亿m³(干、支流分别淤积1.517亿m³和0.131亿m³),占水库防洪库容的0.74%,“十一五”攻关阶段研究得出的多年平均淤积量及排沙比较实测值均偏大,变动回水区冲淤则出现反向的...     

向家坝水库蓄水期间的地震活动分析

根据2008年9月至2013年10月金沙江下游水库地震台网监测资料,分析了蓄水期间水库地震影响区的地震活动和震源机制。向家坝水库蓄水1年多,蓄水后库首区A、B段的地震活动频次和强度都维持在较低水平。蓄水期间地震活动显著增加,集中分布在库区C段,并不是蓄水初期地震活动就增加,与库区2013年6月启动第二阶段蓄水有一定相关性,强度在3级地震活动水平。增加的地震活动位于近南北走向的翼子坝、玛瑙断裂的中段。这些局部地段蓄水后发生的地震,其震源的力学机制多为倾滑或正断型,分析认为部分地震为水库诱发岩溶或塌陷型地震,多数仍属构造地震。本文结果为水库诱发地震的研究提供了资料和震例。     

一座抽水水库中中华窄腹剑水蚤种群动态及其对浮游甲壳类群落结构的影响

中华窄腹剑水蚤是中国河流中发现的土著种类,具有较强的生态入侵能力,通过从西江抽水进入大镜山水库,成为该水库的优势浮游甲壳类,通过2005-2006两年的采样监测,初步分析了该水库中中华窄腹剑水蚤的种群特征及其对浮游甲壳类的群落动态和结构的影响.两年共检出浮游甲壳类13种,其中,桡足类4科7属9种,枝角类3科4属4种,中华窄腹剑水蚤是水库的优势种,两年浮游甲壳类丰度的范围是3.82-364.85ind/L,峰值出现在春季,两年变化趋势相似,水库浮游甲壳类的体长分布范围为0.07-1.25mm,主要由小个体的类群(体长在0.07-0.4mm之间)组成,个体较大(大于0.5mm)的枝角类和桡足类数量很低,中华窄腹剑水蚤和其无节幼体的丰度占甲壳类总丰度的50%以上,其丰度的动态分布与浮游甲壳类的总体丰度的动态分布一致,中华窄腹剑水蚤主要以三种形式(无节幼体Ⅰ期,桡足幼体后期和成体)存在于水体中,存活曲线呈Ⅲ型,无节幼体的丰度远高于后两者之和,该种类能成为水库优势种主要是由水库成度变化剧烈水环境特征和其自身的生态学特点决定的,中华窄腹剑水蚤丰度与叶绿素a浓度具有显著正相关性,与抽水入库量具有极显著的负相关性,作为一个生态入侵种类,中华窄腹剑水蚤降低水体中浮游甲壳类生物多样性,并替代了原来的土著优势种.     

A spatially and temporally adaptive solution of Richards’ equation

Efficient, robust simulation of groundwater flow in the unsaturated zone remains computationally expensive, especially for problems characterized by sharp fronts in both space and time. Standard approaches that employ uniform spatial and temporal discretizations for the numerical solution of these problems lead to inefficient and expensive simulations. In this work, we solve Richards’ equation using adaptive methods in both space and time. Spatial adaption is based upon a coarse grid solve and a gradient error indicator using a fixed-order approximation. Temporal adaption is accomplished using variable order, variable step size approximations based upon the backward difference formulas up to fifth order. Since the advantages of similar adaptive methods in time are now established, we evaluate our method by comparison with a uniform spatial discretization that is adaptive in time for four different one-dimensional test problems. The numerical results demonstrate that the proposed method provides a robust and efficient alternative to standard approaches for simulating variably saturated flow in one spatial dimension.     

A new formulation for the stochastic control of systems with bounded state variables: An application to a single reservoir system

A new formulation is presented for the analysis of reservoir systems synthesizing concepts from the traditional stochastic theory of reservoir storage, moments analysis and reliability programming. The analysis is based on the development of the first and second moments for the stochastic storage state variable. These expressions include terms for the failure probabilities (probabilities of spill or deficit) and consider the storage bounds explicitly. Using this analysis, expected values of the storage state, variances of storage, optimal release policies and failure probabilities — useful information in the context of reservoir operations and design, can be obtained from a nonlinear programming solution. The solutions developed from studies of single reservoir operations on both an annual and monthly basis, compare favorably with those obtained from simulation. The presentation herein is directed to both traditional reservoir storage theorists who are interested in the design of a reservoir and modern reservoir analysts who are interested in the long term operation of reservoirs.     

Building more realistic reservoir optimization models using data mining - A case study of Shelbyville Reservoir

In this paper, we promote a novel approach to develop reservoir operation routines by learning from historical hydrologic information and reservoir operations. The proposed framework involves a knowledge discovery step to learn the real drivers of reservoir decision making and to subsequently build a more realistic (enhanced) model formulation using stochastic dynamic programming (SDP). The enhanced SDP model is compared to two classic SDP formulations using Lake Shelbyville, a reservoir on the Kaskaskia River in Illinois, as a case study. From a data mining procedure with monthly data, the past month’s inflow (Q_t−1), current month’s inflow (Q_t), past month’s release (R_t−1), and past month’s Palmer drought severity index (PDSI_t−1) are identified as important state variables in the enhanced SDP model for Shelbyville Reservoir. When compared to a weekly enhanced SDP model of the same case study, a different set of state variables and constraints are extracted. Thus different time scales for the model require different information. We demonstrate that adding additional state variables improves the solution by shifting the Pareto front as expected while using new constraints and the correct objective function can significantly reduce the difference between derived policies and historical practices. The study indicates that the monthly enhanced SDP model resembles historical records more closely and yet provides lower expected average annual costs than either of the two classic formulations (25.4% and 4.5% reductions, respectively). The weekly enhanced SDP model is compared to the monthly enhanced SDP, and it shows that acquiring the correct temporal scale is crucial to model reservoir operation for particular objectives.     

瀑布沟库区地震活动及应力场特征

通过对瀑布沟水库和成都数字遥测地震台网记录的波形数据的研究发现,四川瀑布沟水库库区大量的所谓小震活动皆是爆破成因,从而改变了此前对该库区蓄水前地震活动密集,蓄水后地震活动骤然平静的认识。2009 年11 月水库蓄水后在蓄水区域周边10km 范围内仅发生ML1. 5 以上地震1 次,即2010 年8 月8 日ML1. 7 地震,蓄水前后小震稀少。水库西侧的2 个地震密集区的走滑型地震占较大比例,区域主压应力为NWW 向。水库东、北部区域的地震震源机制解呈多种类型。瀑布沟水库区小震活动总体受区域应力场控制,NWW 向近水平主压应力有利于库区NNW 向中小断裂发生走滑型错动。     

Improving the performance of data-driven techniques through data pre-processing for modelling daily reservoir inflow

Abstract

The effect of data pre-processing while developing artificial intelligence (AI) -based data-driven techniques, such as artificial neural networks (ANN), model trees (MT) and linear genetic programming (LGP), is studied for Pawana Reservoir in Maharashtra, India. The daily one-step-ahead inflow forecasts are compared with flows generated from a univariate autoregressive integrated moving average (ARIMA) model. For the full-year data series, a large error is found mainly due to the occurrence of zero values, since the reservoir is located in an intermittent river. Hence, all the techniques are evaluated using two data series: 18 years of daily full-year inflow data (from 1 January to 31 December); and 18 years of daily monsoon season inflow data (from 1 June to 31 October) to take into account the intermittent nature of the data. The relevant range of inputs for each category is selected based on autocorrelation and partial autocorrelation analyses of the inflow series. Conventional pre-processing methods, such as transformation and/or normalization of data, do not perform well because of the large variation in magnitudes, as well as the many zero values (65% of the full-year data series). Therefore, the input data are pre-processed into un-weighted moving average (MA) series of 3 days, 5 days and 7 days. The 3-day MA series performs better, maintaining the peak inflow pattern as in the actual data series, while the coarser-scale (5-day and 7-day) MA series reduce the peak inflow pattern, leading to more errors in peak inflow prediction. The results indicate that AI methods are powerful tools for modelling the daily flow time series with appropriate data pre-processing, in spite of the presence of many zero values. The time-lagged recurrent network (TLRN) ANN modelling technique applied in this study maps the inflow forecasting in a better way than the standard multilayer perceptron (MLP) neural networks, especially in the case of the seasonal data series. The MT technique performs equally well for low and medium inflows, but fails to predict the peak inflows. However, LGP outperforms the other AI models, and also the ARIMA model, for all inflow magnitudes. In the LGP model, the daily full-year data series with more zero inflow values performs better than the daily seasonal models.

Citation Jothiprakash, V. & Kote, A. S. (2011) Improving the performance of data-driven techniques through data pre-processing for modelling daily reservoir inflow. Hydrol. Sci. J. 56(1), 168–186.     

Development of stochastic dynamic Nash game model for reservoir operation II. The value of players’ information availability and cooperative behaviors

Stochastic dynamic game models can be applied to derive optimal reservoir operation policies by considering interactions among water users and reservoir operator, their preferences, their levels of information availability and cooperative behaviors. The stochastic dynamic game model with perfect information (PSDNG) has been developed by [Ganji A, Khalili D, Karamouz M. Development of stochastic dynamic Nash game model for reservoir operation. I. The symmetric stochastic model with perfect information. Adv Water Resour, this issue]. This paper develops four additional versions of stochastic dynamic game model of water users interactions based on the cooperative behavior and hydrologic information availability of beneficiary sectors of reservoir systems. It is shown that the proposed models are quite capable of providing appropriate reservoir operating policies when compared with alternative operating models, as indicated by several reservoir performance characteristics. Among the proposed models, the selected model by considering cooperative behavior and additional hydrologic information (about the randomness nature of reservoir operation parameters), as exercised by reservoir operator, provides the highest attained level of performance and efficiency. Furthermore, the selected model is more realistic since it also considers actual behavior of water users and reservoir operator in the analysis.