A Fuzzy Stochastic Dynamic Nash Game Analysis of Policies for Managing Water Allocation in a Reservoir System

In this paper a fuzzy dynamic Nash game model of interactions between water users in a reservoir system is presented. The model represents a fuzzy stochastic non-cooperative game in which water users are grouped into four players, where each player in game chooses its individual policies to maximize expected utility. The model is used to present empirical results about a real case water allocation from a reservoir, considering player (water user) non-cooperative behavior and also same level of information availability for individual players. According to the results an optimal allocation policy for each water user can be developed in addition to the optimal policy of the reservoir system. Also the proposed model is compared with two alternative dynamic models of reservoir optimization, namely Stochastic Dynamic Programming (SDP) and Fuzzy-State Stochastic Dynamic programming (FSDP). The proposed modeling procedures can be applied as an appropriate tool for reservoir operation, considering the interaction among the water users as well as the water users and reservoir operator.     

Multireservoir System Optimization using Fuzzy Mathematical Programming

For a multireservoir system, where the number of reservoirs islarge, the conventional modelling by classical stochastic dynamicprogramming (SDP) presents difficulty, due to the curse ofdimensionality inherent in the model solution. It takes a longtime to obtain a steady state policy and also it requires largeamount of computer storage space, which form drawbacks inapplication. An attempt is made to explore the concept of fuzzysets to provide a viable alternative in this context. Theapplication of fuzzy set theory to water resources systems isillustrated through the formulation of a fuzzy mathematicalprogramming model to a multireservoir system with a number ofupstream parallel reservoirs, and one downstream reservoir. Thestudy is aimed to minimize the sum of deviations of the irrigationwithdrawals from their target demands, on a monthly basis, over ayear. Uncertainty in reservoir inflows is considered by treatingthem as fuzzy sets. The model considers deterministic irrigationdemands. The model is applied to a three reservoir system in theUpper Cauvery River basin, South India. The model clearlydemonstrates that, use of fuzzy linear programming inmultireservoir system optimization presents a potentialalternative to get the steady state solution with a lot lesseffort than classical stochastic dynamic programming.     

QUALIFICATION OF UNCERTAINTY FOR SIMULATING SOLUTE TRANSPORT IN THE HETEROGENEOUS MEDIA WITH SPARSE GRID COLLOCATION METHOD

The sparse grid collocation method is discussed to qualify the uncertainty of solute transport. The Karhunen-Loeve (KL) expansion is employed to decompose the log transformed hydraulic conductivity. The head, velocity and concentration fields are represented by the Lagrange polynomial expansion. A sparse grid collocation method is then used to reduce the original stochastic partial differential equations to a set of deterministic equations which is collocated at selected interpolation (collocation) points. The collocation points are constructed by the Smolyak algorithm. The accuracy, efficiency and convergence property of sparse grid collocation method are investigated by numerical experiments. The analysis shows that stochastic collocation strategy helps to decouple stochastic computations, and all the numerical computation is possible to be implemented by existing deterministic finite element codes. The proposed method provides an efficient way to evaluate the uncertainty of the solute transport in the heterogeneous media.     

Real-Time Operation of Reservoir System by Genetic Programming

Reservoir operation policy depends on specific values of deterministic variables and predictable actions as well as stochastic variables, in which small differences affect water release and reservoir operation efficiency. Operational rule curves of reservoir are policies which relate water release to the deterministic and stochastic variables such as storage volume and inflow. To operate a reservoir system in real time, a prediction model may be coupled with rule curves to estimate inflow as a stochastic variable. Inappropriate selection of this prediction model increases calculations and impacts the reservoir operation efficiency. Thus, extraction of an operational policy simultaneously with inflow prediction helps the operator to make an appropriate decision to calculate how much water to release from the reservoir without employing a prediction model. This paper addresses the use of genetic programming (GP) to develop a reservoir operation policy simultaneously with inflow prediction. To determine a water release policy, two operational rule curves are considered in each period by using (1) inflow and storage volume at the beginning of each period and (2) inflow of the 1^st, 2^nd, 12^th previous periods and storage volume at the beginning of each period. The obtained objective functions of those rules have only 4.86 and 0.44?% difference in the training and testing data sets. These results indicate that the proposed rule based on deterministic variables is effective in determining optimal rule curves simultaneously with inflow prediction for reservoirs.     

三维地下水流随机分析的配点法

本文提出了三维地下水不确定分析的随机配点法，比较了张量积配点法（Tensor Product Collocation Method, TPCM），Smolyak配点法 (Smolyak Collocation Method, SmCM)，Stroud-2配点法 (Stroud-2 Collocation Method, StCM)，以及概率配点法 (Probability Collocation Method, PCM)等四种配点法的基本原理；渗透系数场和水头场分别被表示为Karhunen-Loeve展开和多项式展开的形式；初始随机偏微分方程被转化为在相应配点上的确定性方程；建立了基于Modflow-2000的随机模型；讨论了不同算法的计算精度及计算成本。分析结果表明，配点法可充分继承现有模拟软件或代码的求解功能，配点的选取方法决定了其计算成本，SmCM具有最高的计算精度，但StCM具有最好的性价比。     

基于van Genuchten-Mualem模型的饱和-非饱和介质流动随机数值分析

基于van Genuchten-Mualem非饱和水分特征模型，联合运用Karhunen-Loeve展开法、混沌多项式展开以及摄动方法，对饱和-非饱和流问题进行随机数值分析。将土壤特性参数假定为协方差已知的随机函数，并按Karhunen-Loeve法分解，把压力水头表示为多项式。通过摄动方法得到一系列关于水头展开式的偏微分方程，用有限差分法进行求解，获得了压力水头的随机描述，并计算其均值和方差。应用本文的随机模型研究了二维非饱和以及饱和-非饱和介质流动的实例，结果与动量方法的计算结果一致，而且计算效率高于传统的动量方法。     

STOCHASTIC ANALYSIS OF WATER FLOW IN HETEROGENEOUS MEDIA

A stochastic model for saturated-unsaturated flow is developed based on the combination of the KarhunenLoeve expansion of the input random soil properties with a perturbation method. The saturated hydraulic conductivity ks(x) is assumed to be log-normal random functions, expressed by f(x). f(x) is decomposed as infinite series in a set of orthogonal normal random variables by the Karhunen-Loeve (KL) expansion and the pressure head is expand as polynomial chaos with the same set of orthogonal random variables. With these expansions, the stochastic saturated-unsaturated flow equation and the corresponding initial and boundary conditions are represented by a series of deterministic partial differential equations which can be solved subsequently by a suitable numerical method. Some examples are given to show the reliability and efficiency of the proposed method.     

Multi-Colony Ant Algorithm for Continuous Multi-Reservoir Operation Optimization Problem

Ant Colony Optimization (ACO) algorithms are basically developed for discrete optimization and hence their application to continuous optimization problems require the transformation of a continuous search space to a discrete one by discretization of the continuous decision variables. Thus, the allowable continuous range of decision variables is usually discretized into a discrete set of allowable values and a search is then conducted over the resulting discrete search space for the optimum solution. Due to the discretization of the search space on the decision variable, the performance of the ACO algorithms in continuous problems is poor. In this paper a special version of multi-colony algorithm is proposed which helps to generate a non-homogeneous and more or less random mesh in entire search space to minimize the possibility of loosing global optimum domain. The proposed multi-colony algorithm presents a new scheme which is quite different from those used in multi criteria and multi objective problems and parallelization schemes. The proposed algorithm can efficiently handle the combination of discrete and continuous decision variables. To investigate the performance of the proposed algorithm, the well-known multimodal, continuous, nonseparable, nonlinear, and illegal (CNNI) Fletcher–Powell function and complex 10-reservoir problem operation optimization have been considered. It is concluded that the proposed algorithm provides promising and comparable solutions with known global optimum results.     

Probabilistic Model for the Representation of the Reservoir Water Level of Concrete Dams During Normal Operation Periods

The dam’s reservoir water level varies over the year according to the water inflow and outflow, as a function of environmental events but also of dam exploitation management policy and human decisions. In the Portuguese dam safety regulation (RSB 2007), the normal water level (NWL) is considered as the optimum exploitation level. However, as proved by the continuous monitoring over the lifetime of a set of dams, the NWL is occasionally exceeded for non-negligible time periods. The reservoir water level, to which the water pressure on the upstream face is related, is a fundamental parameter for the safety and reliability analysis of concrete dams. When water-induced actions are considering the leading loads, only the maximum reservoir water level, usually associated with a high-return-period flood, is relevant. However, for other combinations, in particular, earthquake scenarios, the consideration of the variability of the water level over time is crucial. In reliability analysis of concrete dams, the reservoir water level has been considered either as a deterministic variable, once the loading scenario analyzed assume water-induced actions as leading loads (Westberg 2010), or as a random variable defined using hydrological site information of a specific study case (Altarejos et al. Structural Safety 36-37:1–13 2012). This work proposes a probabilistic model of the reservoir water level of any new dam based only on its geometrical properties, which provides a low-cost alternative to in-depth hydrological analysis. The proposed model can be useful in two stages of the lifecycle of dams: (i) initial design and feasibility stages, and (ii) routine safety assessment of existing structures, as in both stages the costs of a complete hydrological analysis is too high for the level of detail required. For that, the recorded reservoir water level of 27 Portuguese large concrete dams is used. A normalized sinusoidal model, with annual period, is adjusted to the reservoir water level annual history of those dams by beta regression. Generally, a good agreement between observations and the proposed model, for most of the annual adjustments, was achieved. The distribution parameters of the random variables were estimated through the maximum likelihood estimation (MLE) method. The physical, model and statistical uncertainties were quantified and can now be included in a reliability analysis procedure.     

应用线性二次高斯控制进行灌区调度

本文研究了随机环境下灌区调度问题，建立了一个水库和田间配水整合随机调度模型。文中引入土壤水库的概念，把灌区调度的问题转化成水库群调度的问题，采用线性二次高斯控制(Linear Quadratic Gaussion control，LQG)对模型进行求解，解决了此类研究中的求解技术难题，从而克服了前人模型中仿真性能不好的问题。此外，本文还对水库群调度中的LQG方法进行了改进。实例计算表明，本文建立的模型能取得良好的效益，LQG算法稳定、计算速度快。     

Stochastic simulation of fluid flow in porous media by the complex variable expression method

A stochastic simulation of fluid flow in porous media using a complex variable expression method (SFCM) is presented in this paper. Hydraulic conductivity is considered as a random variable and is then expressed in complex variable form, the real part of which is a deterministic value and the imaginary part is a variable value. The stochastic seepage flow is simulated with the SFCM and is compared with the results calculated with the Monte Carlo stochastic finite element method. In using the Monte Carlo method to simulate the stochastic seepage flow field, the hydraulic conductivity is assumed in three different probability distributions using random sampling method. The obtained seepage flow field is examined through skewness analysis, and the skewed distribution probability density function is given. The head mode value and the head comprehensive standard deviation are used to represent the statistics of calculation results obtained by the Monte Carlo method. The stochastic seepage flow field simulated by the SFCM is confirmed to be similar to that given by the Monte Carlo method from numerical aspects. The range of coefficient of variation of hydraulic conductivity in SFCM is larger than used previously in stochastic seepage flow field simulations, and the computation time is short. The results proved that the SFCM is a convenient calculating method for solving the complex problems.     

A Modified Constrained State Formulation of Stochastic Soil Moisture for Crop Water Allocation

In response to uncertainty in crop water allocation, several methodologies have been proposed in the literature, most of them considering rainfall as a stochastic variable affecting soil moisture. A methodology considering uncertainties both in irrigation depth and soil moisture is more realistic for irrigated crops as developed here using an explicit stochastic optimization model. This new work is based on an earlier constrained state formulation which did not consider the irrigation depth as stochastic. In constrained state formulation methods, the first and second moments of state variables are developed considering the uncertainties which are then used as constraints in an optimization model. In contrast to alternative methods that are dynamic programming-based, the proposed optimization method can be solved using standard nonlinear optimization tools. Performance of the proposed model is evaluated for the case of two different crops, winter wheat and barley. Model verification is performed by comparing the results with simulation results. The model is quite acceptable and shows considerable improvement over analogous models.     

PAPOA法在黑河综合利用水库优化调度中的应用

对西安市黑河多功能综合利用水库优化调度问题，建立了动态确定性多目标非线性数学模型．利用逐次逼近的逐步优化法，即ＰＡＰＯＡ法求得了模型的最优解．较大地提高了水库效益，获得了较为满意的结果．     

地下水动态预测模型的回顾与展望

回顾了国内外地下水动态预测研究的各种模型,将这些模型划分为确定性模型和随机模型两大类,指出各种模型的适用条件及存在问题。最后提出了提高地下水动态预测的可靠性与精度的两条途径:(1)建立既能描述地下水系统内部特征也能反映其外部特征的确定性-随机耦合模型;(2)建立地下水系统的随机微分方程模型,并将系统内部参数作为随机变量,将其外部环境因素也作为随机时间序列引入地下水流定解问题,并给出解的概率分布。     

Benefit and Risk Balance Optimization for Stochastic Hydropower Scheduling

Limited by inflow forecasting methods, the forecasting results are so unreliable that we have to take their uncertainty and risk into account when incorporating stochastic inflow into reservoir operation. Especially in the electricity market, punishment often happens when the hydropower station does not perform as planned. Therefore, focusing on the risk of power generation, a benefit and risk balance optimization model (BRM) which takes stochastic inflow as the major risk factor is proposed for stochastic hydropower scheduling. The mean-variance theory is firstly introduced into the optimal dispatching of hydropower station, and a variational risk coefficient is employed to give service to managers’ subjective preferences. Then, the multi-period stochastic inflow is simulated by multi-layer scenario tree. Moreover, a specific scenario reduction and reconstruction method is put forward to reduce branches and computing time accordingly. Finally, the proposed model is applied to the Three Gorges Reservoir (TGR) in China for constructing a weekly generation scheduling in falling stage. Compared to deterministic dynamic programming (DDP) and stochastic dynamic programming (SDP), BRM achieves more satisfactory performance. Moreover, the tradeoffs for risk-averse decision makers are discussed, and an efficient curve about benefit and risk is formed to help make decision.     

水电站群发电能力的优化方法及应用分析

针对水电站群中长期发电计划编制的调度期末控制水位决策需求,建立以发电能力最大为目标的水电站群联合优化调度模型,并采用改进的逐次逼近与离散微分动态规划组合算法进行求解。以华电贵州区域乌江和北盘江两流域水电站群为例,采用2013年实际来水过程计算,通过与发电量最大、补偿效益最大等传统优化目标对比分析。结果表明:本文模型可根据各电站发电效率变化情况自动确定水库蓄放策略,实现水电站群调度期末控制水位及对应发电计划过程的同步优化,准确体现水电站群调度期内总发电量和调度期末总蓄能量的动态博弈关系。     

Hydropower Optimization for the Lower Seyhan System in Turkey using Dynamic Programming

Abstract

Dynamic programming with successive approximation has been used in the past for optimizing multi-reservoir water resources systems. In this study, the State Incremental Dynamic Programming (SIDP) model is developed for energy optimization of multi-reservoir systems. A random file access method is used for reaching initial and intermediate data to cope with the curse of dimensionality of dynamic programming. A conventional dynamic programming method is used for each single reservoir to find the initial trajectory of the reservoirs. Then, the computer program developed in the study is applied to the multipurpose-multi-reservoir system in Lower Seyhan Basin, which has six reservoirs, some of which are serial and some parallel. First, extended historical flows were used to maximize firm energy in the critical period, and then total energy in the total flows. The program was run with 50-year long segments (20 flow scenarios) of the synthetic flow data generated by using the HEC-4 generalized computer program to take into account the stochastic nature of stream flows. An increment of approximately 20 percent in total energy was obtained by using the model for the Lower Seyhan System, as compared to that calculated previously by conventional methods.     

随机波浪力连续场的降维模拟

根据水质点的水平速度和加速度过程的功率谱密度,利用线性化的Morison方程,从平稳随机过程自相关函数的角度推导了随机波浪力的功率谱密度函数。在平稳随机过程的源谱表达基础上,通过将标准正交随机变量集定义为仅含1个基本随机变量的正交函数形式,实现了随机波浪力连续场的降维模拟。同时,结合快速傅里叶变换（FFT）技术,给出了随机波浪力连续场降维模拟的快速算法。应用该方法对某一单个小尺度直立圆柱桩所受的随机波浪力进行模拟,给出了随机波浪力过程的均值、标准差、自相关函数及互相关函数等数值特征,并与Monte Carlo方法的结果进行对比分析。研究表明,降维模拟方法具有较高的模拟精度与效率。     

Hydropower Optimization for the Lower Seyhan Basin System in Turkey using dynamic programming

Abstract

Dynamic programming with successive approximation has been used in the past for optimizing multi-reservoir water resources systems. In this study, a State Incremental Dynamic Programming (SIDP) model is developed for energy optimization of multi-reservoir systems. A random file access method is used to generate initial and intermediate data and cope with the curse of dimensionality of dynamic programming. The conventional dynamic programming method is used for each single reservoir to find the initial trajectory of the reservoirs. Then, the computer program developed in the study is applied to the multipurpose-multi-reservoir system in lower Seyhan basin, which comprises six reservoirs, some serial and some parallel. Extended historical flows are used to first maximize firm energy in the critical period, and then total energy over the entire period of flow records. The program is run with 50-year long segments (20 flow scenarios) of the synthetic flow data generated by using the hec-4 generalized computer program to account for the stochastic nature of streamflows. A 20% approximate increase in total energy is obtained by using the developed model for the lower seyhan basin system as compared to that calculated previously by conventional methods.     

三峡梯级电站短期优化调度的模糊多目标动态规划

针对三峡梯级电站的短期优化调度是一个具有复杂约束条件的大型、动态、有时滞、模糊非线性的优化问题，本文建立了模糊优化调度模型，并对其进行了求解。由于常规模糊动态规划的方法在库容变化大的时候会出现维数灾，因此本文利用模糊多目标动态规划法，并结合逐次逼近方法，来求解梯级短期优化调度问题。实例仿真结果表明，它不仅可以根据不同的参数选择生成非劣解集，还可以推荐满意解，为决策支持提供依据。