The value of information in a long–term hydro–thermal electrical planning model

This article presents the analysis, comparison, and application of two alternative models to the optimal long–term operation planning of an hydro–thermal power system under conditions of uncertainty. The electrical system considered comprises one large reservoir, with interannual regulation capacity, and several smaller ones. The analyzed models employ stochastic dynamic programming as the solution methodology. The fundamental problem is to decide, on every temporal stage, how much water to use for generating purposes and how much to store, in order to minimize the total thermal and shortage costs. The original version of the studied model, created originally to forecast fuel consumption, assumes that the decision regarding the water release from the main reservoir is taken knowing the future hydrologic conditions. This criterion is known as wait–and–see . On the contrary, the new versions of the model, proposed in this article, consider a here–and–now criterion. Specifically, it is assumed that the future hydrologic conditions are not known at the time of making the operational decisions. The difference between the optimal cost of the proposed models and the original model defines the value of having the information regarding future hydrologic conditions before taking any decision. This value is generally known as the expected value of perfect information.     

Uncertainty management in multiobjective hydro-thermal self-scheduling under emission considerations

In this paper, a stochastic multiobjective framework is proposed for a day-ahead short-term Hydro Thermal Self-Scheduling (HTSS) problem for joint energy and reserve markets. An efficient linear formulations are introduced in this paper to deal with the nonlinearity of original problem due to the dynamic ramp rate limits, prohibited operating zones, operating services of thermal plants, multi-head power discharge characteristics of hydro generating units and spillage of reservoirs. Besides, system uncertainties including the generating units’ contingencies and price uncertainty are explicitly considered in the stochastic market clearing scheme. For the stochastic modeling of probable multiobjective optimization scenarios, a lattice Monte Carlo simulation has been adopted to have a better coverage of the system uncertainty spectrum. Consequently, the resulting multiobjective optimization scenarios should concurrently optimize competing objective functions including GENeration COmpany's (GENCO's) profit maximization and thermal units’ emission minimization. Accordingly, the ɛ-constraint method is used to solve the multiobjective optimization problem and generate the Pareto set. Then, a fuzzy satisfying method is employed to choose the most preferred solution among all Pareto optimal solutions. The performance of the presented method is verified in different case studies. The results obtained from ɛ-constraint method is compared with those reported by weighted sum method, evolutionary programming-based interactive Fuzzy satisfying method, differential evolution, quantum-behaved particle swarm optimization and hybrid multi-objective cultural algorithm, verifying the superiority of the proposed approach.     

A Chance Constrained Optimal Reserve Scheduling Approach for Economic Dispatch Considering Wind Penetration

The volatile wind power generation brings a full spectrum of problems to power system operation and management, ranging from transient system frequency fluctuation to steady state supply and demand balancing issue. In this paper, a novel wind integrated power system day-ahead economic dispatch model, with the consideration of generation and reserve cost is modelled and investigated. The proposed problem is first formulated as a chance constrained stochastic nonlinear programming (CCSNLP), and then transformed into a deterministic nonlinear programming (NLP). To tackle this NLP problem, a three-stage framework consists of particle swarm optimization (PSO), sequential quadratic programming (SQP) and Monte Carlo simulation (MCS) is proposed. The PSO is employed to heuristically search the line power flow limits, which are used by the SQP as constraints to solve the NLP problem. Then the solution from SQP is verified on benchmark system by using MCS. Finally, the verified results are feedback to the PSO as fitness value to update the particles. Simulation study on IEEE 30-bus system with wind power penetration is carried out, and the results demonstrate that the proposed dispatch model could be effectively solved by the proposed three-stage approach.      

Joint determination of order quantity and reorder point of continuous review model under quantity and freight rate discounts

The increased emphasis on transportation costs has enhanced the need to develop models with transportation consideration explicitly. However, in stochastic inventory models, the transportation cost is considered implicitly as part of fixed ordering cost and thus is assumed to be independent of the size of the shipment. As such, the effect of the transportation and purchasing costs are not adequately reflected in final planning decisions. In this paper, transportation and purchasing considerations are integrated with continuous review inventory model. The objective is to view the system as an integrated whole and determine the lot size and reorder point which minimize the expected total cost per unit time. In addition, procedures are developed to solve the proposed models. Numerical experiments are also performed to explore the effect of key parameters on lot size, reorder point and expected total cost. The new models have a significant impact on lot size, reorder point and expected total cost. Savings up to 17.15% of the expected total cost are realized when using the proposed models.     

Two-stage stochastic programming supply chain model for biodiesel production via wastewater treatment

This paper presents a two-stage stochastic programming model used to design and manage biodiesel supply chains. This is a mixed-integer linear program and an extension of the classical two-stage stochastic location-transportation model. The proposed model optimizes not only costs but also emissions in the supply chain. The model captures the impact of biomass supply and technology uncertainty on supply chain-related decisions; the tradeoffs that exist between location and transportation decisions; and the tradeoffs between costs and emissions in the supply chain. The objective function and model constraints reflect the impact of different carbon regulatory policies, such as carbon cap, carbon tax, carbon cap-and-trade, and carbon offset mechanisms on supply chain decisions. We solve this problem using algorithms that combine Lagrangian relaxation and L-shaped solution methods, and we develop a case study using data from the state of Mississippi. The results from the computational analysis point to important observations about the impacts of carbon regulatory mechanisms as well as the uncertainties on the performance of biocrude supply chains.     

Network Expansion to Mitigate Market Power

Constrained transmission capacity in electricity networks may give generators the possibility to game the market by specifically causing congestion and thereby appropriating excessive rents. Investment in network capacity can ameliorate such behavior by reducing the potential for strategic behavior. However, modeling Nash equilibria between generators, which explicitly account for their impact on the network, is mathematically and computationally challenging. We propose a three-stage model to describe how network investment can reduce market power exertion: a benevolent planner decides on network upgrades for existing lines anticipating the gaming opportunities by strategic generators. These firms, in turn, anticipate their impact on market-clearing prices and grid congestion. In this respect, we provide the first model endogenizing the trade-off between the costs of grid investment and benefits from reduced market power potential in short-run market clearing. In a numerical example using a three-node network, we illustrate three distinct effects: firstly, by reducing market power exertion, network expansion can yield welfare gains beyond pure efficiency increases. Anticipating gaming possibilities when planning network expansion can push welfare close to a first-best competitive benchmark. Secondly, network upgrades entail a relative shift of rents from producers to consumers when congestion rents were excessive. Thirdly, investment may yield suboptimal or even disequilibrium outcomes when strategic behavior of certain market participants is neglected in network planning.     

Minimum energy storage for power system with high wind power penetration using p-efficient point theory

Minimum energy storage （ES） and spinning reserve （SR） for day-ahead power system scheduling with high wind power penetration is significant for system operations. A chance-constrained energy storage optimization model based on unit commitment and considering the stochastic nature of both the wind power and load demand is proposed. To solve this proposed chance-constrained model, it is first converted into a deterministic-constrained model using p-efficient point theory. A single stochastic net load variable is developed to represent the stochastic characteristics of both the wind power and load demand for convenient use with the p-efficient point theory. A probability distribution function for netload forecast error is obtained via the Kernel estimation method. The proposed model is applied to a wind-thermal-storage combined power system. A set of extreme scenarios is chosen to validate the effectiveness of the proposed model and method. The results indicate that the scheduled energy storage can effectively compensate for the net load forecast error, and the increasing wind power penetration does not necessarily require a linear increase in energy storage.     

Short-term hydropower production planning by stochastic programming

Within the framework of multi-stage mixed-integer linear stochastic programming we develop a short-term production plan for a price-taking hydropower plant operating under uncertainty. Current production must comply with the day-ahead commitments of the previous day which makes short-term production planning a matter of spatial distribution among the reservoirs of the plant. Day-ahead market prices and reservoir inflows are, however, uncertain beyond the current operation day and water must be allocated among the reservoirs in order to strike a balance between current profits and expected future profits. A demonstration is presented with data from a Norwegian hydropower producer and the Nordic power market at Nord Pool.     

A stochastic programming model for the thermal optimal day-ahead bid problem with physical futures contracts

The reorganization of the electricity industry in Spain completed a new step with the start-up of the Derivatives Market. One main characteristic of MIBEL's Derivatives Market is the existence of physical futures contracts; they imply the obligation to physically settle the energy. The market regulation establishes the mechanism for including those physical futures in the day-ahead bidding of the generation companies. The goal of this work is to optimize coordination between physical futures contracts and the day-ahead bidding which follow this regulation. We propose a stochastic quadratic mixed-integer programming model which maximizes the expected profits, taking into account futures contracts settlement. The model gives the simultaneous optimization for the Day-Ahead Market bidding strategy and power planning production (unit commitment) for the thermal units of a price-taker generation company. The uncertainty of the Day-Ahead Market price is included in the stochastic model through a set of scenarios. Implementation details and some first computational experiences for small real cases are presented.     

Neurally-inspired stochastic algorithm for determining multi-stage multi-attribute sampling inspection plans

In manufacturing industries, sampling inspection is a common practice for quality assurance and cost reduction. The basic decisions in sampling inspection are how many manufactured items to be sampled from each lot and how many identified defective items in the sample to accept or reject each lot. Because of the combinatorial nature of alternative solutions on the sample sizes and acceptance criteria, the problem of determining an optimal sampling plan is NP-complete. In this paper, a neurally-inspired approach to generating acceptance sampling inspection plans is proposed. A Bayesian cost model of multi-stage-multi-attribute sampling inspections for quality assurance in serial production systems is formulated. This model can accommodate various dispositions of rejected lott such as scraping and screening. The model also can reflect the relationships between stages and among attributes. To determine the sampling plans based on the formulated model, a neurally-inspired stochastic algorithm is developed. This algorithm simulates the state transition of a primal-dual stochastic neural network to generate the sampling plans. The simulated primal network is responsible for generation of new states whereas the dual network is for recording the generated solutions. Starting with an arbitrary feasible solution, this algorithm is able to converge to a near optimal or an optimal sampling plan with a sequence of monotonically improved solutions. The operating characteristics and performance of the algorithm are demonstratedvia numerical examples.     

A Nested Benders Decomposition Approach to Locating Distributed Generation in a Multiarea Power System

A nested Generalized Benders decomposition scheme is used to solve a mixed-integer stochastic programming model. The model evaluates central station and distributed power generation, storage, and demand management assets on a linearized electric power transmission network. It considers temporal and spatial variations in the marginal cost of power, which are captured in the Benders cuts in the solution scheme. These variations are caused not only by differences in generating unit operating expenses and capacity expansion costs, but also by physical transmission constraints that can alter minimum cost dispatch and siting of these units. The transmission constraints addressed include limits on MW power flows and both of Kirchhoff's laws via a linearized DC load flow representation. The model consists of three modules: a stochastic linear production costing model for operating central system generation, a nonlinear program for planning central system generation and transmission, and a mixed-integer program for evaluation of local area distributed resources. Generalized Benders decomposition is applied twice to coordinate these modules. The production costing model is a subproblem to the central system planning model, which is in turn a subproblem to the distributed resource model. The coordination scheme is described in detail, including the calculation of marginal costs. An application shows the effects of marginal cost variations on capacity expansion decisions.     

考虑电能与备用联合出清的日前调度优化方法

为解决电力现货市场与辅助服务市场改革不断深入带来的日前调度计划编制模式转变问题,提出了一种考虑电能与备用辅助服务联合出清的日前调度优化方法。首先,剖析了电能与备用辅助服务之间的耦合关系,从机组运行特性和电网承载能力两个维度出发明确了电能与备用联合出清中需要考虑的运行约束项。接着,以综合购电成本最低为优化目标,全面考虑电网运行、电力平衡、机组运行等三方面约束条件,构建了电能与备用辅助服务联合出清下的日前调度优化模型。并根据模型特点,明确了求解方法。最后基于我国某省区电网实际数据构造的算例表明,该方法能够有效提升发电资源调用效率,避免由于备用均摊等方式造成的发电机组中标量超过系统承载能力或发电机组发电能力的问题。     

Multi-objective optimization of stochastic disassembly line balancing with station paralleling

One of the major activities performed in product recovery is disassembly. Disassembly line is the most suitable setting to disassemble a product. Therefore, designing and balancing efficient disassembly systems are important to optimize the product recovery process. In this study, we deal with multi-objective optimization of a stochastic disassembly line balancing problem (DLBP) with station paralleling and propose a new genetic algorithm (GA) for solving this multi-objective optimization problem. The line balance and design costs objectives are simultaneously optimized by using an AND/OR Graph (AOG) of the product. The proposed GA is designed to generate Pareto-optimal solutions considering two different fitness evaluation approaches, repair algorithms and a diversification strategy. It is tested on 96 test problems that were generated using the benchmark problem generation scheme for problems defined on AOG as developed in literature. In addition, to validate the performance of the algorithm, a goal programming approach and a heuristic approach are presented and their results are compared with those obtained by using GA. Computational results show that GA can be considered as an effective and efficient solution algorithm for solving stochastic DLBP with station paralleling in terms of the solution quality and CPU time.     

A soft computing system for day-ahead electricity price forecasting

Hourly energy prices in a competitive electricity market are volatile. Forecast of energy price is key information to help producers and purchasers involved in electricity market to prepare their corresponding bidding strategies so as to maximize their profits. It is difficult to forecast all the hourly prices with only one model for different behaviors of different hourly prices. Neither will it get excellent results with 24 different models to forecast the 24 hourly prices respectively, for there are always not sufficient data to train the models, especially the peak price in summer. This paper proposes a novel technique to forecast day-ahead electricity prices based on Self-Organizing Map neural network (SOM) and Support Vector Machine (SVM) models. SOM is used to cluster the data automatically according to their similarity to resolve the problem of insufficient training data. SVM models for regression are built on the categories clustered by SOM separately. Parameters of the SVM models are chosen by Particle Swarm Optimization (PSO) algorithm automatically to avoid the arbitrary parameters decision of the tester, improving the forecasting accuracy. The comparison suggests that SOM–SVM–PSO has considerable value in forecasting day-ahead price in Pennsylvania–New Jersey–Maryland (PJM) market, especially for summer peak prices.     

Iterative analysis of Markov regenerative models

Conventional algorithms for the steady-state analysis of Markov regenerative models suffer from high computational costs which are caused by densely populated matrices. In this paper, a new algorithm is suggested which avoids computing these matrices explicitly. Instead, a two-stage iteration scheme is used. An extended version of uniformization is applied as a subalgorithm to compute the required transient quantities “on-the fly”. The algorithm is formulated in terms of stochastic Petri nets. A detailed example illustrates the proposed concepts.     

基于多目标两阶段随机规划方法的电热联合系统调度

针对电热综合能源系统由于风电出力的随机性和波动性而难以有效调度的问题,提出了以成本最小化和弃风最小化为目标的一种多目标两阶段随机规划方法(multi-objective and two-stage stochastic programming,MOTSP),其中采用两阶段的随机规划模型对成本最小化部分进行建模分析,第一阶段以火电机组的启停成本为调度目标,第二阶段以机组运行成本为调度目标。最后采用多目标算法NSGA-Ⅱ中对解的筛选机制求解随机规划问题。该方法利用高斯分布描述负荷和风力发电预测误差来解决风电出力的不确定性,采用蒙特卡罗方法生成随机场景,并采用反向缩减技术对场景进行削减。仿真结果表明,所提的MOTSP算法比其他多种智能算法的解集更均匀广泛,收敛性更好,能够最大限度地减少弃风并使机组运营成本最小。     

Decision making in dynamic stochastic Cournot games

In this paper, the Cournot competition is modeled as a stochastic dynamic game. In the proposed model, a stochastic market price function and stochastic dynamic decision functions of the rivals are considered. Since the optimal decision of a player needs the estimation of the unknown parameters of the market and rivals’ decisions, a combined estimation-optimization algorithm for decision making is proposed. The history of the rivals’ output quantities (supplies) and the market clearing price (MCP) are the only available information to the players. The convergence of the algorithm (for both estimation and decision making processes) is discussed. In addition, the stability conditions of the equilibrium points are analyzed using the converse Lyapunov theorem. Through the case studies, which are performed based on the California Independent System Operator (CA-ISO) historical public data, the theoretical results and the applicability of the proposed method are verified. Moreover, a comparative study among the agents using the proposed method, naïve expectation and adaptive expectation in the market is performed to show the effectiveness and applicability of the proposed method.     

基于ARMA的微惯性传感器随机误差建模方法

针对微惯性传感器随机误差建模效果不理想,影响微惯性组合导航系统性能的问题,提出了采用自回归滑动平均(ARMA)对微惯性传感器随机误差进行建模的方法。通过对随机误差模型应用于微惯性器件误差建模的深入分析,将Yule-Walker方程引入线性预测问题中,实现AR功率谱密度的估计,建立了基于随机过程有理功率谱密度的ARMA模型建立方法,并给出了ARMA建模准确性的LDA验证准则。通过微惯性传感器实测数据,对随机误差建模方法进行了有效性验证。该方法为微惯性器件的随机误差建模和分析提供了一种新的途径。     

Asset pricing with dynamic programming

The study of asset price characteristics of stochastic growth models such as the risk-free interest rate, equity premium, and the Sharpe-ratio has been limited by the lack of global and accurate methods to solve dynamic optimization models. In this paper, a stochastic version of a dynamic programming method with adaptive grid scheme is applied to compute the asset price characteristics of a stochastic growth model. The stochastic growth model is of the type as developed by [Brock and Mirman (1972), Journal of Economic Theory, 4, 479–513 and Brock (1979), Part I: The growth model (pp. 165–190). New York: Academic Press; The economies of information and uncertainty (pp. 165–192). Chicago: University of Chicago Press. (1982). It has become the baseline model in the stochastic dynamic general equilibrium literature. In a first step, in order to test our procedure, it is applied to this basic stochastic growth model for which the optimal consumption and asset prices can analytically be computed. Since, as shown, our method produces only negligible errors, as compared to the analytical solution, in a second step, we apply it to more elaborate stochastic growth models with adjustment costs and habit formation. In the latter model preferences are not time separable and past consumption acts as a constraint on current consumption. This model gives rise to an additional state variable. We here too apply our stochastic version of a dynamic programming method with adaptive grid scheme to compute the above mentioned asset price characteristics. We show that our method is very suitable to be used as solution technique for such models with more complicated decision structure.      

Intertemporal asset allocation when the underlying factors are unobservable

The aim of this paper is to develop an optimal long-term bond investment strategy which can be applied to real market situations. This paper employs Merton’s intertemporal framework to accommodate the features of a stochastic interest rate and the time-varying dynamics of bond returns. The long-term investors encounter a partial information problem where they can only observe the market bond prices but not the driving factors of the variability of the interest rate and the bond return dynamics. With the assumption of Gaussian factor dynamics, we are able to develop an analytical solution for the optimal long-term investment strategies under the case of full information. To apply the best theoretical investment strategy to the real market we need to be aware of the existence of measurement errors representing the gap between theoretical and empirical models. We estimate the model based on data for the German securities market and then the estimation results are employed to develop long-term bond investment strategies. Because of the presence of measurement errors, we provide a simulation study to examine the performance of the best theoretical investment strategy. We find that the measurement errors have a great impact on the optimality of the investment strategies and that under certain circumstance the best theoretical investment strategies may not perform so well in a real market situation. In the simulation study, we also investigate the role of information about the variability of the stochastic interest rate and the bond return dynamics. Our results show that this information can indeed be used to advantage in making sensible long-term investment decisions.