Day-ahead market bidding for a Nordic hydropower producer: taking the Elbas market into account

In many power markets around the world the energy generation decisions result from two-sided auctions in which producing and consuming agents submit their price-quantity bids. The determination of optimal bids in power markets is a complicated task that has to be undertaken every day. In the present work, we propose an optimization model for a price-taker hydropower producer in Nord Pool that takes into account the uncertainty in market prices and both production and physical trading aspects. The day-ahead bidding takes place a day before the actual operation and energy delivery. After this round of bidding, but before actual operation, some adjustments in the dispatched power (accepted bids) have to be done, due to uncertainty in prices, inflow and load. Such adjustments can be done in the Elbas market, which allows for trading physical electricity up to one hour before the operation hour. This paper uses stochastic programming to determine the optimal bidding strategy and the impact of the possibility to participate in the Elbas. ARMAX and GARCH techniques are used to generate realistic market price scenarios taking into account both day-ahead price and Elbas price uncertainty. The results show that considering Elbas when bidding in the day-ahead market does not significantly impact neither the profit nor the recommended bids of a typical hydro producer.     

A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants

We propose a stochastic model for the daily operation scheduling of a generation system including pumped storage hydro plants and wind power plants, where the uncertainty is represented by the hourly wind power production. In order to assess the value of the stochastic modeling, we discuss two case studies: in the former the scenario tree is built so as to include both low and high wind power production scenarios, in the latter the scenario tree is built on historical wind speed data covering a time span of one and a half year. The Value of the Stochastic Solution, computed by a modified new procedure, shows that in scenarios with low wind power production the stochastic solution allows the producer to obtain a profit which is greater than the one associated to the deterministic solution. In-sample stability of the optimal function values for increasing number of scenarios is reported.     

The natural hedge of a gas-fired power plant

Electricity industries worldwide have been restructured in order to introduce competition. As a result, decision makers are exposed to volatile electricity prices, which are positively correlated with those of natural gas in markets with price-setting gas-fired power plants. Consequently, gas-fired plants are said to enjoy a “natural hedge.” We explore the properties of such a built-in hedge for a gas-fired power plant via a stochastic programming approach, which enables characterisation of uncertainty in both electricity and gas prices in deriving optimal hedging and generation decisions. The producer engages in financial hedging by signing forward contracts at the beginning of the month while anticipating uncertainty in spot prices. Using UK energy price data from 2006 to 2011 and daily aggregated dispatch decisions of a typical gas-fired power plant, we find that such a producer does, in fact, enjoy a natural hedge, i.e., it is better off facing uncertain spot prices rather than locking in its generation cost. However, the natural hedge is not a perfect hedge, i.e., even modest risk aversion makes it optimal to use gas forwards partially. Furthermore, greater operational flexibility enhances this natural hedge as generation decisions provide a countervailing response to uncertainty. Conversely, higher energy-conversion efficiency reduces the natural hedge by decreasing the importance of natural gas price volatility and, thus, its correlation with the electricity price.     

TIMBER SUPPLY UNDER DEMAND UNCERTAINTY: WELFARE GAINS FROM PERFECT COMPETITION WITH RATIONAL EXPECTATIONS

ABSTRACT. This paper presents a noneconometric approach to estimating the short‐run timber supply function based on optimal harvest decisions. Determination of optimal harvest levels and estimation of supply function coefficients are integrated into one step by incorporating a parametric short‐run timber supply function into the harvest decision model. In this manner we convert the original harvest decision model into a new optimization problem with the supply function coefficients functioning as “decision variables.” Optimal solution to the new decision model gives the coefficients of the short‐run supply function and, indirectly, the optimal harvest levels. This approach enables us to develop stochastic models of the timber market that are particularly useful for forest sector analysis involving comparison of alternative institutional regimes or policy proposals and when the timber market is affected by stochastic variables. For demonstration purposes, we apply this method to compare the performances of two timber market regimes (perfect competition and monopoly) under demand uncertainty, using the Swedish data. The results show that the expected timber price is 22 percent lower and the expected annual timber supply is 43 percent higher in the competitive market than in the monopoly market. This confirms the theoretical result that monopoly reduces supply and increases price. The expected social welfare gain from perfect competition over monopoly is about 24 percent.     

A stochastic aggregate production planning model in a green supply chain: Considering flexible lead times,nonlinear purchase and shortage cost functions

In this paper we develop a stochastic programming approach to solve a multi-period multi-product multi-site aggregate production planning problem in a green supply chain for a medium-term planning horizon under the assumption of demand uncertainty. The proposed model has the following features: (i) the majority of supply chain cost parameters are considered; (ii) quantity discounts to encourage the producer to order more from the suppliers in one period, instead of splitting the order into periodical small quantities, are considered; (iii) the interrelationship between lead time and transportation cost is considered, as well as that between lead time and greenhouse gas emission level; (iv) demand uncertainty is assumed to follow a pre-specified distribution function; (v) shortages are penalized by a general multiple breakpoint function, to persuade producers to reduce backorders as much as possible; (vi) some indicators of a green supply chain, such as greenhouse gas emissions and waste management are also incorporated into the model. The proposed model is first a nonlinear mixed integer programming which is converted into a linear one by applying some theoretical and numerical techniques. Due to the convexity of the model, the local solution obtained from linear programming solvers is also the global solution. Finally, a numerical example is presented to demonstrate the validity of the proposed model.     

Bidding in sequential electricity markets: The Nordic case

For electricity market participants trading in sequential markets with differences in price levels and risk exposure, it is relevant to analyze the potential of coordinated bidding. We consider a Nordic power producer who engages in the day-ahead spot market and the hour-ahead balancing market. In both markets, clearing prices and dispatched volumes are unknown at the time of bidding. However, in the balancing market, the market participant faces an additional risk of not being dispatched. Taking into account the sequential clearing of these markets and the gradual realization of market prices, we formulate the bidding problem as a multi-stage stochastic program. We investigate whether higher risk exposure may cause hesitation to bid into the balancing market. Furthermore, we quantify the gain from coordinated bidding, and by deriving bounds on this gain, assess the performance of alternative bidding strategies used in practice.     

CO2 emissions trading planning in combined heat and power production via multi-period stochastic optimization

The EU emissions trading scheme (ETS) taking effect in 2005 covers CO₂ emissions from specific large-scale industrial activities and combustion installations. A large number of existing and potential future combined heat and power (CHP) installations are subject to ETS and targeted for emissions reduction. CHP production is an important technology for efficient and clean provision of energy because of its superior carbon efficiency. The proper planning of emissions trading can help its potential into full play, making it become a true “winning technology” under ETS. Fuel mix or fuel switch will be the reasonable choices for fossil fuel based CHP producers to achieve their emissions targets at the lowest possible cost. In this paper we formulate CO₂ emissions trading planning of a CHP producer as a multi-period stochastic optimization problem and propose a stochastic simulation and coordination approach for considering the risk attitude of the producer, penalty for excessive emissions, and the confidence interval for emission estimates. In test runs with a realistic CHP production model, the proposed solution approach demonstrates good trading efficiency in terms of profit-to-turnover ratio. Considering the confidence interval for emission estimates can help the producer to reduce the transaction costs in emissions trading. Comparisons between fuel switch and fuel mix strategies show that fuel mix can provide good tradeoff between profit-making and emissions reduction.     

Integrating stochastic programming and decision tree techniques in land conversion problems

This paper is concerned with gradual land conversion problems, placing the main focus on the interaction between time and uncertainty. This aspect is extremely relevant since most decisions made in the field of natural resources and sustainable development are irreversible decisions. In particular, we discuss and develop a scenario-based multi-stage stochastic programming model in order to determine the optimal land portfolio in time, given uncertainty affecting the market. The approach is then integrated in a decision tree framework in order to account for domain specific (environmental) uncertainty that, diversely from market uncertainty, may depend on the decision taken. Although, the designed methodology has many general applications, in the present work we focus on a particular case study, concerning a semi-degraded natural park located in northern Italy.     

Supply chain optimization of petroleum organization under uncertainty in market demands and prices

A multi-period stochastic planning model has been developed and implemented for a supply chain network of a petroleum organization operating in an oil producing country under uncertain market conditions. The proposed supply chain network consists of all activities related to crude oil production, processing and distribution. Uncertainties were introduced in market demands and prices. A deterministic optimization model was first developed and tested. The impact of uncertainty on the supply chain was studied by performing a sensitivity analysis in which ±20% deviations were introduced in market demands and prices of different commodities. A stochastic formulation was then proposed, which is based on the two-stage problem with finite number of realizations. The proposed stochastic programming approach proved to be quite effective in developing resilient production plans in light of high degree of uncertainty in market conditions. The anticipated production plans have a considerably lower expected value of perfect information (EVPI). The main conclusion of this study is that for an oil producing country with oil processing capabilities, the impact of economic uncertainties may be tolerated by an appropriate balance between crude exports and processing capacities.     

考虑模型不确定的TBP养老金的鲁棒最优投资策略研究

以目标收益养老金计划(TBP)模型研究鲁棒最优投资问题, 其中养老金管理者对模型参数不确定带来的风险是模糊风险厌恶的. 养老金管理者为规避风险和增加收益将投资于无风险资产和风险资产. 考虑连续时间情形, 假设养老金计划参保人的缴费是确定的, 而参保人的收益给付是确定目标收益给付, 资金账户的收益风险由不同代际的参保人共同承担, 同时考虑随机工资及其与金融市场的相关性. 以参保人退休后养老金给付偏离目标的风险和代际之间风险分担的组合最小化为投资决策目标, 并采用指数函数的形式描述实际给付与目标给付的偏离, 利用随机最优控制方法, 建立相应的HJB方程并求解得到最优投资收益策略和最优给付策略的解析解. 通过数值示例分析了模型参数对最优投资和最优给付策略的影响.     

Optimal investment and reinsurance of an insurer with model uncertainty

We introduce a novel approach to optimal investment–reinsurance problems of an insurance company facing model uncertainty via a game theoretic approach. The insurance company invests in a capital market index whose dynamics follow a geometric Brownian motion. The risk process of the company is governed by either a compound Poisson process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal investment–reinsurance problems with model uncertainty are formulated as two-player, zero-sum, stochastic differential games between the insurance company and the market. We provide verification theorems for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) solutions to the optimal investment–reinsurance problems and derive closed-form solutions to the problems.     

Robust Optimal Portfolio Choice Under Markovian Regime-switching Model

We investigate an optimal portfolio selection problem in a continuous-time Markov-modulated financial market when an economic agent faces model uncertainty and seeks a robust optimal portfolio strategy. The key market parameters are assumed to be modulated by a continuous-time, finite-state Markov chain whose states are interpreted as different states of an economy. The goal of the agent is to maximize the minimal expected utility of terminal wealth over a family of probability measures in a finite time horizon. The problem is then formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game between the agent and the market. We solve the problem by the Hamilton-Jacobi-Bellman approach.      

Integrated risk management for an electricity producer

Electricity suppliers face two sources of risk: uncertainty of spot prices and uncertainty of production costs. Uncertainty in selling the output is usually “solved” by signing forwards and the two sources of risk are dealt with separately. However, managing the integrated risk optimally is the direction we will suggest. We intend to analyse the problem a power producer is confronted to upon acting in a market where spot and forward agreements are available. Since forwards allow to sell production in advance at a given price but do not hedge against cost volatility, the total risk can be reduced by selling also in the spot market. The analysis is further detailed to encompass the spread option inherent in electricity production. We find a benchmark value for forwards to sign and the optimal spot/forward combinations. The analysis is carried out by accounting for market figures for input and output variables in the German market EEX.     

The path player game

We give an explicit PDE characterization for the solution of the problemof maximizing the utility of both terminal wealth and intertemporal consumption undermodel uncertainty. The underlying market model consists of a risky asset, whosevolatility and long-term trend are driven by an external stochastic factor process. Therobust utility functional is defined in terms of a HARA utility function with risk aversionparameter 0 < α < 1 and a dynamically consistent coherent risk measure, whichallows for model uncertainty in the distributions of both the asset price dynamics andthe factor process. Ourmethod combines recent results by Wittmüß (Robust optimizationof consumption with random endowment, 2006) on the duality theory of robustoptimization of consumption with a stochastic control approach to the dual problemof determining a ‘worst-case martingale measure’.     

Implementing a parametric maximum flow algorithm for optimal open pit mine design under uncertain supply and demand

Conventional open pit mine optimization models for designing mining phases and ultimate pit limit do not consider expected variations and uncertainty in metal content available in a mineral deposit (supply) and commodity prices (market demand). Unlike the conventional approach, a stochastic framework relies on multiple realizations of the input data so as to account for uncertainty in metal content and financial parameters, reflecting potential supply and demand. This paper presents a new method that jointly considers uncertainty in metal content and commodity prices, and incorporates time-dependent discounted values of mining blocks when designing optimal production phases and ultimate pit limit, while honouring production capacity constraints. The structure of a graph representing the stochastic framework is proposed, and it is solved with a parametric maximum flow algorithm. Lagragnian relaxation and the subgradient method are integrated in the proposed approach to facilitate producing practical designs. An application at a copper deposit in Canada demonstrates the practical aspects of the approach and quality of solutions over conventional methods, as well as the effectiveness of the proposed stochastic approach in solving mine planning and design problems.     

Incorporating model uncertainty into optimal insurance contract design

In stochastic optimization models, the optimal solution heavily depends on the selected probability model for the scenarios. However, the scenario models are typically chosen on the basis of statistical estimates and are therefore subject to model error. We demonstrate here how the model uncertainty can be incorporated into the decision making process. We use a nonparametric approach for quantifying the model uncertainty and a minimax setup to find model-robust solutions. The method is illustrated by a risk management problem involving the optimal design of an insurance contract.     

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??In this paper, we investigate a robust optimal portfolio and reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries about uncertainty in model parameters. We assume that the AAI is allowed to purchase proportional reinsurance and invest his/her wealth in a financial market which consists of a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal expected power utility of terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategies are obtained.     

随机波动率市场存在股票误价时的最优投资策略

本文研究了随机波动率市场中存在股票误价(mispricing)时的最优投资组合选择问题.假设投资者的目标是最大化终端财富的期望幂效用;其可投资于无风险资产、市场指数和两支相同权益或近似度极高的股票,其中至少有一支股票存在误价;市场收益的波动率和股票系统风险由Heston随机波动率模型刻画.运用动态规划方法和Lagrange乘子法,分别得到不存在/存在有限卖空约束时,投资者的最优投资策略及最优值函数的解析式,并通过理论分析和数值算例,阐述了投资时间水平和价格随机误差对最优投资策略的影响.     

Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility.     

Bidding under price uncertainty in multi-unit pay-as-bid procurement auctions for power systems reserve

In this paper a methodology for profit maximized bidding under price uncertainty in a day-ahead, multi-unit and pay-as-bid procurement auction for power systems reserve is proposed. Within this novel methodology a bidder is considered to follow a Bayes-strategy. Thereby, one bidder is assumed to behave strategically and the behavior of the remaining is summarized in a probability distribution of the market price and a reaction function to price dumping by the strategic bidder. With this approach two problems arise: First, as a pay-as-bid auction is considered, no uniform price and therefore no single probability distribution of the market price is readily available. Second, if historic bidding data of all participants are used to estimate such a distribution and market power is a relevant factor, the bid of the strategically behaving bidder is likely to influence the distribution. Within this paper for both of the problems solutions are presented. It is shown that by estimating a probability of acceptance the optimal bidding price with respect to a given capacity can be calculated by maximizing a stochastic non-linear objective function of expected profit. Taking the characteristics of recently established markets in Germany into account, the methodology is applied using exemplary data. It is shown that the methodology helps to manage existing price uncertainties and hence supports the trading decisions of a bidder. It is inferred that the developed methodology may also be used for bidding on other auction markets with a similar market design.