Weekly self-scheduling,forward contracting,and pool involvement for an electricity producer. An adaptive robust optimization approach

This paper addresses the optimization under uncertainty of the self-scheduling, forward contracting, and pool involvement of an electricity producer operating a mixed power generation station, which combines thermal, hydro and wind sources, and uses a two stage adaptive robust optimization approach. In this problem the wind power production and the electricity pool price are considered to be uncertain, and are described by uncertainty convex sets. To solve this problem, two variants of a constraint generation algorithm are proposed, and their application and characteristics discussed. Both algorithms are used to solve two case studies based on two producers, each operating equivalent generation units, differing only in the thermal units’ characteristics. Their market strategies are investigated for three different scenarios, corresponding to as many instances of electricity price forecasts. The effect of the producers’ approach, whether conservative or more risk prone, is also investigated by solving each instance for multiple values of the so-called budget parameter. It was possible to conclude that this parameter influences markedly the producers’ strategy, in terms of scheduling, profit, forward contracting, and pool involvement. These findings are presented and analyzed in detail, and an attempted rationale is proposed to explain the less intuitive outcomes. Regarding the computational results, these show that for some instances, the two variants of the algorithms have a similar performance, while for a particular subset of them one variant has a clear superiority.     

Robust Capacity Assignment in Telecommunications

In telecommunications, the demand is a key data that drives network planning. The demand exhibits considerable variability, due to customers movement and introduction of new services and products in the present competitive markets. To deal with this uncertainty, we consider capacity assignment problem in telecommunications in the framework of robust optimization proposed in Ben-Tal and Nemcrovski (Math Oper Res 23(4):769–805, 1998, MPS-SIAM series on optimization, 2001) and Kouvelis and Yu. We propose a decomposition scheme based on cutting plane methods. Some preliminary computational experiments indicate that the Elzinga–Moore cutting plane method (Elzinga and Moore in Math Program 8:134–145, 1975) can be a valuable choice. Since in some situations different possible uncertainty sets may exist, we propose a generalization of these models to cope at a time with a finite number of plausible uncertainty sets. A weight is associated with each uncertainty set to determine its relative importance or worth against another.     

Semi-Lagrangean approach for price discovery in markets with non-convexities

From standard economic theory, the market clearing price for a commodity is set where the demand and supply curves intersect. Convexity is a property that economic models require for a competitive equilibrium, which is efficient and well-behaved and provides equilibrium prices. However, some markets present non-convexities due to their cost structure or due to some operational constraints that need to be addressed. This is the case for electricity markets where the electricity producers incur costs for shutting down a generating unit and then bringing it back on. Non-convex cost structures can be a challenge for the price discovery process, since the supply and demand curves may not intersect, or if they intersect, the price found may not be high enough to cover the total cost of production. We apply a Semi-Lagrangean approach to find a price that can be applied in the electricity pool markets where a central system operator decides who produces and how much they should produce. By applying the model to an example from the literature, we found prices that are high enough to cover the producer’s total costs, and follows the optimal solution for achieving mining cost in production. The prices are an alternative solution to the price discovery problem in non-convexities economies; in addition, they provide nonnegative profits to all the generators without the use of side-payments or up-lifts, and closes the integrality gap.     

On stochastic auctions in risk-averse electricity markets with uncertain supply

This paper studies risk in a stochastic auction which facilitates the integration of renewable generation in electricity markets. We model market participants who are risk averse and reflect their risk aversion through coherent risk measures. We uncover a closed form characterization of a risk-averse generator’s optimal pre-commitment behaviour for a given real-time policy, both with and without risk trading.     

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.     

Designing robust liner shipping schedules: Optimizing recovery actions and buffer times

It is important for liner shipping companies to maintain cost efficient and robust liner shipping networks. Regularly, they set up pro-forma schedules, yet it is difficult to stay on time. We consider the problem of managing the delays. Therefore, we need to determine an optimal recovery policy and buffer time allocation to the ship route in order to minimize the total costs associated with delays and recovery actions, such as increasing sailing speed. We introduce a general framework consisting of a mixed integer programming formulation to solve discrete stochastic decision problems with short and long term decisions and apply this framework to the above described problem. Furthermore, we propose and test four heuristics for this problem. We compared the results of our method with an existing liner shipping route schedule and found a cost decrease of 28.9% after optimizing the buffer time distribution compared to the cost of sailing the current route schedule at constant speed.     

A stochastic multiscale model for electricity generation capacity expansion

Long-term planning for electric power systems, or capacity expansion, has traditionally been modeled using simplified models or heuristics to approximate the short-term dynamics. However, current trends such as increasing penetration of intermittent renewable generation and increased demand response requires a coupling of both the long and short term dynamics. We present an efficient method for coupling multiple temporal scales using the framework of singular perturbation theory for the control of Markov processes in continuous time. We show that the uncertainties that exist in many energy planning problems, in particular load demand uncertainty and uncertainties in generation availability, can be captured with a multiscale model. We then use a dimensionality reduction technique, which is valid if the scale separation present in the model is large enough, to derive a computationally tractable model. We show that both wind data and electricity demand data do exhibit sufficient scale separation. A numerical example using real data and a finite difference approximation of the Hamilton–Jacobi–Bellman equation is used to illustrate the proposed method. We compare the results of our approximate model with those of the exact model. We also show that the proposed approximation outperforms a commonly used heuristic used in capacity expansion models.     

Block order restrictions in combinatorial electric energy auctions

In Europe, the auctions organized by “power exchanges” one day ahead of delivery are multi-unit, double-sided, uniformly priced combinatorial auctions. Generators, retailers, large consumers and traders participate at the demand as well as at the supply side, depending or whether they are short or long in electric energy. Because generators face nonconvex costs, in particular startup costs and minimum run levels, the exchanges allow “block orders” that are all-or-nothing orders of a given amount of electric energy in multiple consecutive hours, while the standard order consists of an amount for a single hour that can be curtailed. All exchanges restrict the size (MWh/h), the type (span in terms of hours) or the number (per participant per day) of blocks that can be introduced. This paper discusses the rationale of block order restrictions. Based on simulations with representative scenarios, it is argued that the restrictions could be relaxed, which some exchanges have already started doing.     

Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules

The deregulation of electricity markets increases the financial risk faced by retailers who procure electric energy on the spot market to meet their customers’ electricity demand. To hedge against this exposure, retailers often hold a portfolio of electricity derivative contracts. In this paper, we propose a multistage stochastic mean-variance optimisation model for the management of such a portfolio. To reduce computational complexity, we apply two approximations: we aggregate the decision stages and solve the resulting problem in linear decision rules (LDR). The LDR approach consists of restricting the set of recourse decisions to those affine in the history of the random parameters. When applied to mean-variance optimisation models, it leads to convex quadratic programs. Since their size grows typically only polynomially with the number of periods, they can be efficiently solved. Our numerical experiments illustrate the value of adaptivity inherent in the LDR method and its potential for enabling scalability to problems with many periods.     

基于SLQ控制器的时滞微分系统的鲁棒镇定

该文基于随机线性二次控制问题, 讨论了多时滞、且具有马尔可夫跳变参数的微分系统的最优控制的鲁棒性及可镇定问题.应用了Lyapunov-Krasovskii型的泛函、伊藤(Ito)公式、及Schur补等工具, 分析了该随机多时滞、具有马尔可夫过程的微分系统的均方指数稳定性.得到了时滞相关与时滞无关的充分性的代数判据.     

Robust placement of sensors in dynamic water distribution systems

Designing a robust sensor network to detect accidental contaminants in water distribution systems is a challenge given the uncertain nature of the contamination events (what, how much, when, where and for how long) and the dynamic nature of water distribution systems (driven by the random consumption of consumers). We formulate a set of scenario-based minimax and minimax regret models in order to provide robust sensor-placement schemes that perform well under all realizable contamination scenarios, and thus protect water consumers. Single-and multi-objective versions of these models are then applied to a real water distribution system. A heuristic solution method is applied to solve the robust models. The concept of “sensitivity region” is used to visualize trade-offs between multiple objectives.     

Adaptive and robust radiation therapy optimization for lung cancer

A previous approach to robust intensity-modulated radiation therapy (IMRT) treatment planning for moving tumors in the lung involves solving a single planning problem before the start of treatment and using the resulting solution in all of the subsequent treatment sessions. In this paper, we develop an adaptive robust optimization approach to IMRT treatment planning for lung cancer, where information gathered in prior treatment sessions is used to update the uncertainty set and guide the reoptimization of the treatment for the next session. Such an approach allows for the estimate of the uncertain effect to improve as the treatment goes on and represents a generalization of existing robust optimization and adaptive radiation therapy methodologies. Our method is computationally tractable, as it involves solving a sequence of linear optimization problems. We present computational results for a lung cancer patient case and show that using our adaptive robust method, it is possible to attain an improvement over the traditional robust approach in both tumor coverage and organ sparing simultaneously. We also prove that under certain conditions our adaptive robust method is asymptotically optimal, which provides insight into the performance observed in our computational study. The essence of our method – solving a sequence of single-stage robust optimization problems, with the uncertainty set updated each time – can potentially be applied to other problems that involve multi-stage decisions to be made under uncertainty.     

Heuristic algorithms for siting alternative-fuel stations using the Flow-Refueling Location Model

This paper presents three heuristic algorithms that solve for the optimal locations for refueling stations for alternative-fuels, such as hydrogen, ethanol, biodiesel, natural gas, or electricity. The Flow-Refueling Location Model (FRLM) locates refueling stations to maximize the flow that can be refueled with a given number of facilities. The FRLM uses path-based demands, and because of the limitations imposed by the driving range of vehicles, longer paths require combinations of more than one station to refuel round-trip travel. A mixed-integer linear programming (MILP) version of the model has been formulated and published and could be used to obtain an optimal solution. However, because of the need for combinations of stations to satisfy demands, a realistic problem with a moderate size network and a reasonable number of candidate sites would be impractical to generate and solve with MILP methods. In this research, heuristic algorithms—specifically the greedy-adding, greedy-adding with substitution and genetic algorithm—are developed and applied to solve the FRLM problem. These algorithms are shown to be effective and efficient in solving complex FRLM problems. For case study purposes, the heuristic algorithms are applied to locate hydrogen-refueling stations in the state of Florida.     

Role of specific energy in decomposition of time-invariant least-cost reservoir filling problem

A mathematical analysis highlighting the decomposition structure of the least-cost reservoir filling problem under time–invariant conditions is provided. It is shown, without loss of generality, that time invariance and unidimensionality of the state variable (for describing the evolution of the hydrodynamic system) are sufficient in order to achieve full (spatial and temporal) decomposition. Using this result, the role of specific energy in finding least–cost operational schedules for reservoir filling in a general “physically meaningful” hydrodynamic system is discussed.     

Robust Solution of Nonconvex Global Optimization Problems

The concept of ɛ-approximate optimal solution as widely used in nonconvex global optimization is not quite adequate, because such a point may correspond to an objective function value far from the true optimal value, while being infeasible. We introduce a concept of essential ɛ-optimal solution, which gives a more appropriate approximate optimal solution, while being stable under small perturbations of the constraints. A general method for finding an essential ɛ-optimal solution in finitely many steps is proposed which can be applied to d.c. programming and monotonic optimization.     

Models and algorithms to improve earthwork operations in road design using mixed integer linear programming

In road construction, earthwork operations account for about 25% of the construction costs. Existing linear programming models for earthwork optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. As such, current models may result in inaccurate estimates of optimal earthwork costs, leading to poor choices in road design. In this research, we extend the classical linear program model of earthwork operations to a mixed integer linear program model that accounts for blocks. We examine the economic impact of incorporating blocks via mixed integer linear programming, and find significant savings for most road designs in our test-set. However, the resulting model is considerably harder to solve than the original linear program. Based on structural observations, we introduce a set of algorithms that theoretically reduce the solving time of the model. We confirm this reduction in solve time with numerical experiments.     

Robust optimal dynamic production/pricing policies in a closed-loop system

Hybrid manufacturing/remanufacturing systems play a key role in implementing closed-loop production systems which have been considered due to increasingly environmental concerns and latent profit of used products. Manufacturing and remanufacturing rates, selling price of new products, and acquisition price of used products are the most critical variables to optimize in such hybrid systems. In this paper, we develop a dynamic production/pricing problem, in which decisions should be made in each period confronting with uncertain demand and return. The manufacturer is able to control the demand and return by adjusting selling price and acquisition price respectively, also she can stock inventories of used and new products to deal with uncertainties. Modeling a nominal profit maximization problem, we go through robust optimization approach to reformulate it for the uncertain case. Final robust optimization model is obtained as a quadratic programming model over discrete periods which can be solved by optimization packages of QP. A numerical example is defined and sensitivity analysis is performed on both basic parameters and parameters associated with uncertainty to create managerial views.     

Robust stability in distribution of Boolean networks under multi-bits stochastic function perturbations

In this work, robust stability in distribution of Boolean networks (BNs) is studied under multi-bits probabilistic and markovian function perturbations. Firstly, definition of multi-bits stochastic function perturbations is given and an identification matrix is introduced to present each case. Then, by viewing each case as a switching subsystem, BNs under multi-bits stochastic function perturbations can be equivalently converted into stochastic switching systems. After constructing respective transition probability matrices which can unify multi-bits probabilistic and markovian function perturbations in a consolidated framework, robust stability in distribution can be handled. On such basis, necessary and sufficient conditions for robust stability in distribution of BNs under stochastic function perturbations are given respectively. Finally, two numerical examples are presented to verify the validity of our theoretical results.     

Multi-period traffic routing in satellite networks

We introduce a traffic routing problem over an extended planning horizon that appears in geosynchronous satellite networks. Unlike terrestrial (e.g., fiber optic) networks, routing on a satellite network is not transparent to the customers. As a result, a route change is associated with significant monetary penalties that are usually in the form of discounts (up to 40%) offered by the satellite provider to the customer that is affected. The notion of these rerouting penalties requires the network planners to explicitly consider these penalties in their routing decisions over multiple time periods and introduces novel challenges that have not been considered previously in the literature. We develop a branch-and-price-and-cut procedure to solve this problem and describe an algorithm for the associated pricing problem. Our computational work demonstrates that the use of a multi-period optimization procedure as opposed to a myopic period-by-period approach can result in cost reductions up to 13% depending on problem characteristics and network size considered. These cost reductions correspond to potential savings of several hundred million dollars for large satellite providers.     

Measuring energy efficiency in the context of an emerging economy: The case of indian manufacturing

In recent years the Indian economy has achieved a rapid growth rate. However, one of the challenges facing the economy is the large energy supply that would be needed to sustain this growth path. Hence attaining energy efficiency is crucial for the economy. This paper examines the Indian manufacturing sector and utilizes a variant of the directional distance function to obtain comprehensive measures of performance of the major manufacturing states, given the joint goals of achieving energy conservation and output growth. The results of the empirical application reveal that given the prevailing technology, the average manufacturing firm across the states could reduce energy input and increase output simultaneously by an annual average of 3.84%, by improving technical efficiency. To accommodate a larger simultaneous increase in output and reduction in energy use, India should adopt superior technologies so as to shift the efficient frontier outward.