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This paper provides a tutorial overview of robust optimization in power systems, including robust optimization and adaptive robust optimization. We also introduce distributionally robust optimization. For illustration purposes, we describe and analyze a short-term operation problem and a long-term planning one. The operation problem allows identifying the transmission line whose failure has the higher impact on the operation of the system (worst contingency). From a planning perspective, we describe and analyze the problem of identifying which are the most critical transmission lines (vulnerabilities) to be protected against intentional attacks or natural disasters. We provide as well a distributionally robust version of this problem. The operation problem is a robust optimization one, while the planning problem is an adaptive robust optimization one, including a distributionally robust variant. [ABSTRACT FROM AUTHOR]

This paper presents an optimization model to help distribution system planners to site and size battery energy storage systems (BESSs) and to site isolation devices. The purpose is twofold: (i) to enhance system reliability and (ii) to generate revenue by taking advantage of energy arbitrage. Our work fills a research gap regarding the optimal coordination of the siting of BESSs and isolation devices. Input data for the model includes utility historical load and fault data. This data is used to generate multiple scenarios pertaining to normal and faulted operating conditions. The proposed optimization model is recast as a mixed-integer linear programming (MILP) problem and solved using a commercial solver. It is successfully applied in a real-world case study. [ABSTRACT FROM AUTHOR]

This paper compares risk-averse optimization methods to address the self-scheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO; we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO; (2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters; (3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters. Summary of Contribution: The work presented in this manuscript is at the intersection of operations research and computer science, which are intrinsically related with the scope and mission of IJOC. From the operations research perspective, two methodologies for optimization under uncertainty are studied: risk-averse stochastic programming and adaptive robust optimization. These methodologies are illustrated using an energy scheduling problem. The study includes a comparison from the point of view of uncertainty modeling, formulations, decomposition methods, and analysis of solutions. From the computer science perspective, a careful implementation of decomposition methods using parallelization techniques and a sample average approximation methodology was done. A detailed comparison of the computational performance of both methods is performed. Finally, the conclusions allow establishing links between two alternative methodologies in operations research: stochastic programming and robust optimization. [ABSTRACT FROM AUTHOR]

We propose an approach to solving the AC network-constrained unit commitment problem (AC-NCUC) of large-scale power systems. This approach relies on solving convex (continuous and mixed-integer) optimization problems. First, we formulate a second-order conic relaxation of the original optimization problem, which renders a mixed-integer second-order conic optimization problem. To solve this problem, we use Benders’ decomposition, which decomposes it into a master problem, which is mixed-integer linear, and a subproblem, which is continuous and second-order conic. To improve the convergence of the Benders’ algorithm, we add several linear feasibility constraints to the master problem. Since the second-order relaxation of the original problem cannot guarantee AC feasibility of the solution found, we ensure such feasibility by solving a sequence of continuous convex optimization problems. We numerically validate the proposed approach using a Texas 2000-bus system. [ABSTRACT FROM AUTHOR]

Due to the increasing penetration of large-scale renewable energy generation at transmission level, hybrid AC/DC transmission expansion is becoming more relevant by the day, especially if the considered renewable generation is far away from existing AC grid infrastructure. This paper presents a multi-period hybrid AC/DC network expansion planning model that incorporates node voltages, reactive power and network losses. Since the operation of a high voltage DC link is influenced considerably by the converter stations, converter transformer, filter, and phase reactor are modeled in each converter station. Since the AC power flow equations render a non-convex problem whose solution algorithms do not always guarantee convergence or optimality, a second-order conic relaxation is utilized. The resulting model is a mixed-integer convex problem with a convergence guarantee that embodies a better approximation of system operation than its DC linear counterpart. Three case studies are considered to illustrate the proposed model. [ABSTRACT FROM AUTHOR]

The power and natural-gas industries have narrowed the time differences between daily power market clearing and natural-gas market clearing. This facilitates the gas-power coordination in day-ahead scheduling. However, using natural-gas-fired power units to balance the variability of renewable energy production may cause fluctuations in natural-gas demand, which result in risks to the natural-gas system in actual operation. As such, gas-power coordination at all time is highly desirable. We analyze the coordinated operation of power and natural-gas systems at the scheduling stage and in actual operation. We assume that the two systems are operated independently, but with information interchange. We use a Lagrangian decomposition approach to identify an operational equilibrium involving the two systems. Numerical results on two test systems verify the value of gas-power coordination in actual operation. [ABSTRACT FROM AUTHOR]

We consider a significant problem that arises in the planning of many projects. Project companies often use outsourced providers that require capacity reservations that must be contracted before task durations are realized. We model these decisions for a company that, given partially characterized distributional information, assumes the worst-case distribution for task durations. Once task durations are realized, the project company makes decisions about fast tracking and outsourced crashing, to minimize the total capacity reservation, fast tracking, crashing, and makespan penalty costs. We model the company's objective using the target-based measure of minimizing an underperformance riskiness index. We allow for correlation in task performance, and for piecewise linear costs of crashing and makespan penalties. An optimal solution of the discrete, nonlinear model is possible for small to medium size projects. We compare the performance of our model against the best available benchmarks from the robust optimization literature, and show that it provides lower risk and greater robustness to distributional information. Our work thus enables more effective risk minimization in projects, and provides insights about how to make more robust capacity reservation decisions. Summary of Contribution: This work studies a financially significant planning problem that arises in project management. Companies that face uncertainties in project execution may need to reserve capacity with outsourced providers. Given that decision, they further need to plan their operational decisions to protect against a bad outcome. We model and solve this problem via adjustable distributionally robust optimization. While this problem involves two-stage decision making, which is computationally challenging in general, we develop a computationally efficient algorithm to find the exact optimal solution for instances of practical size. [ABSTRACT FROM AUTHOR]

This paper proposes a methodology to identify the best locations to install battery energy storage systems (BESS) in radial distribution networks. Such batteries are mainly intended for improving the reliability of the distribution network. The methodology uses utility historical load, outage and project-cost data, and relies on a mixed-integer linear programming optimization tool. The power and energy capacities of any of the BESS considered can be pre-specified based on the system's load and outage data. On the other hand, such parameters can be optimally determined, which generally results in a large number of small BESSs distributed throughout the distribution system. We analyze both alternatives. The optimal siting of BESSs is illustrated and discussed via sensitivity analysis using a small distribution system. In addition, a case study based on a real-worlddistribution network is analyzed. Outcomes show the validity and efficacy of the proposed methodology. [ABSTRACT FROM AUTHOR]

We study the equilibria reached in electricity, natural-gas, and carbon-emission markets, where power producers simultaneously participate in these three markets. We use a conjectural-variations equilibrium model to represent both electricity and natural-gas markets, in which power/natural gas-producers play a quantity game and adjust their production decisions in reaction to the production levels of their rivals. We use a cap-and-trade mechanism to model the carbon-emission market. We develop a direct approach to identify equilibria involving the three markets, in which the optimality conditions of all players and market clearing conditions are gathered and solved jointly. Numerical results from two test systems illustrate the impact of carbon-emission trading and transmission constraints on the equilibria reached. Additionally, we compare equilibrium results under different types of competition. [ABSTRACT FROM AUTHOR]

A supply-function equilibrium model fits well current electricity market practice, but entails the solution of an equilibrium problem composed of bi-level problems, which is computationally very demanding. To ease such computational burden, we propose a computationally efficient single-level equilibrium model that identifies equilibria similar to those obtained by equilibrium models of bi-level problem. In our model, producers/consumers offer/bid both energy and prices. We use a simple example and a case study to illustrate and characterize the performance of the proposed single-level equilibrium model. [ABSTRACT FROM AUTHOR]

We propose a transmission expansion planning model that integrates thyristor-controlled series compensators (TCSCs) to enhance line transmission capacity, and superconducting fault current limiters (SFCLs) to control short-circuit levels. The harmonious interplay between TCSCs and SFCLs results in effective and economically attractive optimal expansion plans. This multi-stage planning model translates into a complex mixed-integer nonlinear programming problem, which is hard to solve. To solve it, we propose a successive linearization technique within a Benders’ decomposition scheme that proves effective in finding optimal solutions and efficient in terms of computational burden. We illustrate the methodology proposed using the IEEE 39-bus system. [ABSTRACT FROM AUTHOR]

One important task in power distribution system expansion planning is feeder routing (FR), which is to determine the optimal route from a medium voltage substation to load points and the optimal size of the conductors to be installed. This paper proposes a novel graph-based model for the resilient feeder routing problem using geographical information system (GIS). We show that incorporating GIS data enhances the FR solution optimality. Moreover, by introducing a representing graph, radiality of the planned network is guaranteed. A second-order cone programming (SOCP) model is used to model power flows through feeders. Since the representing graph ensures radiality, the SOCP model is exact. The uncertainty of rooftop solar generations and demand forecasting errors are taken into account, and a stochastic programming-based solution algorithm is developed. The proposed model represents practical aspects such as economic objectives (installation cost, power losses, resiliency), technical constraints (voltage limits, radiality constraint, reliability), and geographical constraints. The efficiency of the algorithm is demonstrated using three case studies: a small test system, a realistic one, and a synthetic large test system. [ABSTRACT FROM AUTHOR]

We study investment equilibria in electricity and gas markets wherein electricity producers and natural gas suppliers behave strategically. We consider also hybrid producers that own both generating units and gas sources. Each strategic producer determines its investment decisions in gas-fired units, and its offering and bidding strategies to maximize its own profit, anticipating electricity and gas market-clearing outcomes. Producers owning gas-fired units submit bids to the gas market to procure fuel and offers to the electricity market to sell electricity. The resulting model is recast as an equilibrium problem with equilibrium constraints that we solve using a direct approach. Numerical results from two test systems illustrate the proposed methodology. [ABSTRACT FROM AUTHOR]

We study market equilibria that are achieved by strategic firms that participate in electricity and natural gas markets. Strategic firms submit their offers and bids to both markets with the aim of maximizing profit or utility and we consider firms that can include a combination of electricity and natural gas supply and demand. The strategic actions of these firms are represented by upper-level problems that are optimized subject to shared lower-level problems that represent the clearing of electricity and natural gas markets. This market structure and our modeling approach yields a multiple-leader/two-follower complementarity problem. We develop a modeling approach that can find equilibria with different characteristics, e.g., maximized social welfare, producer profits, or consumer welfare. We demonstrate numerically that producers aim typically to increase market prices while consumers seek to decrease them. [ABSTRACT FROM AUTHOR]

The work reported in this paper aims at developing an algorithm for system operators to solve a robust security-constrained AC optimal power flow (R-SC-ACOPF) problem, which also optimizes the settings of discrete controllers including load tap changers (LTCs) and shunt capacitor banks (SCBs). An iterative algorithm involving a master problem and a number of sub-problems is proposed. The master problem selects the optimal discrete controllers’ settings by taking into account primal SC-ACOPF constraints based on the single-component worst contingencies identified by the sub-problems. Each sub-problem is a bi-level max-min problem used to find the current worst contingency. The overall objective is to minimize the total cost, including generation and load shedding cost, while satisfying all relevant constraints. The master problem and sub-problems allow representing LTC tap positions, SCB status, and contingencies using binary variables. Two case studies are presented as applications of this novel technique. [ABSTRACT FROM AUTHOR]

Electric power and natural gas systems typically are operated independently. However, their operations are interrelated due to the proliferation of natural gas-fired generating units. We analyze the independent but interrelated day-ahead operation of the two systems. We use a direct approach to identify operational equilibria involving these two systems, in which the Karush-Kuhn-Tucker conditions of both electric power and natural gas operational models are gathered and solved jointly. We characterize the equilibria that are obtained under different levels of temporal and spatial granularity in conveying information between the two system operators. Numerical results from the Belgian system are used to examine the impacts of different levels of information interchange on prices, operational costs, and decisions in the two systems. [ABSTRACT FROM AUTHOR]

System expansion planning (SEP) models do not generally represent voltage levels, bundled conductors in transmission lines, and short-circuit limits. These modeling assumptions may result in suboptimal planning outcomes. To overcome this potential flaw, we propose a short-circuit constrained dynamic SEP (SC-SEP) that allows for investment decisions at different stages of the planning horizon and includes a detailed representation of voltage levels, alternative bundled conductor options per line and short-circuit limits. An effective linearization technique is used to transform the resulting mixed-integer nonlinear model into a mixed-integer linear one. The accuracy of the proposed model and the effectiveness of the proposed linearization technique are illustrated using the IEEE 39-bus system. [ABSTRACT FROM AUTHOR]

The interdependency of electric power and natural gas systems requires co-ordinated operational planning. We propose a unit commitment model that integrates a second-order-cone relaxation of a non-convex nonlinear natural-gas flow model that considers pipeline line pack. The model is enhanced by using convex envelopes of bilinear terms, which tighten the relaxation. By fixing the binary variables at their optimal values and linearizing the natural gas-flow-balance equations around the solution that is obtained, we obtain electricity and natural gas locational marginal prices as the dual variables of electricity- and natural gas-flow-balance equations, respectively. The interdependence between these sets of prices is discussed. Numerical results from two test systems validate the solution-quality and computational-efficiency benefits of the proposed modeling methodology. [ABSTRACT FROM AUTHOR]

Restructuring and liberalisation of the electricity industry creates opportunities for investment in energy storage, which could be undertaken by a profit-maximising merchant storage operator. Because such a firm is concerned solely with maximising its own profit, the resulting storage-investment decision may be socially suboptimal (or detrimental). This paper develops a bi-level model of an imperfectly competitive electricity market. The modelling framework assumes electricity-generation and storage-operations decisions at the lower level and storage investment at the upper level. Our analytical results demonstrate that a relatively high (low) amount of market power in the generation sector leads to low (high) storage-capacity investment by the profit-maximising storage operator relative to a welfare maximiser. This can result in net social welfare losses with a profit-maximising storage operator compared to a no-storage case. Moreover, there are guaranteed to be net social welfare losses with a profit-maximising storage operator if the generation sector is sufficiently competitive. Using a charge on generation ramping between off- and on-peak periods, we induce the profit-maximising storage operator to invest in the same level of storage capacity as the welfare-maximising firm. Such a ramping charge can increase social welfare above the levels that are attained with a welfare-maximising storage operator. [ABSTRACT FROM AUTHOR]

Increasing use of natural gas for electricity production places added strains on pipeline systems that are used for transporting fuel. Pipeline constraints require power system operators to account for natural gas-supply restrictions in their operational processes. This paper proposes separate optimization models for clearing day-ahead wholesale markets for scheduling power and natural gas systems. We then develop a market-based mechanism that allows for efficient co-ordination of the two systems. Importantly, the co-ordination mechanism only requires the exchange of fuel-price, -supply, and -demand information between the two markets. This can be contrasted with other co-ordination mechanisms that require operations of the two systems by a single entity. Thus, we provide a computationally tractable co-ordination mechanism that does not require the exchange of proprietary information between natural gas and electricity system operators. We demonstrate the effectiveness and scalability of the technique using a numerical example. [ABSTRACT FROM AUTHOR]

The recent leak of the Aliso Canyon natural gas storage facility calls for coordinated planning of natural gas and electric power systems with specific consideration of electrical energy storage. This paper proposes a coordinated planning model that fills this need. The model is formulated as a two-stage stochastic optimization problem, in which electricity- and natural gas-demand growth and natural gas-supply uncertainties are represented. We analyze the tradeoff between building electrical, natural gas, and electrical energy storage infrastructure. The sensitivity of electrical energy storage investment to the modeling of the operating conditions is also studied. The model is tested using a California-based case study. [ABSTRACT FROM AUTHOR]

This paper focuses on the estimation of the rotor speeds of synchronous machines by means of phasor measurement units. This estimation is aimed at online monitoring of electromechanical transients and transient stability analysis. The proposed technique is based on the concept of frequency divider formula recently published by the first two authors in these Transactions. The dynamic state estimation is formally stated as a convex optimization problem and a thorough discussion of the sensitivity analysis of the optimal solution is provided. The case study considers a dynamic 1479-bus model of the all-island Irish system and discusses the effect of bad data, noise, and latency on the proposed estimation technique. [ABSTRACT FROM AUTHOR]

In this paper, the coordinated investment in transmission and storage systems under uncertainty is addressed from the perspective of a central planner. The problem is formulated with the objective of achieving an effective expansion of both transmission and storage that minimizes the investment cost, while achieving an effective operation of a power system with high penetration of renewable power. The problem is stated within an adaptive robust optimization framework that incorporates stochastic modeling. Future available generation capacity and peak load in the target year (long-term uncertainty) are modeled using a robust set, while daily operating conditions (short-term uncertainty) are represented with a scenario set. The expansion tradeoff between transmission and storage systems is analyzed using the IEEE RTS 24-bus and IEEE 118-bus networks. [ABSTRACT FROM AUTHOR]

The work reported in this paper aims at developing an algorithm for identifying the worst contingencies to be used in a security-constrained optimal power flow problem. A bilevel max-min optimization model is proposed to find the worst contingency scenario while solving an ac optimal power flow problem and considering corrective actions. The objective is to minimize the total cost including generation and load shedding cost while satisfying all relevant constraints. The upper-level problem allows representing contingencies using binary variables. We formulate a second-order cone problem in the lower-level to represent the ac power flow equations. Duality is used to merge the upper- and lower-level problems into a single-level one. The resulting mixed-integer second-order conic problem can be efficiently solved. Three case studies are presented as applications of this novel technique. [ABSTRACT FROM AUTHOR]

This two-paper series introduces a new multistage robust approach for solving the transmission expansion planning (TEP) problem under the uncertainty of loads and wind power productions. The proposed multistage robust TEP model is formulated using mixed binary and linear decision rules, which aims at finding a solution withstanding any realization of the uncertain parameters belonging to a set-based uncertainty model. The robustness of the optimal expansion plan can be adjusted by the budget of uncertainty. The first part of this two-paper series presents the extended and compact formulations for a deterministic TEP problem. Then, the compact formulation of the TEP problem under the uncertainty of loads and wind power productions is introduced as a multistage robust optimization problem, which is reformulated using mixed binary and linear decision rules. The extended formulation of these decision rules, as functions of uncertain parameters, is presented in this part as well. [ABSTRACT FROM AUTHOR]

The second part of this two-paper series presents the extended formulation for the lifted multiyear multistage robust transmission expansion planning model introduced in the first part. The proposed planning approach is successfully tested on the Garver 6-bus and on a 118-bus+118-bus test system with different input settings. [ABSTRACT FROM AUTHOR]

The interdependence between natural gas and electric power systems is becoming increasingly tight, as the share of natural gas-fired units increases. Within this context, this paper addresses the coordinated expansion planning of natural gas and power systems. We analyze the tradeoff of building natural gas-related facilities (e.g., natural gas pipelines and natural gas-fired generation units) versus electric-power-related facilities (e.g., transmission lines and other generation units). We use a two-stage stochastic optimization model that provides an appropriate balance between accuracy and computational tractability and represents uncertainty pertaining to electricity and natural gas demands. We show the importance of representing uncertainty by computing the value of stochastic solution, which is significant. We analyze the functioning of the model through a small example and a case study based on the IEEE 118-bus system. Computational results confirm the need for detailed representation of both the natural gas and power systems to achieve investment decisions that are well coordinated and optimal. [ABSTRACT FROM PUBLISHER]

Power system capacity-expansion models are typically intractable if every operating period is represented. This issue is normally overcome by using a subset of representative operating periods. For instance, representative operating hours can be selected by discretizing the load-duration curve, which captures the effect of load levels on system-operation costs. This approach is inappropriate if system-operating costs depend on parameters other than load (e.g., renewable-resource availability) or if there are important intertemporal operating constraints (e.g., generator-ramping limits). This paper proposes the use of representative operating days, which are selected using clustering, to surmount these issues. We propose two hierarchical clustering techniques, which are designed to capture the important statistical features of the parameters (e.g., load and renewable-resource availability), in selecting representative days. This includes temporal autocorrelations and correlations between different locations. A case study, which is based on the Texan power system, is used to demonstrate the techniques. We show that our proposed clustering techniques result in investment decisions that closely match those made using the full unclustered dataset. [ABSTRACT FROM PUBLISHER]

Distribution expansion planning (DEP) is used to determine the best expansion plan of a distribution system. Considering the uncertainties of loads and of power productions of wind distributed energy resources (DERs) and incorporating AC power flow equations make the DEP problem increasingly challenging. In this context, this paper presents a new adaptive robust distribution expansion planning model to identify the timing of feeder reinforcements in addition to the location, capacity, and installation time of dispatchable and wind DERs. The proposed approach characterizes the uncertain nature of loads and power productions of wind DERs through polyhedral uncertainty sets and the robustness of the solution can be controlled by means of a budget of uncertainty. Since AC power flow equations rather than DC ones are considered, the proposed model is a nonlinear and non-convex min–max–min optimization problem, which cannot be solved directly by commercial optimization packages. Hence, a tri-level decomposition algorithm using both primal and dual cutting planes is introduced to solve the problem. Additionally, to attain a smoother optimization problem, the non-convex AC power flow equations are convexified. The working of the proposed model is illustrated using 33-bus and 123-bus distribution networks. [ABSTRACT FROM AUTHOR]

This paper addresses the transmission expansion planning problem under long- and short-term uncertainty. Long-term uncertainty pertains to changes across years (demand growth and future increase in production capacity), whereas short-term uncertainty pertains to changes within a year (demand and wind/solar power production variability, and equipment availability). This expansion problem is formulated as an adaptive robust optimization problem that provides protection against long-term uncertainty while carefully representing short-term uncertainty via scenarios. The problem is solved via a tailored implementation of the primal Benders’ decomposition algorithm that focuses on an efficient solution of the subproblem. The effectiveness of the proposed algorithm to identify robust expansion plans and its computational efficiency are illustrated through a realistic case study. [ABSTRACT FROM AUTHOR]

We propose a multistage multiscale linear stochastic model to optimize electricity generation, storage, and transmission investments over a long planning horizon. The multiscale structure captures “large-scale” uncertainties, such as investment and fuel-cost changes and long-run demand-growth rates, and “small-scale” uncertainties, such as hour-to-hour demand and renewable-availability uncertainty. The model also includes a detailed treatment of operating periods so that the effect of dispatch decisions on long-term investments are captured. The proposed model can be large in size. The progressive hedging algorithm is applied to decompose the model by scenario, greatly reducing computation times. We also derive bounds on the optimal objective-function value, to assess solution quality. We use a case study based on the state of Texas to demonstrate the model and show the benefits of its detailed representation of the operating periods in making investment decisions. [ABSTRACT FROM PUBLISHER]

This paper presents a novel model for two-stage adaptive robust transmission expansion planning (AR-TEP) problem considering the uncertainty of future load demand and future wind power production. Linear decision rules (LDRs) are utilized to reformulate the AR-TEP model as a tractable mixed-integer linear programming problem, which can be directly solved by off-the-shelf optimization packages. The proposed LDR-based AR-TEP model is illustrated using the Garver 6-bus test system, the IEEE 24-bus reliability test system, and a 236-bus test system under different conditions and compared with a decomposition-based AR-TEP model that uses primal cutting planes. Also, the robustness of the expansion plans obtained from these two AR-TEP models is evaluated by carrying out an ex-post out-of-sample analysis for different values of the budget of uncertainty. Simulation results demonstrate the effectiveness of the proposed LDR-based AR-TEP model. [ABSTRACT FROM PUBLISHER]

The work reported in this paper aims at developing a practical and efficient tool for utility transmission planners to protect critical transmission assets from potential physical attacks. A trilevel min–max–min optimization model is proposed to represent actions of a planner, an attacker, and an operator. These three agents share a common objective function, which is the system load shedding. The strong duality theorem is used to merge the middle- and lower-level problems into a single-level one. The resulting mixed-integer bilevel model is solved by a Benders decomposition technique using primal cuts. Two case studies are presented as applications of this novel technique. Additionally, the proposed technique is compared with an implicit enumeration algorithm. [ABSTRACT FROM AUTHOR]

As the power industry considers moving from a deterministic paradigm to a stochastic one to clear the day-ahead market in systems with significant stochastic production, the following question arises: how should a stochastic clearing algorithm be formulated? Our analyses indicate that a three-stage stochastic approach is clearly superior to a two-stage one. To show this, we propose a model that includes three stages: the first one represents the day-ahead market, the second stage, the intraday market, and the third one, the real-time operation. The objective is to clear the day-ahead market, but with a prognosis of the future: the intraday market and the real-time operation. To assess the impact of the intraday market on the day-ahead outcomes, we compare the results obtained from the proposed model with those of a simpler but more common two-stage model, which represents the day-ahead market and the real-time operation. [ABSTRACT FROM AUTHOR]

The cost of a generating unit as a function of its power output is generally non‐linear and discontinuous. Therefore, the objective function of a unit commitment (UC) problem is generally non‐linear and discontinuous as well. The authors describe two efficient piecewise‐linear formulations of the objective function of an UC problem that are not found in the power literature. These formulations allow representing non‐convexities and discontinuities. They compare these formulations with a common one in the power literature. They show the advantage of the proposed formulations for both deterministic and stochastic UC problems. [ABSTRACT FROM AUTHOR]

In highly polluted regions, it might be necessary to operate power plants enforcing appropriate emission limits to help ensuring air quality. If the air quality is low and pollutants are difficult to diffuse, few additional emissions should be allowed to avoid further deteriorating the air quality. Wind power helps reducing pollution levels as it substitutes polluting thermal production. However, wind and air pollution are generally anti‐correlated. Storage can be used to shift wind power production from hours when higher pollution levels are allowed to hours when lower pollution levels are allowed to help meeting emission constraints at critical hours. Storage availability also achieves a comparatively lower production cost for the system as a whole. To represent the stochastic nature of wind power production and weather‐dependent emission limits, the authors propose a two‐stage stochastic programming model to analyse the trade‐off emission versus cost in an electric energy system with storage facilities. The first stage represents the day‐ahead scheduling, while the second one represents the real‐time operation under different scenarios. The authors analyse the impact of emission limits and/or storage on generation scheduling by comparing different models using an illustrative example and a case study based on the IEEE 118‐node system. [ABSTRACT FROM AUTHOR]

This paper analyzes the impacts of flexible demands on day-ahead market outcomes in a system with significant wind power production. We use a two-stage stochastic market-clearing model, where the first stage represents the day-ahead market and the second stage represents the real-time operation. On one hand, flexibility of demands is beneficial to the system as a whole since such flexibility reduces the operation cost, but on the other hand, shifts in demands from peak periods to off-peak periods may influence prices in such a way that demands may not be willing to provide flexibility. Specifically, we investigate the impacts of different degree of demand flexibility on day-ahead prices. A number of scenarios modeling the uncertainty associated with wind production at the operation stage, and nonconvexities due to start-up costs of generators and their minimum power outputs are taken into account. [ABSTRACT FROM AUTHOR]

We propose a two-stage mixed-integer linear stochastic optimization model to analyze the scheduling of electricity-production units under natural gas-supply uncertainty due to pipeline congestion and natural gas-price variability. The first stage of this stochastic optimization model represents the day-ahead scheduling (i .e., unit commitment) stage, while the second stage represents actual real-time operations through a number of scenarios. We use this model to analyze the effect on unit commitment and dispatch of two types of natural gas-supply conditions. First, we analyze a case involving low-cost natural gas supply with natural gas-transmission issues related to potential gas-pipeline congestion. We then examine a case involving higher-cost natural gas, which is used solely to attain feasibility with fast-ramping events. The first case mimics situations in the ISO New England system, in which relatively low-cost natural gas supply is uncertain in cold-weather conditions due to natural gas-transmission bottlenecks. The second case is reminiscent of situations in the California ISO system, in which relatively expensive but flexible natural gas-fired units need to be used to handle rapid changes in net demand in the early mornings and late afternoons. [ABSTRACT FROM AUTHOR]

This paper proposes a pricing scheme for the day-ahead market in power systems with a large percentage of renewable stochastic production. To clear the day-ahead market, instead of a simplistic deterministic model, we use a two-stage stochastic programming model that embodies a prognosis of future operating conditions. Non-convexities due to start-up costs and the on/off status of generators and their minimum power outputs are properly taken into account. Our goal is to obtain uniform day-ahead clearing prices that deviate in the least possible manner from marginal prices and that allow producers to recover their costs without uplifts. The proposed methodology is illustrated using a simple example and a realistic case study. [ABSTRACT FROM AUTHOR]