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One of the most important daily decisions that a Genco has to make is to allocate generation assets between the forward and spot markets. That is, how much capacity should be contracted in the forward market and how much should be kept to bid in spot market? This paper focuses on generation asset allocation between monthly forward contracts, such as bilateral contracts, futures contracts, options contracts, and daily spot markets, considering operating costs and constraints of generating units, as well as spot price risk. The problem is to find the optimal hedging position based on the known forward price and the forecasted hourly spot prices and is formulated based on the model of PJM market. The double dynamic programming method is applied to solve this optimal asset allocation problem with all the short-term operating constraints of the generating unit satisfied. Three types of forward contracts that are commonly used in practice are considered and their impact on generation asset allocation are analyzed and compared. The analytic relationship between the optimal contract quantity of a particular type and spot generation is established. Based on this relationship, the applicability and characteristics of a particular type of contract are discussed. Furthermore, with the solution to the generation asset allocation problem as a basis, the pricing strategy in the forward market is analyzed. A Nash game model is established and an iterative process is developed to determine the equilibrium pricing of futures contracts. Numerical testing shows that the method for the generation asset allocation is effective. Various factors influencing the decision of generation asset allocation and the relationship between futures contract price and spot price are tested and analyzed. Index Terms--Dynamic programming, forward contracts, generation asset allocation, Nash game, power markets, unit commitment.

The core of solving security-constrained unit commitment (SCUC) problems within the Lagrangian relaxation framework is how to obtain feasible solutions. However, due to the existence of the transmission constraints, it is very difficult to determine if feasible solutions to SCUC problems can be obtained by adjusting generation levels with the commitment states obtained in the dual solution of Lagrangian relaxation. The analytical and computational necessary and sufficient conditions are presented in this paper to determine the feasible unit commitment states with grid security constraints. The analytical conditions are proved rigorously based on the feasibility theorem of the Benders decomposition. These conditions are very crucial for developing an efficient method for obtaining feasible solutions to SCUC problems. Numerical testing results show that these conditions are effective. Index Terms--Benders decomposition, Lagrangian relaxation (LR), power generation scheduling, security constrained unit commitment (SCUC).

The purchase allocation problem is one of the most important problems faced by an electric energy service provider under new market environments. The optimal purchase allocation problem for dual electric power markets and demand bid generation are discussed in this paper. The price volatility is explicitly considered in purchase allocation problems and the sequential nature is modeled by conditional stochastic characteristics. An analytical solution for the optimal allocation is derived with given demand and statistical characteristics of the market prices. The method for generating demand bids for purchaser based on market allocation and price forecasting is then developed. The numerical simulations for market allocation and bid generation are demonstrated based on the actual data of the U.S. California power market. Index Terms--Bid curve, electric power market, price forecast.

When the Lagrangian relaxation based methods are applied to solve power system unit commitment, the identical solutions to the subproblems associated with identical units may cause the dual solution to be far away from the optimal solution and serious solution oscillations. As a result, the quality of the feasible solution obtained may be very unsatisfactory. This issue has been long recognized as an inherent disadvantage of Lagrangian relaxation based methods. In this paper, the homogeneous solution issue is identified and analyzed through a simple example. Based on this analysis, a successive subproblem solving method is developed. The new method combines the concepts of augmented Lagrangian relaxation and surrogate subgradient to produce a good search direction at the high level. The low level subproblems including those corresponding to the identical units are solved successively so that the commitments of the identical units may not be homogeneous in the dual solution. Compared with the standard Lagrangian relaxation method, the new method can obtain better dual solutions and avoid the solution oscillations. Numerical testing shows the new method is efficient and the quality of the feasible solution is greatly improved. Index Terms--Homogeneous solution oscillations, Lagrangian relaxation, surrogate subgradient, unit commitment.

An optimization-based algorithm is presented for scheduling hydrothermal power systems with cascaded and head-dependent reservoirs. Within the Lagrangian relaxation framework, the hydro river catchment subproblems are difficult to solve because of the continuous reservoir dynamics and constraints, discontinuous operating regions, discrete operating states and hydraulic coupling of cascaded reservoirs. The head-dependent water-power conversion adds another dimension of difficulty since the objective functions of hydro subproblems are no longer stage-wise additive with respect to water discharge. It is difficult to solve the subproblems by relaxing the reservoir limits or the hydraulic coupling among units as in previous work. A new algorithm with a novel relaxation structure is presented in this paper to solve hydro river catchment subproblems. The key idea is to use another set of multipliers to relax capacity and minimum generation constraints of individual hydro units. A river catchment subproblem can be further decomposed into two sets of subproblems: a continuous-variable optimization problem determining the generation levels of all units in the entire river catchment, and a number of pure integer problems determining the hydro commitment states, one for each unit. The continuous problem is solved by a nonlinear network flow algorithm, and the integer problems are solved by dynamic programming with a small number of states and well-structured transitions. The two sets of subproblems are coordinated through the multipliers that are updated at the intermediate dual level by using a modified subgradient algorithm. After the dual problem converges, the feasible hydro schedule is obtained by using the same network flow algorithm with operating states obtained in the dual solution and possibly adjusted by heuristics. Numerical testing based on the data sets of a practical system shows that this method is efficient and effective to deal with hydrothermal systems with cascaded and head-dependent reservoirs, and discrete hydro constraints. The results indicate that near-optimal schedules are obtained. Key Words: Hydrothermal generation scheduling, Cascaded reservoirs, Head-dependent reservoirs, Mixed-integer programming, Nonlinear network flow algorithm

Energy trading in a competitive electricity market can be modeled as a two-level optimization. At the top level a Centralized Economic Dispatch (CED) uses a priority list method to solve the fundamental problem of reliable market clearing with price discovery. The lower level consists of a set of Decentralized Bidding (DB) subproblems. The DB model uses a self-unit scheduling simulator based on parametric dynamic programming to produce hourly bid curves for the central dispatch coordinator. Unit operating constraints and costs such as the unit minimum-up and minimum-down times, ramp rates, and the unit start-up, no-load and sunk capital costs are internalized in the bid curves through the simulator. A special algorithm is presented to solve the revenue adequacy problem for marginal units. Both CED and DB models are based on the revenue maximization in contrast with the cost minimization criteria used in the conventional Unit Commitment (UC). The proposed method has been tested in a study case and some interesting results have been demonstrated. Keywords: Energy markets, Bidding strategies, Unit commitment, Parametric dynamic programming.

An optimization-based algorithm is presented for scheduling hydro power systems with restricted operating zones and discharge ramping constraints. Hydro watershed scheduling problems are difficult to solve because many constraints, continuous and discrete, including hydraulic coupling of cascaded reservoirs have to be considered. Restricted or forbidden operating zones as well as minimum generation limits of hydro units result in discontinuous preferred operating regions, and hinder direct applications of efficient continuous optimization methods such as network flow algorithms. Discharge ramping constraints due to navigational, environmental and recreational requirements in a hydro system add another dimension of difficulty since they couple generation or water discharge across time horizon. Integrated consideration of the above constraints is very challenging. The key idea of this paper is to use additional sets of multipliers to relax discontinuous operating region and discharge ramping constraints on individual hydro units so that a two-level optimization structure is formed. The low level consists of a continuous discharge scheduling subproblem determining the generation levels of all units in the entire watershed, and a number of pure integer scheduling subproblems determining the hydro operating states, one for each unit. The discharge subproblem is solved by a network flow algorithm, and the integer scheduling problems are solved by dynamic programming with a small number of states and well-structured transitions. The two sets of subproblems are coordinated through multipliers updated at the high level by using a modified subgradient algorithm. After the dual problem converges, a feasible hydro schedule is obtained by using the same network flow algorithm with the operating states obtained, and operating ranges modified to guarantee satisfaction of ramping constraints. Numerical testing based on the data sets of the PG&E power system shows that this method is efficient and effective in dealing with cascaded hydro systems with restricted operating zones and discharge ramping constraints. Key Words: Power generation scheduling, Hydrothermal scheduling, Restricted operating zones, Mixed-integer programming, Network flow algorithm

An optimization-based algorithm is presented for the short-term scheduling of hydrothermal power systems using the Lagrangian relaxation technique. This paper concentrates on the solution methodology for hydro subproblems with cascaded reservoirs and discrete hydro constraints. Continuous reservoir dynamics and constraints, discontinuous operating regions, discrete operating states, and hydraulic coupling of cascaded reservoirs are considered in an integrated fashion. The key idea is to substitute out the reservoir dynamics and to relax the reservoir level constraints by using another set of multipliers, making a hydro subproblem unitwise and stage-wise decomposable. The optimal generation level for each operating state at each hour can be obtained simply by minimizing a single variable function. Dynamic programming is then applied to optimize the operating states across the planning horizon with a small number of well-structured transitions. A modified subgradient algorithm is used to update multipliers. After the dual problem converges, the feasible solution to the hydro power subsystem is obtained by using a network flow algorithm, with operating states obtained in the dual solutions, and possibly adjusted by heuristics. Numerical testing based on practical system data sets show that this method is efficient and effective for dealing with hydrothermal systems with cascaded reservoirs and discrete hydro constraints. Key Words - Scheduling of hydrothermal power systems, Scheduling of cascaded reservoirs, Mixed-integer programming

The modeling of constraints is an important issue in power system scheduling. Constraints can be generally classified into two categories: 1) physical limits and 2) operating limits. A schedule violating physical limit or constraint would not be acceptable. An operating limit, however, is often imposed to enhance system security but does not represent a physical bound. This kind of 'soft limits' can be temporarily violated 'a little bit' if necessary, but not 'too much.' These constraints are therefore 'fuzzy' in nature, and crisp treatment of them may lead to over conservative solutions. In this paper, a fuzzy optimization-based method is developed to solve power system scheduling problem with fuzzy reserve requirements. The problem is first converted to a crisp and separable optimization problem. Lagrange multipliers are then used to relax system-wide constraints and decompose the problem into a number of unit-wise subproblems and a membership subproblem. These subproblems can be efficiently solved, and the multipliers are updated by using a subgradient method. Heuristics are then used to construct a feasible schedule based on subproblem solutions. Numerical testing results show that near optimal schedules are obtained, and the method can provide a good balance between reducing costs and satisfying reserve requirements. Keywords: Power System Scheduling, Fuzzy Optimization, Unit Commitment.

Conventional Optimal Power Flow (OPF) solutions utilize standard optimization techniques. These techniques limit the practical value and scope of OPF applications. Realistic OPF solutions require special attention to the constraint enforcement and control action curtailment. From a practical point of view, an OPF does not need to find a rigid minimum solution. Certain trade-off among (a) minimizing of objective function, (b) satisfying constraints, and (3) moving control variables would be desirable. In this paper, a fuzzy set method is applied to the OPF problem to model these considerations. The method has been implemented in a successive linear programming OPF package. Test results based on the PG&E network are promising. Keywords : Optimal Power Flow, Fuzzy Optimization, Energy Management Systems, On-Line Controls.

This paper presents an optimization-based method for scheduling hydrothermal systems based on the Lagrangian relaxation technique. After system-wide constraints are relaxed by Lagrange multipliers, the problem is converted into the scheduling of individual units. This paper concentrates on the solution methodology for pumped-storage units. A pumped-storage unit can be operated in generation, pumping or idle states. It can smooth peak loads and provide reserve, therefore plays an important role in reducing total generation costs. There are, however, many constraints limiting the operation of a pumped-storage unit, such as pond level dynamics and constraints, and discontinuous generation and pumping regions. Moreover, according to the current practice, the dynamic transitions among operating states (generation, pumping and idle) are not arbitrary. The most challenging issue in solving pumped-storage subproblems within the Lagrangian relaxation framework is the integrated consideration of these constraints. The basic idea of our method is to relax the pond level dynamics and constraints by using another set of multipliers. The subproblem is then converted into the optimization of generation or pumping levels for each operating state at individual hours, and the optimization of operating states across hours. The optimal generation or pumping level for a particular operating state at each hour can be obtained by optimizing a single variable function without discretizing pond levels. Dynamic programming is then used to optimize operating states across hours with only a few number of states and transitions. A subgradient algorithm is used to update the pond level Lagrangian multipliers. This method provides an efficient way to solve a class of subproblems involving continuous dynamics and constraints, discontinuous operating regions, and discrete operating states. Testing results based on Northeast Utilities power system show that this algorithm is efficient, and near optimal solutions are obtained.

Power transactions are important activities among electric utility companies. Effective transaction decisions can result in significant savings since marginal generation costs of neighboring utilities could be quite different. Transactions, however, are coupled with the scheduling of units through system demand and reserve requirements, and are confined by many accustomed rules. How to make effective transaction decisions is therefore a very difficult problem. In this paper, an optimization-based method for the integrated consideration of power purchase transactions and the scheduling of thermal units is presented based on the augmented Lagrangian decomposition and coordination method. After the system-wide demand and reserve requirements are relaxed by using Lagrange multipliers and penalty coefficients, the overall problem is decomposed into purchase and thermal subproblems. For a purchase subproblem, the optimal purchase level for each purchase interval is first determined. The subproblem is then efficiently solved by using the dynamic programming approach without discretizing purchase levels. Thermal subproblems are solved by extending our previous method recently reported in the literature. The multipliers and penalty coefficients are then updated at the high level so that system demand and reserve requirements are gradually satisfied over iterations. The augmented Lagrangian decomposition and coordination method avoids the solution oscillation difficulties associated with linear cost functions of purchase subproblems and speeds up algorithm convergence. Numerical testing results based on modified Northeast Utility's data show that the algorithm is efficient, and significant savings can be obtained.

This paper presents a method for scheduling hydrothermal power systems based on the Lagrangian relaxation technique. By using Lagrange multipliers to relax system-wide demand and reserve requirements, the problem is decomposed and converted into a two-level optimization problem. Given the sets of Lagrange multipliers, a hydro unit subproblem is solved by a merit order allocation method, and a thermal unit subproblem is solved by using dynamic programming without discretizing generation levels. A subgradient algorithm is used to update the Lagrange multipliers. Numerical results based on Northeast Utilities data show that this algorithm is efficient, and near-optimal solutions are obtained. Comparing with our previous work where thermal units were scheduled by using the Lagrangian relaxation technique and hydro units by heuristics, the new coordinated hydro and thermal scheduling generates lower total costs and requires less computation times.

Many challenging issues arise under the newly deregulated competitive electric power markets. Instead of centralized decision-making in a vertically integrated environment as in the past, decision-making is now decentralized and driven by market forces. Gaming and price spikes have been observed in almost every electricity market but explicit analysis of these phenomena is rare. In this paper we study historical bidding behavior to see how power suppliers and demand service providers were actually bidding in the California day-ahead energy market. Based on our observations we formulate a Prisoner's dilemma matrix game and introduce the notion of 'opportunistic tacit collusion' to explain strategic bidding behaviors in which suppliers withhold generation capacity from the market to drive up prices. This explanation is applicable with or without market power, transmission constraints, and insufficient supply, and is only enhanced by these factors. Our analysis is generally applicable to any uniform price electricity market in which there is significant insensitivity to price on the demand side. Index Terms--Electric power industry deregulation, game theory, market clearing price, market power.

The generation schedules obtained in traditional hydrothermal scheduling or unit commitment programs are in hourly generation levels. In the new deregulated power market, the power transactions are processed in terms of hourly energy delivery. Failing to fulfill scheduled energy delivery may result in penalty to power producers. This paper shows that although ramp-rate constraints are satisfied in hydrothermal scheduling, taking a generation level schedule as an energy delivery schedule may not be realizable. Based on the Maximum Principle in optimal control theory, the energy delivery capacity across the scheduling horizon is established as a set of recursive equations with given ramp-rate constraints. A sufficient and necessary condition is obtained to check if an energy delivery schedule is realizable. Based on this condition, two cases, where ramp-rate constraints are both satisfied, are analyzed and an unrealizable energy delivery schedule is observed. Index Terms--Electric energy bidding, electric power industry deregulation, optimal control, power generation scheduling, ramp-rate constraints.

The deregulation and reconstruction of the electric power industry worldwide raises many challenging issues related to the economic and reliable operation of electric power systems. Traditional unit commitment or hydrothermal scheduling problems have been integrated with generation resource bidding, but the development of optimization based bidding strategies is only at a very preliminary stage. This paper presents a bidding strategy based on the theory of ordinal optimization that the ordinal comparisons of performance measures are robust with respect to noise and modeling error, and the problems become much easier if the optimization goal is softened from asking for the 'best' to 'good enough' solution. The basic idea of our approach is to use a approximate model that describes the influence of bidding strategies on the market clearing prices (MCP). A nominal bid curve is obtained by solving optimal power generation for a given set of MCPs via Lagrangian relaxation. Then N bids are generated by perturbing the nominal bid curve. The ordinal optimization method is applied to isolate a good enough set S that contains some good bids with high probability by performing rough evaluation. The best bid is then selected by solving full hydrothermal scheduling or unit commitment problems for each of the bids in S. Using ordinal optimization approach we are able to obtain a good enough bidding strategy with reasonable computational effort. Numerical results using historical MCPs from the California market and a generation company with 10 units show that the ordinal optimization based method is efficient and good bidding strategies are obtained. Index Terms--Bidding strategies, electric energy bidding, electric power market deregulation, hydrothermal scheduling.