Gary A. Stern

Payment cost minimization auction for deregulated electricity markets using surrogate optimization

Deregulated electricity markets use an auction mechanism to select offers and their power levels for energy and ancillary services. A settlement mechanism is then used to determine the payments resulting from the selected offers. Currently, most independent system operators (ISOs) in the United States use an auction mechanism that minimizes the total offer costs but determine payment costs using a settlement mechanism that pays uniform market clearing prices (MCPs) to all selected offers. Under this setup, the auction and settlement mechanisms are inconsistent since minimized costs are different from payment costs. Illustrative examples in the literature have shown that for a given set of offers, if an auction mechanism that directly minimizes the payment costs is used, then payment costs can be significantly reduced as compared to minimizing offer costs. This observation has led to discussions among stakeholders and policymakers in the electricity markets as to which of the two auction mechanisms is more appropriate for ISOs to use. While methods for minimizing offer costs abound, limited approaches for minimization of payment costs have been reported. This paper presents an effective method for directly minimizing payment costs. In view of the specific features of the problem including the nonseparability of its objective function, the discontinuity of offer curves, and the maximum term in defining MCPs, our key idea is to use augmented Lagrangian relaxation and to form and solve offer and MCP subproblems by using the surrogate optimization framework. Numerical testing results demonstrate that the method is effective, and the resulting payment costs are significantly lower than what are obtained by minimizing the offer costs for a given set of offers.

Payment Cost Minimization Auction for Deregulated Electricity Markets With Transmission Capacity Constraints

Deregulated electricity markets in the U.S. currently use an auction mechanism that minimizes total supply bid costs to select bids and their levels. Payments are then settled based on market-clearing prices. Under this setup, the consumer payments could be significantly higher than the minimized bid costs obtained from auctions. This gives rise to ldquopayment cost minimization,rdquo an alternative auction mechanism that minimizes consumer payments. We previously presented an augmented Lagrangian and surrogate optimization framework to solve payment cost minimization problems without considering transmission. This paper extends that approach to incorporate transmission capacity constraints. The consideration of transmission constraints complicates the problem by entailing power flow and introducing locational marginal prices (LMPs). DC power flow is used for simplicity and LMPs are defined by ldquoeconomic dispatchrdquo for the selected supply bids. To characterize LMPs that appear in the payment cost objective function, Karush-Kuhn-Tucker (KKT) conditions of economic dispatch are established and embedded as constraints. The reformulated problem is difficult in view of the complex role of LMPs and the violation of constraint qualifications caused by the complementarity constraints of KKT conditions. Our key idea is to extend the surrogate optimization framework and use a regularization technique. Specific methods to satisfy the ldquosurrogate optimization conditionrdquo in the presence of transmission capacity constraints are highlighted. Numerical testing results of small examples and the IEEE Reliability Test System with randomly generated supply bids demonstrate the quality, effectiveness, and scalability of the method.

Simultaneous Optimal Auction and Unit Commitment for Deregulated Electricity Markets

Is the cost function minimized in the unit commitment problem comparable to the costs incurred in a market-clearing price auction mechanism? The authors present an alternative minimization problem: a simultaneous optimal auction that is consistent with deregulated electricity markets using a market-clearing price auction.

Convergence of the Surrogate Lagrangian Relaxation Method

Studies have shown that the surrogate subgradient method, to optimize non-smooth dual functions within the Lagrangian relaxation framework, can lead to significant computational improvements as compared to the subgradient method. The key idea is to obtain surrogate subgradient directions that form acute angles toward the optimal multipliers without fully minimizing the relaxed problem. The major difficulty of the method is its convergence, since the convergence proof and the practical implementation require the knowledge of the optimal dual value. Adaptive estimations of the optimal dual value may lead to divergence and the loss of the lower bound property for surrogate dual values. The main contribution of this paper is on the development of the surrogate Lagrangian relaxation method and its convergence proof to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations, a stepsizing formula that guarantees convergence without requiring the optimal dual value has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. Testing results demonstrate that non-smooth dual functions can be efficiently optimized, and the new method leads to faster convergence as compared to other methods available for optimizing non-smooth dual functions, namely, the simple subgradient method, the subgradient-level method, and the incremental subgradient method.

What objective function should be used for optimal auctions in the ISO/RTO electricity market?

In this paper, we provide mathematical formulations for the offer cost and MCP payment cost minimizations for optimal auctions in the ISO/RTO electricity market, and summarize the newly developed solution methodology using augmented Lagrangian relaxation and surrogate optimization for solving the optimal auction with the MCP payment objective function. Data has been used to test the method based on a simplified energy market, and for a given set of offers, the testing result demonstrates significant potential savings for electricity consumers if the MCP payment cost minimization is implemented in the ISO/RTO electricity markets. More importantly, this paper addresses economic implications of the objective function choice, including whether maximizing social welfare should be one of objectives of electricity industry deregulation. We conclude that an objective to maximize social welfare, even if it were determined to be desirable, is not achievable based on current bidding rules after moving from traditional vertically integrated utilities to a market approach, and is certainly not achieved by the offer cost minimization approach in use today. Other implications such as the inconsistency between the actual payment and the cost function minimized, and bidding behaviors are also discussed

Payment cost minimization with transmission capacity constraints and losses using the objective switching method

Deregulated electricity markets in the U.S. currently minimize total bid costs to select bids and their generation levels but determine payments based on market clearing prices. The inconsistency between auction and settlement mechanisms can lead to a significantly higher consumer payment. This gives rise to the “payment cost minimization,” an alternative auction mechanism that minimizes consumer payments directly. This paper formulates payment cost minimization problems with transmission capacity constraints and losses. DC power flow is used to model the transmitted power. The locational marginal prices are defined by “economic dispatch” and characterized by using the Karush-Kuhn-Tucker conditions. The formulation is converted to linear to be solved by the branch-and-cut method in standard commercial solver CPLEXs MIP. Specific methods for the linear conversion are highlighted. The efficiency for solving this linear payment cost minimization model in CPLEXs MIP is still low. The difficulties are studied by comparing the convex hulls of the two auction problems. To overcome the difficulties and improve the efficiency, the new “objective switching method” is developed which can be also used for solving other NP hard problems. Performance cuts are first generated to reduce the feasible region. The infeasibilities of originally discrete variables are then minimized within the reduced region to find one of many feasible near-optimal solutions with quantifiable quality. Numerical testing results of small examples and IEEE Reliability Test System demonstrate the effectiveness and efficiency of the model and the method.

An efficient surrogate subgradient method within Lagrangian relaxation for the Payment Cost Minimization problem

Studies have shown that for a given set of bids, Payment Cost Minimization leads to lower customer payments as compared to Bid Cost Minimization. In order to provide a thorough analysis of the two mechanisms an efficient solution methodology is required. It has previously been shown that the surrogate optimization within the Lagrangian relaxation framework can lead to savings in the CPU time while ensuring a high-quality solution. This paper develops an efficient methodology to solve Payment Cost Minimization using the surrogate optimization framework and the branch-and-cut method. In the presented methodology the problem structure is exploited using Lagrangian relaxation and the relaxation of the integrality constraints is exploited using branch-and-cut. The resulting method is further improved by using additional cutting planes that reduce the search space and by the advanced start to reinitialize the decision variables at each iteration. For large Payment Cost Minimization problems, the method can find significantly better feasible solutions within less CPU time than that obtained by standard branch-and-cut methods implemented in commercial MIP solver. The methodology developed in this paper is generic and can be used for solving other optimization problems.

Payment versus bid cost [The Business Scene]

This article summarizes the recent developments in the solution methodology of payment-cost minimization and the economic analysis of the two auction methods. Topics such as revenue adequacy implications are brought into discussion. Generally speaking, the research on the appraisal of the two auction methods is still at the early stages, and that a comprehensive study of the two auction methods is highly valuable for both researchers and industrial practitioners. This article can initiate more serious debate among researchers and stakeholders as to which objective should be used in ISO markets.

An efficient approach for solving mixed-integer programming problems under the monotonic condition

Many important integer and mixed-integer programming problems are difficult to solve. A representative example is unit commitment with combined cycle units and transmission capacity constraints. Complicated transitions within combined cycle units are difficult to follow, and system-wide coupling transmission capacity constraints are difficult to handle. Another example is the quadratic assignment problem. The presence of cross-products in the objective function leads to nonlinearity. In this study, building upon the novel integration of surrogate Lagrangian relaxation and branch-and-cut, such problems will be solved by relaxing selected coupling constraints. Monotonicity of the relaxed problem will be assumed and exploited and nonlinear terms will be dynamically linearised. The linearity of the resulting problem will be exploited using branch-and-cut. To achieve fast convergence, guidelines for selecting stepsizing parameters will be developed. The method opens up directions for solving nonlinear mixed-in...

Novel exploitation of convex hull invariance for solving unit commitment by using surrogate Lagrangian relaxation and branch-and-cut

Many important problems in power systems, including unit commitment and economic dispatch (UCED), are modeled as MILP problems which are computationally intensive. Such problems can often be viewed as subsystems coupled by system-wide constraints. These structures have been efficiently exploited by surrogate Lagrangian relaxation and branch-and-cut to solve UCED problems with combined cycle units by relaxing system-wide coupling constraints, decomposing the relaxed problem into subproblems, and solving each subproblem by branch-and-cut. However, while complicated features of a subproblem are handled locally within that subproblem, there is no guarantee that each subproblem can be efficiently solved because corresponding convex hulls are generally difficult to obtain, and the overall computational effort may be significant. In this paper, this difficulty is alleviated by the novel exploitation of the invariance of subproblem constraints and the associated convex hulls with respect to updating of Lagrange multipliers. Consequently, cuts remain valid throughout the entire iterative process. When cuts are retained, solving subproblems in subsequent iterations becomes easier than starting from scratch. This idea is operationalized in CPLEX by using C Concert Technology to extract, save and load CPLEX-generated cuts by using callable libraries. Numerical results demonstrate that the new approach is computationally efficient and generates good feasible solutions.