Nonlinear Interior Point Algorithm Research Articles

Parameterization of linear supply functions in nonlinear AC electricity market equilibrium models - Part II: Case studies

This study utilizes the primal-dual nonlinear interior point algorithm formulated in Part I of this two-series paper, to obtain electricity market supply function equilibrium solutions for the bi-level market problem of nonlinear power systems with AC meshed networks. The strategic offers of the market players are modeled by parameterizing the generation marginal cost functions using any of the four available parameterization methods to obtain profit-maximizing linear supply function bids. The analysis takes into account a variety of test systems and examines the relation between the equilibrium solutions obtained from the different parameterization methods and the role of network complexity. An analysis of multiple equilibria in AC network constrained systems is also provided.

Parameterization of Linear Supply Functions in Nonlinear AC Electricity Market Equilibrium Models—Part I: Literature Review and Equilibrium Algorithm

The linear supply function equilibrium (SFE) model can be used to investigate bid-based electricity pool markets. Several studies assume that the market players construct optimal strategic bids by parameterizing the linear marginal cost function of their generating units. Four different types of parameterization are possible, for which the slope and/or the intercept of the marginal cost functions are varied. This paper reviews the existing literature and proposes a primal-dual nonlinear interior point algorithm to solve a bi-level market problem with AC network representation, using any of the parameterization methods. Part II of this paper uses the proposed algorithm to examine the effects of the different parameterizations on the resulting SFE solutions, the role of network complexity and the nature of the equilibrium points.

Reliability-Based Design Optimization of Uncertain Stochastic Systems: Gradient-Based Scheme

A method to carry out reliability-based optimization of uncertain structural systems under stochastic excitation is presented in this paper. The approach is based on a nonlinear interior point algorithm and a line search strategy. The associated reliability problems to be solved during the optimization process are high-dimensional (thousands or more random variables). An advanced Monte Carlo simulation is adopted for the purpose of estimating the corresponding failure probabilities. The gradients of the failure probability functions needed during the design process are estimated by an approach based on the local behavior of the normalized demand functions that define the failure domains. Numerical results show that only a small number of reliability estimates has to be performed during the entire design process. By construction, the design scheme is monotonically convergent; that is, it generates a sequence of steadily improved feasible designs. Example problems that consider a resistant element of a shea...

Electricity market equilibrium analysis based on nonlinear interior point algorithm with complementarity constraints

This paper presents an interior point algorithm based on a.c. network model for determining the Nash supply function equilibrium (SFE) of bid-based electricity markets. The SFE problem is considered as a bi-level game. At the first level, the problem begins with the formulation of an optimal power flow (OPF) interior point-based algorithm to handle the independent system operator (ISO) problem for maximising social welfare. This algorithm is based on the OPF with a.c. network transmission model taking into account all the operating aspects such as the generation capacity limits, bus voltage limits, transmission line constraints, network losses and especially the effect of the reactive power. The resulting Karush-Kuhn-Tucker (KKT) conditions of the problem at the first level are then reformulated using nonlinear complementarity constraints and incorporated as equality constraints in the second-level formulation for maximising the individual profit for each strategic generating firm. By employing a special nonlinear complementarity function, the complementarity constraints are then transformed into nonlinear algebraic expressions, thus the KKT conditions of the resulting combined problem can be derived. The final problem is then solved iteratively based on the solution techniques of the interior point algorithm. Numerical examples of a three-bus system, the IEEE 14-bus system and the IEEE 30-bus system, show that the algorithm can successfully determine the electricity market equilibrium with the a.c. network model.

Parameter estimation in active plate structures using gradient optimisation and neural networks

Two non-destructive methods for elastic and piezoelectric parameter estimation in active plate structures with surface bonded piezoelectric patches are presented. The first of these solves the inverse problem through gradient based optimisation techniques, minimising the difference between experimental and numerical finite element eigenfrequencies for a test plate. Such minimisation is conducted with feasible arc interior point algorithm (FAIPA), a non-linear interior point algorithm. The second method relies on building a metamodel of the inverse problem, using artificial neural networks (ANNs). The training data set is obtained through the same numerical model as in the first approach. The simulation of the network is then used with the experimental eigenfrequency data set in order to produce an estimate for the material parameters. Results from both approaches are compared and discussed through a simulated identification.

Initialisation, decoupling, hot start, and warm start in direct nonlinear interior point algorithm for optimal power flows

Effective approaches dealing with four important issues in a direct nonlinear interior point algorithm (DNIPA) for optimal power flows (OPF) are presented. The first issue, initialisation, is important for the robustness of DNIPA; the second issue, decoupling, is important for OPF applications. The other two issues are particularly important to OPF performance in energy management systems. The approaches enhance the performance of DNIPA for OPF problems in terms of iteration counts and CPU times. Extensive numerical tests are performed on two different sizes of real-life systems: 244- and 2423-bus systems. Based on these results, the initialisation approach improves PCPDIPA by 24% in term of CPU times. The decoupling approach enhances PCPDIPA (with initialisation) by 27%. The hot start and warm start solve the problems about 2.0 and 2.7 times, respectively, faster than PCPDIPA with initialisation, and 2.5 and 3.3 times, respectively, faster than PCPDIPA without using initialisation.

A New Scenario Decomposition Method for Large-Scale Stochastic Optimization

A novel parallel decomposition algorithm is developed for large, multistage stochastic optimization problems. The method decomposes the problem into subproblems that correspond to scenarios. The subproblems are modified by separable quadratic terms to coordinate the scenario solutions. Convergence of the coordination procedure is proven for linear programs. Subproblems are solved using a nonlinear interior point algorithm. The approach adjusts the degree of decomposition to fit the available hardware environment. Initial testing on a distributed network of workstations shows that an optimal number of computers depends upon the work per subproblem and its relation to the communication capacities. The algorithm has promise for solving stochastic programs that lie outside current capabilities.