Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis

A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap.This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector

Smoothing Methods for Nonlinear Complementarity Problems

International audienceIn this paper, we present a new smoothing approach to solve general nonlinear complementarity problems. Under the P0 condition on the original problems, we prove some existence and convergence results . We also present an error estimate under a new and general monotonicity condition. The numerical tests confirm the efficiency of our proposed methods

Implementation of a continuation method for nonlinear complementarity problems via normal maps

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references.In this thesis, a continuation method for nonlinear complementarity problems via normal maps that is developed by Chen, Harker and Pinar [8] is implemented. This continuation method uses the smooth function to approximate the normal map reformulation of nonlinear complementarity problems. The algorithm is implemented and tested with two different plussmoothing functions namely interior point plus-smooth function and piecewise quadratic plus-smoothing function. These two functions are compared. Testing of the algorithm is made with several known problems.Erkan, AliM.S

Moving from Linear to Conic Markets for Electricity

We propose a new forward electricity market framework that admits heterogeneous market participants with second-order cone strategy sets, who accurately express the nonlinearities in their costs and constraints through conic bids, and a network operator facing conic operational constraints. In contrast to the prevalent linear-programming-based electricity markets, we highlight how the inclusion of second-order cone constraints enables uncertainty-, asset- and network-awareness of the market, which is key to the successful transition towards an electricity system based on weather-dependent renewable energy sources. We analyze our general market-clearing proposal using conic duality theory to derive efficient spatially-differentiated prices for the multiple commodities, comprising of energy and flexibility services. Under the assumption of perfect competition, we prove the equivalence of the centrally-solved market-clearing optimization problem to a competitive spatial price equilibrium involving a set of rational and self-interested participants and a price setter. Finally, under common assumptions, we prove that moving towards conic markets does not incur the loss of desirable economic properties of markets, namely market efficiency, cost recovery and revenue adequacy. Our numerical studies focus on the specific use case of uncertainty-aware market design and demonstrate that the proposed conic market brings advantages over existing alternatives within the linear programming market framework.Comment: Manuscript with electronic companion; submitted to Operations Researc

Price-based optimal control of electrical power systems

During the past decade, electrical power systems have been going through some major restructuring processes. From monopolistic, highly regulated and one utility controlled operation, a system is being restructured to include many parties competing for energy production and consumption, and for provision of many of the ancillary services necessary for system operation. With the emergence of competitive markets as central operational mechanisms, the prime operational objective has shifted from a centralized, utility cost minimization objective to decentralized, profit maximization objectives of competing parties. The market-based (price-based) operation is shown to be practically the only approach that is capable to simultaneously provide incentives to hold the prices at marginal costs and to minimize the costs. As a result, such an operational structure inherently tends to maximize the social welfare of the system during its operation, and to accelerate developments and applications of new technologies. Another major change that is taking place in today’s power systems is an increasing integration of small-scale distributed generation (DG) units. Since in future power systems, a large amounts of DG will be based on renewable, intermittent energy sources, e.g. wind and sun, these systems will be characterized by significantly larger uncertainties than those of the present power systems. Power markets significantly deviate from standard economics since the demand side is largely disconnected from the market, i.e. it is not price responsive, and it exhibits uncertain, stochastic behavior. Furthermore, since electrical energy cannot be efficiently stored in large quantities, production has to meet these rapidly changing demands in real-time. In future power systems, efficient real-time power balancing schemes will become crucial and even more challenging due to the significant increase of uncertainties by large-scale integration of renewable sources. Physical and security limits on the maximal power flows in the lines of power transmission networks represent crucial system constraints, which must be satisfied to protect the integrity of the system. Creating an efficient congestion management scheme for dealing with these constraints is one of the toughest problems in the electricity market design, as the line power flows are characterized by complex dependencies on nodal power injections. Efficient congestion control has to account for those limits by adequately transforming them into market signals, i.e. into electricity prices. One of the main contributions of this thesis is the development of a novel dynamic, distributed feedback control scheme for optimal real-time update of electricity prices. The developed controller (which is called the KKT controller in the thesis) reacts on the network frequency deviation as a measure of power imbalance in the system and on measured violations of line flow limits in a transmission network. The output of the controller is a vector of nodal prices. Each producer/consumer in the system is allowed to autonomously react on the announced price by adjusting its production/consumption level to maximize its own benefit. Under the hypothesis of global asymptotic stability of the closed-loop system, the developed control scheme is proven to continuously balance the system by driving it towards the equilibrium where the transmission power flow constraints are satisfied, and where the total social welfare of the system is maximized. One of the advantageous features of the developed control scheme is that, to achieve this goal, it requires no knowledge of marginal cost/benefit functions of producers/consumers in the system (neither is it based on the estimates of those functions). The only system parameters that are explicitly included in the control law are the transmission network parameters, i.e. network topology and line impedances. Furthermore, the developed control law can be implemented in a distributed fashion. More precisely, it can be implemented through a set of nodal controllers, where one nodal controller (NC) is assigned to each node in the network. Each NC acts only on locally available information, i.e. on the measurements from the corresponding node and on the information obtained from NC’s of the adjacent nodes. The communication network graph among NC’s is therefore the same as the graph of the underlying physical network. Any change is the network topology requires only simple adjustments in NC’s that are local to the location of the change. To impose the hard constraints on the level to which the transmission network lines are overloaded during the transient periods following relatively large power imbalances in the system, a novel price-based hybrid model predictive control (MPC) scheme has been developed. The MPC control action adds corrective signals to the output of the KKT controller, i.e. to the nodal prices, and acts only when the predictions indicate that the imposed hard constraint will be violated. In any other case, output of the MPC controller is zero and only the KKT controller is active. Under certain hypothesis, recursive feasibility and asymptotic stability of the closed-loop system with the hybrid MPC controller are proven. Next contribution of this thesis is formulation of the autonomous power networks concept as a multilayered operational structure of future power systems, which allows for efficient large-scale integration of DG and smallscale consumers into power and ancillary service markets, i.e. markets for different classes of reserve capacities. An autonomous power network (AN) is an aggregation of networked producers and consumers, whose operation is coordinated/controlled with one central unit (AN market agent). By performing optimal dispatching and unit commitment services, the main goals of an AN market agent is to efficiently deploy the AN’s internal resources by its active involvement in power and ancillary service markets, and to optimally account for the local reliability needs. An autonomous power network is further defined as a major building block of power system operation, which is capable of keeping track of its contribution to the uncertainty in the overall system, and is capable of bearing the responsibility for it. With the introduction of such entities, the conditions are created that allow for the emergence of novel, competitive ancillary service market structures. More precisely, in ANs based power systems, each AN can be both producer and consumer of ancillary services, and ancillary service markets are characterized by double-sided competition, what is in contrast to today’s single-sided ancillary service markets. One of the main implications of this novel operational structure in that, by facilitating competition, it creates the strong incentive for ANs to reduce the uncertainties and to increase reliability of the system. On a more technical side, the AN concept is seen as decentralization and modularization approach for dealing with the future, large scale, complex power systems. As additional contribution of this thesis, motivated by the KKT controller for price-based real-time power balancing and congestion management, the general KKT control paradigm is presented in some detail. The developed control design procedure presents a solution to the problem of regulating a general linear time-invariant dynamical system to a time-varying economically optimal operating point. The system is characterized with a set of exogenous inputs as an abstraction of time-varying loads and disturbances. Economic optimality is defined through a constrained convex optimization problem with a set of system states as decision variables, and with the values of exogenous inputs as parameters in the optimization problem. A KKT controller belongs to a class of dynamic complementarity systems, which has been recently introduced and which has, due to its wide applicability and specific structural properties, gained a significant attention in systems and control community. The results of this thesis add to the list of applications of complementarity systems in control