Kuhn Tucker Optimality Conditions Research Articles

Strategic Offering for Wind Power Producers Considering Energy and Flexible Ramping Products

The increasing deployments of renewable generation methods, such as wind, affects the flexibility of electric power system operating due to their inherent variability and uncertainty. To mitigate this, power systems need flexible resources. This paper investigates the potential for wind power to provide flexible ramping products in the real-time market, an additional value stream to the energy it provides. The proposed model for wind power’s strategic offering is formulated as a bi-level optimization problem with wind profit maximization at the upper level and the independent system operator’s economic dispatch—considering both the energy balance and the flexible ramping requirement to counter uncertainty—at the lower level. This bi-level model is converted to a mathematical program with equilibrium constraints (MPEC) by recasting the lower level problem with its Karush–Kuhn–Tucker optimality conditions. Then, through strong duality theory and the big-M method, the MPEC model is converted to a mixed-integer linear programming model. The opportunity cost and the price for wind power-providing ramping products are analyzed. Numerical examples based on a 5-bus network are presented to verify the proposed model and concept.

Optimal power allocation scheme for distributed antenna system with OFDM in frequency‐selective fading channels

Optimal power allocation (PA) scheme for distributed antenna systems with orthogonal frequency division multiplexing (DAS-OFDM) over frequency-selective fading channels is presented. The PA design is to optimise the transmit powers of subcarriers to maximise the capacity subject to the total power constraint. Based on the Karush–Kuhn–Tucker optimality conditions, the optimal PA to the distributed nodes for a subcarrier can be simplified to allocating the power to the node of the largest channel gain. With this result, an efficient algorithm is derived to obtain the optimal PA to the nodes for all the subcarriers of the DAS-OFDM. Compared with an existing suboptimal scheme, the developed scheme can obtain higher capacity and has lower complexity. By computer simulations, the effectiveness of the proposed scheme is verified.

An ADMM with continuation algorithm for non-convex SICA-penalized regression in high dimensions

The smooth integration of counting and absolute deviation (SICA) penalty has been demonstrated theoretically and practically to be effective in non-convex penalization for variable selection. However, solving the non-convex optimization problem associated with the SICA penalty when the number of variables exceeds the sample size remains to be enriched due to the singularity at the origin and the non-convexity of the SICA penalty function. In this paper, we develop an efficient and accurate alternating direction method of multipliers with continuation algorithm for solving the SICA-penalized least squares problem in high dimensions. We establish the convergence property of the proposed algorithm under some mild regularity conditions and study the corresponding Karush–Kuhn–Tucker optimality condition. A high-dimensional Bayesian information criterion is developed to select the optimal tuning parameters. We conduct extensive simulations studies to evaluate the efficiency and accuracy of the proposed algorithm, while its practical usefulness is further illustrated with a high-dimensional microarray study.

Research on Optimizing Trading Strategy of an Electricity Retailer Considering Users’ Response under New Strategy of Electric Power System Reformation

Under New Strategy of Electric Power System Reformation (NSEPSR), an electricity retailer decides its level of involvement in the contract market and in the spot market as well as the selling price offered to its clients with the goal of maximizing the expected profit at its acceptable risk level. Under time-of-use pricing, users will adjust their needs according to the price changes. How to determine the time-of-use electricity price and the power purchase strategy in the contract market and in the spot market while accounting for the profit of the retailer and users becomes an important topic. Based on the above reasons, this paper formulates the optimal pricing and dispatch problem of an electricity retailer as a bilevel programming model, in which the upper level maximizes the retailer’s benefit, while the lower level minimizes the cost of each electricity user. The model is transformed into a mixed integer linear program by jointly using the Karush- Kuhn-Tucker (KKT) optimality condition as well as the duality theorem of linear programming. Finally, a case is studied to analyze effect of risk preference and market price volatility on electricity retailer profit. The result can provide reference for the electricity retailer to make the purchase and sale strategy.

A novel neural network model for solving a class of nonlinear semidefinite programming problems

In this paper, we describe a dynamic optimization technique for solving a class of nonlinear semidefinite programming based on Karush–Kuhn–Tucker optimality conditions. By employing Lyapunov function approach, it is investigated that the suggested neural network is stable in the sense of Lyapunov and globally convergent to an exact optimal solution of the original problem. The effectiveness of the proposed method is demonstrated by two numerical simulations.

Optimal Allocation Model for EV Charging Stations Coordinating Investor and User Benefits

Deploying charging stations (CSs) catering to the demand from electric vehicles plays an important role in modernizing energy infrastructures. In this paper, a bi-level optimal allocation model for allocating fast CSs is proposed aiming to maximize the CS investor benefit (upper layer sub-problem) by optimally allocating CSs with the coordinated determination of the expected efficiency of charging service supply (lower layer sub-problem). The efficiency is formulated as a charging performance index in terms of user satisfaction degree, which mathematically couples the upper level and lower level sub-problems. The proposed nonlinear bi-level model is reduced to a single-layer optimization model under the Karush–Kuhn–Tucker optimality conditions. An improved dynamic differential evolution algorithm with an adaptive update strategy is proposed to solve this single-layer optimization model. The proposed method is validated using a realistic case. The test results show that the proposed methodology is able to co-ordinate both objectives of interest, i.e., allocating CSs network with the maximized benefits and optimizing the fast charging service efficiency at the same time.

Inverse Parametric Optimization in a Set-Membership Error-in-Variables Framework

In this study, we aim to recover the interval hull of the set of feasible cost functions that can make uncertain observations optimal for a class of nonlinear constrained parametric optimization problems when all uncertainty and disturbances acting on observations or modeling are taken bounded but otherwise unknown. Fostering on inverse Karush–Kuhn–Tucker optimality conditions, we first state the solving equations as a constraint satisfaction problem, then show how to derive a safe overapproximation of the feasible solution set combining standard numerical tools and a posteriori validation with guaranteed methods based on interval analysis. The approach is evaluated on two well-tuned numerical examples: A discrete unicycle robot model and a planar elastica model, respectively.

Quasi-variational equilibrium models for network flow problems

We consider a formulation of a network equilibrium problem given by a suitable quasi-variational inequality where the feasible flows are supposed to be dependent on the equilibrium solution of the model. The Karush–Kuhn–Tucker optimality conditions for this quasi-variational inequality allow us to consider dual variables, associated with the constraints of the feasible set, which may receive interesting interpretations in terms of the network, extending the classic ones existing in the literature.

Particle Swarm Optimization Based on Smoothing Approach for Solving a Class of Bi-Level Multiobjective Programming Problem

Abstract As a metaheuristic, Particle Swarm Optimization (PSO) has been used to solve the Bi-level Multiobjective Programming Problem (BMPP). However, in the existing solving approach based on PSO for the BMPP, the upper level and the lower level problem are solved interactively by PSO. In this paper, we present a different solving approach based on PSO for the BMPP. Firstly, we replace the lower level problem of the BMPP with Kuhn-Tucker optimality conditions and adopt the perturbed Fischer-Burmeister function to smooth the complementary conditions. After that, we adopt PSO approach to solve the smoothed multiobjective programming problem. Numerical results show that our solving approach can obtain the Pareto optimal front of the BMPP efficiently.

A Stochastic Bilevel Model for the Energy Hub Manager Problem

A bilevel stochastic programming problem (BSPP) model of the decision-making of an energy hub manager is presented. Hub managers seek ways to maximize their profit by selling electricity and heat. They have to make decisions about: 1) the level of involvement in forward contracts, electricity pool markets, and natural gas networks and 2) the electricity and heat offering prices to the clients. These decisions are made under uncertainty of pool prices, demands as well as the prices offered by rival hub managers. On the other hand, the clients try to minimize the total cost of energy procurement. This two-agent relationship is presented as a BSPP in which the hub manager is placed in the upper level and the clients in the lower one. The bilevel scheme is converted to its equivalent single-level scheme using the Karush–Kuhn–Tucker optimality conditions although there are two bilinear products related to electricity and heat. The heat bilinear product is replaced by a heat price-quota curve and the electricity bilinear product is linearized using the strong duality theorem. In addition, conditional value at risk is used to reduce the unfavorable effects of the uncertainties. The effectiveness of the proposed model is evaluated in various simulations of a realistic case study.

Operational scheduling of electric vehicles parking lot integrated with renewable generation based on bilevel programming approach

With increasing the share of electric vehicles in electricity market, it is important to investigate their impact on electricity trading and their interactions with other market entities involved in the system. This paper provides a methodology to develop the interaction between parking lot and distribution system operator in energy and reserve market while considering load and wind power uncertainty. To this end, a bilevel approach is applied to model inherently conflicting objective between distribution system operator and parking lot and interactions between the two agents. In the proposed model, upper-level problem represents the total operation cost minimization from the distribution system operator’s perspective while the lower-level problem represents the scheduling energy and reserve from the parking lot owner’s point of view with the objective of minimizing the parking cost. The method is capable of finding the equilibrium point for decision making conflict between the objective of the upper and lower level. The proposed bilevel problem is reduced to a single level optimization problem by implementing dual theorem and the Karush–Kuhn–Tucker optimality conditions. The numerical results illustrate the effectiveness of the proposed method.

New Second-Order Karush–Kuhn–Tucker Optimality Conditions for Vector Optimization

In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Frechet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem.

Tractable Transmit MIMO Beampattern Design Under a Constant Modulus Constraint

Multiple-input multiple-output (MIMO) radar systems allow each antenna element to transmit a different waveform. This waveform diversity can be exploited to enhance the beampattern design, in particular, effective management of radar radiation power in directions of interest. We address the problem of designing a beampattern for MIMO radar, which in turn is determined by the transmit waveform. While unconstrained design is straightforward, a key open challenge is enforcing the constant modulus constraint on the radar waveform. It is well known that the problem of minimizing deviation of the designed beampattern from an idealized one subject to the constant modulus constraint constitutes a hard nonconvex problem. Existing methods that address constant modulus invariably lead to a stiff tradeoff between analytical tractability (achieved by relaxations and approximations) and realistic design that exactly achieves constant modulus but is computationally burdensome. A new approach is proposed in our paper, which involves solving a sequence of convex equality constrained quadratic programs, each of which has a closed form solution and such that constant modulus is achieved at convergence. We further prove that the converged solution satisfies the Karush–Kuhn–Tucker optimality conditions of the aforementioned hard nonconvex problem. We evaluate the proposed successive closed forms (SCF) algorithm against the state-of-the art MIMO beampattern design techniques in both narrowband and wideband setups and show that the SCF breaks the tradeoff between desirable performance and the associated computation cost.

Dynamic real‐time optimization with closed‐loop prediction

Process plants are operating in an increasingly global and dynamic environment, motivating the development of dynamic real‐time optimization (DRTO) systems to account for transient behavior in the determination of economically optimal operating policies. This article considers optimization of closed‐loop response dynamics at the DRTO level in a two‐layer architecture, with constrained model predictive control (MPC) applied at the regulatory control level. A simultaneous solution approach is applied to the multilevel DRTO optimization problem, in which the convex MPC optimization subproblems are replaced by their necessary and sufficient Karush–Kuhn–Tucker optimality conditions, resulting in a single‐level mathematical program with complementarity constraints. The performance of the closed‐loop DRTO strategy is compared to that of the open‐loop prediction counterpart through a multi‐part case study that considers linear dynamic systems with different characteristics. The performance of the proposed strategy is further demonstrated through application to a nonlinear polymerization reactor grade transition problem. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3896–3911, 2017

A Robust Linear Approach for Offering Strategy of a Hybrid Electric Energy Company

This paper presents a new approach for determining the day-ahead bidding strategies of a large-scale hybrid electric energy company. The company has both energy generation and energy retailing businesses in a competitive electricity market. Demand response programs are also considered in the retail side of the company in order to hedge the risk of participation in wholesale market. This paper introduces a max-min bilevel mathematical programming with equilibrium constraint model for offering a strategy that manages the risk of uncertain forecasted rivals’ bids by robust optimization. The max-min bilevel model is converted to its equivalent single-level optimization using Karush–Kuhn–Tucker optimality conditions. The duality theory is utilized to find the equivalent ordinary maximization model of the max-min problem. Strong duality theory and big M method are also used to linearize the final model of offering strategy. Applicability of the proposed approach is shown by implementing it on the IEEE 118-bus test system.

Stochastic–multiobjective market equilibrium analysis of a demand response program in energy market under uncertainty

In the electricity market, demand response programs are designed to shift peak demand and enhance system reliability. A demand response program can reduce peak energy demand, power transmission congestion, or high energy-price conditions by changing consumption patterns. The purpose of this research is to analyze the impact of a demand response program in the energy market, under demand uncertainty. A stochastic–multiobjective Nash–Cournot competition model is formulated to simulate demand response in an uncertain energy market. Then, Karush–Kuhn–Tucker optimality conditions and a linear complementarity problem are derived for the stochastic Nash–Cournot model. Accordingly, the linear complementarity problem is solved and its stochastic market equilibrium solution is determined by using a general algebraic modeling system. Additionally, the case of the Taiwanese electric power market is taken up here, and the results show that a demand response program is capable of reducing peak energy consumption, energy price, and carbon dioxide emissions. The results show that demand response program decreases electricity price by 2–10%, total electricity generation by 0.5–2%, and carbon dioxide emissions by 0.5–2.5% in the Taiwanese power market. In the simulation, demand uncertainty leads to an 2–7% increase in energy price and supply risk in the market. Additionally, tradeoffs between cost and carbon dioxide emissions are presented.

Higher-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization with nonsolid ordering cones

In this paper, we consider higher-order Karush–Kuhn–Tucker optimality conditions in terms of radial derivatives for set-valued optimization with nonsolid ordering cones. First, we develop sum rules and chain rules in the form of equality for radial derivatives. Then, we investigate set-valued optimization including mixed constraints with both ordering cones in the objective and constraint spaces having possibly empty interior. We obtain necessary conditions for quasi-relative efficient solutions and sufficient conditions for Pareto efficient solutions. For the special case of weak efficient solutions, we receive even necessary and sufficient conditions. Our results are new or improve recent existing ones in the literature.

Congestion Management in Modern Power System Containing Active Distribution Network Nodes

This article presents a novel methodology to determine a transmission congestion management strategy for a power system containing active distribution network nodes. Constrained scheduling, used in traditional pools, has been utilized. A bi-level optimization model is proposed to obtain the optimal congestion management strategy by rescheduling the distributed generators inside active distribution network nodes. The upper-level optimization model determines the generation rescheduling in the main grid and the load adjustment of each active distribution network node to relieve the congestion. The lower-level optimization model aims at minimizing load adjustment cost in the upper level by rescheduling the distributed generators. The proposed bi-level optimization model is transformed into an equivalent single-level optimization model through application of Karush–Kuhn–Tucker optimality conditions. Three heuristic algorithms—particle swarm optimization, cat swarm optimization, and clonal selection algorithm—have been applied to solve the equivalent single-level optimization problem. The effectiveness of the proposed methodology has been tested on a modified system using the IEEE 30-bus system as the main network and the IEEE 14-bus system as the active distribution network node. The results obtained by the particle swarm optimization algorithm have been compared with those obtained by cat swarm optimization and clonal selection algorithms.

Multi-material topology optimization using ordered SIMP interpolation

In this paper an ordered multi-material SIMP (solid isotropic material with penalization) interpolation is proposed to solve multi-material topology optimization problems without introducing any new variables. Power functions with scaling and translation coefficients are introduced to interpolate the elastic modulus and the cost properties for multiple materials with respect to the normalized density variables. Besides a mass constraint, a cost constraint is also considered in compliance minimization problems. A heuristic updating scheme of the design variables is derived from the Kuhn-Tucker optimality condition (OC). Since the proposed method does not rely on additional variables to represent material selection, the computational cost is independent of the number of materials considered. The iteration scheme is designed to jump across the discontinuous point of interpolation derivatives to make stable transition from one material phase to another. Numerical examples are included to demonstrate the proposed method. Because of its conceptual simplicity, the proposed ordered multi-material SIMP interpolation can be easily embedded into any existing single material SIMP topology optimization codes.

Strategic bidding for price-maker producers in predominantly hydroelectric systems

This paper proposes an equilibrium model to determine price and quantity strategic bids for generation companies participating in a day-ahead electricity market, with predominantly hydroelectric generation. Each agent aims to maximize its profit by solving a bi-level optimization problem, where the upper level represents the producer revenue maximization problem, while the lower one consists of minimizing the cost of system operation, faced by the independent system operator. Price-maker companies operating both hydro and thermal plants are considered in a cascade hydro system with reservoirs managed by different owners. Consequently, a novel approach is proposed to decouple the first and second levels of the problem. We present an individual plant modeling, where the main constraints related to a hydrothermal system are considered. Through the utilization of the Karush–Kuhn–Tucker optimality conditions, the bi-level optimization model is converted to a single level nonlinear problem, known in literature as a mathematical program with equilibrium constraints (MPEC). To face the difficulties of this nonlinear, nonconvex, multi-stage problem, the MPEC complementarity conditions are replaced by the strong duality condition. Moreover, competition among several leaders is modeled as an iterative procedure, and the methodology is applied in two systems with data and configurations derived from the Brazilian hydrothermal system.