Kuhn Tucker Optimality Conditions Research Articles

Strategic scheduling of energy storage for load serving entities in locational marginal pricing market

Cost-effective approaches of storing electrical energy on a large-scale can help the grid operate flexibly and reliably. Public utility commissions view energy storage (ES) as a vital yet complementary part of other clean technologies such as renewable generation. As a coherent framework is being mapped out to increase the penetration of renewables in total electricity generation, load serving entities (LSEs) have been encouraged to procure ES systems to hedge against the intermittence and uncertainty of renewable power such as wind. Owing to the increasing penetration of wind generation, it becomes more complex for LSEs to obtain a deterministic ES scheduling decision, i.e. charging/discharging portfolio. To address this challenge, a bi-level strategic scheduling model is proposed in which the primary objective is to maximise the LSE’s profit by optimally scheduling ES charging/discharging profile. The sub-problem is Independent System Operator’s (ISO’s) economic dispatch for generation cost minimisation, assuming foreseeable impact in market-clearing price from ES. This bi-level model is converted to a stochastic mathematic programme with equilibrium constraints by recasting the lower-level problem as its Karush–Kuhn–Tucker optimality conditions. Numerical examples based on the PJM 5-bus and IEEE 118- bus systems are presented to demonstrate and validate the proposed approach.

Incorporating price-responsive customers in day-ahead scheduling of smart distribution networks

Demand response and real-time pricing of electricity are key factors in a smart grid as they can increase economic efficiency and technical performances of power grids. This paper focuses on incorporating price-responsive customers in day-ahead scheduling of smart distribution networks under a dynamic pricing environment. A novel method is proposed and formulated as a tractable mixed integer linear programming optimization problem whose objective is to find hourly sale prices offered to customers, transactions (purchase/sale) with the wholesale market, commitment of distribution generation units, dispatch of battery energy storage systems and planning of interruptible loads in a way that the profit of the distribution network operator is maximized while customers’ benefit is guaranteed. To hedge distribution network operator against financial risk arising from uncertainty of wholesale market prices, a risk management model based on a bi-level information-gap decision theory is proposed. The proposed bi-level problem is solved by recasting it into its equivalent single-level robust optimization problem using Karush–Kuhn–Tucker optimality conditions. Performance of the proposed model is verified by applying it to a modified version of the IEEE 33-bus distribution test network. Numerical results demonstrate the effectiveness and efficiency of the proposed method.

Incorporating Price-Responsive Demand in Energy Scheduling Based on Consumer Payment Minimization

In the restructured power industry, demand responsiveness is a key factor whose importance will be boosted due to the impetus provided by the development of smart grids. Within the context of pool-based electricity markets, this paper addresses the incorporation of price-responsive demand in multiperiod energy scheduling driven by consumer payment minimization. Although consumer payment minimization has drawn considerable attention mainly in an operational setting and also recently under a planning framework, available models and solution approaches typically neglect demand-side participation. The proposed scheduling model considers a marginal pricing scheme as well as the effects of both network constraints and intertemporal constraints associated with generation operation. Modeling demand-side participation leads to bilinear payment terms that significantly increase the mathematical complexity of the optimization process. The resulting problem is formulated as a mixed-integer nonlinear bilevel program for which no exact solution technique is currently available. This paper presents a novel methodology by which the original bilevel and bilinear problem is converted into an equivalent single-level mixed-integer linear program suitable for efficient off-the-shelf software. This transformation is based on the application of duality theory of linear programming, integer algebra, and Karush–Kuhn–Tucker optimality conditions. The proposed approach has been successfully applied to the IEEE 118-bus system.

Group search optimiser-based optimal bidding strategies with no Karush–Kuhn–Tucker optimality conditions

General strategic bidding procedure has been formulated in the literature as a bi-level searching problem, in which the offer curve tends to minimise the market clearing function and to maximise the profit. Computationally, this is complex and hence, the researchers have adopted Karush–Kuhn–Tucker (KKT) optimality conditions to transform the model into a single-level maximisation problem. However, the profit maximisation problem with KKT optimality conditions poses great challenge to the classical optimisation algorithms. The problem has become more complex after the inclusion of transmission constraints. This paper simplifies the profit maximisation problem as a minimisation function, in which the transmission constraints, the operating limits and the ISO market clearing functions are considered with no KKT optimality conditions. The derived function is solved using group search optimiser (GSO), a robust population-based optimisation algorithm. Experimental investigation is carried out on IEEE 14 as well as IEEE 30 bus systems and the performance is compared against differential evolution-based strategic bidding, genetic algorithm-based strategic bidding and particle swarm optimisation-based strategic bidding methods. The simulation results demonstrate that the obtained profit maximisation through GSO-based bidding strategies is higher than the other three methods.

Dynamic User Clustering and Power Allocation for Uplink and Downlink Non-Orthogonal Multiple Access (NOMA) Systems

Non-orthogonal multiple access (NOMA) has recently been considered as a key enabling technique for 5G cellular systems. In NOMA, by exploiting the channel gain differences, multiple users are multiplexed into transmission power domain and then non-orthogonally scheduled for transmission on the same spectrum resources. Successive interference cancellation (SIC) is then applied at the receivers to decode the message signals. In this paper, first, we briefly describe the differences in the working principles of uplink and downlink NOMA transmissions in a cellular wireless system. Then, for both uplink and downlink NOMAs, we formulate a sum-throughput maximization problem in a cell such that the user clustering (i.e., grouping users into a single cluster or multiple clusters) and power allocations in NOMA clusters can be optimized under transmission power constraints, minimum rate requirements of the users, and SIC constraints. Due to the combinatorial nature of the formulated mixed integer non-linear programming problem, we solve the problem in two steps, i.e., by first grouping users into clusters and then optimizing their respective power allocations. In particular, we propose a low-complexity sub-optimal user grouping scheme. The proposed scheme exploits the channel gain differences among users in an NOMA cluster and groups them into a single cluster or multiple clusters in order to enhance the sum-throughput of the system. For a given set of NOMA clusters, we then derive the optimal power allocation policy that maximizes the sum-throughput per NOMA cluster and in turn maximizes the overall system throughput. Using Karush–Kuhn–Tucker optimality conditions, closed-form solutions for optimal power allocations are derived for any cluster size, considering both uplink and downlink NOMA systems. Numerical results compare the performances of NOMA and OMA and illustrate the significance of NOMA in various network scenarios.

Network utility maximization in multipath lossy wireless networks

SummaryThe end‐to‐end rate control in multipath wireless multihop networks has been widely studied so as to exploit path diversity and improve network performance. However, it has been recognized that the hop‐by‐hop congestion control is stable and spatially smoothed in the absence and presence of delay, respectively. As a result, we are motivated to consider the lossy features of wireless links and propose a novel utility maximization problem for multipath communication networks. In view of the fact that the considered optimization problem formulated as a multipath network utility maximization problem is in non‐convex form and hard to tackle, we first cast the underlying problem into a convex one and then solve the convex approximation problem by the Lagrangian duality technique. What is more, an efficient hop‐by‐hop rate control algorithm is proposed. In addition, we prove that the proposed two‐loop algorithm converges to the optimal solution satisfying the Karush–Kuhn–Tucker optimality conditions of the original optimization problem. We finally validate the effectiveness of the proposed algorithm through extensive simulations and compare the performance of the proposed algorithm with a hop‐by‐hop single‐path rate control algorithm and also with an end‐to‐end multipath algorithm. Copyright © 2015 John Wiley & Sons, Ltd.

Medium-term retailer's planning and participation strategy considering electricity market uncertainties

This paper presents a risk-constrained programming approach to solve a retailer's medium-term planning problem. A retailer tries to maximize its profit via determining the optimal price offered to the customers as well as optimal strategy of participating in futures and pool markets. The uncertainty of pool prices is modeled by an envelope-bound information-gap model. Another source of uncertainty in this problem is the clients' demand, which is considered via a scenario generation method. The proposed method is formulated as a bi-level stochastic programming problem based on the information-gap decision theory. The Karush–Kuhn–Tucker optimality conditions are used to convert the bi-level problem into a single-level robust optimization problem. The performance of the proposed method is demonstrated using a case study of the New England market, and results are discussed. Copyright © 2015 John Wiley & Sons, Ltd.

A hybrid optimization model for proficient bidding strategies with no Karush–Kuhn–Tucker optimality conditions

SummaryThe computational complexity behind the bi‐level optimization problem has led the researchers to adopt Karush–Kuhn–Tucker (KKT) optimality conditions. However, the problem function has more number of complex constraints to be satisfied. Classical optimization algorithms are impotent to handle the function. This paper presents a simplified minimization function, in which both the profit maximization problem and the ISO market clearance problem are considered, but with no KKT optimality conditions. Subsequently, this paper solves the minimization function using a hybrid optimization algorithm. The hybrid optimization algorithm is developed by combining the operations of group search optimizer (GSO) and genetic algorithm (GA). The hybridization enables the dispersion process of GSO to be a new mutated dispersion process for improving the convergence rate. We evaluate the methodology by experimenting on IEEE 14 and IEEE 30 bus systems. The obtained results are compared with the outcomes of bidding strategies that are based on GSO, PSO, and GA. The results demonstrate that the hybrid optimization algorithm solves the minimization function better than PSO, GA, and GSO. Hence, the profit maximization in the proposed methodology is relatively better than that of the conventional algorithms. Copyright © 2015 John Wiley & Sons, Ltd.

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush---Kuhn---Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.

Integrating Goal Programming, Taylor Series, Kuhn-Tucker Conditions, and Penalty Function Approaches to Solve Linear Fractional Bi-level Programming Problems

In this paper, we integrate goal programming (GP), Tay- lor Series, Kuhn-Tucker conditions and Penalty Function approaches to solve linear fractional bi-level programming (LFBLP)problems. As we know, the Taylor Series is having the property of transforming fractional functions to a polynomial. In the present article by Taylor Series we obtain polynomial objective functions which are equivalent to fractional objective functions. Then on using the Kuhn-Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complemen- tary and slackness condition of the lower level problem is appended to the upper level objective with a penalty, that can be reduce to a single objective function. In the other words, suitable transformations can be applied to formulate FBLP problems. Finally a numerical example is given to illustrate the complexity of the procedure to the solution.

Rigorous design of distillation columns using surrogate models based on Kriging interpolation

The economic design of a distillation column or distillation sequences is a challenging problem that has been addressed by superstructure approaches. However, these methods have not been widely used because they lead to mixed‐integer nonlinear programs that are hard to solve, and require complex initialization procedures. In this article, we propose to address this challenging problem by substituting the distillation columns by Kriging‐based surrogate models generated via state of the art distillation models. We study different columns with increasing difficulty, and show that it is possible to get accurate Kriging‐based surrogate models. The optimization strategy ensures that convergence to a local optimum is guaranteed for numerical noise‐free models. For distillation columns (slightly noisy systems), Karush–Kuhn–Tucker optimality conditions cannot be tested directly on the actual model, but still we can guarantee a local minimum in a trust region of the surrogate model that contains the actual local minimum. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2169–2187, 2015

An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems

This paper presents an interactive approach based on a discrete differential evolution algorithm to solve a class of integer bilevel programming problems, in which integer decision variables are controlled by an upper-level decision maker and real-value or continuous decision variables are controlled by a lower-level decision maker. Using the Karush--Kuhn–Tucker optimality conditions in the lower-level programming, the original discrete bilevel formulation can be converted into a discrete single-level nonlinear programming problem with the complementarity constraints, and then the smoothing technique is applied to deal with the complementarity constraints. Finally, a discrete single-level nonlinear programming problem is obtained, and solved by an interactive approach. In each iteration, for each given upper-level discrete variable, a system of nonlinear equations including the lower-level variables and Lagrange multipliers is solved first, and then a discrete nonlinear programming problem only with inequality constraints is handled by using a discrete differential evolution algorithm. Simulation results show the effectiveness of the proposed approach.

Optimal Offering Strategy of a Virtual Power Plant: A Stochastic Bi-Level Approach

This paper addresses the optimal bidding strategy problem of a commercial virtual power plant (CVPP), which comprises of distributed energy resources (DERs), battery storage systems (BSS), and electricity consumers, and participates in the day-ahead (DA) electricity market. The ultimate goal of the CVPP is the maximization of the DA profit in conjunction with the minimization of the anticipated real-time production and the consumption of imbalance charges. A three-stage stochastic bi-level optimization model is formulated, where the uncertainty lies in the DA CVPP DER production and load consumption, as well as in the rivals’ offer curves and real-time balancing prices. Demand response schemes are also incorporated into the virtual power plant (VPP) portfolio. The proposed bi-level model consists of an upper level that represents the VPP profit maximization problem and a lower level that represents the independent system operator (ISO) DA market-clearing problem. This bi-level optimization problem is converted into a mixed-integer linear programing model using the Karush–Kuhn–Tucker optimality conditions and the strong duality theory. Finally, the risk associated with the VPP profit variability is explicitly taken into account through the incorporation of the conditional value-at-risk metric. Simulations on the Greek power system demonstrate the applicability and effectiveness of the proposed model.

Irregular optimization models and p-order Kuhn-Tucker optimality conditions

A problem with irregular constraints in the calculus of variations is investigated. Optimality conditions for a degenerate case are obtained that make it possible to reject false solutions. The reasoning is conducted for the finite dimensional case; however, the main results are valid for Banach spaces.

Optimal stiffener layout of plate/shell structures by bionic growth method

Bionic growth method, which is based on the growth mechanism of branch systems in nature, has been used as a new approach for structural topology design optimization. Currently, its application is limited because the iterative scheme in the optimization process is heuristic. This paper suggests a new approach combined with the bionic branch model and optimality criteria. Based on the Kuhn–Tucker optimality condition, an analytical iterative formula is derived. The minimum compliance problem with multi-loading condition, the maximum fundamental frequency problem and the multi-objective optimization problem are studied. Typical design examples are demonstrated to validate the effectiveness of the suggested approach. Compared with the current growth technique, the suggested approach is more effective, practicable and applicable. The results show that the new bionic growth method can effectively and flexibly deal with optimum stiffener layout design of plate/shell structures to achieve various design objectives, thus it provides a new solution approach for structural topology design optimization.

An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems

Bilevel programming techniques deal with decision processes involving two decision makers with a hierarchical structure. In this paper, an augmented Lagrangian multiplier method is proposed to solve nonlinear bilevel programming (NBLP) problems. An NBLP problem is first transformed into a single level problem with complementary constraints by replacing the lower level problem with its Karush–Kuhn–Tucker optimality condition, which is sequentially smoothed by a Chen–Harker–Kanzow–Smale (CHKS) smoothing function. An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program to obtain an approximate optimal solution of the NBLP problem. The asymptotic properties of the augmented Lagrangian multiplier method are analyzed and the condition for solution optimality is derived. Numerical results showing viability of the approach are reported.

Optimal algorithm for the demand routing problem in multicommodity flow distribution networks with diversification constraints and concave costs

Distribution problems are of high relevance within the supply chain system. In real life situations various different commodities may flow in the distribution process. Furthermore, the connection between production and demand centres makes use of complex mesh networks that can include diversification constraints to avoid overcharged paths. In addition, the consideration in certain situations of economies of scale gives rise to non-linear cost functions that make it difficult to deal with an optimal routing scheme. This problem is well represented by the multicommodity flow distribution networks with diversification constraints and concave costs (MFDCC) problem. Here we present an optimal algorithm based on the Kuhn–Tucker optimality conditions of the problem and capable of supplying optimal distribution routes in such complex networks. The algorithm follows an iterative procedure. Each iteration constructive solutions are checked with respect to the Kuhn–Tucker optimality conditions. Solutions consider a set of paths transporting all the demand allowed by its diversification constraint (saturated paths), a set of empty paths, and an indicator path transporting the remaining demand to satisfy the demand equation. The algorithm reduces the total cost in the network in a monotonic sequence to the optimum. The algorithm was tested in a trial library and the optimum was reached for all the instances. The algorithm showed a major dependency with respect to the number of nodes and arcs of the graph, as well as the density of arcs in the graph.

Duality and Weak Efficiency in Vector Variational Problems

We establish weak, strong, and converse duality results for weakly efficient solutions in vector or multiobjective variational problems, which extend and improve recent papers. For this purpose, we consider Kuhn–Tucker optimality conditions, weighting variational problems, and some classes of generalized convex functions, recently introduced, which are extended in this work. Furthermore, a related open question is discussed.

Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz---Robinson---Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush---Kuhn---Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz---Robinson---Zowe regularity assumption.

Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems

Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on theF-convex,ρ-convex, and(F,ρ)-convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving(F,ρ)-convex,(F,α,ρ,d)-convex, and generalized(F,α,ρ,d)-convex functions.