Strong Duality Theorems Research Articles

Optimality Conditions and Dualities for Robust Efficient Solutions of Uncertain Set-Valued Optimization with Set-Order Relations

In this paper, we introduce a second-order strong subdifferential of set-valued maps, and discuss some properties, such as convexity, sum rule and so on. By the new subdifferential and its properties, we establish a necessary and sufficient optimality condition of set-based robust efficient solutions for the uncertain set-valued optimization problem. We also introduce a Wolfe type dual problem of the uncertain set-valued optimization problem. Finally, we establish the robust weak duality theorem and the robust strong duality theorem between the uncertain set-valued optimization problem and its robust dual problem. Several main results extend to the corresponding ones in the literature.

A Cooperative Energy Transaction Model for VPP Integrated Renewable Energy Hubs in Deregulated Electricity Markets

This article proposes a cooperative energy transaction model for a virtual power plant (VPP) developed as a bilevel optimization program based on the Stackelberg game in the deregulated electricity markets. The VPP integrates multiple renewable energy hubs consisting of electric vehicle charging stations with photovoltaic and battery energy storage systems. The proposed model derives co-optimized strategic decisions for the VPP operator in the day-ahead (joint active power, reserve, and reactive power) markets while considering the balancing market operation. The proposed model is reformulated as a mathematical program with equilibrium constraints (MPECs) using the Karush–Kuhn–Tucker conditions and the strong duality theorem. For an efficient solution, the MPEC is approximated as a mixed-integer linear program by applying the Fortuny-Amat transform on the complementarity and slackness conditions and the binary expansion on the bilinear terms. The uncertainties of the renewables, loads, and prices are modeled using stochastic scenarios. Finally, comparative case studies of a VPP in an IEEE 33 node active distribution network show that the proposed model improves the profit by strategic cooperative energy transactions.

Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative

AbstractIn this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, control of space structures, control of production and inventory, advertising investment, impulsive control problems, mechanics, and several other engineering and scientific problems. The proposed work considers the newly derived Caputo–Fabrizio (CF) fractional derivative operator. It is actually a convolution of the exponential function and the first-order derivative. The significant characteristic of this fractional derivative operator is that it provides a non-singular exponential kernel, which describes the dynamics of a system in a better way. Moreover, the proposed work also presents various weak, strong, and converse duality theorems under the diverse generalized invexity conditions in view of the CF fractional derivative operator.

Robust optimality conditions for multiobjective programming problems under data uncertainty and its applications

ABSTRACT In this article, we employ advanced techniques of convex analysis and C -differentiation to examine KKT-type robust necessary and sufficient optimality conditions and robust duality for an uncertain multiobjective programming problem under uncertainty sets, where C denotes the set of all G -derivatives which are positively homogeneous and convex with respect to the second argument. We first provide the robust constraint qualification of the (RCQ) type via the C -derivatives of uncertain constraint functions. We second establish KKT-type robust necessary conditions for robust (weakly) efficient solutions via the subdifferentials of C -derivatives to such problems. We third propose two new kinds of generalized C -convex functions via the subdifferentials of C -derivatives involving max-functions. Under suitable assumptions on the generalized C -convexity, KKT-type robust necessary optimality conditions become robust sufficient optimality conditions. Furthermore, we formulate as some applications a dual multiobjective programming problem to the underlying programming and examine weak, strong, and converse duality theorems for the same. Some illustrative examples are also provided for our findings.

Market-based hosting capacity maximization of renewable generation in power grids with energy storage integration

In an attempt to achieve net zero, the operation and planning of the energy system face techno-economic challenges brought by integrating large-scale distributed energy resources (DERs) with low carbon footprints. Previous work has analyzed the technical challenges including hosting capacity (HC) for DERs. In light of the deregulation of the power industry and the transition to power system with renewables at its center, this article takes the lead to maximizing renewable integration in power grids from a market viewpoint. It solves a significant problem brought forth by the fall in electricity prices, resulting from increasing renewable penetration that jeopardizes investment cost recovery and prevents sustainable grid integration of renewables. To this end, a novel bi-level optimization model is formulated, where the upper-level problem aims to maximize the HC of renewables ensuring the recovery of investment, and the lower-level problem describes the market clearing process considering network constraints. The optimal solution of devised bi-level problem can be found after reformulating it to a single-level mixed-integer linear problem (MILP) using the strong duality theorem and a special ordered set-type 1 (SOS1) founded linearization approach. Case studies confirm the significance of the devised model and quantitatively analyze the impact of different network capacities, renewable subsidies, and energy storage, respectively, on the market-based HC obeying its profitability constraint.

A game-theoretic model for wind farm planning problem: A bi-level stochastic optimization approach

As the electric network progressively shifts away from fossil fuel-fired generators to inverter-based generators, including most renewable energy sources (RESs), considering the system strength measure in the planning and operation problems is essential to maintaining the power system stability. To this end, this paper aims to introduce a new game-based bi-level framework for the wind farm planning problem (WFPP). The upper level seeks to maximize each investor's profit, and the lower level aims to clear the electricity market by considering the optimal investment decisions of investors. One innovative contribution of this paper is to incorporate the equivalent short circuit ratio (ESCR) as an indicator for the system strength measure in the WFPP. The bi-level model is converted into its equivalent single-level mixed-integer linear programming (MILP) employing the Karush-Kuhn-Tucker (KKT) conditions and strong duality theorem. This will yield a generalized Nash equilibrium problem (GNEP) containing binary decision variables, which is further transformed into a MILP. The intermittency of wind-demand scenarios is considered in the mathematical formulation of the presented bi-level model through scenario-based stochastic programming. Numerical studies are carried out to validate the effectiveness of the proposed model.

Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity

This paper deals with multiobjective semi-infinite programming problems on Hadamard manifolds. We establish the sufficient optimality criteria of the considered problem under generalized geodesic convexity assumptions. Moreover, we formulate the Mond-Weir and Wolfe type dual problems and derive the weak, strong and strict converse duality theorems relating the primal and dual problems under generalized geodesic convexity assumptions. Suitable examples have also been given to illustrate the significance of these results. The results presented in this paper extend and generalize the corresponding results in the literature.

Using a Bi-Level Optimization Model for Assessing the Impact of Demand Forecast Uncertainty on the Maximum Profit of Power Distribution Companies Owning Energy Storage Systems

This paper proposes a bi-level optimization model to maximize the net revenue of a power distribution company owning energy storage systems. Furthermore, the proposed model also takes into account the uncertainty of load forecast based on a set of multiple scenarios in order to assess its impact on the maximum net revenue of the power distribution company. This bi-level optimization model is transformed into the single-level mixed-integer linear programming model by using the Karush-Kuhn-Tucker optimality conditions and strong duality theorem. This single-level optimization formulation can be effectively solved by using standard commercial solvers such as CPLEX. The proposed model and the effects of the load forecast uncertainty on the net revenue of the power distribution company are validated and analyzed on an IEEE 24-bus meshed transmission grid.

Investigating How the Equilibria of the Electricity Market are Affected by Modeling the Strategic Behavior of Consumers

The paper proposes a formulation for a generalized Nash equilibrium model which incorporates the strategic biddings of some consumers capable of providing reserve and balancing services in the day-ahead and balancing markets, respectively. The strategic bidding in the electricity market is modelled using a bilevel optimization programming, where the electricity cost of consumers is minimized at the Upper Level (UL), and the energy and reserve costs in the market clearing process are co-optimized at the lower level (LL). By means of the strong duality theorem, the original bilevel model is transformed into an equivalent single-level Mathematical Problem with Equilibrium Constraints (MPEC). The joint problem of all MPECs, one per consumer, constitutes an Equilibrium Problem with Equilibrium Constraints (EPEC). The resulting EPEC is finally formulated as an auxiliary Mixed-Integer Linear Programming (MILP) problem. To this end, an exact linearization technique and Fortuny-Amat transformation are adopted to substitute the nonlinear terms and the complementarity conditions. In addition, a parametrization technique is used to replace the dual variable associated with the strong duality equation which appears in the EPEC problem. The diagonalization method is also adopted in this part as an ex-post analysis to verify the obtained solutions of the resulted MILP. Finally, a 3-bus illustrative example and the IEEE RTS 24-Bus System and 118-Bus System are considered to investigate the performance of the proposed approach.

Mangasarian-type second- and higher-order duality for mathematical programs with complementarity constraints

ABSTRACT Based on the strong stationary condition, we formulate second-order and higher order duals for a mathematical program with complementary constraints (MPCC). Under suitable generalized convexity assumptions, we then propose several duality theorems for second and higher order duality in MPCC problems, including the weak duality, strong duality, converse duality, and strict converse duality theorems. Moreover, we present examples to illustrate these results, where one of them reveals that a lower bound provided by our proposed second-order duality is tighter than the one given by the first-order duality.

An Optimal Dispatching Model for Integrated Energy Microgrid Considering the Reliability Principal–Agent Contract

As the increasing penetration of sustainable energy brings risks and opportunities for energy system reliability, at the same time, considering the multi-dimensional differentiation of users’ reliability demands can further explore the potential value of reliability resources in Integrated Energy Microgrid (IEM). To activate the reliability resources in a market-oriented perspective and flexibly optimize the operational reservation in dispatch, an optimal dispatching model in IEM considering reliability principal–agent contracts is proposed. We establish the reliability principal–agent mechanism and propose a cooperative gaming model of Integrated Energy Operator (IEO) and Integrated Energy User (IEU) based on the optimal dispatching model. At the upper level, the economic dispatching model of IEO is established to optimize the operation reservation, and the reliability principal–agent contract from users in the lower level would influence reliability improvement. Each IEU in the lower level maximizes its energy utilization and gives the corresponding reliability principal–agent incentives according to the reliability improvement degree and its actual demand. The bi-level model is solved by the KKT condition and strong duality theorem. A case study verifies the effectiveness of the proposed model in reducing the energy dispatch cost, improving the economic benefits of each participant, realizing the optimal allocation of reliability resources and optimizing the IEM energy structure, and the sensitivity analysis of dispatch cost with the user’s energy-using benefits is discussed.

An accelerated Stackelberg game approach for distributed energy resource aggregator participating in energy and reserve markets considering security check

With increasing distributed energy resources (DERs) integration, the strategic behavior of a DER aggregator in electricity markets will significantly affect the secure operation of the distribution system. In this paper, the interactions among the DER aggregator, energy and reserve markets, and distribution system are investigated through a single-leader-multi-follower Stackelberg game model with the DER aggregator as the leader and the independent system operator and distribution system operator as the followers. To guarantee the operation security of the distribution system, security check problems under three different scenarios are involved in the follower level, which is linearized using a mixed-integer linearized power flow model. Then, using the strong duality theorem, the proposed model is converted into a bi-level mixed-integer linear (BMILP) programming model with only mixed-integer linear follower-level problems. Next, an accelerated relaxation-based bi-level reformulation and decomposition algorithm is proposed to solve the BMILP problem. Finally, case studies are carried out on a constructed integrated transmission and distribution (T&D) system and a practical integrated T&D system to verify the effectiveness of the proposed model and algorithm. The simulation results indicate that the available downward reserve of the DER aggregator will decrease with the security limitation of the distribution system.

Leveraging the flexibility of electric vehicle parking lots in distribution networks with high renewable penetration

The ongoing rapid increase in the integration of variable and uncertain renewable energy sources calls for enhancing the ways of providing flexibility to power grids. To this end, we propose an optimal approach for utilizing electric vehicle parking lots to provide flexibility at the distribution level. Accordingly, we present a day-ahead scheduling model for distribution system operators, where they can offer discounts on the network tariff to electric vehicle parking lot operators. This way, they will be encouraged to exploit the potential flexibility of electric vehicle batteries to assist in alleviating the steep ramps of system net-load. To determine the optimal discounts, the distribution system operator minimizes the network operating costs considering the network operational constraints, while the electric vehicle parking lot operators try to maximize their profits. Due to the contradictory objectives and decision hierarchy, the problem is an instance of Stackelberg games and can be formulated as a bi-level program, which is linearized and converted to a single-level mixed-integer linear program using strong-duality theorem and Karush–Kuhn–Tucker conditions. To validate the proposed model, comprehensive simulation studies are performed on a test distribution network. The simulation results show that implementing the model can reduce the peak-off-peak difference and peak-to-average ratio of the network net-load by up to 15% and 24%, respectively.

Distributionally robust optimization of non-fossil fuels processing network under uncertainty

Under the ever more prominent climate issues concerning greenhouse gas emission reduction, governments have sped up the pace of “Decarbonization while meeting demand”, and traditional energy companies have also begun to advance strategy layout and transformation, of which gasoline substitute, diesel substitute, hydrogen and ammonia are the highly potential candidates in the future. This paper investigates the design of non-fossil fuels processing network with renewable feedstocks under various demand uncertainties in the framework of two-stage distributionally robust optimization (TSDRO) model. To handle the difficulties of estimating the exact probability distribution of the considered uncertainties, 1-norm Wasserstein metric is applied to construct the ambiguity set, and the resulting tractability issue is recast in terms of sufficiently expensive recourse and strong duality theorem. Compared with two-stage stochastic programming and two-stage robust optimization, optimal design decisions of the Wasserstein TSDRO of base case and net-zero emission case exhibit the balance between the optimality and robustness, which lay a foundation to ensure an orderly transition towards sustainable and resilient network systems under various demand uncertainties, as well as a pathway to alleviate the abandon or redundancy power phenomena existing in renewable power through employing non-fossil fuels as the storage medium.

On symmetric gH-derivative: Applications to dual interval-valued optimization problems

This paper provides a complete study on properties of symmetric gH-derivative. More precisely, a necessary and sufficient condition for the symmetric gH-differentiability of interval-valued functions is presented. Further, we clarify the relationship between the symmetric gH-differentiability and gH-differentiability. Moreover, quasi-mean value theorem, chain rule and some operations of symmetric gH-differentiable interval-valued functions are established. As applications, we develop the Mond–Weir duality theory for a class of symmetric gH-differentiable interval-valued optimization problems. Weak, strong and strict converse duality theorems are formulated and proved. Also, several examples are presented in order to support the corresponding theoretical results.

Robust duality for the uncertain multitime control optimization problems

AbstractThis article gives some new duality results for the multitime control optimization problem with data uncertainty (MCOPU). We formulate Wolfe and Mond‐Weir type robust dual problems for (MCOPU) and prove the weak robust duality theorem which states that the duality gap is positive and the strong robust duality theorem which shows the zero duality gap under convexity assumptions. Further, we also prove the converse robust duality result. Moreover, we give some nontrivial examples to verify the established results in this article.

Bidding behaviors of GENCOs under bounded rationality with renewable energy

With the worldwide restructuring of power markets, it is more meaningful to analyze the bidding behaviors of generation companies (GENCOs). A realistic bidding behavior model can help to design better power market mechanisms. However, most existing studies usually use strong assumptions, where all GENCOs behave perfectly rationally when participating in power markets. Many GENCOs' observed bidding behaviors are proved to deviate from the simulated perfectly rational ones. Thus, it is significant to introduce individual subjective perspectives into the bidding behavior modeling to obtain more realistic models. Although several studies have been working on this topic by introducing bounded rationality, they still have some problems in modeling bounded rationality, power system operation, power market rules, etc. This paper proposes a novel bidding behavior model for GENCOs under bounded rationality with renewable energy to overcome these problems. The introduced bounded rationality features include prospect theory in the form of framing effects and the concept of fairness constraints on profit-making. A Stackelberg game model is formulated to analyze the interaction between a GENCO with renewables and a day-ahead power market. The proposed model is nonlinear, which is further transformed into a mixed-integer linear formulation by applying Karush–Kuhn–Tucker theorem and strong duality theorem. Based on a modified IEEE-30 bus system with high renewable, an illustrative example is proposed to demonstrate the effectiveness of the proposed model and analyze the bidding behaviors of bounded rational GENCOs and the corresponding effects on power markets.

Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness

Abstract In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption.

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

Second-order symmetric duality in vector optimization involving (K,η)-pseudobonvexity

This paper presents new classes generalizations of second-order (K,η)-pseudobonvexity functions, strongly second-order (K,η)-pseudobonvexity functions, and establishes a pair Mond-Weir type duality multiobjective nonlinear programs over these new classes generalizations. The weak duality, strong duality, converse duality, and self-duality theorems are presents under these generalizations functions. Finally, we introduced two nontrivial numerical examples to illustrating results of the weak duality theorem and strong duality theorem.