Strong Duality Research Articles

RETRACTED: Robust bidding strategy of interconnected multi-carrier systems in the electricity markets under the uncertainty in electricity load

This paper presents a robust bi-level model for optimal bidding of interconnected multi-carrier systems (MCSs) in the wholesale electricity markets under the uncertainty in electricity load. The proposed model implements a robust optimization method (ROM) to model the risk-based bidding of interconnected MCSs under the uncertainty in load in the upper-level problem while including the energy and reserve markets in the lower-level problem. The studied MSCs include combined heat and power (CHP) units, boilers, central air conditioning (CAC) systems, and battery storage units to provide local energy and make the MCSs’ operator capable of participating in the reserve market to gain profit. ROM models the electricity demand and provides different bidding strategies (i.e., risk-seeking, risk-neutral, and risk-averse) for the participation of MCSs’ operator in the energy and reserve markets. The studied bi-level model is converted to a single-level mixed-integer linear programming problem using the Karush–Kuhn–Tucker conditions and strong duality and then solved using the CPLEX solver of the General Algebraic Modeling System. Numerical experiments reveal that the interconnection of MSCs, along with the integration of storage units, enables the MSCs’ operator to control the uncertainty in the electricity load and increase the energy efficiency in order to optimally bid in the day-ahead energy and reserve markets. Results show that by enabling the MCSs to locally trade energy, the total operation cost of MCSs in the risk-seeking, risk-neutral, and risk-averse modes decreases by 5.65%, 4.28%, and 3.49%. Moreover, with the integration of battery storage systems into the interconnected operation of MCSs, the total operation cost of systems in the risk-seeking, risk-neutral, and risk-averse modes decreases by 7.65%, 6.99%, and 6.40%, respectively.

Multi-objective distributionally robust optimization for hydrogen-involved total renewable energy CCHP planning under source-load uncertainties

Hydrogen is one type of clean renewable energy. By introducing hydrogen production, storage and conversion devices into the combined cooling, heating and power (CCHP) system to replace its fossil fuel components, the hydrogen-involved total renewable energy CCHP (H-RE-CCHP) system is formed, thus bringing an effective way to promote energy transition towards low carbon. However, the conflict between economy and environment, and interference of source-load uncertainties hinder the low-cost construction as well as economic, stable, and low-carbon operation of H-RE-CCHP. In view of this, this paper proposes a multi-objective distributionally robust optimization (DRO) model for H-RE-CCHP planning. Firstly, a moment-based ambiguity set is constructed to define the possible range for the joint distribution of solar irradiation and multiple loads. Secondly, considering the “worst-case” distribution within the ambiguity set, the economic and environmental objective functions are presented for minimizing system construction, maintenance and operation costs while reducing carbon emission. Furthermore, the sizing- and operation-related constraints are formulated according to such principles as maintaining multi-energy balance when the forecasting errors of uncertainties occur. Finally, by applying strong duality and augmented ε-constraint for model solving, Pareto optimal solution set is obtained, and from which final planning and operation scheme is selected according to decision preference and interactions between objectives. Simulations mainly verify: 1) due to considering multiple objectives, H-RE-CCHP planning and operation reach the reasonable trade-off between economy and environment, that is, carbon emission/economic cost can be reduced by 99.81 %/66.70 % comparing with single economic/environmental objective; 2) with applying DRO in resisting uncertainties, 100 % renewable energy accommodation is achieved while meeting stable H-RE-CCHP operation; etc. Accordingly, this paper provides an effective approach for renewable energy system planning, thus contributing to the low carbon development of economic society.

Low-Latency Hybrid NOMA-TDMA: QoS-Driven Design Framework

Enabling ultra-reliable and low-latency communication services while providing massive connectivity is one of the major goals to be accomplished in future wireless communication networks. In this paper, we investigate the performance of a hybrid multi-access scheme in the finite blocklength (FBL) regime that combines the advantages of both non-orthogonal multiple access (NOMA) and time-division multiple access (TDMA) schemes. Two latency-sensitive application scenarios are studied, distinguished by whether the queuing behaviour has an influence on the transmission performance or not. In particular, for the latency-critical case with one-shot transmission, we aim at a certain physical-layer quality-of-service (QoS) performance, namely the optimization of the reliability. And for the case in which queuing behaviour plays a role, we focus on the link-layer QoS performance and provide a design that maximizes the effective capacity. For both designs, we leverage the characterizations in the FBL regime to provide the optimal framework by jointly allocating the blocklength and transmit power of each user. In particular, for the reliability-oriented design, the original problem is decomposed and the joint convexity of sub-problems is shown via a variable substitution method. For the effective-capacity-oriented design, we exploit the method of Lagrange multipliers to formulate a solvable dual problem with strong duality to the original problem. Via simulations, we validate our analytical results of convexity/concavity and show the advantage of our proposed approaches compared to other existing schemes.

A proactive strategy for virtual power plants including multiple private owners equipped with energy storages

This paper presents a novel framework for the optimal operation strategy of a virtual power plant (VPP) comprises a set of various stakeholders such as renewable energy sources, distributed energy resources with private owners, and flexible loads in day-ahead and real-time energy markets. In order to model the strategic behavior of private owners within the VPP geographic location, a proactive stochastic bi-level model is developed, where the upper level assigns to minimize expected operation cost of the technical VPP and the lower level associates with maximizing the private owner's profits. Note that, the uncertainty set encompasses the real-time energy market price as well as renewables power production. Afterward, the proposed bi-level model is converted to a mixed-integer linear programming problem through Karush–Kuhn–Tucker optimality conditions along with strong duality theory. Moreover, the impacts of implementation of three different pricing schemes including fix pricing, time of use, and real-time pricing on the market trading strategy of VPP and also its interactions with private owners is investigated through comparative case studies. The obtained results demonstrate that the proposed approach could allow both VPP aggregator and private owners to take the advantages of the market conditions.

Global Convergence of Policy Gradient Primal–Dual Methods for Risk-Constrained LQRs

While the techniques in optimal control theory are often model-based, the policy optimization (PO) approach directly optimizes the performance metric of interest. Even though it has been an essential approach for reinforcement learning problems, there is little theoretical understanding of its performance. In this paper, we focus on the risk-constrained linear quadratic regulator (RC-LQR) problem via the PO approach, which requires addressing a challenging non-convex constrained optimization problem. To solve it, we first build on our earlier result that an optimal policy has a time-invariant affine structure to show that the associated Lagrangian function is coercive, locally gradient dominated, and has local Lipschitz continuous gradient, based on which we establish strong duality. Then, we design policy gradient primal-dual methods with global convergence guarantees in both model-based and sample-based settings. Finally, we use samples of system trajectories in simulations to validate our methods.

Optimal Offering Strategy of GenCo With Joint Participation in FTR Auction and Day-Ahead Market Considering Virtual Bidding

Nodal price separations in Day-Ahead (DA) market caused by transmission congestion create congestion charges/surplus that are reallocated to the market participants through the financial transmission right (FTR) auction. From a market participant's viewpoint, these two markets are interrelated because the revenue of market participant in FTR auction is determined based on the day-ahead locational marginal prices. Furthermore, virtual transactions which are designed to improve price convergence between the day-ahead and real-time markets can directly impact day-ahead prices. This impact through virtual transactions may be utilized by a market participant to increase its FTR value and improve its overall strategy in participating in both FTR auction and day-ahead market. To this end, this paper is the first attempt to reveal this tactic in literature and proposes an offer strategy framework for a price-maker generating company participating in both FTR auction and day-ahead market with the consideration of virtual bidding. First, the possibility of FTR value manipulation is conceptually demonstrated by placing virtual bids in the day-ahead market. Second, a two-stage bi-level offering strategy model is formulated for strategic GenCos, which is further converted to a single level optimization problem by using Karush-Kuhn-Tucker conditions and strong duality theory. Numerical tests on a 5-bus system and IEEE reliability test system (RTS) demonstrate the effectiveness and applicability of the proposed approach.

A joint clearing market for coupled electricity and gas system considering bilateral energy contracts

In recent years, with the progress of new energy conversion technology, especially an increasing number of gas turbine units applied to the market, the connection between the natural gas system and the electric system has become increasingly close. Initially the gas and electricity markets operated relatively independently. Therefore, it is necessary to establish the coupling architecture of natural gas market and electricity market to meet the energy demand of users. This paper attempts to propose a power and gas coupling system based on clearing market structure. Firstly, the difference between natural gas–electricity coupling market and joint settlement market is discussed. The combined market considered in this paper is somewhat different from the associated electricity market and natural gas market. The calculations are easier and the results more reliable than those of the traditional linked gas markets and electric power market. Then, a mathematical model of joint clearing market is established based on bilateral energy contracts. Then, the piecewise linearization approximation (PLA) algorithm is proposed to solve the nonlinear problem. The locational piecewise price (LMEP) and locational marginal gas price (LMGP) are identified respectively, and the bilateral contracts are processed. The solution proposed in this paper guarantees strong duality and clear meaning of dual variables, making PLA easy to apply and solve market clearing problems based on dual variables or Lagrange multiplier pricing, as well as easy to deal with bilateral contract problems. Finally, the proposed joint clearing market is verified by an improved 3-electric bus and 6-gas node system.

Tri-level expansion planning for transmission, energy storage, and renewable energy considering carbon emission limitation

In recent years, renewable energy (RE) has developed rapidly due to its environmentally friendly characteristics. However, RE has strong intermittency and volatility, and a high proportion of RE integration will have a huge impact on power systems. Moreover, some power systems also put forward requirements for RE utilization, which requires more flexible resources. Conventional units are limited in development under the carbon emission reduction policies. The energy storage system (ESS) can stabilize the volatility of RE power and alleviate transmission congestion. Therefore, to promote the energy transformation of power systems, it is necessary to jointly consider transmission network, ESS, and RE in power system planning. Considering that the planning decision-makers are different, this paper proposes a tri-level expansion planning model considering carbon emission limitation and RE development. In the upper-level (UL) problem, the transmission system operator constructs transmission lines and grid-side ESS to ensure load supply and carbon emission requirements. On the premise that the RE curtailment rate is within a certain range, the middle-level (ML) problem constructs RE units to maximize the RE installed capacity. Based on the construction results of the UL and ML problems, lower-level (LL) problems simulate economic dispatching from the perspective of the independent system operator. To solve the tri-level model, an equivalent bi-level model is transformed according to strong duality theory, and then the column and constraint generation (C&CG) algorithm is adopted to solve the bi-level problem. In case studies, the numerical results analyze the impacts of carbon emissions, RE curtailment rate, and ESS parameters on the planning results, which verifies the rationality of the approach proposed in this paper.

An interval-based nested optimization framework for deriving flexibility from smart buildings and electric vehicle fleets in the TSO-DSO coordination

Emerging renewable-based transmission and distribution systems, despite many environmental and economic benefits, due to the intermittent nature of their production resources, compared to traditional systems, need more flexibility capacities, which necessitates the need for more suppliers of flexibility. To deal with these challenges, a nested framework is presented to derive the required flexibility of the transmission system operator (TSO) from distributed energy resources (DERs) and active end-users such as smart buildings (SBs) and electric vehicle (EV) fleets at the distribution level. To this end, a novel mechanism to design the demand response program (DRP) is introduced in which tariffs with time-varying rewards are built based on flexibility requirements. The coordination between TSO and distribution system operator (DSO) is initially modeled as a bi-level non-linear programming (NLP) problem, in which the upper-level is day-ahead (DA) operational planning of DS considering the schedules received from SBs, while the lower-level is DA operational planning of the TS. The bi-level NIL problem is transformed into a single-level linear programming (LP) problem by Krush Kuhn Tucker (KKT) conditions, Big-M method and Strong Duality Theory (SDT), which makes it computationally tractable. Finally, a two-stage interval-based algorithm solves the obtained single-level problem to secure the planning against uncertainties where battery energy storage systems (BESSs) are responsible for dealing with extreme conditions. The simulation results testify that the proposed interval-based nested framework has improved the economic, technical and security aspects of the TSO-DSO coordination since it has reduced the daily costs of the energy and flexibility markets, relieved lines congestion and improved voltage characteristics.

Bi-level transmission expansion planning considering prohibited operating zones and multi-fuel units

Transmission expansion planning (TEP) provides a solution to economically and securely expand the transmission network. In this context, the bi-level frameworks are receiving increased attention considering a DC or (linear/convex) AC economic dispatch at the lower level. However, the prohibited operating zones (POZ) and multi-fuel units (MFU) are ignored to be modeled at the lower level, while it plays a vital role in the output results. To this end, this article presents a bi-level TEP model that considers POZ and MFU. Incorporating these two issues into the lower level’s economic dispatch problem introduces more complexity to the optimization problem but provides more realistic optimal plans. Unlike the typical approaches to solving bi-level programming (BP), the formulated problem cannot be solved by Karush–Kuhn–Tucker (KKT) conditions or strong duality theory (SDT) due to binary variables at the lower level. To this end, an iterative-based reformulation-and-decomposition (R&D) technique is utilized to find the optimal solutions to the bi-level mixed-integer programming (BMIP) problem. The presented methodology is applied to the IEEE 118-bus system. The numerical results corroborate the effectiveness of the proposed structure and highlight that modeling POZ and MFU is an essential task in the TEP problems.

Nonsmooth semi-infinite roughly B-invex multi-objective programming problems

In this paper, we will study a nonsmooth semi-infinite multi-objective programming problem involving roughly B-invex functions. Sufficient optimality conditions for the primal problem are derived. We shall formulate Mond–Weir type dual and establish weak and strong duality theorems under various roughly B-invexity assumptions.

Strategic Participation of Integrated Thermal and Electrical Energy Service Provider in Natural Gas and Wholesale Electricity Markets

The accelerating interest in energy system models has influenced numerous revolutionized alterations in the wholesale gas and electricity market. This article proposes a novel bilevel multifollower optimization framework for the strategic behavior observation of an integrated (thermal and electrical) energy service provider (IESP) as a pricemaker in the wholesale electricity market (WEM) and natural gas market (NGM). At the upper level, the IESP submits offers/bids in WEM and NGM to procure electricity/gas to the customers. To this end, the IESP endeavors to minimize operational costs and influences the market-clearing price (MCP) by deploying demand-side flexibilities, i.e., elastic electrical and thermal loads. At the lower level, the WEM operator and NGM operator receive offers/bids from all market participants and clear the market with the goal of maximizing social welfare. The IESP is modeled via IEEE-33 bus active distribution system and an 8-node district heating system, whereas the WEM and NGMs are embodied by a 6-bus transmission network and a 21-node natural gas network, respectively. Karush–Kuhn–Tucker conditions are introduced to transform the multifollower bilevel optimization problem into a single-level problem, whereas the inherent nonlinearities of the problem are linearized using the theory of strong duality. Moreover, the intrinsic intermittencies of the renewable energy sources is dealt with by the risk-averse information gap decision theory. The results confirm that flexible electrical and thermal demands can diminish the MCP by as much as 4.1%.

Revisiting degeneracy, strict feasibility, stability, in linear programming

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs, LPs. Unlike general conic programs, LPs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent to stability and a compact dual optimal set. This lack of concern is also true for other types of degeneracy of basic feasible solutions in LP. In this paper we discuss that the specific degeneracy that arises from lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, we show that the lack of strict feasibility implies that every basic feasible solution, BFS, is degenerate; thus conversely, the existence of a nondegenerate BFS implies that strict feasibility (regularity) holds. We prove the results using facial reduction and simple linear algebra. In particular, the facially reduced system reveals the implicit non-surjectivity of the linear map of the equality constraint system. As a consequence, we emphasize that facial reduction involves two steps where, the first guarantees strict feasibility, and the second recovers full row rank of the constraint matrix. This illustrates the implicit singularity of problems where strict feasibility fails, and also helps in obtaining new efficient techniques for preproccessing. We include an efficient preprocessing method that can be performed as an extension of phase-I of the two-phase simplex method. We show that this can be used to avoid the loss of precision for many well known problem sets in the literature, e.g., the NETLIB problem set.

PBW theory for quantum affine algebras

Let U_q'(\mathfrak{g}) be a quantum affine algebra of arbitrary type and let \mathscr{C}_\mathfrak{g}^0 be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor \mathcal{F}_\mathcal{D} to a duality datum \mathcal{D} in \mathscr{C}_\mathfrak{g}^0 . In this paper, we introduce the notion of a strong (complete) duality datum \mathcal{D} and prove that, when \mathcal{D} is strong, the induced duality functor \mathcal{F}_\mathcal{D} sends simple modules to simple modules and preserves the invariants \Lambda , \tilde\Lambda and \Lambda^\infty introduced by the authors. We next define the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k acting on strong duality data \mathcal{D} . We prove that if \mathcal{D} is a strong (resp.\ complete) duality datum, then \mathscr{S}_k(\mathcal{D}) and \mathscr{S}_k^{-1}(\mathcal{D}) are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in \mathscr{C}_\mathfrak{g}^0 by using the duality functor \mathcal{F}_\mathcal{D} , and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When \mathcal{D} is complete, we show that all simple modules in \mathscr{C}_\mathfrak{g}^0 can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.

Wolfe-Type Duality for Mathematical Programs with Switching Constraints

The mathematical program with switching constraints (MPSC), which has been introduced recently, is a difficult class of optimization problems. The reason for the difficulty is that the existence of the switching constraint makes the standard constraint qualifications invalid at local minimizers. This paper proposes the Wolfe-type dual (WD) model of this problem without switching constraints. Under the assumptions of convexity and strict convexity, we derive the weak, strong, converse, restricted converse, and strict converse duality results between MPSC and WD.

An optimal transport problem with storage fees

We establish basic properties of a variant of the semi-discrete optimal transport problem in a relatively general setting. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support of the target measure, and a “storage fee” function. The goal is to find a map for which the total transport cost plus the storage fee evaluated on the masses of the pushforward of the source measure is minimized. We prove existence and uniqueness for the problem, derive a dual problem for which strong duality holds, and give a characterization of dual maximizers and primal minimizers. Additionally, we find some stability results for minimizers and a Γ-convergence result as the target set becomes denser and denser in a continuum domain.

Control and optimization of abstract continuous time evolution inclusions

Abstract Abstract controlled evolution inclusions are revisited in the Banach spaces setting. The existence of solution is established for each selected control. Then, the input–output (or, control-states) multimap is examined and the Lipschitz continuous well posedness is derived. The optimal control of such inclusions handled in terms of a Bolza problem is investigated by means of the so-called P ℱ format of optimization. A strong duality is provided, the existence of an optimal pair is given and the system of optimalty is derived. A Fenchel duality is built and applied to optimal control of convex process of evolution. Finally, it will be shown how the general theory we provided can be applied to a wide class of controled integrodifferental inclusions.

Exploiting DERs’ Flexibility Provision in Distribution and Transmission Systems Interface

This paper presents a new market-based framework to exploit distributed energy resources’ (DERs) flexibility in distribution and transmission systems. This framework targets intra-hourly flexibility requirements of energy systems. In this regard, a local market is operated in distribution network to exploit DERs’ capabilities. Local market is cleared in coordination with wholesale market to optimize DERs’ flexibility procurement for meeting intra-hourly variability and uncertainty in distribution and transmission systems. In the proposed framework, a technical virtual power plant (TVPP) operates day-ahead (DA) local energy and flexibility markets in which DER aggregators take part. Furthermore, the TVPP participates in DA wholesale energy and flexibility markets which are cleared by the market operator and the transmission system operator, respectively. In this framework, all agents’ preferences and their transactions are addressed using a bilevel model with multiple lower levels. The proposed model is reformulated into a mixed integer linear programming problem by utilizing strong duality theory, Karush-Kuhn-Tucker optimality conditions and primal-dual counterpart. The model is implemented on an IEEE standard test system. The results demonstrate effectiveness of the framework in utilizing DERs’ capabilities to meet flexibility requirements as well as satisfying agents’ preferences.

Stochastic Strategic Participation of Active Distribution Networks With High-Penetration DERs in Wholesale Electricity Markets

With the increasing penetration of distributed energy resources (DERs), traditional distribution networks as load-serving entities in wholesale electricity markets, now evolve towards active distribution networks (ADNs) which can proactively participate in wholesale markets by optimally controlling the DERs in their networks. A stochastic bilevel optimization model is proposed in this paper for the strategic participation of ADNs and DERs to provide energy and grid services in wholesale electricity markets. The bilevel optimization model can capture the interactions between the ADN and the wholesale energy and ancillary service markets, considering the uncertainties of DERs in the ADN. In the upper-level model, the ADN makes optimal decisions on energy and reserve bidding considering the availability, uncertainties, and flexibility of DERs. The joint energy and reserve market-clearing of the independent system operator (ISO) is modeled as the lower-level problem. Using strong duality theory and Karush-Kuhn Tucker (KKT) conditions, the proposed bilevel optimization problem is reformulated as mathematical programming with equilibrium constraints (MPEC) problem and further converted into a computationally-solvable mixed-integer second-order-cone programming (MISOCP) model. The simulation results demonstrate the effectiveness of the model and the interactions between an ADN and wholesale electricity markets.

Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems

Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.