Search Results (65)

With the large-scale integration of renewable energy into the power system, meteorological uncertainty poses challenges to the safe and stable operation of the system. Traditional uncertainty optimization methods struggle to balance robustness and economy. This paper proposes a Wasserstein distance-based distributionally robust optimization strategy that considers covariate factors for a renewable energy power grid considering meteorological uncertainty. By introducing covariate factors to construct the Wasserstein ambiguity set, the intrinsic connection between weather uncertainty and the output of new energy is effectively depicted. The optimization problem is transformed into a solvable form of mixed integer linear programming by using linear decision rules and duality theorems, and the distributionally robust optimization scheduling problem is solved based on the improved cross optimization algorithm. Simulation results based on the IEEE 33 system show that under the same worst-case expected energy shortage of 20 kWh, the proposed method achieves an expected total dispatch cost of approximately CNY 0.534 million, reducing cost by about 0.4%, 0.9%, and 1.8% compared with conventional Wasserstein DRO, KL-divergence DRO, and Moment Information DRO; when the radius of the Wasserstein ball is 1, using the CSO algorithm reduces the runtime by 59.4% compared with the solver. It effectively reduces operating costs and solution speed while ensuring system security, offering a new approach for the optimal dispatch of power systems with a high penetration of renewable energy. Full article

Calculations testing can be effectively used in the construction of discrete self-checking devices. Calculations testing is based on the parity and self-duality of the calculated functions. This can be used for modern blocks and nodes of control systems for responsible technological processes. However, its use has a number of features that must be considered when building concurrent error-detection circuits. The authors used methods of discrete mathematics and Boolean algebra as well as technical diagnostics of discrete systems to investigate the problem of ensuring the testability of the parity encoder. Theorems on the testability of convolution functions modulo 2 are proved. Considering these theorems allowed the authors of the article to propose a method for synthesizing CED circuits. This method increases the testability of the encoder for parity. This method is based on the use of two diagnostic signs at once. The first sign is that the code words belong to the parity code. The second is the self-dual control function in the concurrent error-detection circuit. This method is guaranteed to increase the testability of the parity coder compared to using one of the diagnostic signs for calculations testing. Experiments with testing discrete devices have shown the effectiveness of the organization structure of the concurrent error-detection circuit that we developed. The theorems that we proved form the basis of proof of similar provisions for the use of other linear codes in the synthesis of concurrent error-detection circuits. Our proposed solutions with calculations testing based on two diagnostic signs should be used in the synthesis of discrete systems. Discrete systems should be self-checking and have improved testability indicators. Full article

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

In this paper, for the considered multi-cost variational problem (P), we associate a dual model (D) in order to study and state the connections between the solution sets of these control problems. Thus, under S-type I assumptions associated with the integral functionals involved, we formulate and prove various reciprocal results, such as weak, strong, and converse-type dualities. Full article

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras to the category of finite dimensional Yang–Baxter structures. In the same manner, we attempt to unify the Tzitzeica–Johnson theorem and its dual version, obtaining a new theorem about circle configurations. Full article

As energy and carbon markets evolve, it has emerged as a prevalent trend for multiple virtual power plants (VPPs) to engage in market trading through coordinated operation. Given that these VPPs belong to diverse stakeholders, a competitive dynamic is shaping up. To strike a balance between the interests of the distribution system operator (DSO) and VPPs, this paper introduces a bi-level energy–carbon coordination model based on the Stackelberg game framework, which consists of an upper-level optimal pricing model for the DSO and a lower-level optimal energy scheduling model for each VPP. Subsequently, the Karush-Kuhn-Tucker (KKT) conditions and the duality theorem of linear programming are applied to transform the bi-level Stackelberg game model into a mixed-integer linear program, allowing for the computation of the model’s global optimal solution using commercial solvers. Finally, a case study is conducted to demonstrate the effectiveness of the proposed model. The simulation results show that the proposed game model effectively optimizes energy and carbon pricing, encourages the active participation of VPPs in electricity and carbon allowance sharing, increases the profitability of DSOs, and reduces the operational costs of VPPs. Full article

With the ongoing integration of renewable energy and energy storage into the power grid, the voltage safety issue has become a significant challenge for the distribution power system. Therefore, this study proposes a coordinated operation for energy storage systems with reactive power compensators. Taking into account the benefits of energy storage equipped with reactive power compensators and the market clearing process, a bi-level optimization model is formulated. In the proposed model, the upper-level model formulates the bidding strategy for energy storage and aims at maximizing the energy storage revenue; the lower-level model carries out a market-clearing process that takes into account various constraints for ensuring the safe operation of the grid. Afterward, by applying several math tricks such as the KKT optimality condition, the strong duality theorem, and the big M method, the bi-level equilibrium programming problem is transformed into an equivalent and tractable single-level mixed-integer linear programming problem. The results show that the coordinated operation of energy storage and reactive power compensators increases the benefit of energy storage by 3.47%. The benefit increment and security improvement brought by the collaborative operation of energy storage and reactive power compensators are verified. Full article

In order to consider the impact of multiple uncertainties on the interaction between the distribution network operator (DNO) and photovoltaic powered charging stations (PVCSs), this paper proposes a regulation strategy for a distribution network with a PVCS based on bi-level stochastic optimization. First, the interaction framework between the DNO and PVCS is established to address the energy management and trading problems of different subjects in the system. Second, considering the uncertainties in the electricity price and PV output, a bi-level stochastic model is constructed with the DNO and PVCS targeting their respective interests. Furthermore, the conditional value-at-risk (CVaR) is introduced to measure the relationship between the DNO’s operational strategy and the uncertain risks. Next, the Karush–Kuhn–Tucker (KKT) conditions and duality theorem are utilized to tackle the challenging bi-level problem, resulting in a mixed-integer second-order cone programming (MISCOP) model. Finally, the effectiveness of the proposed regulation strategy is validated on the modified IEEE 33-bus system. Full article

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond–Weir-type weak and strong duality theorems, and illustrate our results with an example. Full article

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ${\sum }_{a+b=r-1}{(-1)}^a{\zeta }^{★}(a+2,{\left\{2\right\}}^{p-1}){\zeta }^{★}({\left\{1\right\}}^{b+1},{\left\{2\right\}}^q)$ equals ${\zeta }^{★}({\left\{2\right\}}^p,{\left\{1\right\}}^r,{\left\{2\right\}}^q)+{(-1)}^{r+1}{\zeta }^{★}({\left\{2\right\}}^q,r+2,{\left\{2\right\}}^{p-1})$. With the help of this equation and Zagier’s ${\zeta }^{★}({\left\{2\right\}}^p,3,{\left\{2\right\}}^q)$ formula, we can easily determine ${\zeta }^{★}({\left\{2\right\}}^p,1,{\left\{2\right\}}^q)$ and several interesting expressions. Full article

Multiobjective programming refers to a mathematical problem that requires the simultaneous optimization of multiple independent yet interrelated objective functions when solving a problem. It is widely used in various fields, such as engineering design, financial investment, environmental planning, and transportation planning. Research on the theory and application of convex functions and their generalized convexity in multiobjective programming helps us understand the essence of optimization problems, and promotes the development of optimization algorithms and theories. In this paper, we firstly introduces new classes of generalized $(F,\alpha ,\rho ,d)-I$ functions for semi-preinvariant convex multiobjective programming. Secondly, based on these generalized functions, we derive several sufficient optimality conditions for a feasible solution to be an efficient or weakly efficient solution. Finally, we prove weak duality theorems for mixed-type duality. Full article

In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin’s theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if $\left\{{\Lambda }_n\right\}$ is a sequence of open sets and ${\Lambda }_n\searrow \varnothing$, then ${\displaystyle \underset{n\to \infty }{lim}\mathcal{M}\left\{{\Lambda }_n\right\}=0}$ are equivalent in compact metric spaces. It is shown that Lusin’s theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin’s theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin’s theorem for uncertain variables. As far as we know, our results are new in uncertainty theory. Full article

Reducing procurement risks to ensure the supply of emergency supplies is crucial for mitigating the losses caused by mine thermodynamic disasters. The risk preference of decision-makers and supply chain collaboration are the important aspects for this reductiom. In this study, a novel P-CVaR (Piecewise conditional risk value) distributionally robust optimization model is proposed to accurately assist the decision-makers’ decision of risk preference for reducing procurement risks. Meanwhile, the role of cooperation between procurement and reserves are considered for the weakening procurement risks. A risk-averse bi-level optimization model is proposed to obtain the optimal procurement strategy. Furthermore, by applying the Lagrange duality theorem, the complexity of the bi-level optimization model is simplified then solved using a PSO algorithm. Using empirical analysis, it has been verified that the model presented in this paper serves as a valuable guideline for mine thermodynamic pre-disaster emergency material procurement strategies for the prevention of thermodynamic disasters. Full article

Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved. Full article

In the future, the active load of the distribution network side will be dominated by electric vehicles (EVs), showing that the charging power demand of electric vehicles will change with the change in charging electricity price. With the popularity of electric vehicles in the distribution network, their aggregation operators will play a more prominent role in pricing management and charging behavior, and setting an appropriate charging price can achieve a win–win situation for operators and electric vehicle users. At the same time, the proportion of scenery in the distribution network is relatively high, and the uncertainty of self-output has a certain impact on the pricing strategy of operators and the charging behavior of electric vehicle users, which has become an important research topic. Based on the above background, an EV operator pricing strategy considering the landscape uncertainty is proposed, a Stackelberg game model is established to maximize the respective benefits of operators and EV users, and the two-layer model is further transformed into a single-layer model through the Karush–Kuhn–Tucker (KKT) condition and duality theorem. Finally, the IEEE 33 system is simulated with the CPLEX solver, and the global optimal pricing strategy is obtained. Simulation results prove that electric vehicle operators experience a maximum profit increase of 2.6% due to the impact of maximum capacity of energy storage equipment and the uncertainty of renewable energy output can result in electric vehicle operators losing approximately 20% of their profits at most. Full article