A novel dual-decomposition method for non-convex mixed integer quadratically constrained quadratic problems

In this paper, we propose the novel p-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by non-convex mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal non-convex MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and non-anticipativity conditions are met in the optimal solution. The p-branch-and-bound method's efficiency has been tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi. This paper also presents a comparative analysis of the p-branch-and-bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank-Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive hedging.Comment: 19 pages, 5 table

Optimal transmission expansion planning in the context of renewable energy integration policies

This paper assesses the extent to which a renewables-driven expansion of the transmission system infrastructure impacts the total generation mix in the decentralised energy market. For that, we employ an optimisation bi-level model in which a welfare-maximizing transmission system operator makes investments in transmission lines at the upper level while considering power market dynamics at the lower level. To account for the deregulated energy market structure, we assume that the generation companies at the lower level make capacity investments as price-takers in perfect competition. Considering alternative transmission infrastructure expansion budgets, carbon emission taxes and monetary incentives for renewable generation capacity expansion, we study how alternative compositions of these factors affect the share of renewable generation in the generation mix. We apply the proposed modelling assessment to an illustrative three-node instance and a case study considering a simplified representation of the energy system of the Nordic and Baltic countries. Our results suggest the limited efficiency of the three measures when applied individually. Nevertheless, when applied together, these three measures demonstrated a positive impact on Nordics' and Baltics' energy system welfare, renewable share, and total generation. However, the amplitude of this impact differs depending on the composition of values used for the three measures.Comment: 31 pages, 20 Figures, 12 Table

Modelling and solution methods for renewables-driven energy markets

Responding to the alarming climate change consequences, many countries are paying significant attention to the energy systems' transition towards environmental sustainability. As an example, European Union established an ambitious goal to become climate-neutral by 2050 compared to 10 levels, and South Korea aims to reduce greenhouse gas emissions by 37% below business-as-usual by 2030. Considering the essential role of energy markets in modern economies such targets pose a fundamental challenge to finding a potential solution that would ensure furthering human-kind well-being and decarbonisation. One of the commonly exploited crucial tools for planning energy systems transition and understanding its effect on the economy and social welfare is energy systems modelling. However, modelling techniques undergo criticism regarding the insufficient level of precision provided for police makers. In particular, two of the main challenges are associated with i) a limited number of attempts to integrate multiple energy-sector stakeholders into a single-model formulation and ii) a trade-off between the model complexity and its numerical tractability. This dissertation addresses both of these challenges. First, it formulates a modelling framework allowing one to represent energy systems operations involving multiple generation companies and transmission system planning. Additionally, the energy system models formulated in this dissertation allow for the consideration of renewable supporting policies such as carbon tax and investment subsidies. These models provide insights into how a welfare-maximising unit may impact the increase of renewable share in the generation mix without harming the total welfare. Such a study was conducted for Nordic and Baltic countries. Second, this dissertation provides a solution method formulation that can be applied to solving proposed mathematical models. The solution algorithm was developed in two stages: i) combining Lagrangian decomposition with mixed-integer relaxation allowing one to obtain an arbitrary precise solution in case of duality gap absence and ii) embedding these techniques within a duality-based branching strategy that would ensure solving to optimality even in the presence of duality gap. The models in this dissertation serve as a support for decision-makers trying to understand how and to which extent they can exploit the influence of the transmission system operator on the energy system with or without other supportive policies in the context of decarbonising the energy system. The solution algorithms proposed in this dissertation are generally applicable to a wide range of two-stage stochastic mixed integer problems appearing in such sectors as, for example, the design of water networks, modelling refinery processes and transportation systems

The p-Lagrangian relaxation for separable nonconvex MIQCQP problems

This paper presents a novel technique to compute Lagrangian bounds for nonconvex mixed-integer quadratically constrained quadratic programming problems presenting a separable structure (i.e., a separable problems) such as those arising in deterministic equivalent representations of two-stage stochastic programming problems. In general, the nonconvex nature of these models still poses a challenge to the available solvers, which do not consistently perform well for larger-scale instances. Therefore, we propose an appealing alternative algorithm that allows for overcoming computational performance issues. Our novel technique, named the p-Lagrangian decomposition, is a decomposition method that combines Lagrangian decomposition with mixed-integer programming-based relaxations. These relaxations are obtained using the reformulated normalised multiparametric disaggregation technique and can be made arbitrarily precise by means of a precision parameter p. We provide a technical analysis showing the convergent behaviour of the approach as the approximation is made increasingly precise. We observe that the proposed method presents significant reductions in computational time when compared with a previously proposed techniques in the literature and the direct employment of a commercial solver. Moreover, our computational experiments show that the employment of a simple heuristic can recover solutions with small duality gaps.Peer reviewe