Research on Unit Commitment Optimization Based on Discrete Differential Evolution and Quadratic Programming

In a chance constrained program (CCP), decision makers seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk ([Formula: see text]) has been known to be the best for more than a decade. This paper studies and generalizes the [Formula: see text], originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017 , for solving a CCP. We first show that the [Formula: see text] resembles a bilevel optimization, where the upper-level problem is to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, [Formula: see text] always outperforms the [Formula: see text] approximation. We further show (i) sufficient conditions under which [Formula: see text] can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance [Formula: see text] with a convergent alternating minimization method ([Formula: see text]); and (iii) an extension of [Formula: see text] and [Formula: see text] to distributionally robust chance constrained programs (DRCCPs) under the [Formula: see text]Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.2225 .

The protection of critical facilities has been attracting increasing attention in the past two decades. Critical facilities involve physical assets such as bridges, railways, power plants, hospitals, and transportation hubs among others. In this study we introduce a bilevel optimization problem for the determination of the most critical depots in a vehicle routing context. The problem is modeled as an attacker–defender game (Stackelberg game) from the perspective of an adversary agent (the attacker) who aims to inflict maximum disruption on a routing network. We refer to this problem as the r‐interdiction selective multi‐depot vehicle routing problem (RI‐SMDVRP). The attacker is the decision maker in the upper level problem (ULP) who chooses r depots to interdict with certainty. The defender is the decision maker in the lower level problem (LLP) who optimizes the vehicle routes in the wake of the attack. The defender has to satisfy all customer demand either using the remaining depots or through outsourcing to a third party logistics service provider. The ULP is solved through exhaustive enumeration, which is viable when the cardinality of interdictions does not exceed five among nine depots. For the LLP we implement a tabu search heuristic adapted to the selective multi‐depot VRP. Our results are obtained on a set of RI‐SMDVRP instances synthetically constructed from standard MDVRP test instances.

In this paper, we consider the problem of scheduling N robots interacting with a moving target. Both, the sequence of the robots and their trajectories are unknown and subject to optimization. Such type of problems appear in highly automated production plants and in the simulation of virtual factories. The purpose of the paper is to provide a mathematical model and to suggest a numerical solution approach. Our approach is based on the formulation of the problem as a bilevel optimization problem, where the lower level problem is an optimal control problem, while the upper level problem is a finite dimensional mixed-integer optimization problem. We approach the problem by exploitation of necessary optimality conditions for the lower level problem and by application of a Branch & Bound method for the resulting single level optimization problem. Two settings are taken into account. Firstly, no state constraints are assumed on the lower level problem, thus the local minimum principle applies directly. Secondly, the problem setting is augmented by pure state constraints, which are being handled by virtual controls in order to regularize the problem.

Optimal monopoly pricing with congestion and random utility via partial mass transport

In this research, we formulate the emergency evacuation and rescue network reconstruction problem as a bi-level programming network design model. The upper level problem is a decision of minimized system cost under limited resources. The lower level problem is one of the constraints for the upper level problem that considers users equilibrium route choice behaviors. We also consider the multi-class users' route choice behaviors in the lower level problem. In solving the problem, the variational inequality sensitivity analysis method, generalized inverse matrix method, diagonalization method, and gradient projection method are adopted to develop the solution algorithms for the bi-level model. The numerical test results indicated that the Stackelberg equilibrium solution of the emergency evacuation and rescue network reconstruction for natural disasters bi-level programming model do exist. More significantly, the optimal network reconstruction planning, disaster victim emergency evacuation and rescue route planning under multi-class user route choice conditions can be achieved.

The multiobjective bilevel program is a sequence of two optimization problems, with the upper-level problem being multiobjective and the constraint region of the upper level problem being determined implicitly by the solution set to the lower-level problem. In the case where the Karush-Kuhn-Tucker (KKT) condition is necessary and sufficient for global optimality of all lower-level problems near the optimal solution, we present various optimality conditions by replacing the lower-level problem with its KKT conditions. For the general multiobjective bilevel problem, we derive necessary optimality conditions by considering a combined problem, with both the value function and the KKT condition of the lower-level problem involved in the constraints. Most results of this paper are new, even for the case of a single-objective bilevel program, the case of a mathematical program with complementarity constraints, and the case of a multiobjective optimization problem.

The electrolytic aluminum accounts for about 7% of China’s total electricity use and has great potential for demand-side response (DR) with smart grid technologies. This paper focuses on the optimal scheduling of the electrolytic aluminum with DR function. The bi-level optimization model is proposed to guide electrolytic aluminum load for providing DR. In the proposed optimization model, the upper-level problem is to minimize the system operational cost, while the lower-level problem is to maximize the profit of the aluminum plant. The Karush-Kuhn-Tucker optimality conditions for the lower-level problem is derived, and the bi-level problem is reformulated as the mix-integer quadratic programming problem. The simulation results demonstrate that the proposed bi-level programming can not only reduce the operational cost of the power system, but also increase the profit of the electrolytic aluminum plant when it provides DR services.

Transit assignment, a problem of predicting passengers’ route choices and determining the usage for transit services, is an important research field because the results of transit assignment models are crucial for evaluating the performance of transit networks. Essentially, the passengers’ route choices in transit assignment models depend on the supply of transit networks and a principle describing how passengers select transit routes. This thesis contributes to the literature of transit assignment by incorporating the stochasticity of the supply side of transit networks and proposing the reliability-based user equilibrium conditions that define the passengers’ route choice principle in uncertain transit networks.
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\nThis thesis proposes three general frameworks based on the linear programming approach, the complementarity problem approach, and the variational inequality problem approach for modeling and analyzing reliability-based user equilibrium problems. Based on these frameworks, this thesis develops mean standard-deviation and mean-variance formulations wherein either hard capacity constraints or soft capacity constraints are incorporated. Within each of these frameworks, the existence of a solution is guaranteed given continuous cost functions and a convex compact solution set. The properties of these formulations are discussed and illustrated.
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\nTo solve the proposed formulations, this thesis develops different algorithms. For the mean standard-deviation formulations, the column generation technique is adopted, while for the mean-variance formulations, the dynamic programming method is used. In particular, this thesis proposes a revised extragradient method. This method only requires a mild assumption for convergence and its computational advantages are demonstrated in a realistic size network.
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\nThis thesis also initially investigates the supply side paradox issues in transit area. Two types of paradoxes are pointed out. One is a Braess-like paradox, which is a phenomenon that the system performance deteriorates, in terms of the total effective travel cost, after providing a new line or increasing the frequency of a transit line. The other is a capacity paradox phenomenon that the network throughout reduces after adding a new line or increasing the frequency of a transit line.
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\nFinally, one of the proposed reliability-based transit assignment models is applied into a bilevel framework for solving a transit network design problem. The upper level problem is to determine the frequency setting of transit lines, while the lower level problem is the mean-variance formulation of the transit assignment problem with hard capacity constraints. A decent direction method is developed to solve the bilevel problem. Numerical studies are also provided to investigate the properties of the problem. Future research directions are also pointed out.

The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bilinear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This paper contains a compilation of theoretical and algorithmic results on the network Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and for the development of algorithms: establishing NP-hardness, approximability, special cases solvable in polynomial time, and an efficient exact branch-and-bound algorithm.

The unit commitment(UC) problem is a critical problem in economic dispatch of power system, and it is the key to planning for short-term power generation. Its economic benefit is generally greater than the benefit of economic distribution of load. However, unit commitment is a high-dimensional, nonconvex, discrete, nonlinear mixed integer optimization problem, and is difficult to find the optimal theoretical solution. Therefore, people have been actively studying to solve this problem. Social learning particle swarm optimizer is a recent proposed metaheuristic algorithm specialized in solving the high-dimensional problem. In this paper, a novel binary social learning particle swarm optimizer (BSLPSO) is proposed for solving the UC problem associated with lambda iteration method. In order to verify the effectiveness of the algorithm for UC problem, a comprehensive numerical study of 10 to 100 units has been conducted, and compared with other related algorithms. The result shows that this algorithm performs well in UC problem and is better than other algorithms.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

ABSTRACTIn this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward–Forward algorithm (PFFSA) and the Forward–Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field.

The Great Britain (GB) government is paving the path to decarbonisation by actively promoting the integration of wind power into its generation mix. This sharp transition to renewable energy, however, introduces specific challenges. The characteristics of non-synchronous wind turbines have the potential to impact grid stability and frequency security due to the reduction in system inertia. In response to these challenges, the deployment of virtual power plants (VPPs) is envisioned in deregulated power systems, which allow for the aggregation and coordinated control of diverse distributed energy resources, enhancing grid flexibility, reliability, and efficiency. Moreover, VPPs offer the provision of system inertia and frequency response services at the national level. This study drops this assumption and leverages the localised flexibility inherent in VPPs to meet the ancillary service requirements of the low-carbon power system. The formulated problem is constructed as a non-convex bi-level optimisation problem, where the upper-level problem represents the operation of a frequency-constrained unit commitment model, and the lower-level problem embodies the energy dispatches and frequency responses of a group of VPPs, guided by the dual price signals of cleared energy and ancillary services. To address the non-convex nature of the unit commitment problem, a two-fold approach is employed. First, binary commitment status decision variables are relaxed to continuous versions. Subsequently, the duality gap between the original non-convex unit commitment problem and its relaxed dual form is minimised, yielding a solution closely approximating the optimal solution of the original problem. Second, the relaxed bi-level optimisation problem is transformed into a single-level mathematical programs with equilibrium constraints by replacing the lower-level VPP problems with their equivalent Karush-Kuhn–Tucker optimality conditions. The case studies conducted in this work encompass a comprehensive scope. Initially, the study evaluates the effectiveness of VPPs in delivering ancillary services within the projected context of the GB power system for 2030. The findings reveal a substantial 28.28% decrease in system operation costs when compared to the benchmark case lacking VPP flexibility. Additionally, the integration of VPPs leads to significantly lower price signals for inertia, primary frequency response, and enhanced frequency response, reduced by 81.21%, 79.13%, and 86.27%, respectively, in contrast to the benchmark case. The analysis then delves into a sensitivity exploration, investigating the inherent flexibility of VPPs. The profit of local VPPs is expected to increase with VPP flexibility. Finally, the study illustrates how VPPs can adapt to and derive benefits from an increasing level of wind power penetration by providing a higher amount of frequency response.

Unified solution of a non-convex SCUC problem using combination of modified Branch-and-Bound method with Quadratic Programming

Most direct methods solve finite time horizon optimal control problems with nonlinear programming solver. In this paper, we propose a numerical method for solving nonlinear optimal control problem with state and control inequality constraints. This method used quasilinearization technique and Haar wavelet operational matrix to convert the nonlinear optimal control problem into a quadratic programming problem. The linear inequality constraints for trajectories variables are converted to quadratic programming constraint by using Haar wavelet collocation method. The proposed method has been applied to solve Optimal Control of Multi-Item Inventory Model. The accuracy of the states, controls and cost can be improved by increasing the Haar wavelet resolution.