Scheduling double round-robin tournaments with divisional play using constraint programming
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Carlsson, Mats↗
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Job Scheduling for HPC Clusters: Constraint Programming vs. Backfilling Approaches
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Goponenko, Alexander V.↗
Performing fully parallel constraint logic programming on a quantum annealer
A quantum annealer exploits quantum effects to solve a particular type of optimization problem. The advantage of this specialized hardware is that it effectively considers all possible solutions in parallel, thereby potentially outperforming classical computing systems. However, despite quantum annealers having recently become commercially available, there are relatively few high-level programming models that target these devices. In this article, we show how to compile a subset of Prolog enhanced with support for constraint logic programming into a two-local Ising-model Hamiltonian suitable for execution on a quantum annealer. In particular, we describe the series of transformations one can apply to convert constraint logic programs expressed in Prolog into an executable form that bears virtually no resemblance to a classical machine model yet that evaluates the specified constraints in a fully parallel manner. We evaluate our efforts on a 1,095-qubit D-Wave 2X quantum annealer and describe the approach's associated capabilities and shortcomings.
97 MATHEMATICS AND COMPUTING↗
Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints
Here, we propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence.
97 MATHEMATICS AND COMPUTING↗
A Hybrid Method to Estimate Specific Differential Phase and Rainfall With Linear Programming and Physics Constraints
A hybrid method of combining linear programming (LP) and physical constraints is developed to estimate specific differential phase (K DP ) and to improve rain estimation. Moreover, the hybrid K DP estimator and the existing estimators of LP, least squares fitting, and a self-consistent relation of polarimetric radar variables are evaluated and compared using simulated data. Our simulation results indicate the new estimator's superiority, particularly in regions where backscattering phase (δ hv ) dominates. Further, a quantitative comparison between auto-weather-station rain-gauge observations and K DP -based radar rain estimates for a Meiyu event also demonstrate the superiority of the hybrid K DP estimator over existing methods.
54 ENVIRONMENTAL SCIENCES↗
A stochastic biomass blending problem in decentralized supply chains
Blending biomass materials of different physical or chemical properties provides an opportunity to adjust the quality of the feedstock to meet the specifications of the conversion platform. We propose a model which identifies the right mix of biomass to optimize the performance of the thermochemical conversion process at the mini-mum cost. This is a chance-constraint programming (CCP) model which takes into account the stochastic nature of biomass quality. The proposed CCP model ensures that process requirements, which are impacted by physical and chemical properties of biomass, are met most of the time. We consider two problem settings, a centralized and a decentralized supply chain. We propose a mixed-integer linear program to model the blending problem in the centralized setting and a bilevel program to model the blending problem in the decentralized setting. We use the sample average approximation method to approximate the chance constraints, and propose solution algorithms to solve this approximation. We develop a case study for South Carolina using data provided by the Billion Ton Study. Based on our results, the blends identified consist mainly of pine and softwood residues. The blends identified and the suppliers selected by both models are different. The cost of the centralized supply chain is 2%–6% lower. The implications of these results are twofold. First, these results could lead to improved collaborations in the supply chain. Second, these results provide an estimate of the approximation error from assuming centralized decision making in the supply chain.
09 BIOMASS FUELS↗
Graphical models for optimal power flow
Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary tree-structured distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. In conclusion, numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Stochastic Models for an Optimal Blending of Biomass under Cost, Quality and Uncertainty Considerations
Blending biomass materials of different physical or chemical properties provides an opportunity to passively adjust the quality of the feedstock to meet the specifications of the conversion platform. We propose a model which identifies the right mix of biomass to optimize the performance of the Thermochemical conversion process at the minimum cost. This is a chance-constraint programming (CCP) model which takes into account the stochastic nature of biomass availability and quality. The proposed CCP model takes as inputs the physical and chemical properties, and biomass availability. We use the sample average approximation (SAA) method to solve the problem. We develop a case study for South Carolina by using data provided by the Billion Ton Study. We conduct extensive numerical analysis to evaluate the quality of the solutions obtained.
09 BIOMASS FUELS↗
Approximating Joint Chance Constraints in Two-Stage Stochastic Programs
We present various approximations to joint chance constraints arising in two-stage stochastic programming models. Our approximations are derived from three classical inequalities: Markov's inequality, Chebysev's inequality, and Chernoff's bound. We provide preliminary computational results illustrating the quality of our approximation using a two-stage joint-chance-constrained stochastic program from the literature. We also briefly introduce other alternatives for constructing approximations for joint-chance-constrained two-stage programs.
97 MATHEMATICS AND COMPUTING↗
On optimal control of hybrid dynamical systems using complementarity constraints
Optimal control for switch-based dynamical systems is a challenging problem in the process control literature. In this study, we model these systems as hybrid dynamical systems with finite number of unknown switching points and reformulate them using non-smooth and non-convex complementarity constraints as a mathematical program with complementarity constraints (MPCC). We utilize a moving finite element based strategy to discretize the differential equation system to accurately locate the unknown switching points at the finite element boundary and achieve high-order accuracy at intermediate non-collocation points. We propose a globalization approach to solve the discretized MPCC problem using a mixed NLP/MILP-based strategy to converge to a non-spurious first-order optimal solution. The method is tested on three dynamic optimization examples, including a gas–liquid tank model and an optimal control problem with a sliding mode solution.
97 MATHEMATICS AND COMPUTING↗
A constraint-reduced MPC algorithm for convex quadratic programming, with a modified active set identification scheme
A constraint-reduced Mehrotra-predictor-corrector algorithm for convex quadratic programming is proposed. (At each iteration, such algorithms use only a subset of the inequality constraints in constructing the search direction, resulting in CPU savings.) The proposed algorithm makes use of a regularization scheme to cater to cases where the reduced constraint matrix is rank deficient. Global and local convergence properties are established under arbitrary working-set selection rules subject to satisfaction of a general condition. A modified active-set identification scheme that fulfills this condition is introduced. Numerical tests show great promise for the proposed algorithm, in particular for its active-set identification scheme. While the focus of the present paper is on dense systems, application of the main ideas to large sparse systems is briefly discussed.
97 MATHEMATICS AND COMPUTING↗
Uncertainty-Informed Operation Coordination in a Water-Energy Nexus
The widespread deployment of smart heterogeneous technologies and the growing complexity in our modern society calls for effective coordination of the interdependent lifeline networks. In particular, operation coordination of electric power and water infrastructures is urgently needed as the water system is one of the most energy-intensive networks, an interruption in which may quickly evolve into a dramatic societal concern. This paper develops a novel analytic for uncertainty-aware day-ahead operation optimization of the interconnected power and water systems (PaWS). Joint probabilistic constraint (JPC) programming is employed to capture the uncertainties in wind resources and water demand forecasts. The proposed integrated stochastic model is presented as a non-linear non-convex optimization problem, where the non-linear hydraulic constraints in the water network are linearized using piece-wise linearization technique, and the non-convexity is efficiently tackled with a solution methodology to convert the proposed model with JPCs to a tractable mixed-integer linear programming (MILP) formulation that can be quickly solved to optimality. Here, the suggested framework is applied to a 15-node commercial-scale water network jointly operated with a power transmission system using a modified IEEE 57-bus test system. The numerical results demonstrate the of the proposed stochastic framework, resulting in cost reduction (13% on average when compared to the traditional setting) and energy saving of the integrated model under different realizations of uncertain renewable energy sources (RESs) and water demand scenarios. Additionally, the scalability of the proposed model is tested on a modified IEEE 118-bus test system connected to five water networks.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Two-Stage Distributionally Robust Conic Linear Programming over 1-Wasserstein Balls
Here, this paper studies two-stage distributionally robust conic linear programming under constraint uncertainty over type-1 Wasserstein balls. We present optimality conditions for the dual of the worst-case expectation problem, which characterizes worst-case uncertain parameters for its inner maximization problem. This condition offers an alternative proof, a counterexample, and an extension to previous works. Additionally, the condition highlights the potential advantage of a specific distance metric for out-of-sample performance, as exemplified in a numerical study on a facility location problem with demand uncertainty. Furthermore, cutting-plane-based algorithms, equipped with a unified scenario generation framework, are proposed for addressing both unbounded support and second-stage dual feasible regions, with a finite convergence proof under less stringent assumptions.
Wasserstein↗
Viscosities of the Baryon-Rich Quark-Gluon Plasma from Beam Energy Scan Data
Here, this work presents the first Bayesian inference study of the (3 + 1)D dynamics of relativistic heavy-ion collisions and quark-gluon plasma viscosities using an event-by-event (3 + 1)D hydrodynamics + hadronic transport theoretical framework and data from the Relativistic Heavy Ion Collider Beam energy scan program. Robust constraints on initial state nuclear stopping and the baryon chemical potential-dependent shear viscosity of the produced quantum chromodynamic (QCD) matter are obtained. The specific bulk viscosity of the QCD matter is found to exhibit a preferred maximum around $\sqrt{s_{NN}}$ = 19.6 GeV . This result allows for the alternative interpretation of a reduction (and/or increase) of the speed of sound relative to that of the employed lattice-QCD based equation of state for net baryon chemical potential μ B ~0.2(0.4) GeV .
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching
This paper studies mixed-integer nonlinear programs featuring disjunctive constraints and trigonometric functions and presents a strengthened version of the convex quadratic relaxation of the optimal transmission switching problem. We first characterize the convex hull of univariate quadratic on/off constraints in the space of original variables using perspective functions. Next, we introduce new tight quadratic relaxations for trigonometric functions featuring variables with asymmetrical bounds. These results are used to further tighten recent convex relaxations introduced for the optimal transmission switching problem in power systems. Using the proposed improvements, along with bound propagation, on 23 medium-sized test cases in the PGLib benchmark library with a relaxation gap of more than 1%, we reduce the gap to less than 1% on five instances. The tightened model has promising computational results when compared with state-of-the-art formulations.
97 MATHEMATICS AND COMPUTING↗
A Mixed integer linear programming‐based distributed energy management for networked microgrids considering network operational objectives and constraints
Abstract Mixed integer linear programming (MILP)–based distributed energy management for networked microgrids embedded modern distribution systems is proposed. Considering the diverse ownership of microgrids, distributed energy resources (DERs) that interface directly with utilities and responsive loads, an alternating direction method of multipliers–based distributed framework was formulated for the scheduling of networked microgrids embedded modern distribution systems by adjusting nodal price signals iteratively. In addition, to make the formulated optimization problems resolvable through more accessible and popular MILP solvers, different linearisation techniques were employed to transform the nonlinear terms into linear or mixed integer linear formats. The proposed MILP‐based distributed method preserves all participants' autonomy (e.g., microgrids, DERs that interface directly with utilities and responsive loads), while incentivising them to actively participate in the distribution system operation with price signals. The proposed method is validated with results of numerical simulation using a modern distribution system consisting of multiple networked microgrids, DERs that interface directly with utilities, as well as responsive loads.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Bi-Level Arbitrage Potential Evaluation for Grid-Scale Energy Storage Considering Wind Power and LMP Smoothing Effect
Our paper deals with extended-term energy storage (ES) arbitrage problems to maximize the annual revenue in deregulated power systems with high penetration wind power. The conventional ES arbitrage model takes the locational marginal prices (LMP) as an input and is unable to account for the impacts of ES operations on system LMPs. This paper proposes a bi-level ES arbitrage model, where the upper level maximizes the ES arbitrage revenue and the lower level simulates the market clearing process considering wind power and ES. The bi-level model is formulated as a mathematical program with equilibrium constraints (MPEC) and then recast into a mixed-integer linear programming (MILP) using strong duality theory. Wind power fluctuations are characterized by the GARCH forecast model and the forecast error is modeled by forecast-bin based Beta distributions. Case studies are performed on a modified PJM 5-bus system and an IEEE 118-bus system with a weekly time horizon over an annual term to show the validity of the proposed bi-level model. The results from the conventional model and the bi-level model are compared under different ES power and energy ratings, and also various load and wind penetration levels.
17 WIND ENERGY↗
Real-Time Procurement Strategies of a Proactive Distribution Company With Aggregator-Based Demand Response
In this paper, we present a real-time trading framework for distribution networks where a rational aggregator is identified as a broker by contracting with individual demands and dealing with the distribution company. Demand response capability is characterized by the coexistence of elastic and inelastic demand components. A one-leader multi-follower bi-level model is proposed to derive the procurement strategies, i.e., the upperlevel problem intends to maximize the profit of the proactive distribution company, while the lower-level expresses the profit maximization per rational aggregator. The proposed model is then transformed into a solvable mathematical program with equilibrium constraints through a primal-dual approach. A modified 33-bus distribution network is utilized to demonstrate the effectiveness of the proposed model.
demand response (DR)↗