McCormick Relaxation Research Articles

Robust wind power capacity planning under fuel price uncertainty using conic duality theory and piecewise McCormick relaxation

Increasing the proportion of renewable energy sources (RESs) in power generation is crucial due to fossil fuel depletion and rising environmental pollution. In this regard, identifying the most lucrative sites and sizes for installing wind farms, as one of the fastest-growing types of RESs, is imperative. This paper presents a robust profit-oriented wind power capacity planning (WPCP) considering long- and short-term uncertainty. The model forms a two-stage min-max-min hierarchical structure. The first stage minimizes the investment cost plus maximum regret, while the second stage maximizes the profit under the worst-case uncertainty realization. Unlike the existing approaches where uncertainty in fossil fuel prices is neglected, we model fuel price uncertainty using both polyhedral and ellipsoidal uncertainty sets. In this respect, the third level is formulated as a bi-level program, with the upper level being the profit maximization and the lower level being the locational marginal price (LMP) calculation. In the case of the ellipsoidal set, the conic duality theory is employed to dualize the lower level. The piecewise McCormick relaxation (PMR) technique linearizes the bilinear terms. The nested column-and-constraint generation (NCCG) technique solves the formulated problem. A clarifying case study is employed to demonstrate the efficacy of the proposed model.

Interval Assessment Method for Distribution Network Hosting Capacity of Renewable Distributed Generation

The traditional fixed value assessment of the renewable distributed energy hosting capacity of a distribution network cannot accurately and comprehensively reflect the change in hosting capacity; therefore, we propose the interval assessment method for the renewable distributed energy hosting capacity of a distribution network. The renewable distributed energy hosting capacity interval consists of an optimistic upper boundary and a pessimistic lower boundary. First, the optimistic upper bound is described by a deterministic model that takes into account the constraints of safe system operation. Second, the pessimistic lower bound is portrayed by a two-layer robust assessment model that accounts for the DG temporal uncertainty, DG spatial uncertainty, and active distribution network flexible resource dispatch uncertainty. Each pessimistic sub-model was constructed in turn, and then the model was solved by linear simplification using pairwise transformation, as well as McCormick relaxation. Finally, simulations were carried out in the IEEE 135 system, and the results validated the effectiveness and practicality of the proposed method.

An integrated planning framework for optimal power generation portfolio including frequency and reserve requirements

AbstractElectricity system decarbonisation poses several challenges to network stability and supply security, given renewables' intermittency and possible reduction of system inertia. This manuscript presents a novel integrated system framework to determine optimal generation investments for addressing decarbonisation challenges and achieving cost‐effective electricity systems while ensuring frequency stability and reserve requirements are met at the operational level in a net‐zero system. The novel planning framework is a mixed‐integer bilinear programming problem accurately modelling clustered variables for the on/off status of generation units and seconds‐timescale frequency requirements at an operational and planning level. The benefits of the decision framework and effects of dispatch decisions in a year are illustrated using the Great Britain case study. The results provide optimal trade‐offs and cost‐effective investment portfolios for including detailed modelling of unit‐commitment and frequency stability constraints versus not including them in the planning model. Making investment decisions for a net‐zero electricity system without these constraints can lead to very high system costs due to significant demand curtailment. Although the model's computation burden was increased by these constraints, complexity was managed by formulating them tightly and compactly. Non‐convex quadratic nadir constraints were efficiently solvable to global optimality by applying McCormick relaxations and branching techniques in an advanced solver.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

Optimal dispatch of integrated electricity and heating systems considering the quality-quantity regulation of heating systems to promote renewable energy consumption

The integrated electricity and heating system (IEHS) can improve energy efficiency and promote renewable energy consumption. When quality-quantity regulation (QQR) is adopted, namely adjusting hydraulic and thermal conditions of heating system, the increase in operation flexibility is more remarkable. However, present IEHS dispatch models considering QQR overlook hydraulic characteristics of valves and variable frequency pumps resulting in inexecutable dispatch results easily and hindering IEHS's flexibility potential. Furthermore, loop networks and the access of multiple heat sources in heating systems may lead to flow reversal, but current research adopts a fixed flow direction which suppresses the potential of flow reversal to improve flexibility. In this paper, we propose an IEHS dispatch model considering QQR to promote renewable energy consumption, which considers hydraulic characteristics of valves and variable frequency pumps, and flow reversal. Specifically, we introduce two binary flow direction labels for every branch in heating system model and then the IEHS dispatch model which can deal with flow reversal is established. A sequential solution process combining linear and nonlinear optimization is formulated to overcome the non-convex feature of IEHS dispatch model. Specifically, piecewise linearization and piecewise McCormick relaxation are combined to handle complex nonlinear terms in the dispatch model. Therefore, a mixed integer linear programming model is obtained and solved, of which results are used as initial values for nonlinear optimization. Results in the case study show that operation cost is decreased by 0.97 % and renewable power consumption rate is increased from 83.31 % to 84.42 % after considering valve adjustment. Operation cost is further decreased by 6.06 % and renewable power consumption rate is increased to 94.04 % after considering flow reversal.

Convexification of Bilinear Terms over Network Polytopes

It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where xi represents the arc-flow variable in a network polytope, and yj is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques. Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 .

Optimal Energy Configuration of Integrated Energy Community Considering Carbon Emission

An integrated energy community with a distributed utilization of renewable energy and complementary electricity–gas–cold–heat integrated energy will play an important role in energy conservation and emission reduction. In addition, compared with traditional thermoelectric power equipment, solid oxide fuel cells have many advantages, such as a high energy utilization rate, good waste heat quality, and low carbon emissions. Therefore, the SOFC-based multi-energy and energy storage sharing operation model of an integrated energy community with an electricity–gas–cooling–heat integrated energy system is constructed, and a bi-objective optimal configuration model considering the carbon emission index is established. Considering the economic objective of the smallest annual total operating cost as the most important objective in optimizing the planning model, the ε-constraint method is used to transform the environmental objective function with the smallest annual total carbon emission into a constraint condition under the decision making of an economic single objective function, and then the planning model is linearized and solved by using the Big-M method and the McCormick relaxation method. By calculating and analyzing the energy allocation results in five scenarios, the effectiveness and rationality of the model built in this article are verified. At the same time, the calculation example analysis results show that as the ε value decreases, the energy configuration of the integrated energy community will shift from natural gas to clean energy. From this perspective, the energy equipment configuration and operating costs will increase. However, the heat storage system and power storage system sharing can effectively reduce the energy allocation capacity and costs.

Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

Optimal operation of lithium-ion batteries in microgrids using a semidefinite thermal model

The growing adoption of microgrids necessitates efficient management of electrical energy storage units to ensure reliable and sustainable power supply. This paper investigates a thermal management system (TMS) for maintaining the longevity of large-scale batteries. To streamline the thermal modeling of batteries, the McCormick relaxation method is employed to linearize a nonlinear and interdependent heat generation model. The thermal model of the battery follows a nonlinear behavior where the generated heat makes the battery system temperature soar, thereby affecting the thermal performance of the battery. To showcase the efficacy of the proposed approach, four distinct case scenarios are studied, highlighting the critical importance of batteries within microgrid operations. A comparative analysis is conducted between linear and nonlinear models for TMS performance. A quantitative assessment based on simulation results demonstrates the precision of the linearized model, particularly in a multitemporal optimal power flow and day-ahead scheduling of microgrids incorporating energy storage units. Controlling the battery temperature within a permissible range (from 15 °C to 40 °C) is achieved by using a heating, ventilation, and air conditioning (HVAC) system. The paper explores the economic implications of energy storage units in microgrids by extracting and comparing daily operational costs with and without battery integration. The findings reveal that the inclusion of energy storage units yields substantial economic benefits, with potential profit margins of approximately 20 % during typical working days and 60 % on weekends.

BEELINE: BilevEl dEcomposition aLgorithm for synthesis of Industrial eNergy systEms

Energy transition to a more sustainable basis is the most significant and complex challenge facing industry. On most industrial sites, the largest single energy user is the utility system that produces the heat and power necessary for the site. The heavy reliance of current utility systems on fossil fuels and the requirement of strategic measures to ensure a sustainable future has prompted researchers to explore different energy sources, technologies and pathways for evolving existing systems to a sustainable basis for future utility systems. Whether based on renewable or non-renewable energy sources, it is essential to minimize energy demand and mitigate emissions. The present work focuses on developing cost-effective solutions for the synthesis of process utility systems, considering site-wide energy integration. This will provide a platform in future work for the efficient introduction of renewable energy sources in the transition to sustainable systems. Utility system performance is generally determined by system configuration and operational load. Steam mains selection, in terms of pressures and superheating have a critical role in the utility system performance and site energy integration. Therefore, the synthesis of energy-efficient utility systems involves optimizing the configuration of utility components and number and operating conditions of steam mains simultaneously. Due to nonlinearities and non-convexities from underlying physics and binary decisions involved, the resulting Mixed-Integer Non-Linear Problem (MINLP) presents challenges for advanced state-of-art solvers to solve real-world problems. In past work, a number of important practical issues have been oversimplified in order to make the solution tractable. However, the oversimplifications also lead to misleading results. In this research, a mathematical formulation for simultaneous optimization of comprehensive utility system configurations and operating conditions is for the first time combined with more realistic steam system configurations and operating conditions to represent the utility systems. Its framework is constructed via a bilevel decomposition algorithm based on piecewise MILP relaxation, McCormick relaxation, and linearization of steam properties. In addition, the solution pool feature of the CPLEX solver is incorporated to enhance the performance and convergence of the algorithm. This work presents the fundamental problem formulation that has not been sufficiently addressed previously. Indeed, this methodology sets out the basis for synthesizing energy-efficient utility systems for the future and allows for many energy conversion technologies and sources to be added to the framework that have previously not been possible to include.

Automatic differentiation rules for Tsoukalas–Mitsos convex relaxations in global process optimization

With increasing digitalization and vertical integration of chemical process systems, nonconvex optimization problems often emerge in chemical engineering applications, yet require specialized optimization techniques. Typical global optimization methods proceed by progressively refining bounds on the unknown optimal value, by strategically employing convex relaxations. This article constructs a general closed-form expression for the convex subdifferentials of recent “multivariate McCormick” convex relaxations of nontrivial composite functions, by solving a previous duality formulation in all cases using nonsmooth Karush–Kuhn–Tucker conditions. Based on this subdifferential expression, new automatic differentiation rules are developed to compute gradients and subgradients for multivariate McCormick relaxations, to ultimately generate useful bounds in global optimization. Unlike established differentiation techniques for these relaxations, our new rules are expressed in closed form, do not require solving separate dual optimization problems, are efficiently carried out, and are compatible with the reverse/adjoint mode of algorithmic differentiation. Our formulations become more straightforward when the relevant functions are either smooth or piecewise smooth.

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.

An Enhanced McCormick Envelopes to Represent Kron's Loss Formula

Recently, some researchers have employed the McCormick envelopes method to convexify some NP-hard optimization problems with bilinear terms. However, few publications concentrate on its variants to derive a more tight convex relaxation for practical applications. This paper proposes a new viewpoint on Kron’s loss formula, also known as the B-matrix formula, as an equation having bilinear terms. Relying on the perspective, we transform the loss equation to some linear constraints using an enhanced McCormick relaxation. In the technique, the domain of bilinear variables is divided into some smaller parts to improve the relaxation tightness. Some case studies with different nonconvex terms are considered to verify the effectiveness of the enhanced envelopes for capturing Kron’s loss formula. The findings from the numerical simulations suggest that the proposed approach can represent Kron’s loss equation precisely. Moreover, the method performs more effectively than the other methods available in the literature as it usually converges to more optimal solutions.

On piecewise linear approximations of bilinear terms: structural comparison of univariate and bivariate mixed-integer programming formulations

Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an varepsilon -approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error varepsilon , univariate varepsilon -approximations require fewer simplices than bivariate varepsilon -approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.

Optimal strategy of electricity and natural gas aggregators in the energy and balance markets

This paper presents a stochastic two-stage model for energy aggregators (EAs) in the energy and balancing markets to supply electricity and natural gas to end-users equipped with combined heat and power (CHP) units. The suggested model takes into account the battery energy storage (BES) as a self-generating unit of EA. The upper and lower subproblems determine the optimal energy supply strategy of EA and consumption of consumers, respectively. In the lower subproblem, the McCormick relaxation is used to linearize the cost function of the CHP unit. To solve the proposed model, the two-stage problem is transformed into a linear single-stage problem using the KKT conditions of the lower subproblem, the Big M method, and the strong duality theory. The performance and efficiency of the proposed model are evaluated using a case study and three scenarios. According to the simulation results, adding CHP units to the energy-scheduling problem of BES-owned aggregators increases the profit of EA by 5.96% and decreases the cost of consumers by 1.57%.

Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk

In this article, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model-predictive control (MPC) method for limiting the risk of unsafety even when the true distribution of the obstacles’ movements deviates, within an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ambiguity set</i> , from the empirical distribution obtained using a limited amount of sample data. By choosing the ambiguity set as a statistical ball with its radius measured by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wasserstein metric</i> , we achieve a probabilistic guarantee of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">out-of-sample risk</i> , evaluated using new sample data generated independently of the training data. To resolve the infinite-dimensionality issue inherent in the distributionally robust MPC problem, we reformulate it as a finite-dimensional nonlinear program using modern distributionally robust optimization techniques based on the Kantorovich duality principle. To find a globally optimal solution in the case of affine dynamics and output equations, a spatial branch-and-bound algorithm is designed using McCormick relaxation. The performance of the proposed method is demonstrated and analyzed through simulation studies using nonlinear dynamic and kinematic vehicle models and a linearized quadrotor model. The simulation results indicate that, even when the sample size is small, the proposed method can successfully avoid randomly moving obstacles with a guarantee of out-of-sample risk, while its sample average approximation counterpart fails to do so.

Sizing battery energy storage and PV system in an extreme fast charging station considering uncertainties and battery degradation

This paper presents mixed integer linear programming (MILP) formulations to obtain optimal sizing for a battery energy storage system (BESS) and solar generation system in an extreme fast charging station (XFCS) to reduce the annualized total cost. The proposed model characterizes a typical year with eight representative scenarios and obtains the optimal energy management for the station and BESS operation to exploit the energy arbitrage for each scenario. Contrasting extant literature, this paper proposes a constant power constant voltage (CPCV) based improved probabilistic approach to model the XFCS charging demand for weekdays and weekends. This paper also accounts for the monthly and annual demand charges based on realistic utility tariffs. Furthermore, BESS life degradation is considered in the model to ensure no replacement is needed during the considered planning horizon. Different from the literature, this paper offers pragmatic MILP formulations to tally BESS charge/discharge cycles using the cumulative charge/discharge energy concept. McCormick relaxations and the Big-M method are utilized to relax the bi-linear terms in the BESS operational constraints. Finally, a robust optimization-based MILP model is proposed and leveraged to account for uncertainties in electricity price, solar generation, and XFCS demand. Case studies were performed to signify the efficacy of the proposed formulations.

Optimal Operation of Integrated Heat and Electricity Systems: A Tightening McCormick Approach

Combined heat and electricity operation with variable mass flow rates promotes flexibility, economy, and sustainability through synergies between electric power systems (EPSs) and district heating systems (DHSs). Such combined operation presents a highly nonlinear and nonconvex optimization problem, mainly due to the bilinear terms in the heat flow model—that is, the product of the mass flow rate and the nodal temperature. Existing methods, such as nonlinear optimization, generalized Benders decomposition, and convex relaxation, still present challenges in achieving a satisfactory performance in terms of solution quality and computational efficiency. To resolve this problem, we herein first reformulate the district heating network model through an equivalent transformation and variable substitution. The reformulated model has only one set of nonconvex constraints with reduced bilinear terms, and the remaining constraints are linear. Such a reformulation not only ensures optimality, but also accelerates the solving process. To relax the remaining bilinear constraints, we then apply McCormick envelopes and obtain an objective lower bound of the reformulated model. To improve the quality of the McCormick relaxation, we employ a piecewise McCormick technique that partitions the domain of one of the variables of the bilinear terms into several disjoint regions in order to derive strengthened lower and upper bounds of the partitioned variables. We propose a heuristic tightening method to further constrict the strengthened bounds derived from the piecewise McCormick technique and recover a nearby feasible solution. Case studies show that, compared with the interior point method and the method implemented in a global bilinear solver, the proposed tightening McCormick method quickly solves the heat–electricity operation problem with an acceptable feasibility check and optimality.

Deterministic global optimization with Gaussian processes embedded

Gaussian processes (Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the free variables and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the “MeLOn—MachineLearning Models for Optimization” toolbox (https://git.rwth-aachen.de/avt.svt/public/MeLOn).