Mixed-integer Semidefinite Programming Research Articles

Distributionally robust chance-constrained optimization for the integrated berth allocation and quay crane assignment problem

In this paper, we are the first to propose a distributionally robust chance-constrained (DRCC) optimization model for the integrated berth allocation and quay crane assignment problem (BACAP) in container terminals. In contrast to the classical deterministic BACAP model, we consider the arrival time to be uncertain due to the frequent arrival delays in ports. We then impose a chance constraint that the service time must start after the uncertain arrival time with a probability of at least 1−ϵ, where 1−ϵ represents the target service level in container terminals (ϵ represents the target risk tolerance of the port manager). Under the moment-based ambiguity set, we reformulate the DRCC model into a mixed integer semi-definite programming (MISDP) model. Additionally, we develop an efficient decomposition branch-and-bound algorithm to solve the MISDP model and obtain the exact solution. Fortunately, a special case of the DRCC model arises when the mean and covariance utilized in the ambiguity set are precise, allowing for the transformation of the DRCC model into a mixed integer programming (MIP) model. This conversion significantly reduces the complexity of the problem. Impressively, the solving time of the MISDP model with the decomposition branch-and-bound algorithm is comparable to that of the transformed MIP model. The numerical results show that our model can achieve a schedule with high service at low cost. Meanwhile, we have made an intriguing discovery that the correlation between the target risk tolerance and the actual service level can be depicted as a staircase function regardless of the datasets. This finding offers crucial insights for port management, enabling them to strike a balance between the cost and actual service level by determining an appropriate target service level.

Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

The systematic design of exact optimal designs of experiments is typically challenging, as it results in nonconvex optimization problems. The literature on the computation of model-based exact optimal designs of experiments via mathematical programming, when the covariates are categorical variables, is still scarce. We propose mixed-integer semidefinite programming formulations, to find exact D-, A- and I-optimal designs for linear models, and locally optimal designs for nonlinear models when the design domain is a finite set of points. The strategy requires: (i) the generation of a set of candidate treatments; (ii) the formulation of the optimal design problem as a mixed-integer semidefinite program; and (iii) its solution, employing appropriate solvers. For comparison, we use semidefinite programming-based formulations to find equivalent approximate optimal designs. We demonstrate the application of the algorithm with various models, considering both unconstrained and constrained setups. Equivalent approximate optimal designs are used for comparison.

Co-design of active vibration control and optimal sensor and actuator placement for a flexible wing using reinforcement learning

This paper presents applying reinforcement learning to find the optimal sensor/actuator placement (OSAP) policy and optimal control for the flexible wing. The “co-design” objective is to find the OSAP and its associate controller to render the optimal closed-loop performance. The nonlinear vibration dynamics of the flexible wing are modeled in the linear parameter varying (LPV) approach so that LPV- H ∞ controllers can be designed. The co-design problem is formulated into mixed-integer semi-definite programming (MISDP). As a special form of combinatorial optimization, MIDSP solves integer optimization for sensor/actuator selection and convex optimization for controller design. A modified reinforcement learning algorithm is applied to solve this NP-hard optimization problem and obtain a converged solution. In addition, RL is compared with the greedy algorithm and genetic algorithm to demonstrate its strengths and drawbacks in solving high-dimensional MISDP. The solutions obtained by RL and the greedy algorithm are verified and compared in the high-fidelity simulation with the full-order model.

Optimal Micro-PMU Placement for Improving State Estimation Accuracy via Mixed-integer Semidefinite Programming

Micro-phasor measurement units (µPMUs) with a micro-second resolution and milli-degree accuracy capability are expected to play an important role in improving the state estimation accuracy in the distribution network with increasing penetration of distributed generations. Therefore, this paper investigates the problem of how to place a limited number of µPMUs to improve the state estimation accuracy. Combined with pseudo-measurements and supervisory control and data acquisition (SCADA) measurements, an optimal µPMU placement model is proposed based on a two-step state estimation method. The E-optimal experimental criterion is utilized to measure the state estimation accuracy. The nonlinear optimization problem is transformed into a mixed-integer semidefinite programming (MISDP) problem, whose optimal solution can be obtained by using the improved Benders decomposition method. Simulations on several systems are carried out to evaluate the effective performance of the proposed model.

Optimal pump scheduling in multi-phase distribution networks using Benders decomposition

With the growing adoption of renewable energy resources in the power grid, co-optimizing the operation of water and power distribution networks (WDNs and PDNs) increases the flexibility of both systems. The optimal water power flow problem (OWPF) is formulated to manage resource utilization across the WDNs and multi-phase PDNs. The nonlinear WDN hydraulic constraints and PDN ac power flow equations render the OWPF problem nonconvex. This paper formulates a convex optimal water power flow (C-OWPF) problem adopting successive approximations in WDNs and branch-flow relaxations in PDNs. The C-OWPF problem is a mixed-integer semidefinite program, which is computationally challenging even for small instances. Additionally, privacy considerations of WDN and PDN operators motivate the need for solving the C-OWPF in a distributed fashion. This paper develops a C-OWPF solver based on Benders decomposition that overcomes computational complexity challenges and preserve the privacy of the respective operators. The merits of the Benders decomposition-based solver are demonstrated on the IEEE 4-bus PDN coupled with a 3-node WDN and the IEEE 123-bus PDN coupled with the 36-node WDN.

A Framework for Optimized Topology Design and Leader Selection in Affine Formation Control

This paper studies the problem of topology design and leader selection to activate affine formation control schemes. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">affine formation control</i> enjoys a distinguishing feature from other control methods in that the whole formation can be dynamically maneuvered by controlling a small number of agents called leaders. This relies on the stress matrix which defines the inter-agent communication/sensing topology determining the dynamic performance for an autonomous system. In the first step, a topology design algorithm based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixed integer semi-definite programming</i> (MISDP) is proposed to obtain the stress matrix with low communication cost, fast convergence speed and high tolerance to time-delay. Secondly, the proposed leader selection algorithm focuses on two objectives of practical significance: convergence speed and control energy, which are tailored to two optimization problems based on MISDP. At last, the collective agents are driven to the target formation via their local interactions and leaders’ external control inputs. The effectiveness of the overall framework is validated by both simulations and physical experiments.

Presolving for Mixed-Integer Semidefinite Optimization

This paper provides a discussion and evaluation of presolving methods for mixed-integer semidefinite programs. We generalize methods from the mixed-integer linear case and introduce new methods that depend on the semidefinite condition. The methods considered include adding linear constraints, deriving bounds relying on 2 × 2 minors of the semidefinite constraints, tightening of variable bounds based on solving a semidefinite program with one variable, and scaling of the matrices in the semidefinite constraints. Tightening the bounds of variables can also be used in a node presolving step. Along the way, we discuss how to solve semidefinite programs with one variable using a semismooth Newton method and the convergence of iteratively applying bound tightening. We then provide an extensive computational comparison of the different presolving methods, demonstrating their effectiveness with an improvement in running time of about 22% on average. The impact depends on the instance type and varies across the methods. Funding: This work was supported by the EXPRESS II project within the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) priority program CoSIP (DFG-SPP 1798). It was also partly supported by the DFG within Project A4 in the SFB 805.

Identification of best discrimination surface by mixed-integer semi-definite programming for support vector machine

This paper proposes two improvements to the support vector machine (SVM): (i) extension to a semi-positive definite quadratic surface, which improves the discrimination accuracy; (ii) addition of a variable selection constraint. However, this model is formulated as a mixed-integer semi-definite programming (MISDP) problem, and it cannot be solved easily. Therefore, we propose a heuristic algorithm for solving the MISDP problem efficiently and show its effectiveness by using corporate credit rating data.

A Transactive Retail Market Mechanism for Active Distribution Network Integrated With Large-Scale Distributed Energy Resources

The burgeoning integration of distributed energy resources (DER) poses new challenges for the economic and safe operation of the electricity system. The current distribution-side policy is largely based on mandatory regulations and incentives, rather than the design of a competitive market mechanism to arouse DERs to freely compete in the retail market. To address this issue, we proposed a transactive retail market mechanism to attract the active participation of profit-driven DER retailers in a deregulated way. Considering the limited competitive property of the retail market, a bi-level DSO-dominated framework is constructed to simulate the virtual game between distribution system operator (DSO) and DER retailers. Specifically, the flexible interval pricing of retailers is modeled as a series of binary revenue constraints to influence the DSO's centralized economic dispatch decisions, where the time-varying distribution locational marginal price is adopted to settle energy activities at different locations over the time horizon. Due to the uncertain power flow path during the coordinated decision-making process, we proposed a general undirected second-order cone-based AC radial power flow model and provided sufficient conditions to ensure its exactness. Also, we applied a series of approximation and relaxation techniques to transform the bi-level mixed-integer quadratic framework with highly discrete induced domains into a solvable mixed-integer semidefinite programming problem. It is demonstrated in the case study that the proposed mechanism not only improves market efficiency but also eliminates market power and the resulting market failures.

Cost-effective communication network planning considering performance of pinning-based secondary control in microgrids

Driven by the fact that the structural topology of the communication network has a significant effect on the dynamical performance of the related physical network, this paper addresses the communication network planning problem for islanded microgrids (MGs). The optimal communication topology with the lowest communication cost, which supports the desired performance of pinning-based distributed control is designed in the MG planning stage. To achieve that, firstly, pinning synchronization control (PSC) is developed to regulate the voltage and frequency at each distributed generator (DG) to the reference state. Then criteria of the algebraic connectivity of communication network to fulfill the convergence requirement of PSC protocols are derived analytically using spectral graph properties and Lyapunov theory. A communication network planning model is formulated as Mixed Integer Semidefinite Program (MISDP) by using Linear Matrix Inequalities (LMI) techniques. The convergence speed requirement of PSC, communication time delay and the importance of connections between DGs are considered as constraints. The optimal communication network with the lowest communication cost can be acquired and meanwhile the satisfactory dynamical performance of PSC protocols can be guaranteed. The case studies are designed to evaluate how the planning model can practically facilitate the pinning-based control in the islanded MG. The simulation results effectively verify that the dynamical control convergence can be improved under the properly designed communication network. The control performance of the proposed protocol is validated experimentally based on a two-DG MG prototype.

Resilience in Multirobot Multitarget Tracking With Unknown Number of Targets Through Reconfiguration

We address the problem of maintaining resource availability in a networked multirobot team performing distributed tracking of an unknown number of targets in a bounded environment. Robots are equipped with sensing and computational resources, enabling them to cooperatively track a set of targets using a distributed probability hypothesis density (PHD) filter. We use the trace of a robot's sensor measurement noise covariance matrix to quantify its sensing quality. While executing the tracking task, if a robot experiences sensor quality degradation, the team's communication network is reconfigured such that the robot with the faulty sensor may share information with other robots to improve the team's target-tracking ability without enforcing a large change in the number of active communication links. A central system monitor executes the network reconfiguration computations. We consider two different PHD fusion methods and propose four different mixed-integer semi-definite programming (MISDP) formulations (two formulations for each PHD fusion method) to accomplish our objective. All MISDP formulations are validated in simulation.

Optimal Reactive Power Dispatch With Discrete Controllers Using a Branch-and-Bound Algorithm: A Semidefinite Relaxation Approach

In this paper, a methodology to solve the optimal reactive power dispatch (ORPD) in electric power systems (EPS), considering discrete controllers, is proposed. Discrete controllers, such as the tap position of on-load tap changing (OLTC) transformers and switchable reactive shunt compensation, are optimized by the proposed method. A semidefinite relaxation (SDR) of the ORPD problem and a branch-and-bound (B&B) algorithm have been fully deployed. A new formulation is presented for the OLTC transformers to obtain a connected structure of the semidefinite programming (SDP) matrices. The customized B&B algorithm deals with the discrete nature of the binary control variables. Moreover, in order to enhance the convexification, valid inequalities called lifted nonlinear cuts (NLC) are implemented in the SDR. Additionally, a chordal decomposition technique is used to improve the computational performance. Finally, the B&B algorithm is used to solve the mixed-integer semidefinite programming problem. Several benchmarks have been used to show the accuracy and scalability of the proposed method, and convergence analysis shows that near-global optimal solutions are generated with small relaxation gaps.

An MI-SDP Model for Optimal Location and Sizing of Distributed Generators in DC Grids That Guarantees the Global Optimum

This paper deals with a classical problem in power system analysis regarding the optimal location and sizing of distributed generators (DGs) in direct current (DC) distribution networks using the mathematical optimization. This optimization problem is divided into two sub-problems as follows: the optimal location of DGs is a problem, with those with a binary structure being the first sub-problem; and the optimal sizing of DGs with a nonlinear programming (NLP) structure is the second sub-problem. These problems originate from a general mixed-integer nonlinear programming model (MINLP), which corresponds to an NP-hard optimization problem. It is not possible to provide the global optimum with conventional programming methods. A mixed-integer semidefinite programming (MI-SDP) model is proposed to address this problem, where the binary part is solved via the branch and bound (B&amp;B) methods and the NLP part is solved via convex optimization (i.e., SDP). The main advantage of the proposed MI-SDP model is the possibility of guaranteeing a global optimum solution if each of the nodes in the B&amp;B search is convex, as is ensured by the SDP method. Numerical validations in two test feeders composed of 21 and 69 nodes demonstrate that in all of these problems, the optimal global solution is reached by the MI-SDP approach, compared to the classical metaheuristic and hybrid programming models reported in the literature. All the simulations have been carried out using the MATLAB software with the CVX tool and the Mosek solver.

Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets

We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in Delage and Ye (Oper Res 58(3):595–612, 2010) to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.

Addressing the Conditional and Correlated Wind Power Forecast Errors in Unit Commitment by Distributionally Robust Optimization

In this paper, a study of the day-ahead unit commitment problem with stochastic wind power generation is presented, which considers conditional, and correlated wind power forecast errors through a distributionally robust optimization approach. Firstly, to capture the characteristics of random wind power forecast errors, the least absolute shrinkage, and selection operator (Lasso) is utilized to develop a robust conditional error estimator, while an unbiased estimator is used to obtain the covariance matrix. The conditional error, and the covariance matrix are then used to construct an enhanced ambiguity set. Secondly, we develop an equivalent mixed integer semidefinite programming (MISDP) formulation of the two-stage distributionally robust unit commitment model with a polyhedral support of random variables. Further, to efficiently solve this problem, a novel cutting plane algorithm that makes use of the extremal distributions identified from the second-stage semidefinite programming (SDP) problems is introduced. Finally, numerical case studies show the advantage of the proposed model in capturing the spatiotemporal correlation in wind power generation, as well as the economic efficiency, and robustness of dispatch decisions.

Topology-Finding of Tensegrity Structures Considering Global Stability Condition

This study proposes a general framework for topology-finding or topology optimization of tensegrity structures. The existing topology-finding formulation of tensegrity structures based on mixed-integer linear programming (MILP) was improved and transformed into a formulation based on mixed-integer semidefinite programming (MISDP) which considers the global stability condition of tensegrity. We illustrated and analyzed two undesirable phenomena, loss of prestress stability and loss of integrity, caused by the missing global stability condition in previous MILP-based approaches. A branch-and-bound algorithm combined with a primal-dual interior-point algorithm is employed to solve the proposed MISDP model. Some numerical examples illustrated the improvements and effectiveness of the proposed approach. The proposed approach successfully can avoid the two undesirable phenomena and ensure the global stability of the found tensegrity structures. By using different stability conditions in the topology-finding process, the proposed approach can find general stable tensegrity structures and superstable tensegrity structures.

Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method

Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.

A mixed-integer SDP solution approach to distributionally robust unit commitment with second order moment constraints

A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of stochastic parameters can be inferred from historical data, and then employed to model the set of probability distributions. The resulting problem is a two-stage distributionally robust unit commitment with second order moment constraints, and we show that it can be recast as a mixed-integer semidefinite programming (MI-SDP) with finite constraints. The solution algorithm of the problem comprises solving a series of relaxed MI-SDPs and a subroutine of feasibility checking and vertex generation. Based on the verification of strong duality of the semidefinite programming (SDP) problems, we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce a SDP relaxation for the feasibility checking problem, which is an intractable biconvex optimization. Experimental results on a IEEE 6-bus system are presented, showing that without any tunings of parameters, the real-time operation cost of distributionally robust UC method outperforms those of deterministic UC and two-stage robust UC methods in general, and our method also enjoys higher reliability of dispatch operation.

A regularized state estimation scheme for a robust monitoring of the distribution grid

In this paper, we propose a regularized state estimation (SE) scheme for the distribution grid. The ultimate goal is to track accurately the system state at a faster time scale according to the requirements of the new operational environment. The SE algorithm operates at two different time scales in which the set of available measurements are different. At the main time instants (every 15 min) the set of observations comprises SCADA measurements, pseudomeasurements and micro Phasor Measurement Units (μPMUs). In this case, we resort to a Regularized version of the Normal Equations based SE (R-NESE). In the intermediate time instants, only a reduced number of μPMU measurements is available. To circumvent observability issues, we exploit the fact that the voltage drop in adjacent buses is limited and, on that basis, a regularized weighted total variation estimation (WTVSE) problem is formulated. Then, the impact of in-line voltage regulators (IVLRs) is explicitly taken into consideration and that, forces us to decompose and solve the original SE problem for a number of smaller regions (D-WTVSE). The latter can be iteratively solved by resorting to the Alternating Direction Method of Multipliers (ADMM). Complementarily, we also present a μPMU placement method (μPP) in order to improve the conditioning of the R-NESE problem. This problem can be posed as a mixed integer semidefinite programming model (MISDP) and, thus, can be efficiently solved. The performance of the proposed scheme is numerically assessed on (mostly) a 95-bus distribution system for a number of realistic conditions of noise, load and photovoltaic generation profiles. A number of benchmarks are provided, as well.

Coordinating Multiple Sources for Service Restoration to Enhance Resilience of Distribution Systems

When a major outage occurs on a distribution system due to extreme events, microgrids, distributed generators, and other local resources can be used to restore critical loads and enhance resiliency. This paper proposes a decision-making method to determine the optimal restoration strategy coordinating multiple sources to serve critical loads after blackouts. The critical load restoration problem is solved by a two-stage method with the first stage deciding the post-restoration topology and the second stage determining the set of loads to be restored and the outputs of sources. In the second stage, the problem is formulated as a mixed-integer semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced three-phase power flow constraint and operational constraints are considered. An iterative algorithm is proposed to deal with integer variables and can attain the global optimum of the critical load restoration problem by solving a few semidefinite programs under two conditions. The effectiveness of the proposed method is validated by numerical simulation with the modified IEEE 13-node test feeder and the modified IEEE 123-node test feeder under plenty of scenarios. The results indicate that the optimal restoration strategy can be determined efficiently in most scenarios.