Mixed-integer Linear Constraints Research Articles

Deep Transfer Learning Based Surrogate Modeling for Optimal Investment Decision of Distribution Networks

At present, the distribution network investment is faced with complicated operation strategies and various new investment elements. It's difficult to evaluate the impact of different investment measures on the final benefits due to the complicated physical modeling and potential coupling correlation, which contributes to that the traditional model-driven decision-making method may not work. This paper proposes a novel deep transfer learning based surrogate-assisted method to address the challenge. The key idea is to develop a surrogate model to replace the input-output correlation between the certain investment measure and its benefits, which can be further reformulated into mixed-integer linear constraints and embedded into the classical investment decision-making model for the target distribution network. As the model is trained through a designed two-stage adaptive transfer learning algorithm, which can extract the similar rules from other distribution networks, the dilemma of insufficient effective samples is alleviated, while a neural network constrained algorithm is designed to linear the surrogate model for simplifying the optimization. Numerical cases prove the effectiveness of the surrogate model on correlation mining and show the superiority of the surrogate-assisted method in achieving the precision investment for target distribution network.

Voltage Stability Constrained Operation Optimization: An Ensemble Sparse Oblique Regression Tree Method

Voltage stability remains a main challenge for high renewable penetrated power systems. In future power systems, the dispatch needs to consider the voltage stability margin. However, considering static and transient voltage stability is challenging because a complex relationship exists between voltage stability and operation state variables. This paper proposes an ensemble sparse oblique regression tree method for voltage stability constrained operation optimization. First, we train multiple oblique regression trees on voltage stability simulation datasets. The trained trees are then grouped into ensemble using a boosting method to extract accurate and understandable voltage stability rules. Finally, the rules are embedded as mixed-integer linear programming constraints in a linear optimal power flow model. Validation on the IEEE-14 case shows the interpretability and efficiency of the proposed algorithm. Case studies on the Qinghai power grid and IEEE-118 further show that the proposed strategy can effectively improve the voltage stability margin of optimized operation states.

Conservative Sparse Neural Network Embedded Frequency-Constrained Unit Commitment With Distributed Energy Resources

The increasing penetration of distributed energy resources (DERs) will decrease the rotational inertia of the power system and further degrade the system frequency stability. To address the above issues, this paper leverages the advanced neural network (NN) to learn the frequency dynamics and incorporates NN to facilitate system reliable operation. This paper proposes the conservative sparse neural network (CSNN) embedded frequency-constrained unit commitment (FCUC) with converter-based DERs, including the learning and optimization stages. In the learning stage, it samples the inertia parameters, calculates the corresponding frequency, and characterizes the stability region of the sampled parameters using the convex hulls to ensure stability and avoid extrapolation. For conservativeness, the positive prediction error penalty is added to the loss function to prevent possible frequency requirement violation. For the sparsity, the NN topology pruning is employed to eliminate unnecessary connections for solving acceleration. In the optimization stage, the trained CSNN is transformed into mixed-integer linear constraints using the big-M method and then incorporated to establish the data-enhanced model. The case study verifies 1) the effectiveness of the proposed model in terms of high accuracy, fewer parameters, and significant solving acceleration; 2) the stable system operation against frequency violation under contingency.

Deep-Quantile-Regression-Based Surrogate Model for Joint Chance-Constrained Optimal Power Flow With Renewable Generation

Joint chance-constrained optimal power flow (JCC-OPF) is a promising tool for managing distributed renewable generation uncertainties. However, existing works are usually based on power flow equations, which require accurate network parameters that may be unobservable in many distribution systems. To address this issue, this paper proposes a learning-based surrogate model for JCC-OPF with renewable generation. This model equivalently converts joint chance constraints into quantile-based forms. Two multi-layer perceptrons are trained based on special loss functions to predict the quantile of constraint violations and expected power loss. By reformulating these two MLPs into mixed-integer linear constraints, we can replicate the JCC-OPF without network parameters. Two pre-processing steps, i.e., data augmentation and calibration, are further developed to improve its performance. The former trains a simulator to generate more training samples for enhancing the prediction accuracy of MLPs. The latter designs a positive parameter based on empirical prediction errors to calibrate the outputs of MLPs so that feasibility can be guaranteed. Numerical experiments based on the IEEE 33- and 123-bus systems validate that the proposed model can achieve desirable feasibility and optimality simultaneously with no need for network parameters.

Error bounds for maxout neural network approximations of model predictive control

Neural network (NN) approximations of model predictive control (MPC) are a versatile approach if the online solution of the underlying optimal control problem (OCP) is too demanding and if an exact computation of the explicit MPC law is intractable. The drawback of such approximations is that they typically do not preserve stability and performance guarantees of the original MPC. However, such guarantees can be recovered if the maximum error with respect to the optimal control law and the Lipschitz constant of that error are known. We show in this work how to compute both values exactly when the control law is approximated by a maxout NN. We build upon related results for ReLU NN approximations and derive mixed-integer (MI) linear constraints that allow a computation of the output and the local gain of a maxout NN by solving an MI feasibility problem. Furthermore, we show theoretically and experimentally that maxout NN exist for which the maximum error is zero.

Quantifying frequency containment reserve using cross-entropy frequency-constrained contingency-state-analysis model

With the increasing penetration of converter-interfaced generators, the frequency containment reserve (FCR) from conventional generators keeps going down, leading to a potential risk of frequency instability under contingencies. Consequently, Converter-interfaced generators are required to provide FCR and participate in the corrective rescheduling. Nevertheless, how to assess the FCR and quantify the adequacy of FCR under contingencies is a big challenge in modern new power system. To address this challenge, a cross-entropy-based frequency-constrained contingency-state-analysis (FC-CSA) model is proposed in this paper. Notably, both frequency control (FC) of units (i.e., conventional synchronous generators and converter-interfaced generators), and under frequency load shedding (UFLS) are incorporated in the primary frequency response. Then a unified system frequency response (SFR) function representing frequency dynamic is derived. This SFR function is extracted and reformulated as a group of mixed-integer linear constraints and participates in the traditional CSA model. Moreover, a set of frequency dynamic indexes, i.e., Expectation of UFLS risk, Expectation of FCR from conventional and converter-interfaced generators, is extended to depict the FCR that the power system requires. These indexes are calculated by the FC-CSA in a cross-entropy-based monte carlo simulation (CE-MCs). Case studies on a modified IEEE 6-bus test system and IEEE 118-bus test system are carried out to demonstrate the effectiveness of the proposed FC-CSA model.

Encoding Frequency Constraints in Preventive Unit Commitment Using Deep Learning With Region-of-Interest Active Sampling

With the increasing penetration of renewable energy, frequency response and its security are of significant concerns for reliable power system operations. Frequency-constrained unit commitment (FCUC) is proposed to address this challenge. Despite existing efforts in modeling frequency characteristics in unit commitment (UC), current strategies can only handle oversimplified low-order frequency response models and do not consider wide-range operating conditions. This paper presents a generic data-driven framework for FCUC under high renewable penetration. Deep neural networks (DNNs) are trained to predict the frequency response using real data or high-fidelity simulation data. Next, the DNN is reformulated as a set of mixed-integer linear constraints to be incorporated into the ordinary UC formulation. In the data generation phase, all possible power injections are considered, and a region-of-interests active sampling is proposed to include power injection samples with frequency nadirs closer to the UFLC threshold, which significantly enhances the accuracy of frequency constraints in FCUC. The proposed FCUC is verified on the the IEEE 39-bus system. Then, a full-order dynamic model simulation using PSS/E verifies the effectiveness of FCUC in frequency-secure generator commitments.

Data-Driven Natural Gas Compressor Models for Gas Transport Network Optimization

The fuel cost minimization problem (FCMP) for natural gas transport is important because of the immense energy consumed by compressors to satisfy increasing natural gas demands. Current approaches to the FCMP use inaccurate simplified models, or more complex and computationally challenging models, to describe compressor performance. This paper develops two novel data-driven surrogate models, namely, the dimensionless group based model and the deep neural network (DNN) model. The DNN involves rectified linear units as activation functions, so it can be reformulated into mixed-integer linear constraints in the FCMP. The case study results show that both the dimensionless group based and the DNN models achieve better accuracy than two typical surrogate models in the literature, and they are also computationally more efficient for optimization. The computational performance of the dimensionless based model is sensitive to gas supply and demand data, while that of the DNN model is robust.

Optimizing behavioral interventions to regulate gestational weight gain with sequential decision policies using hybrid model predictive control

Excessive gestational weight gain is a significant public health concern that has been the recent focus of control systems-based interventions. Healthy Mom Zone (HMZ) is an intervention study that aims to develop and validate an individually-tailored and “intensively adaptive” intervention to manage weight gain for pregnant women with overweight or obesity using control engineering approaches. This paper presents how Hybrid Model Predictive Control (HMPC) can be used to assign intervention dosages and consequently generate a prescribed intervention with dosages unique to each individuals needs. A Mixed Logical Dynamical (MLD) model enforces the requirements for categorical (discrete-level) doses of intervention components and their sequential assignment into mixed-integer linear constraints. A comprehensive system model that integrates energy balance and behavior change theory, using data from one HMZ participant, is used to illustrate the workings of the HMPC-based control system for the HMZ intervention. Simulations demonstrate the utility of HMPC as a means for enabling optimized complex interventions in behavioral medicine, and the benefits of a HMPC framework in contrast to conventional interventions relying on “IF–THEN” decision rules.

Weekly hydropower scheduling of cascaded reservoirs with hourly power and capacity balances

A medium/long-term hydropower scheduling routinely assumes the energy produced in each week will be fully utilized in peak-shaving the power demands during every day in that week, ignoring the fluctuation of the hourly power loads. This work improves the traditional weekly hydropower scheduling by integrating the hourly power and capacity balances (HPCB), which has rarely been investigated, mainly due to great challenges imposed in solving the model. The HPCB are formulated into mixed integer linear constraints, involving the spare, maintenance, disabled, reserve and working capacities, as well as the order and levels of hydroplants in peak-shaving the hourly power load curve, all to be optimized. The formulation is then improved with successive strategies in updating water heads and peak-shaving hours, aiming to boost the solution efficiency by excluding more binary variables. The case studies involving 6 cascaded hydropower reservoirs in Lancang River reveal that the traditional weekly hydropower scheduling overestimates the benefits significantly, strongly suggesting the necessity to include the HPBC into a long/mid-term hydropower scheduling. The experiments also recommend an improved model formulation that achieves very consistent results on the top three prioritized objectives, while taking less than 0.1 s to solve the problem involving all six cascaded reservoirs, demonstrating a great prospect in solving problems in large scale.

Vehicle Speed and Gear Position Co-Optimization for Energy-Efficient Connected and Autonomous Vehicles

This work proposes a real-time implementable control strategy to optimize vehicle speed and transmission gear position simultaneously for connected and autonomous vehicles (CAVs). Co-optimization of vehicle speed and transmission gear position has the advantage to maximize the fuel benefits. Drivability is considered during the optimization to satisfy the acceleration requirement and avoid shift busyness. The target vehicle's speed and gear position are controlled intelligently using predicted future traffic conditions based on information enabled by connectivity. The optimal control problem is a hybrid one with both continuous (vehicle speed) and discrete (gear position) control inputs. The problem is formulated and simplified to a mixed-integer programming problem with a convex quadratic objective function and mixed-integer linear constraints. The optimal control solutions are obtained in real time using an efficient numerical solver in the model predictive control (MPC) fashion. Future traffic conditions are anticipated using a traffic prediction method based on a traffic flow model. The traffic prediction method can be applied to scenarios where both connected vehicles and nonconnected vehicles share the road. As a case study, a vehicle platooning scenario on an urban road is evaluated in both simulation and experiment. The target vehicle is at the end of the vehicle platoon and follows the preceding vehicle. The average computational time of the optimization is 0.44 s. By co-optimizing vehicle speed and gear position, the target vehicle can achieve 10.6% fuel benefits compared with the immediate preceding vehicle and 8.9% energy benefits compared with a human-driven vehicle (driven by VISSIM's car-following model). The proposed control strategy can be potentially extended to various CAV applications and traffic scenarios as well.

A MILP-based restoration planning method for generator start-up considering flexible re-energizing times of transmission lines

During power system restoration, the planning of the generator start-up sequence (GSUS) can significantly affect the restoration efficiency. However, a GSUS optimization model based on mixed integer linear programming (MILP) cannot satisfy the need for flexible re-energizing times of transmission lines and the serial restoration of generators. To solve this issue, a new MILP-based GSUS optimization model is proposed in this paper that can generate a serial GSUS scheme with flexible re-energizing times of transmission lines. The requirements of flexible re-energizing times of transmission lines and the serial restoration of the generators are formulated as restoration time constraints for the transmission lines, start-up time constraints for the buses on the restoration paths, and a serial restoration constraint, which are transformed into mixed integer linear constraints by means of auxiliary binary (0–1) decision variables. Meanwhile, the restart time constraints of the generators and the transient frequency requirements are modeled as mixed integer linear constraints. Finally, the performance of the proposed method is verified on the IEEE 39-bus system.

Simulation and Evaluation of Zonal Electricity Market Designs

Zonal pricing with countertrading (a market-based redispatch) gives arbitrage opportunities to the power producers located in the export-constrained nodes. They can increase their profit by increasing the output in the day-ahead market and decrease it in the real-time market (the inc-dec game). For a six-node and a 24-node system, we numerically show that this leads to large inefficiencies in a standard zonal market. We also show how the inefficiencies can be significantly mitigated for these two examples by changing the design of the real-time market. We consider a two-stage game with oligopoly producers, wind-power shocks and real-time shocks. The game is formulated as a two-stage stochastic Equilibrium Problem with Equilibrium Constraints (EPEC), which we recast into a two-stage stochastic Mixed-Integer Linear Constraints (MILC).

Distributed Communication-Aware Motion Planning for Networked Mobile Robots Under Formal Specifications

Control and communication are often tightly coupled for networked mobile robots; motions of robots impact communication quality and communication quality of service (QoS), in turn, affecting coordination performance of robots. In this article, we propose a theoretical motion planning control framework for a team of networked mobile robots to accomplish high-level spatial and temporal motion specifications while optimizing communication QoS. Desired motion specifications are formulated as signal temporal logic (STL), whereas the communication QoS to be optimized is captured by spatial temporal reach and escape logic (STREL) formulas. Both the STL and STREL specifications are encoded as mixed integer linear constraints posed on the system and environment state variables of the mobile robot network, where satisfactory control strategies can be computed by exploiting a distributed model predictive control (MPC) approach. A two-layer hierarchical MPC procedure is proposed to efficiently solve the co-optimization problem, whose recursive feasibility is formally ensured. The effectiveness of the proposed framework is validated by simulation.

Robust chance-constrained programming approach for the planning of fast-charging stations in electrified transportation networks

In this paper, a bi-level programming model is established to address the planning issues of fast-charging stations in electrified transportation networks with the consideration of uncertain charging demands. The capacitated flow refueling location model is considered in the upper level to minimize the planning cost of fast-charging stations while the traffic assignment model is utilized in the lower level to determine the spatial and temporal distribution of plug-in electric vehicle flows over entire transportation networks. Such bi-level model unveils the inherent relationship among charging demands, electrical demands and the spatial and temporal distribution of plug-in electric vehicle flows. Robust chance constraints are formulated to characterize the service abilities of fast-charging stations under distribution-free uncertain charging demands, where the ambiguity set is constructed to estimate the potential values of the uncertainties based on their moment-based information, such that the robust chance constraints can exactly be reduced to mixed integer linear constraints. By introducing new variables, the bi-level model is then reformulated into a single-level mixed integer second-order cone programming model so as to be solved via off-the-shelf solvers, which guarantee the optimality of the solution. A case study is conducted to illustrate the effectiveness of the proposed planning model, which reveals three critical factors that significantly impact the planning outcomes.

Optimization-based Motion Planning and Runtime Monitoring for Robotic Agent with Space and Time Tolerances

Abstract We present an optimization-based approach for robot planning, monitoring and self-correction problems under signal temporal logic specifications (STL). The STL specifications are translated into mixed-integer linear constraints, and we generate the reference trajectory by solving a mixed-integer-linear-programming (MILP) to maximize the overall space and time tolerances. During runtime execution, a prediction module is constantly evaluating the robustness degree of the predicted trajectory, and a self-correction module based on event-triggered model predictive control (MPC) has been designed to predict and correct possible future violations of the specifications. Simulation results show that with our approach, the robotic agent is able to generate a path that satisfies the STL specifications while maximizing space and time tolerances, and able to make corrections when there are possible violations of the specifications during runtime execution.

Optimal Dispatch of Coupled Electricity and Heat System With Independent Thermal Energy Storage

Notable benefit can be brought by combined operation of a coupled electricity and heat system (CEHS), and be enhanced by introducing thermal energy storage (TES). Existing literature researching on this topic either neglects the heat network, or requires TES system being located and operated together with a combined heat and power unit. This paper aims at removing such requirement whilst taking heat network into account. With prevalent constant-flow variable temperature control strategy of heat network, the optimal dispatch model of CEHS considering independent TES (ITES) system with networked electric and heat energy distribution systems is presented. The non-convex parts in the heat network, mainly caused by the MFR reversal when ITES changes its operation state, are exactly reformulated to mixed-integer linear (MIL) constraints, which leads to a mixed-integer second-order programming formulation of the problem. Case studies demonstrate that introducing ITES not only brings reduction in the total operation cost, but also ameliorates the system reliability and its capability in accommodating renewables.

Indirect Heat Integration across Plants: Novel Representation of Intermediate Fluid Circles

Interplant heat integration (IPHI) using intermediate fluid circles provides an additional opportunity for energy saving and emission reduction, beyond traditional intraplant heat integration. However, the number of intermediate fluid circles as well as their heat capacity and temperatures are all variables. It is difficult to find promising solutions, since such conditions generally lead to nonlinearity. To deal with this problem, a transshipment type model, which holistically optimizes the parameters and interconnectivity patterns of intermediate fluid circles and keeps the constraints linear, is developed. New formulations are developed to identify exact distributions of intermediate fluid circles (namely heat transfer available intervals) among all temperature intervals, with unknown temperatures. Additionally, mixed-integer linear constraints are developed to describe interconnectivity patterns of intermediate fluid circles so as to calculate piping and pumping cost. The performances of different int...

A Three-Phase Microgrid Restoration Model Considering Unbalanced Operation of Distributed Generation

Recent severe outages highlight the urgency of improving grid resiliency in the U.S. microgrid formation schemes are proposed to restore critical loads after outages occur. Most distribution networks have unbalanced configurations that are not represented in sufficient detail by single-phase models. This paper provides a microgrid formation plan that adopts a three-phase network model to represent unbalanced distribution networks. The problem formulation has a quadratic objective function with mixed-integer linear constraints. The three-phase network model enables us to examine the three-phase power outputs of distributed generators (DGs), preventing unbalanced operation that might trip DGs. Because the DG unbalanced operation constraint is non-convex, an iterative process is presented that checks whether the unbalanced operation limits for DGs are satisfied after each iteration of optimization. We also develop a relatively conservative linear approximation on the unbalanced operation constraint to handle larger networks. Compared with the iterative solution process, the conservative linear approximation is able to accelerate the solution process at the cost of sacrificing optimality to a limited extent. Simulation in the IEEE 34 node and IEEE 123 test feeders indicate that the proposed method yields more practical microgrid formations results. In addition, this paper explores the coordinated operation of DGs and energy storage (ES) installations. The unbalanced three-phase outputs of ESs combined with the relatively balanced outputs of DGs could supply unbalanced loads. The case study also validates the DG-ES coordination.

Nonintrusive Load Monitoring Algorithm Using Mixed-Integer Linear Programming

This paper presents a nonintrusive load monitoring (NILM) algorithm based on mixed-integer linear programming. The formulation deals with the problem of multiple switching that arises when disaggregating individual appliance’s consumptions from a compound power measurement. Mixed-integer linear constraints are used to efficiently represent the load signatures of each appliance. Also, a window-based strategy is used to enhance the computational performance of the proposed NILM algorithm. The disaggregation can be made using only active power measurements at a low sampling rate, which is available in most energy meters. Moreover, if available, other signatures can be added to the model to improve its accuracy, such as reactive power signatures or harmonics. The performance of the algorithm is evaluated using three test cases from the almanac of minutely power dataset. The proposed method is also compared with a disaggregation method called aided linear integer programming. Results demonstrate the ability of the proposed method to accurately identify and allocate individual energy signatures in a computationally efficient manner, which makes it suitable for inexpensive home energy management.