Multi-parametric Programming Research Articles

Integrated stochastic transmission network and wind farm investment considering maximum allowable capacity

• A stochastic model is presented for transmission and wind power investment. • The model considers demand and wind generation uncertainty. • The model considers maximum allowable capacity. • The model is formulated as a stochastic bilevel optimization problem. • Multi-parametric programming is used to solve the problem. This research presents a novel mathematical methodology for integrated transmission network and wind farm investment (ITWI) considering maximum allowable capacity (MAC). The joint problem of transmission and wind farm investment planning is carried out under a central planner perspective, where load and wind power uncertainties are managed using scenario-based stochastic programming. Distinct from the existing models in which the installation capacity of wind farms is restricted only by the available investment budget, the wind power capacity is limited by MAC considering strength measures. In this regard, the investment problem is addressed as a bi-level programming model where the upper level seeks to minimize the investment cost associated with transmission lines and wind farms plus the operation cost while the lower level determines the MAC of wind farms. The existence of integer decision variables in the lower level renders Karush-Kuhn-Tucker (KKT) conditions invalid. Thus, the multi-parametric programming (MPP) method is utilized to solve the mixed-integer bi-level linear programming (MIBLP). Numerical tests on two different power systems corroborate the efficiency of the proposed model. The comparable results demonstrate that ignoring the MAC of wind farms leads to inefficient solutions due to additional unnecessary investment in the wind energy sector.

A Review of the Optimization and Control Techniques in the Presence of Uncertainties for the Energy Management of Microgrids

This paper reviews the current techniques used in energy management systems to optimize energy schedules into microgrids, accounting for uncertainties for various time frames (day-ahead and real-time operations). The current uncertainties affecting applications, including residential, commercial, virtual power plants, electric mobility, and multi-carrier microgrids, are the main subjects of this article. We outline the most recent modeling approaches to describe the uncertainties associated with various microgrid applications, such as prediction errors, load consumption, degradation, and state of health. The modeling approaches discussed in this article are probabilistic, possibilistic, information gap theory, and deterministic. Then, the paper presents and compares the current optimization techniques, considering the uncertainties in their problem formulations, such as stochastic, robust, fuzzy optimization, information gap theory, model predictive control, multiparametric programming, and machine learning techniques. The optimization techniques depend on the model used, the data available, the specific application, the real-time platform, and the optimization time. We hope to guide researchers to identify the best optimization technique for energy scheduling, considering the specific uncertainty and application. Finally, the most challenging issues to enhance microgrid operations, despite uncertainties by considering new trends, are discussed.

Real time Energy Management System of a photovoltaic based e-vehicle charging station using Explicit Model Predictive Control accounting for uncertainties

This paper proposes an Explicit Model Predictive Control (eMPC) for the energy management of an e-vehicle charging station fueled by a photovoltaic plant (PV), a battery energy storage system (BESS), and the electrical grid. The method computes offline an explicit solution of the MPC, which is stored and then used for real time control. Multiparametric programming is used to calculate the explicit solution by mapping the MPC laws as a function of uncertain parameters. In this paper, the uncertainties introduced into the multiparametric programming are the photovoltaic power production, the electricity price, the battery’s state of charge, and the electric vehicle power consumption. Moreover, the environmental impact of the charging station operation is considered through the C O 2 emissions level. The explicit solution is computed for a specific range of uncertain parameters. Then, during the real-time control, their current values are measured to evaluate the control laws saved beforehand. The proposed approach, consisting of an offline MPC-based determination of the control laws followed by their online use, reduces the computational burden without affecting the MPC performance. Numerical simulations and experimental results confirm the eMPC’s performance for the proposed application. • EV charging station based on PV/BESS uses eMPC control as its EMS. • PV and EV power, CO 2 , price, and BESS’s SoC are included as uncertainties. • eMPC creates offline the control laws as function of the uncertain parameters. • The online evaluation of the control laws takes less than one second.

Parameter Estimation of DC Motor using Multiparametric Programming

Parameter estimation is important in aspects for applied in many process models in control system design, chemistry, and other engineering application for expectation models of statistical observation. The focus of this work is on a nonlinear ordinary differential equation (ODEs) system which is a direct current (DC) motor model to estimate the model parameters using multiparametric programming. In this work, the ODE model is discretized by implementing Euler’s method to obtain the algebraic equation. The model parameters of the DC motor will be derived as an explicit function of measurements to estimate the parameter values. The applicability of the proposed method and accuracy of parameter estimation is demonstrated.

Strategic Generation Investment in Energy Markets: A Multiparametric Programming Approach

An investor has to carefully select the location and size of new generation units it intends to build, since adding capacity in a market affects the profit from units this investor may already own. To capture this closed-loop characteristic, strategic investment (SI) of generation can be posed as a bilevel optimization. By analytically studying a small market, we first show that its objective function can be non-convex and discontinuous. Realizing that existing mixed-integer problem formulations become impractical for larger markets and number of instances, this work put forth two SI solvers: a grid search to handle setups where the candidate investment locations are few, and a stochastic gradient descent approach for otherwise. Both solvers leverage powerful results of multiparametric programming (MPP), each in a unique way. The grid search entails finding the primal/dual solutions for a large number of optimal power flow (OPF) problems, which nonetheless can be efficiently computed several at once thanks to the properties of MPP. The same properties facilitate the rapid calculation of gradients in a mini-batch fashion, thus accelerating the implementation of a stochastic (sub)-gradient descent search. Tests on the IEEE 30- and 118-bus systems using real-world data corroborate the advantages of the novel solvers.

Nonparametric Probabilistic Optimal Power Flow

With the increasing penetration of renewable energy, accurate and efficient probabilistic optimal power flow (POPF) calculation becomes more and more important to provide decision support for secure and economic operation of power systems. This paper develops a novel nonparametric probabilistic optimal power flow (N-POPF) model describing the probabilistic information by quantiles, which avoids any parametric probability distribution assumptions of random variables. A novel critical region integral method (CRIM) which combines the multiparametric programming theory and discrete integral is proposed to efficiently solve the N-POPF problem. In the CRIM, the critical region partitioning algorithm is firstly introduced into the POPF model to directly establish the mapping relationship from wind power to optimal solutions of the POPF problem. Besides, a discrete integral method is developed in the CRIM to achieve the probability convolution calculation based on quantiles. Comprehensive numerical experiments verify the superior performance of the proposed CRIM in estimation accuracy and computational efficiency, and demonstrate that N-POPF model significantly improves the accuracy of uncertainty analysis. In general, the proposed N-POPF model and CRIM form a new framework of POPF problem for power system analysis and operation.

Differentiable predictive control: Deep learning alternative to explicit model predictive control for unknown nonlinear systems

We present differentiable predictive control (DPC) as a deep learning-based alternative to the explicit model predictive control (MPC) for unknown nonlinear systems. In the DPC framework, a neural state-space model is learned from time-series measurements of the system dynamics. The neural control policy is then optimized via stochastic gradient descent approach by differentiating the MPC loss function through the closed-loop system dynamics model. The proposed DPC method learns model-based control policies with state and input constraints, while supporting time-varying references and constraints. In embedded implementation using a Raspberry-Pi platform, we experimentally demonstrate that it is possible to train constrained control policies purely based on the measurements of the unknown nonlinear system. We compare the control performance of the DPC method against explicit MPC and report efficiency gains in online computational demands, memory requirements, policy complexity, and construction time. In particular, we show that our method scales linearly compared to exponential scalability of the explicit MPC solved via multiparametric programming.

A smart manufacturing strategy for multiparametric model predictive control in air separation systems

AbstractRecent trends in digitization and automation of information systems have led to the Industry 4.0 revolution in manufacturing systems. With the emergence of integrated “smart” systems that communicate through the cloud, collecting and manipulating the system data became a key yet, challenging component for developing optimal control strategies for these complex systems. In this work, we propose a strategy to address this problem with the case study on an air separation unit (ASU). Our approach involves developing an ASU's controllers via high‐fidelity modeling, studies in data‐driven reduced‐order models, and providing implementable control policies for the high‐fidelity model. Connecting the high‐fidelity model to a smart manufacturing platform allows integration into other smart manufacturing tools and applications. Since the high‐fidelity model is computationally challenging for online optimization tasks, such as model predictive control, surrogate models are generated that represent the high‐fidelity model's behavior. The derived reduced‐order models are then embedded into a model predictive control formulation for the optimal control of the whole process through multiparametric programming. A multiparametric approach based on solving a small portion of the multiparametric program is proposed to reduce the computational overhead. We then close the loop by deploying the developed controllers on the high‐fidelity model for tuning with prospects of employing them on the real industrial plant.

Active Jet Noise Control of Turbofan Engine Based on Explicit Model Predictive Control

The active jet noise control received significant attention due to the little influence it has on the engine performance. The active jet noise control is a multivariable problem because it needs to achieve the simultaneous closed-loop control of jet noise and engine performance. Model predictive control (MPC) has great application potentials in the field of multivariable control of aero-engines, but the real-time performance of MPC is intractable. This paper proposed an active jet noise controller of a turbofan engine, based on explicit model predictive control (EMPC). An integrated model of turbofan engine and jet noise, which calculates the engine parameters and jet noise in real time, was established. The online computational burden of MPC was transferred to offline computation using multi-parametric quadratic programming (MPQP). To improve the efficiency of the online positioning algorithm, the sequence search method was replaced by a binary search tree. Step simulations were performed to test the effectiveness of the proposed controller. The results show that the proposed EMPC controller not only achieves the simultaneous control of jet noise and the turbofan engine, but also improve the real-time performance greatly.

Optimal Design and Competitive Spans of Timber Floor Joists Based on Multi-Parametric MINLP Optimization.

This study investigates the optimization of the design of timber floor joists, taking into account the self-manufacturing costs and the discrete sizes of the structure. This non-linear and discrete class of optimization problem was solved with the multi-parametric mixed-integer non-linear programming (MINLP). An MINLP optimization model was developed. In the model, an accurate objective function of the material and labor costs of the structure was subjected to design, strength, vibration and deflection (in)equality constraints, defined according to Eurocode regulations. The optimal design of timber floor joists was investigated for different floor systems, different materials (sawn wood and glulam), different load sharing systems, different vertical imposed loads, different spans, and different alternatives of discrete cross-sections. For the above parameters, 380 individual MINLP optimizations were performed. Based on the results obtained, a recommended optimal design for timber floor joists was developed. Engineers can select from the recommendations the optimal design system for a given imposed load and span of the structure. Economically suitable spans for timber floor joists structures were found. The current knowledge of competitive spans for timber floor joists is extended based on cost optimization and Eurocode standards.

Integrated capacity configuration and control optimization of off-grid multiple energy system for transient performance improvement

The off-grid multiple energy system offers a promising way for energy supply due to its advantages of independency, multi energy co-generation, high efficiency and local utilization of renewable energy. A key issue of the off-grid multiple energy system is the operating performance during the dynamic transition because it is isolated from the utility grid. The conventional steady-state configuration methods ignore the significant impact of configuration scheme on the transient performance of the multiple energy system, which can easily lead to poor dynamic performance, thus are not suited for the off-grid multiple energy system with high penetration of intermittent renewable energy. It is necessary to consider the closed-loop dynamic control behavior of the system early in the configuration stage. To this end, this paper proposes a novel integrated capacity configuration and control optimization method of the multiple energy system, in which both the economic costs and closed-loop dynamic performance are fully considered. The multi-parametric programming is applied to construct a design-perceptive predictive tracking controller to bridge the gap between configuration and dynamic operation of the system. Case study on a typical off-grid combined heat and power multiple energy system shows that the proposed approach can reduce the dynamic thermal and electrical deviations of the multiple energy system by 84.45% and 28.11% respectively at the expense of only 1.86% increase of total economic costs. In-depth validations are carried out under extreme weather conditions and low carbon requirements, which further demonstrate the effectiveness and applicability of the proposed approach.

A Multi-parametric Programming Based Analytic Method to Compute Consumer Offer Curve for Reserves

An analytic method is proposed to compute the price-reserve offer curve at the consumer level in hierarchical direct load control. The convexification of the consumer reserve provision is examined, and the analytic expression of the optimal solution within each critical region is derived. Then, based on multi-parametric programming, a combinatorial enumeration method in conjunction with efficient reduction and pruning strategy is proposed to compute the optimal response of consumers in the whole price space. Numerical tests along with an application example in the bi-level aggregator pricing problem demonstrate the merit of this method.

Explicit Model Predictive Control for a Highly Interacting System

Explicit Model Predictive Control (MPC) of highly interacting systems using multiparametric programming is challenging as the offline solution to the Optimal Control Problem (OCP) typically entails calculation of a large number of regions of the uncertainty space. This can result in the case in which the point location problem is computationally more expensive than solving the OCP online. Hence, in this paper, with an aim to reduce computational costs of explicit MPC, we reformulate the OCP and study the computational and control performance of the reformulated explicit MPC compared to the conventional explicit MPC. As a case study, we consider a highly interacting quadruple tank process. The closed-loop simulation results show that between conventional MPC and reformulated MPC, in the online case, the total computational times are comparable, whereas in the explicit MPC case, the reformulation results in significant reduction in the total computational time by 44%.

AN AUTOMATED SYSTEM FOR SOLVING MULTIPARAMETRIC PROGRAMMING PROBLEMS

An automated system for solving multiparametric programming problems has been developed. The system is designed to solve multiparametric linear and quadratic programming problems. The parameters can exist in the coefficients of objective function as well as in the coefficients of constraint and right-hand sides. Functional dependencies of parameters can be both linear and non-linear. It is assumed that the partial derivatives of the coefficients with parametric dependencies, right-hand sides, as well as the functional dependencies of the solutions which are sought, do exist in respect to all parameters. The system prompts the user to select the class of the problem, the number of variables, the number of parameters and the number of differential images. Then it suggests entering the symbolic notations of the parameters, the interval of change. After that, text fields are created, in which the coefficients of the objective function of the optimization problem, the coefficients of the constraints and the right-hand sides are entered symbolically. For the solution of the problem, the system uses developed models using multidimensional differential transforms. The solutions to the problem are given by change intervals of the parameter, in each of them by the functional dependencies of the objective function and the variables, as well as graphical forms (in the case of one or two parameters). The computational operations are performed in Python 3 programming language using NumPy, SymPy, SciPy and Matplotlib libraries. The modules for calculating differential transforms are integrated in the system. The paper presents the results of solving a two-parameter programming problem by using the developed system which includes 24 intervals of change of the parameter, the functional dependencies of the solutions in each of them. The solutions were presented both in functional and graphical forms.

A Linear Programming Method Based on Proximal-Point Iterations With Applications to Multi-Parametric Programming

A Linear Programming Method Based on Proximal-Point Iterations With Applications to Multi-Parametric Programming

Scaled Experiment of the Detonation Control System for the High-Speed Penetration on Concrete

The design scheme and fabrication technology of the detonation control system for the high-speed deep penetration need to be tested for reliability and effectiveness through shooting range tests. However, the shooting range tests of the high-speed deep penetration are so demanding and expensive that it is difficult for the detonation control system to be tested many times. This paper focuses on penetration characteristics of the detonation control system to put forward a laboratory-scaled experiment method with the low impact velocity. Independent parameters of projectile and target affecting the penetration characteristics are effectively analyzed and extracted. A multi-parameter programming method of the scaled experiment for high-speed deep penetration is established. By adjusting the key parameters, the loading conditions of the scaled experiment can be obtained, which can get the comparable deceleration curve with those of the high-speed deep penetration. Finally, the extreme working environment for the detonation control system in the high-speed deep penetration is simulated through the scaled experiment in the laboratory. The scaled experiment method can get the comparable deceleration peak and time history. It is highly economical, and the experimental process is also repeatable, which can provide a reliable reference for the protection design into the projectile.

A feasible region evaluation method of renewable energy accommodation capacity

Power systems operate under energy and climate policies with the targets of carbon neutrality and CO2 emissions reduction in recent years. To mitigate environmental problems, it is important to promote the accommodation of renewable energy with strong fluctuations through hosting capacity evaluation of the distribution network. A feasible region evaluation method is proposed to characterize an accurate region of renewable energy accommodation capacity in the distribution network on short-time scales. The evaluation model based on the optimal power flow model is established with the objective of minimizing the operating cost of the distribution network, satisfying the operational constraints. Through multi-parametric programming, the constraints of the optimal power flow model are projected onto the boundary of the feasible region. Each operation point (the renewable generator output) in the feasible region can be directly obtained, which can serve for real-time dispatch with safety. Simulation results show that the proposed method can accurately obtain the feasible region of renewable energy accommodation capacity and the computational efficiency has been greatly improved compared with the Monte Carlo method.

Incorporating External Flexibility in Generation Expansion Planning

Exploiting flexibilities (e.g., demand response, energy storage, and deep peak regulation.) in generation expansion planning (GEP) is significant for coping with the increase of renewable energy (RE). This letter presents a GEP model with novel consideration of the flexibility of the interconnected external network without data leakage. External flexibility (detailed information) is represented as a feasible region with generation cost functions related to tie-line power, by using multi-parametric programming. Numerical results demonstrate the effectiveness of the proposed model.

Learning the Optimal Strategy of Power System Operation With Varying Renewable Generations

Optimal dispatch of modern power systems often entails efficiently solving large-scale optimization problems, especially when generators have to respond to the fast fluctuation of renewable generation. This paper develops a method to learn the optimal strategy from a mixed-integer quadratic program with time-varying parameters, which can model many power system operation problems such as unit commitment and optimal power flow. Different from existing machine learning methods that learn a map from the parameter to the optimal action, the proposed method learns the map from the parameter to the optimal integer solution and the optimal basis, forming a discrete pattern. Such a framework naturally gives rise to a classification problem: the parameter set is partitioned into polyhedral regions; in each region, the optimal 0-1 variable and the set of active constraints remain unchanged, and the optimal continuous variables are affine functions in the parameter. The outcome of classification is compared with analytical results derived from multi-parametric programming theory, showing interesting connections between traditional mathematical programming theory and the interpretability of the learning-based method. Tests on a small-scale problem demonstrate the partition of the parameter set learned from data meets the theoretical outcome. More tests on the IEEE 57-bus system and a real-world 1881-bus system validate the performance of the proposed method with a high-dimensional parameter for which the analytical method is intractable.

Multi-Parameter Quadratic Programming Explicit Model Predictive Based Real Time Turboshaft Engine Control

The traditional model predictive control (tMPC) algorithms have a large amount of online calculation, which makes it difficult to apply them directly to turboshaft engine–rotor systems because of real time requirements. Therefore, based on the theory of the perturbed piecewise affine system (PWA) and multi-parameter quadratic programming explicit model predictive control (mpQP-eMPC) algorithm, we develop a controller design method for turboshaft engine–rotor systems, which can be used for engine steady-state, transient state and limit protection control. This method consists of two steps: controller offline design and online implementation. Firstly, the parameter space of the PWA system is divided into several partitions offline based on the disturbance and performance constraints. Each partition has its own control law, which is in the form of piecewise affine linear function between the controller and the parameters. The control laws for those partitions are also obtained in this offline step. After which, for the online control implementation step, the corresponding control law can be obtained by a real-time query of a corresponding partition, which the current engine state falls into. This greatly reduces the amount of online calculation and thus improves the real-time performance of the MPC controller. The effectiveness of the proposed method is verified by simulating the steady-state and transient process of a turboshaft engine–rotor system with a limit protection requirement. Compared with tMPC, an mpQP-eMPC based controller can not only guarantee good steady-state, dynamic control performance and limit protection, but can also significantly improve the real-time performance of the control system.