Multi-parametric Programming Technique Research Articles

Power system operational reliability assessment based on the data center energy consumption elastic space

In the era of big data, data centers with high energy consumption, interconnectivity, and load flexibility have developed rapidly. However, due to data privacy issues, the traditional power-system operational reliability assessment (ORA) struggles to precisely consider the load flexibility of data centers, leading to inaccurate evaluation. To this end, this article proposes an ORA method considering the load flexibility of data centers via the energy consumption elastic space. By transforming the inner operation constraints of data centers into an equivalent elastic space, the ORA does not require any private data to complete the evaluation. Specifically, the energy consumption model of data centers is established to accurately describe the load flexibility. Then, based on multi-parametric programming techniques, the energy consumption elastic space of data centers is characterized by data centers’ power demand constraints, which do not involve privacy data, and no privacy concerns exist. Finally, the ORA model and the evaluation method based on the energy consumption elastic space can be constructed. With a lot of data center operation constraints being replaced by power demand constraints, the proposed method can complete an evaluation faster without accuracy loss. Its effectiveness is validated through simulations using the IEEE RTS 24-bus system and a provincial 661-bus system.

Optimal bidding strategies of advanced adiabatic compressed air energy storage based energy hub in electricity and heating markets

Integration of power and heating systems can not only improve energy efficiency but also reduce the peak generation capacity by narrowing the gap between peak and valley demands. Advanced adiabatic compressed air energy storage (AA-CAES) is a large-scale and environmental-friendly storage technology that can supply heat and power. It can be adopted as an energy hub that integrates electricity and heating systems. This paper studies the strategic behavior of an AA-CAES based energy hub in deregulated electricity and heat markets. Firstly, a heat market clearing model with elastic thermal demands is developed, and the impact of demand elasticity on the stability of the heat market is analyzed. Secondly, based on the proposed heat market model, a bi-level optimization model is established to explore the strategic bidding and scheduling of the merchant AA-CAES based energy hub in both markets. Through exploiting the weakly coupling structure of the bi-level problem, a special decomposition method is proposed to compute the market equilibrium. The optimal response of the lower-level power/heat market clearing problem is analytically expressed by multi-parametric programming techniques, where the bidding strategies are parameters. With the lower-level parameterized solution, the original complex joint optimization problem is transformed into a small-scale mixed integer quadratic program with only a few binary variables, and thus can be solved efficiently. Case studies verify the proposed model and method. Results show that the AA-CAES based energy hub/heat market is conducive to an 8.6%/22.5% reduction in the operation cost of the heat market/energy hub.

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.

Multiparametric Programming in Process Systems Engineering: Recent Developments and Path Forward

The inevitable presence of uncertain parameters in critical applications of process optimization can lead to undesirable or infeasible solutions. For this reason, optimization under parametric uncertainty was, and continues to be a core area of research within Process Systems Engineering. Multiparametric programming is a strategy that offers a holistic perspective for the solution of this class of mathematical programming problems. Specifically, multiparametric programming theory enables the derivation of the optimal solution as a function of the uncertain parameters, explicitly revealing the impact of uncertainty in optimal decision-making. By taking advantage of such a relationship, new breakthroughs in the solution of challenging formulations with uncertainty have been created. Apart from that, researchers have utilized multiparametric programming techniques to solve deterministic classes of problems, by treating specific elements of the optimization program as uncertain parameters. In the past years, there has been a significant number of publications in the literature involving multiparametric programming. The present review article covers recent theoretical, algorithmic, and application developments in multiparametric programming. Additionally, several areas for potential contributions in this field are discussed, highlighting the benefits of multiparametric programming in future research efforts.

Real-Time Self-Dispatch of a Remote Wind-Storage Integrated Power Plant Without Predictions: Explicit Policy and Performance Guarantee

This paper investigates real-time self-dispatch of a remote wind-storage integrated power plant connecting to the main grid via a transmission line with a limited capacity. Because prediction is a complicated task and inevitably incurs errors, it is a better choice to make real-time decisions based on the information observed in the current time slot without predictions on the uncertain electricity price and wind generation in the future. To this end, the operation problem is formulated under the Lyapunov optimization framework to maximize the long-term time-average revenue of the wind-storage plant. Inter-temporal storage dynamics are represented by a virtual queue which is mean rate stable. An online method for real-time dispatch is proposed based on Lyapunov drift algorithm via a drift-minus-revenue function. The upper bound of such a function, which does not depend on future uncertainty, is minimized in each time slot. Explicit dispatch policies are obtained through multi-parametric programming technique so that no optimization problem is solved online. It is proved that the online algorithm can maintain all the constraints across the entire horizon and the expected optimality gap compared to the deterministic offline optimum with perfect uncertainty information is inversely proportional to the weight coefficient in the drift-minus-revenue function. Numerical tests using real wind and electricity price data validate the effectiveness and performance of the proposed method.

Explicit MPC for column‐type EPS systems

For decades, electric power steering (EPS) systems have replaced hydraulic-assist steering systems. The main purpose of EPS systems is to generate an assist torque using an electric motor for the improvement of the driving comfort. The proposed controller is designed against a model predictive control (MPC) and linear quadratic regulator (LQR) for a comparison in terms of generating a desired value of the assist torque. The motion equations of a column-type EPS system are derived in this study. Meanwhile, MPC has been regarded as an effective control method for constrained problems because this method can anticipate future events of the system. However, the repeated online optimisation imposes huge computational complexity. The proposed scheme explicitly obtains the current control law by adopting a multi-parametric quadratic programming technique. The proposed controller is evaluated in MATLAB/Simulink and a hardware-in-the-loop simulation using the Infineon automotive TriCore AURIX TC277 microcontroller. The simulation and experimental results confirmed that the proposed controller successfully tracks the reference value while reducing the computational burden. It is also shown by conducting experiments with various vehicle speeds that the tracking ability of the explicit model predictive controller (MPC controller) is superior to that of the LQR controller.

PAROC—An integrated framework and software platform for the optimisation and advanced model-based control of process systems

In this paper we present the main foundations and features of an integrated framework and software platform that enables the use of model-based tools in design, operational optimisation and advanced control studies. A step-wise procedure is outlined involving (i) the development of a high-fidelity dynamic model, and its validation and model analysis, (ii) a model approximation step, including system identification, model reduction and global sensitivity analysis, (iii) a receding horizon modelling step for model-predictive control (MPC) and reactive scheduling, (iv) a suite of multi-parametric programming techniques for optimisation under uncertainty, explicit/multi-parametric MPC and state-estimation and (v) an ‘in-silico’ validation step for the derived optimisation, control and/or scheduling strategies to be analysed within the original high-fidelity model. The proposed software platform, PAROC, is also introduced and demonstrated in three different classes of process systems engineering applications; a combined heat and power energy system, a distillation column and a periodic purification process for biopharmaceuticals.

Explicit Model Predictive Control Approach for Low-Thrust Spacecraft Proximity Operations

The key role of autonomous systems in future space missions has made model predictive control a very attractive guidance and control technique. However, the capability of low-power spacecraft processors to handle the real-time computational load of this technique still needs to be fully established, especially for complex control problems. This paper introduces a method to improve the computational efficiency of model predictive control when applied to the problem of autonomous rendezvous and proximity maneuvering using low-thrust propulsion. To ensure safe trajectories in this scenario, a long control horizon is required and the control problem must be solved at a relatively fast sampling rate. The proposed design addresses such requirements by parameterizing the thrust profile with a set of Laguerre functions. In this setting, the number of control variables can be made significantly smaller than the length of the control horizon, as opposed to standard design methods. By exploiting this property, in combination with multiparametric programming techniques, an explicit control law is derived that is suitable for real-time implementation on simple hardware. The performance of this approach is demonstrated on a small spacecraft mission and compared with that of other control techniques.

Reactive Scheduling by a Multiparametric Programming Rolling Horizon Framework: A Case of a Network of Combined Heat and Power Units

In this work, we introduce an approach for the reactive scheduling of production systems with bounded uncertain parameters. The proposed method follows a state-space representation for the scheduling problem, and relies on the use of a rolling horizon framework and multiparametric programming techniques. We show that by considering as uncertain parameters the set of variables that describe the state of the system at the beginning of the prediction horizon, we can effectively formulate a set of state-space multiparametric programming problems that are solved just once and off-line. In contrast to existing approaches, the repetitive solution of a new multiparametric problem after each disruptive event is avoided. The results of the parametric optimization are used in a rolling horizon basis without the need for online optimization. The proposed multiparametric programming rolling horizon (mp-RH) approach is applied in the scheduling problem of a network of combined heat and power units (i.e., a unit commitment problem type). Several case studies are solved, potential extensions of the proposed method are provided, and challenging areas wherein research is necessary are discussed.

Robust Explicit Solution of Multirate Predictive Control System with External Disturbances

This paper considers the problem of robust predictive control for a class of uncertain multirate systems, in which the output sampling period is several times larger than the one of input updating. By means of dynamic programming and multiparametric quadratic programming (mp-QP) techniques, this work proposes a novel robust explicit model predictive control (REMPC) algorithm such that the higher on-line updating speed of input can be attained. Firstly, the optimization problem is decomposed into several subproblems. For each subproblem, a constraint condition only involving current state and input is constructed. Then taking current state and future inputs as parameter variables we can obtain a robust explicit solution for the new reformulated optimization subproblem by mp-QP method. Especially, by choosing a maximal robust positive invariant set as the terminal constraint set of the optimization problem, closed-loop robust stability of the multirate control system subject to external disturbance can be guaranteed. Finally, a numerical simulation is given to show the effectiveness of the proposed method.

Explicit/multi-parametric model predictive control (MPC) of linear discrete-time systems by dynamic and multi-parametric programming

This work presents a new algorithm for solving the explicit/multi-parametric model predictive control (or mp-MPC) problem for linear, time-invariant discrete-time systems, based on dynamic programming and multi-parametric programming techniques. The algorithm features two key steps: (i) a dynamic programming step, in which the mp-MPC problem is decomposed into a set of smaller subproblems in which only the current control, state variables, and constraints are considered, and (ii) a multi-parametric programming step, in which each subproblem is solved as a convex multi-parametric programming problem, to derive the control variables as an explicit function of the states. The key feature of the proposed method is that it overcomes potential limitations of previous methods for solving multi-parametric programming problems with dynamic programming, such as the need for global optimization for each subproblem of the dynamic programming step.

Multiparametric programming based algorithms for pure integer and mixed-integer bilevel programming problems

This work introduces two algorithms for the solution of pure integer and mixed-integer bilevel programming problems by multiparametric programming techniques. The first algorithm addresses the integer case of the bilevel programming problem where integer variables of the outer optimization problem appear in linear or polynomial form in the inner problem. The algorithm employs global optimization techniques to convexify nonlinear terms generated by a reformulation linearization technique (RLT). A continuous multiparametric programming algorithm is then used to solve the reformulated convex inner problem. The second algorithm addresses the mixed-integer case of the bilevel programming problem where integer and continuous variables of the outer problem appear in linear or polynomial forms in the inner problem. The algorithm relies on the use of global multiparametric mixed-integer programming techniques at the inner optimization level. In both algorithms, the multiparametric solutions obtained are embedded in the outer problem to form a set of single-level (M)(I)(N)LP problems – which are then solved to global optimality using standard fixed-point (global) optimization methods. Numerical examples drawn from the open literature are presented to illustrate the proposed algorithms.

Incorporating uncertainty into a supplier selection problem

Abstract Supplier selection is an important strategic supply chain design decision. Incorporating uncertainty of demand and supplier capacity into the optimization model results in a robust selection of suppliers. A two-stage stochastic programming (SP) model and a chance-constrained programming (CCP) model are developed to determine a minimal set of suppliers and optimal order quantities with consideration of business volume discounts. Both models include several objectives and strive to balance a small number of suppliers with the risk of not being able to meet demand. The SP model is scenario-based and uses penalty coefficients whereas the CCP model assumes a probability distribution and constrains the probability of not meeting demand. Both formulations improve on a deterministic mixed integer linear program and give the decision maker a more complete picture of tradeoffs between cost, system reliability and other factors. We present Pareto-optimal solutions for a sample problem to demonstrate the benefits of the SP and CCP models. In order to describe the tradeoffs between costs and risks in an analytical form, we use multi-parametric programming techniques to more completely analyze the alternative Pareto-optimal supplier selection solutions in the CCP model. This analysis gives insights into the robustness of the solutions with respect to number of suppliers, costs and probability of not meeting demand.

Reference governor for constrained piecewise affine systems

We present a methodology for designing reference tracking controllers for constrained, discrete-time piecewise affine systems. The approach follows the idea of reference governor techniques where the desired set-point is filtered by a system called the “reference governor”. Based on the system current state, set-point, and prescribed constraints, the reference governor computes a new set-point for a low-level controller so that the state and input constraints are satisfied and convergence to the original set-point is guaranteed. In this note we show how to design a reference governor for constrained piecewise affine systems by using polyhedral invariant sets, reachable sets, multiparametric programming and dynamic programming techniques.

Reference governor for constrained systems with time-varying references

This paper proposes constructing a reference governor for constrained linear systems with time-varying references. The main feature of the constructed reference governor is to simultaneously consider fulfillment of state and control constraints, as well as tracking performance by appropriately managing the reference to be inputted. To achieve constraint fulfillment and to evaluate tracking performance, the reference management is reduced into a convex quadratic programming problem using the concept of a maximal output admissible set. The reference governor is finally obtained in the form of a piecewise affine function of state and reference variables by means of a multi-parametric programming technique. In addition, the effectiveness of the reference governor is demonstrated by numerical and experimental examples of a practical DC position servomechanism with the control constraint.

Algorithms for Online Implementations of Explicit MPC Solutions

One of the key problems in Model Predictive Control (MPC) is the inherent on-line computational complexity, which restricts its application to slow dynamic systems. To address this issue, multi-parametric programming technique is introduced in MPC (explicit MPC), where the optimization effort is moved off-line. The optimal solution is given in an explicitly piecewise affine function defined over a polyhedral subdivision of the set of feasible states. Instead of solving an optimization problem, the on-line work is simplified to identify the region the current state belongs to and simply evaluate the piecewise affine function. Hence, identifying of the member of the solution partition that contains a given point (referred to as a point location problem) impacts on the time to implement the explicit controller in real-time, which is one component of the complexity of explicit MPC. In this paper, two simple algorithms for point location problems are proposed to efficiently implement of explicit MPC solutions, which aim at reducing the number of polyhedral sets that are candidates to contain the state at the next time instant.

A multi-parametric programming approach for constrained dynamic programming problems

In this work, we present a new algorithm for solving complex multi-stage optimization problems involving hard constraints and uncertainties, based on dynamic and multi-parametric programming techniques. Each echelon of the dynamic programming procedure, typically employed in the context of multi-stage optimization models, is interpreted as a multi-parametric optimization problem, with the present states and future decision variables being the parameters, while the present decisions the corresponding optimization variables. This reformulation significantly reduces the dimension of the original problem, essentially to a set of lower dimensional multi-parametric programs, which are sequentially solved. Furthermore, the use of sensitivity analysis circumvents non-convexities that naturally arise in constrained dynamic programming problems. The potential application of the proposed novel framework to robust constrained optimal control is highlighted.

An MPC/hybrid system approach to traction control

This paper describes a hybrid model and a model predictive control (MPC) strategy for solving a traction control problem. The problem is tackled in a systematic way from modeling to control synthesis and implementation. The model is described first in the Hybrid Systems Description Language to obtain a mixed-logical dynamical (MLD) hybrid model of the open-loop system. For the resulting MLD model, we design a receding horizon finite-time optimal controller. The resulting optimal controller is converted to its equivalent piecewise affine form by employing multiparametric programming techniques, and finally experimentally tested on a car prototype. Experiments show that good and robust performance is achieved in a limited development time by avoiding the design of ad hoc supervisory and logical constructs usually required by controllers developed according to standard techniques.

Using mathematical programming to compute singlular multivariate normal probablities

This paper describes two new, mathematical programming-based approaches for evaluating general, one- and two-sidedp-variate normal probabilities where the variance-covariance matrix (of arbitrary structure) is singular with rankr(r<pand r and p can be of unlimited dimensions. In both cases, principal components are used to transform the original, ill-definedp-dimensional integral into a well-definedrdimensional integral over a convex polyhedron. The first algorithm that is presented uses linear programming coupled with a Gauss-Legendre quadrature scheme to compute this integral, while the second algorithm uses multi-parametric programming techniques in order to significantly reduce the number of optimization problems that need to be solved. The application of the algorithms is demonstrated and aspects of computational performance are discussed through a number of examples, ranging from a practical problem that arises in chemical engineering to larger, numerical examples.

Flexibility analysis and design of linear systems by parametric programming

AbstractA new, unified theory and algorithms, based on multiparametric programming techniques, for the solution of flexibility analysis and design optimization problems in linear process systems are presented. They are used for the flexibility test and index problems in systems with deterministic parameters, and for the stochastic and expected stochastic flexibility evaluation problems in systems with stochastic parameters. They are computationally efficient and give the explicit dependence of various flexibility metrics on the values of the continuous design variables. The latter feature enables the easy and efficient comparison of design alternatives. It also allows for the compact formulation of design optimization problems that can be solved parametrically to yield the exact algebraic form of the trade‐off curve of economics against flexibility. Key features of the proposed approach are demonstrated through both mathematical and process examples.