Multi-parametric Programming Research Articles

Multi-parametric Analysis for Mixed Integer Linear Programming: An Application to Transmission Upgrade and Congestion Management

Upgrading the capacity of existing transmission lines is essential for meeting the growing energy demands, facilitating the integration of renewable energy, and ensuring the security of the transmission system. This study focuses on the selection of lines whose capacities and by how much should be expanded from the perspective of the Independent System Operators (ISOs) to minimize the total system cost. We employ advanced multi-parametric programming and an enhanced branch-and-bound algorithm to address complex mixed-integer linear programming (MILP) problems, considering multi-period time constraints and physical limitations of generators and transmission lines. To characterize the various decisions in transmission expansion, we model the increased capacity of existing lines as parameters within a specified range. This study first relaxes the binary variables to continuous variables and applies the Lagrange method and Karush-Kuhn-Tucker (KKT) conditions to obtain optimal solutions and identify critical regions associated with active and inactive constraints. Moreover, we extend the traditional branch-and-bound (B&B) method by determining the problem’s upper and lower bounds at each node of the B&B decision tree, helping to manage computational challenges in large-scale MILP problems. We compare the difference between the upper and lower bounds to obtain an approximate optimal solution within the decision-makers’ tolerable error range. In addition, the first derivative of the objective function on the parameters of each line is used to inform the selection of lines for easing congestion and maximizing social welfare. Finally, the capacity upgrades are selected by weighing the reductions in system costs against the expense of upgrading line capacities. The findings are supported by numerical simulations and provide transmission-line planners with decision-making guidance.

A hierarchical multi-parametric programming approach for dynamic risk-based model predictive quality control

In this work, we present a hierarchical batch quality control strategy with real-time process safety management. It features a multi-time-scale decision-making framework augmenting: (i) Risk-aware model predictive controller for short-term set point tracking and dynamic risk control under disturbances; (ii) Control-aware optimizer for long-term quality and safety optimization over the entire batch operation; (iii) Intermediate surrogate model to bridge the timescale gap by readjusting the optimizer operating decisions for the controller. All of the above problems are solved via multi-parametric mixed-integer quadratic programming with a key advantage to generate offline explicit control/optimization laws as affine functions of process and risk variables. This allows for the design of a fit-for-purpose risk management plan prior to real-time implementation, while reducing the need for repetitive online dynamic optimization. A unified process model is used to underpin the consistency of hierarchical operational optimization. The proposed approach offers a flexible strategy to integrate distinct decision-making time scales which can be selected separately tailored to the process-specific need of control, fault prognosis, and end-batch quality control. A T2 batch reactor case study is presented to showcase this approach to systematically address the interactions and trade-offs of multiple decision layers toward improving process efficiency and safety.

Integration and capacity optimization of molten-salt heat storage in coal-fired power plant with carbon capture system

Developing a new generation of economical, low-carbon, and flexible coal-fired power plants (CFPP) is crucial for achieving the low-carbon transformation of the energy industry. Deploying molten salt heat storage (MSHS) in the CFPP can effectively enhance its peak shaving capability and reduce the limitations of carbon capture on wide range operation of the plant. To this end, this paper proposes four integration schemes for MSHS into the power plant-carbon capture system. The thermal efficiencies, exergy efficiencies, and peak shaving capabilities of the entire plant under different integration schemes are evaluated and compared. The results indicate that the optimal integration scheme involves utilizing hot reheat steam to heat the cold molten salt and further provide heat for the carbon capture system during the heat charging process and utilizing hot molten salt to heat the outlet water of the feed water pump during the heat discharging process, which achieves the highest thermal efficiency of 38.00 % and exergy efficiency of 37.09 % across the entire cycle. A novel integrated design, scheduling, and control optimization (IDSCO) method is proposed to find the optimal capacity configuration of MSHS, in which the investment, maintenance, fuel, carbon emission, and long-timescale and short-timescale load tracking deviation penalty costs are fully considered. A design parameter-aware control system is developed based on a multi-parametric programming model predictive control approach to ensure a fair evaluation of closed-loop dynamic performance under different configuration capacities. The simulation results demonstrate that the deployment of MSHS can significantly enhance the flexibility of the power plant-carbon capture system and the proposed IDSCO approach can reduce 17.05 % of load tracking deviation penalty cost and 1.38 % of totalized cost compared to the conventional configuration approach. This paper effectively leverages the potential of MSHS in enhancing the operational flexibility of CFPP-PCC system, thereby providing useful guidance for future deployment of MSHS within CFPP-PCC system.

Explicit machine learning-based model predictive control of nonlinear processes via multi-parametric programming

Machine learning-based model predictive control (ML-MPC) has been developed to control nonlinear processes with unknown first-principles models. While ML models can capture nonlinear dynamics of complex systems, the complexity of ML models leads to increased computation time for real-time implementation of ML-MPC. To address this issue, in this work, we propose an explicit ML-MPC framework for nonlinear processes using multi-parametric programming. Specifically, a self-adaptive approximation algorithm is first developed to obtain a piecewise linear affine function that approximates the behaviors of ML models. Then, multi-parametric quadratic programming (mpQP) problems are formulated to generate the solution map for states in discretized state–space. Furthermore, to accelerate the implementation of explicit ML-MPC, a neighbor-first search algorithm is developed. Finally, an example of a chemical reactor is used to demonstrate the effectiveness of the explicit ML-MPC.

Robust Parametric Programming for Adaptive Piecewise Linear Control of Photovoltaic Inverters to Regulate Voltages in Power Distribution Systems

This paper proposes a robust parametric programming (RPP) method to adaptively obtain piecewise linear control functions of photovoltaic (PV) inverters for real-time voltage regulation in distribution systems. First, the voltage regulation problem is designed as a multi-parameter programming problem, which is transformed into single-parameter linear programming (sp-LP) problems for realizing local control of a PV inverter while considering the maximum voltage deviation caused by other PV units. Second, an alternating iterative algorithm is developed to solve the sp-LP problem of each PV unit and obtain the control functions of all the PV units. Third, considering the information on PV power correlation, a two-level parametric programming problem is established to enhance the voltage regulation performance of PV inverters. A geometric method is developed to solve the inner-level problem, and breakpoint reduction with robust interpolation is developed to improve the computational efficiency in solving the outer-level problem. Four distribution systems integrated with PV units are used to validate the effectiveness of the proposed method. Numerical results show that the performance of the proposed PV inverter control strategy is close to that of the ideal optimal solution and superior to existing piecewise linear control strategies.

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.

Explicit hybrid MPC for the lateral stabilization of electric vehicle system

This paper presents a hybrid Model Predictive Control (hMPC) approach to improve Electric Vehicle (EV) stability when cornering or in high-risk driving conditions. By using approximations of tire force characteristics, the EV system proposed in this study is considered as a Piecewise Affine (PWA) system which is a class of Hybrid Dynamical Systems (HDSs). First, a hybrid modeling of the studied system is developed within the PWA framework. Then, the synthesis of an explicit hybrid MPC law, based on the integration of the Active Front-wheel Steering control method (AFS) and Direct Yaw moment Control method (DYC), can be established. The purpose of the PWA-MPC is to force the yaw rate and sideslip angle of the EV system to follow the given references to guarantee lateral stability. The calculation of the control laws of this proposed approach is performed offline using multiparametric programming. Finally, this paper investigates the effectiveness of the designed controller by performing a set of simulations to analyze the efficiency and robustness of the proposed approach. A comparative study is addressed in this work to demonstrate the efficiency of the proposed hybrid MPC controller compared to the conventional MPC approach which considers the linear model of the system. Another simulations are carried out to assess the robustness of the proposed control, important disturbances and uncertainties were incorporated such as the change of the additional mass, the change of terrain and the high velocity. The simulation results show that the PWA EV system model with the designed controller is able to ensure lateral stability by keeping the sideslip angle and yaw rate tracking the ideal values.

Improved Unit Commitment with Accurate Dynamic Scenarios Clustering Based on Multi-Parametric Programming and Benders Decomposition

Improved Unit Commitment with Accurate Dynamic Scenarios Clustering Based on Multi-Parametric Programming and Benders Decomposition

Control Algorithm for Two-Tank System using Multiparametric Programming

The two-tank system consists of two connected tanks, where the goal is to control the liquid levels in both tanks by manipulating the inflow rates. This study proposes a control algorithm for a two-tank system based on multiparametric programming. In this work, the flow rate is determined using multiparametric programming. The simulation studies demonstrate the proposed work’s effectiveness by comparing its performance with and without noise. Results show accurate and robust regulation of liquid levels, minimizing control effort and making it suitable for systems with dynamic variations or parameter uncertainties. In conclusion, the implementation of multiparametric programming is able to estimate the value of the output for the control algorithm of the two-tank system.

A novel parallel combinatorial algorithm for multiparametric programming

A novel parallel combinatorial algorithm for multiparametric programming

Optimal Operation of an Industrial Dividing Wall Column through Multiparametric Programming.

In this contribution, we present a high-fidelity dynamic model of an industrial dividing wall column and the application of explicit model predictive control for its regulation. Our study involves the separation of methyl methacrylate from a quaternary mixture. The process includes a dividing wall column coupled with a decanter, which results in highly concentrated methyl methacrylate and water streams from the middle side draw of the column and the decanter, respectively. An equation-oriented mathematical model of the process is developed and presented in detail, where non-ideal thermodynamic calculations are adopted to describe the complex nature of the component interactions. The operability of the process is enhanced by the synthesis and application of an explicit model predictive controller, which is used to track the purity specifications of the product. Our results demonstrate that the proposed modeling and control approach can be utilized for the optimal online operation of the studied system.

A time-coupling consideration for evaluation of load carrying capacity in district multi-energy systems

This paper proposes a systematic approach to evaluate the load carrying capability of a district multi-energy system (MES) using the Energy Hub (EH) modeling approach. The EH approach is used to model the system, and the EH steady-state security region concept is introduced as a means of assessing the system's load carrying capacity. To calculate the security region, a multi-parametric programming (MPP) algorithm is proposed, while a standardized matrix modeling approach is utilized to formulate its mathematical representation. To handle the high-dimensional polytope resulting from time coupling, a strategy is employed to approximate the original polytope using a reduced number of lower-dimensional polytopes. These lower-dimensional polytopes are specifically associated with a limited number of time periods, thereby alleviating the computational complexity. The proposed method offers an improved balance between computational accuracy and efficiency compared to existing approaches, while retaining the inherent time coupling characteristics across different period combinations.

Dealing with infeasibility in multi-parametric programming for application to explicit model predictive control

Motivated by explicit model predictive control, we address infeasibility in multi-parametric quadratic programming according to the exact penalty function approach, where some user-chosen parameter-dependent constraints are relaxed and the 1-norm of their violation is penalized in the cost function. We characterize the relation between the resulting multi-parametric quadratic program and the original one and show that, as the penalty coefficient grows to infinity, the solution to the former provides a piecewise affine continuous function, which is an optimal solution for the latter over the feasibility region, while it minimizes the 1-norm of the relaxed constraints violation over the infeasibility region.

Hierarchical MPC for a dynamic process system employing parametric global optimization strategy

The hierarchical decision making in process industries has been traditionally viewed as having a common objective, such as the overall cost, which needs to be optimized. However, a more appropriate approach is to formulate and solve hierarchical optimization and control problems. The solution algorithms for hierarchical optimization problems have been reported in the literature. The idea is to recast each optimization sub-problem in the hierarchy into a multiparametric programming problem, considering the variables of upper-level problems as unknown parameters. In this paper, explicit Model Predictive Control (MPC) and hierarchical optimization techniques, employing multiparametric programming, are combined for hierarchical MPC. The solution algorithm for hierarchical MPC is described in detail. Note that the solution to a hierarchical MPC problem is challenging, even for the simplest case of linear-quadratic objectives. Closed-loop simulations of a thermal mixing process, under two different hierarchical MPC formulations, are performed and the control performance is studied.

Explicit Solution of Parameter Estimate using Multiparametric Programming for Boost Converter

This work proposes an approach to estimate the parameters of capacitance and inductance in a boost converter using an explicit solution. A multiparametric programming (MPP) algorithm is fundamental to the suggested methodology, which aims to develop parameters as explicit functions of measurements. In this method, the generalised mathematical model of the boost converter is discretised into an algebraic equation. The parameter estimation problem is then formulated and solved using Karush-Kuhn-Tucker. Finally, an explicit solution of estimate parameters such as capacitor and inductor is formulated as an explicit function of the inductor currents and capacitor voltage. Finally, the state variables of the boost converter are numerically obtained and used to test the capability of the proposed work. The results presented in this work prove the MPP algorithm can estimate the boost converter’s parameters, which can be extended to other power converters and filters.

Multiparametric Programming-Based Coordinated Economic Dispatch of Integrated Electricity and Natural Gas System

Continuous development and application of gas-fired units worldwide have raised concerns about the inherent interdependency between electricity and natural gas systems. The coordinated dynamic economic dispatch for integrated electricity and natural gas systems (IENs) is increasingly essential. Since electricity and natural gas systems belong to different administrative organizations, we build a coordinated economic dispatch model for IENs and propose a decentralized algorithm to realize collaborative distributed control. A distributed algorithm based on multiparametric programming and benders cuts is proposed to crack the optimization problem in a decentralized way. The quadratic exchanging function is built to accelerate the convergence speed, and the auxiliary variables relaxation method is applied to deal with the infeasible subproblems. The global convergence of the proposed methods is proved in detail. Furthermore, from the point of view of both iteration number and computation time, simulation results demonstrate that the computational performance of the proposed method is far superior to the popular decentralized methods.

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.

Learning assisted column generation for model predictive control based energy management in microgrids

Model predictive control is an effective approach for microgrid energy management. However, the main downside of such method is its expensive online computational cost, which is not amenable to most practical microgrid implementations. To address this issue, we propose a deep neural network assisted column generation approach that can accelerate the solution procedure of model predictive control. In each iteration, our approach leverages different deep neural networks to predict the optimal solutions of all the subproblems in column generation, which can accelerate the computation of all the subproblems and the entire process of column generation. The pre-existing knowledge of the microgrid is also exploited to guarantee the feasibility of the neural network outputs using multi-parametric programming theory. The numerical results show that our approach leads to a reduction in computational time of approximately 50% in a medium-sized microgrid compared with the full mixed integer solution based on traditional branch and bound method, while the optimality loss is only 0.02% in terms of operating costs.

Dynamic risk-based process design and operational optimization via multi-parametric programming

We present a dynamic risk-based process design and multi-parametric model predictive control optimization approach for real-time process safety management in chemical process systems. A dynamic risk indicator is used to monitor process safety performance considering fault probability and severity, as an explicit function of safety–critical process variables deviation from nominal operating conditions. Process design-aware risk-based multi-parametric model predictive control strategies are then derived which offer the advantages to: (i) integrate safety–critical variable bounds as path constraints, (ii) control risk based on multivariate process dynamics under disturbances, and (iii) provide model-based risk propagation trend forecast. A dynamic optimization problem is then formulated, the solution of which can yield optimal risk control actions, process design values, and/or real-time operating set points. The potential and effectiveness of the proposed approach to systematically account for interactions and trade-offs of multiple decision layers toward improving process safety and efficiency are showcased in a real-world example, the safety–critical control of a continuous stirred tank reactor at T2 Laboratories.

Integrating model predictive control and deep learning for the management of an EV charging station

Explicit model predictive control (EMPC) maps offline the control laws as a set of regions as a function of bounded uncertain parameters using multi-parametric programming. Then, in online mode, it seeks the best solution within these areas. Unfortunately, the offline solution can be computationally demanding because the number of regions can grow exponentially. Thus, this paper presents the application of a deep neural network (DNN) to learn the EMPC’s regions for a photovoltaic-based charging station. The main uncertain parameters in this study are the forecast error of photovoltaic power production and the battery’s state of charge. Additionally, the connection or disconnection of an electric vehicle is considered a disruption. The final controller creates the regions at the start of each prediction time or when a disruption occurs, only using the previously created DNN. The obtained solution is validated using data from an e-vehicle charging station installed at the University of Trieste, Italy.